UNIVERSIDAD NACIONAL AUTÓNOMA DE MÉXICO PROGRAMA DE POSGRADO EN ASTROFÍSICA INSTITUTO DE FÍSICA ASTROFÍSICA TEÓRICA METAL FORESTS IN LARGE SCALE STRUCTURE: INSIGHTS FROM IGM METALLICITY USING FIRST-YEAR DESI SURVEY DATA TESIS QUE PARA OPTAR POR EL GRADO DE: DOCTORA EN CIENCIAS (ASTROFÍSICA) PRESENTA: ANDREA MUÑOZ GUTIÉRREZ TUTORES PRINCIPALES: AXEL RICARDO DE LA MACORRA PETTERSSON MORIEL INSTITUTO DE FÍSICA, UNIVERSIDAD NACIONAL AUTÓNOMA DE MÉXICO ALMA XÓCHITL GONZÁLEZ MORALES DIVISIÓN DE CIENCIAS E INGENIERÍAS, UNIVERSIDAD DE GUANAJUATO MIEMBROS DEL COMITÉ TUTOR: JOSÉ OCTAVIO VALENZUELA TIJERINO INSTITUTO DE ASTRONOMÍA, UNAM ALDO ARMANDO RODRÍGUEZ PUEBLA INSTITUTO DE ASTRONOMÍA, UNAM MARIANA VARGAS MAGAÑA INSTITUTO DE FÍSICA, UNAM CIUDAD UNIVERSITARIA, CIUDAD DE MÉXICO, AGOSTO DE 2025 UNAM – Dirección General de Bibliotecas Tesis Digitales Restricciones de uso DERECHOS RESERVADOS © PROHIBIDA SU REPRODUCCIÓN TOTAL O PARCIAL Todo el material contenido en esta tesis esta protegido por la Ley Federal del Derecho de Autor (LFDA) de los Estados Unidos Mexicanos (México). El uso de imágenes, fragmentos de videos, y demás material que sea objeto de protección de los derechos de autor, será exclusivamente para fines educativos e informativos y deberá citar la fuente donde la obtuvo mencionando el autor o autores. Cualquier uso distinto como el lucro, reproducción, edición o modificación, será perseguido y sancionado por el respectivo titular de los Derechos de Autor. Metal Forests in Large Scale Structure: Insights from IGM Metallicity Using First-Year DESI Survey Data Andrea Muñoz Gutiérrez Supervised by: Dr. Axel Ricardo de la Macorra Pettersson Moriel Dr. Alma Xóchitl González Morales August 2025 2 Acknowledgements This material is based upon work supported by the U.S. Department of Energy (DOE), Of- fice of Science, Office of High-Energy Physics, under Contract No. DE-AC02-05CH11231, and by the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility under the same contract. Additional support for DESI was provided by the U.S. Na- tional Science Foundation (NSF), Division of Astronomical Sciences under Contract No. AST- 0950945 to the NSF’s National Optical-Infrared Astronomy Research Laboratory; the Science and Technology Facilities Council of the United Kingdom; the Gordon and Betty Moore Foundation; the Heising-Simons Foundation; the French Alternative Energies and Atomic Energy Commission (CEA); the National Council of Humanities, Science and Technology of Mexico (CONAHCYT)1; the Ministry of Science and Innovation of Spain (MICINN), and by the DESI Member Institutions: https://www.desi.lbl.gov/collaborating-institutions. This work was supported by CONAHCYT through the Fronteras de la Ciencia project No. 281. It also benefited from institutional support at UNAM through the DGAPA projects UNAM DGAPA IN101415, IN103518, IN-101124, UNAM DGAPA IN-105021. I am honored to be permitted to conduct my scientific research on I’oligam Du’ag (Kitt Peak), a mountain with particular significance to the Tohono O’odham Nation. I would like to express my deep gratitude to Andreu Font-Ribera2 and Ignasi Pérez-Ráfols3 for their invaluable guidance and support throughout the development of my thesis research in Barcelona. Their mentorship greatly shaped the work with which I am graduating, and I am espe- cially thankful for their generosity and insight throughout this process. - No tinc prou paraules per agrair tot el suport que m’heu donat durant el meu postgrau, i en particular al llarg del doctorat. Us estic profundament agräıda per ser-hi i per ajudar-me a creure en el següent pas. I acknowledge that ChatGPT (GPT-4) by OpenAI was used to improve grammar and language clarity in sections of this thesis. 1Since 2024, Secretariat of Science, Humanities, Technology and Innovation (SECIHTI). 2Institut de F́ısica d’Altes Enerǵıes 3Universitat de Barcelona/Universitat Politècnica de Catalunya i ii ACKNOWLEDGEMENTS I feel truly grateful for the support, follow-up, and advice of my advisory committee: Octavio Valenzuela, Aldo Rodŕıguez Puebla and Mariana Vargas Magaña. Thank you for being present throughout my post-graduate programs and always having words to motivate and cheer me up. I owe special thanks to my thesis advisory committee: Sébastien Fromenteau and Alfredo Montaña, for all the time you spent scrutinizing even the tiniest details to help improve my work; Anna Lia Longinotti and José Alberto Vázquez, for your feedback, valuable comments, and time spent chasing down typos, alignment issues, and formatting gremlins; and Octavio Valenzuela, for your support, generosity, and trust — your presence has guided both this chapter and the one that follows in more ways than words can capture. I wish I could thank every single member of the DESI Collaboration who has contributed to my work and personal growth. Unfortunately — or rather, fortunately — that’s virtually impossible, as I’ve been lucky to be surrounded by so many generous and supportive people. Still, this achievement would have been thousands of times more difficult without the support of César Ramı́rez-Pérez, Calum Gordon, Andrei Cuceu, and my dear friend Hiram Herrera-Alcántar, whose contributions were invaluable throughout this research. I would also like to thank Julián Bautista for your con- stant feedback and for reminding me of the importance of having fun; Claire Lamman and my friend Samuel Brieden, for your support in my forays into the world of outreach; Paul Martini, for encouraging me as I take the next steps in both my professional path and personal development; and my mentors Joan Najita and Lado Samushia, whose guidance has been deeply valuable — Joan’s kindness, encouragement, and steady support, even during my most difficult moments, have meant more than words can express. With deep appreciation, I thank Biprateep Dey, with whom I had the privilege of co-creating and chairing the Early Career Scientist (ECS) Committee — and whom I’m now lucky to call a friend. Thank you to Namitha Kizhuprakkat and Anthony Kremin for shar- ing in this initiative, and to Kyle Dawson and Nathalie Palanque-Delabrouille for trusting us with the creation and leadership of the ECS Committee. It has been a privilege to serve the Collabo- ration with you by my side, and an honor to receive the Builder Status in recognition of that service. Thank you, Christophe Morisset, Natalia Osorio, Miriam Gudiño, José Antonio Vázquez and Aldebarán López Vargas for your thoughtful input and support in improving my work; Ary Rodŕıguez and Bertha Vázquez for the follow-up and help to acheive this milestone; and Ana Lorena Gutiérrez for your advice and support since I started chasing this dream. It means a lot to me to thank my big outreach family: Astrof́ısicos en Acción, for the astronom- ical adventures lived for so many years. Thank you for reminding me in each event about the big picture of why doing this. Living two solar eclipses with you has been one of the most incredible experiences of my life, and it has been wonderful to live them together. My heartfelt thanks to the projects Tz’unun Ek, Daat Sendero Interior, Terra Luna, Spirale, and Dharma Bellydance, whose support was essential to completing this thesis. Last, but not least, I am grateful to my supervisors, Alma and Axel, for their support throughout my PhD. I am particularly thankful for their support during the challenging times of the COVID-19 health emergency, and for being there for me despite facing their own challenges. I am also thankful for the guidance, time, and resources invested in me that helped initiate this research. iii Para mis padres, quienes me han acompañado y apoyado amorosamente en cada paso, incondicionalmente. Los amo más allá de los confines del Universo. Para mis abuelos, que me vieron construir este sueño desde que naćı, y ahora me ven cumplirlo desde las estrellas. iv ACKNOWLEDGEMENTS Resumen Esta tesis explora el potencial de usar caracteŕısticas de absorción alternativas en espectros de cuásares como nuevos trazadores de la estructura a gran escala (LSS) del Universo. Mientras que el bosque de Lyman-α ha sido una herramienta cosmológica sólida y bien establecida durante más de una década, este trabajo se adentra en territorios menos explorados al investigar la información cosmológica codificada en los bosques de Lyman-β y de metales, como C iv, Si iv y Mg ii. Estas transiciones abren una nueva ventana al análisis del medio intergaláctico, con el objetivo a largo plazo de evaluar su viabilidad como trazadores de las oscilaciones acústicas de bariones (BAO). Esto es particularmente interesante porque las galaxias proporcionan una herramienta observa- cional robusta para obtener información cosmológica a z ≲ 1.5, y el bosque de Lyman-α de z ≳ 2, dejando una brecha en los corrimientos al rojo que potencialmente pueden ser cubiertos por estos nuevos trazadores. Al hacerlo, esta investigación contribuye al esfuerzo más amplio de expandir el conjunto de herramientas de la cosmoloǵıa de LSS, proporcionando nuevos caminos para restringir parámetros cosmológicos fundamentales. Este trabajo presenta las primeras mediciones de correla- ciones cruzadas entre el bosque de Lyman-β y cuásares, junto con el cálculo de su bias y un primer enfoque hacia la medición de funciones de correlación cruzadas entre metales y cuásares a pequeñas escalas (como 60 y 40 Mpc/h). Este avance permitió modelar los dobletes de las transiciones analizadas y ofrece conocimientos fundamentales hacia este territorio poco explorado. El primer caṕıtulo (1) de esta tesis está enfocada en proporcionar al lector una perspectiva amplia sobre el estado del arte de la cosmoloǵıa, aśı como la introducción de las herramientas estad́ısticas y observables para entender el contexto de la investigación de esta tesis. Finalmente, presenta las motivaciones y objetivos de estudiar bosques de metales en la LSS. El segundo caṕıtulo (2) aborda el contexto observacional para esta investigación: el Instrumento Espectroscópico de Enerǵıa Oscura (DESI), un censo de galaxias cuyo objetivo es el mapeo de la historia de expansión del Universo. Describe el diseño y aspectos técnicos clave tanto del instru- mento como del censo, junto con sus principales objetivos cient́ıficos. También se explica el primer conjunto completo de datos del censo, que contiene los datos utilizados para este trabajo de investi- gación: la muestra de cuásares de absorción observados durante el primer año de observada durante el primer año de operaciones (DR1). La naturaleza de esta muestra está explicada en el caṕıtulo (3), que explica en términos generales la naturaleza de los núcleos activos de galaxias (AGN), cuásares y el medio intergaláctico, y los absorbedores que se estudian en esta tesis: los bosques de Lyman-α y β, al igual que los bosques formados por otros absorbedores (también denominados metales), es decir, bosques de C iv, Si iv, y Mg ii. Las herramientas estad́ısticas y computacionales usadas en esta tesis se presentan en el caṕıtulo 4, que proporciona una visión general de los fundamentos teóricos de la metodoloǵıa, incluyendo los códigos y el proceso de análisis utilizado. Estas herramientas se han desarrollado, probado y v vi RESUMEN validado de manera extensiva a lo largo de los años a través de varios censos, principalmente en el contexto del bosque de Lyman-α. La última sección de este caṕıtulo hace una revisión de las publicaciones principales que han demostrado la fiabilidad del proceso. Es importante recalcar que, si bien la misma metodoloǵıa se aplicó a otros absorbedores, como los bosques de Lyman-β y metales, esta extensión no necesariamente es directa e involucró retos adicionales. El caṕıtulo 5 presenta la primera correlación cruzada medida usando el bosque de Lyman-β como trazador de la LSS. Usando los cuásares de absorción de los dato del priemr año de DESI y el proceso de análisis descrito anteriormente, estas correlaciones se calcularon en escalas de 60 Mpc/h. Asimismo se reporta un bias de b = −0.0344± 0.0012. En el futuro cercano se planea extender este análisis a escalas más grandes. Basándonos en esto, el caṕıtulo 6 preesenta un primer paso hacia la misma meta, con ab- sorbedores de metales. Se describen las propiedades atómicas de las transiciones del C iv, Si iv, y Mg ii, recalcando su naturaleza de dobletes y las complicaciones que esto introduce en la medición de la correlaciones. En particular, la estructura de doblete puede producir caracteŕısticas en las funciones de correlación que asemejan señales en distancias no f́ısicas. EStas no tienen un origen cosmológico, sino que surgen de efectos astrof́ısicos que deben ser entendidos y modelados adecuada- mente. Se describen los pasos tomados para abordar estos efectos a escalas pequeñas (40 Mpc/h and 60 Mpc/h, dependiendo de la resolución necesaria para estudiar cada bosque) y posteriormente presentar las correlaciones cruzadas correspondientes. Este caṕıtulo marca un paso importante ha- cia la adaptación de metodoloǵıas consolidadas a estos prometedores nuevos trazadores. En futuras investigaciones este trabajo se centrará en la obtención de parámetros, en particular los bias de cada transición, con la finalidad de extender este análisis a escalas mayores y evaluar el potencial de estos bosques como trazadores de BAO a diferentes corrimientos al rojo. El caṕıtulo 7 concluye la tesis resumiendo los principales hallazgos, discutiendo las limitaciones actuales, y definir las pautas para trabajos futuros. Estos incluyen un pronóstico de la señal de BAO en el bosque de C iv usando los datos completos del quinto año de operaciones de DESI y trabajando en conjunto con colaboradores en un análisis BAO usando los datos del tercer año; un trabajo aún en proceso. Abstract This thesis explores the potential of using alternative absorption features in quasar spectra as novel tracers of the Large Scale Structure (LSS) of the Universe. While the Lyman-α forest has been a well-established cosmological tool for over a decade, this work ventures into less-explored terri- tory by investigating the cosmological information encoded in the Lyman-β and metal absorption forests, such as C iv, Si iv, and Mg ii. These transitions open a new window into the intergalactic medium (IGM), with the long-term goal of evaluating their viability as Baryon Acoustic Oscillation (BAO) tracers. This is particularly interesting because galaxies provide a robust observational tool to obtain cosmological information at z ≲ 1.5, and the Lyman-α forest above z ≳ 2, leaving a gap in redshift that can potentially be covered by these new tracers. By doing so, this research contributes to the broader effort of expanding the toolkit of LSS cosmology, providing new paths to constrain fundamental cosmological parameters. This work presents the first-ever measurements of Lyman-β–quasar cross-correlations, along with the computation of a bias and an initial approach to measuring correlation functions with metals at small scales (such as 60 and 40 Mpc/h). This advancement enables the modeling of doublets and offers foundational insights into this unexplored territory. The first chapter (1) of this thesis is focused on providing the reader with a broad perspective about the state-of-the-art of Cosmology, as well as introducing the main statistical tools and ob- servational probes to understand the theoretical framework and background of this thesis research. It finally introduces the motivations and objectives of studying metal forests in the LSS. The second chapter (2) addresses the observational ground for this research: the Dark Energy Spectroscopic Instrument, a galaxy survey which aims to map the expansion history of the Universe. It describes the design and key technical aspects of the instrument and survey, along with its primary scientific objectives. It also explains the full first data assembly of the survey, which contains the data used for this research work: the absorption quasar sample observed during the first year of operations (also known as DR1). The nature of this sample is explained in Chapter 3, which broadly explains the nature of Active Galactic Nuclei, Quasars, the Intergalactic Medium, and the absorbers studied in this thesis: the Lyman-α and β forests, as well as absorber (also referred to as metal) forests, namely C iv, Si iv, and Mg ii forests. The statistical and computational tools used in this thesis are presented in Chapter 4, which provides a broad overview of the theoretical foundations of the methodology, including the codes and analysis pipeline employed. These tools have been extensively developed, tested, and validated across multiple surveys, primarily in the context of the Lyman-α forest. The final section of the chapter reviews key publications that have established the reliability of this pipeline. It is important to note, however, that while the same methodology is applied here to other absorption features, such as the Lyman-β and metal forests, this extension is not necessarily straightforward and involved vii viii ABSTRACT additional challenges. Chapter 5 presents the first ever measured cross-correlations using the Lyman-β forest as a tracer of the LSS. Using the absorption quasars of the DESI Year 1 data and the analysis pipeline described earlier, these correlations are computed on scales of 60 Mpc/h. A bias measurement of b = −0.0344± 0.0012 is reported, and future work aims to extend this analysis to larger scales. Building on this, Chapter 6 presents a first step into the same goals with metal forests. We describe the atomic properties of the C iv, Si iv, and Mg ii transitions, highlighting their nature as doublets and the complications this introduces for measuring cross-correlations. In particular, the doublet structure can produce features in the correlation function that resemble signals at non- physical distances. These are not cosmological in origin but arise from astrophysical effects that must be properly understood and modeled. We outline the steps taken to address these effects at small scales (40 Mpc/h and 60 Mpc/h, depending on the needed resolution to study each forest) and then present the resulting cross-correlation measurements. This chapter marks an important step toward adapting well-established methodologies to these promising new tracers. Future work will focus on extracting best-fit parameters, particularly the bias of each transition, in order to extend the analysis to larger scales and evaluate the potential of these forests as BAO tracers at different redshifts. Chapter 7 concludes the thesis by summarizing the main findings, discussing the current limi- tations, and outlining directions for future work. These include forecasting the BAO signal in the C iv forest using the full DESI Year 5 data and pursuing a joint BAO analysis with collaborators using Year 3 data, a project that is still in progress. Contents Acknowledgements i Resumen v Abstract vii 1 Cosmological Background 1 1.0.1 History of the Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1 Cosmological Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.1 Alternative models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Cosmological Redshift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4 Statistical Tools and Cosmological Parameters . . . . . . . . . . . . . . . . . . . . . 14 1.4.1 Power Spectrum P (k) and Correlation Functions (ξ) . . . . . . . . . . . . . . 14 1.4.2 Measured Cosmological parameters . . . . . . . . . . . . . . . . . . . . . . . . 16 1.5 Observational Probes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.5.1 Cosmic Microwave Background . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.5.2 Large Scale Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.5.3 Gravitational Lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.5.4 Redshift Space Distorsions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.5.5 Baryon Acoustic Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.5.6 Type Ia Supernovae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.5.7 Galaxy and Galaxy Cluster Kinematics . . . . . . . . . . . . . . . . . . . . . 25 1.6 Motivation and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2 The Dark Energy Spectroscopic Instrument (DESI) 29 2.1 The DESI Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.2 Instrument facilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.3 Scientific Goals of the DESI Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.3.1 Baryon Acoustic Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.3.2 Redshift Space Distorsions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.4 DESI Y1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.4.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.4.2 Cosmological Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ix x CONTENTS 3 Quasars and IGM as Large Scale Structure Tracers 45 3.1 Quasars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.1.1 Active Galactic Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.1.2 AGN spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.1.3 Quasars and QSO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.2 Intergalactic Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.2.1 Lyman-α and Lyman-β Forests . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.2.2 Absorber Forests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.3 Cosmology with the Lyman-α forest . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4 Correlation Functions from QSO and IGM Tracers 53 4.1 IGM Absorption Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.2 Alcock-Paczyński effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.3 The Flux Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.3.1 Auto-correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.3.2 Cross-correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.4 Flux Power Spectrum PF (k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.5 Pipeline Validation and Application to Data . . . . . . . . . . . . . . . . . . . . . . . 63 5 The Lyman β Forest as Tracer of the LSS 67 5.1 The Lyman-β forest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.2 Measurements of Cross-Correlations of Quasars and the Lyman β Forest . . . . . . . 68 5.3 Best Fit Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 6 The SiIV, CIV, and MgII Forests as LSS Tracers 73 6.1 Absorber forests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 6.1.1 CIV (triply ionized carbon) forest . . . . . . . . . . . . . . . . . . . . . . . . 76 6.1.2 SiIV (Triply ionized silicon) forest . . . . . . . . . . . . . . . . . . . . . . . . 76 6.1.3 MgII (Singly ionized magnesium) forest . . . . . . . . . . . . . . . . . . . . . 80 6.2 Cross-Correlations of QSO and Absorber Forests . . . . . . . . . . . . . . . . . . . . 81 6.2.1 CIV forest analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 6.2.2 SiIV forest analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 6.2.3 MgII forest analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 6.3 Best Fit Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 7 Discussion and Conclusions 91 Chapter 1 Cosmological Background Cosmology is the science that investigates the origin, structure, evolution, composition, and fate of the Universe. It is studied from different angles: from the first imprints in temperature in back- ground radiation after recombination to the pattern in which galaxies are distributed nowadays, it seeks to understand the nature of the Universe through theoretical models and observational data. The Lambda Cold Dark Matter (ΛCDM) model, currently the most accepted cosmological paradigm, provides a robust framework for describing the Large Scale Structure (LSS, see Section 1.5.2) of the Universe, dark matter (responsible for the gravitational attraction of different cosmic structures), and dark energy (to which the accelerated expansion of the Universe is attributed). However, alternative models and theories, including modifications to general relativity and the standard model of particle physics, offer different perspectives on cosmic phenomena. Complemen- tary to these theoretical models, a variety of observational probes, such as the Cosmic Microwave Background (CMB), galaxy redshift surveys (explained in Section 2), Baryon Acoustic Oscilla- tions (BAO, defined in Section 2.3.1), and the Lyman-α forest (see Section 3.2.1), offer critical insights into the past and present of the Universe. In this context, the Dark Energy Spectroscopic Instrument (DESI; see Chapter 2) survey, with unprecedented precision, plays a fundamental role. This chapter provides a foundational overview of cosmology, including a discussion of the ΛCDM model and alternative approaches that will be introduced later, setting the stage for a detailed explanation of the role that metal forests, the intergalactic medium (IGM, see Section 3.2.1), and Lyman-α forest contamination play in our understanding of cosmic structures and the evolution of the Universe. Observations indicate that the Universe is conformed by four main components: ⋆ Baryonic matter refers to all known forms of what we call ordinary matter, made by elementary particles. It interacts via the strong nuclear force (which binds atomic nuclei together), the weak nuclear force (which governs certain types of particle decay and nuclear reactions), gravitationally, and electromagnetically. The latter interaction allows it to be studied by diverse instruments, like optic and radio telescopes, interferometers, spectrographs, X-ray and gamma-ray detectors, etc. It accounts for ∼ 5% of the mass-energy of the Universe. ⋆ Radiation refers to all forms of energy that decrease with the expansion of the Universe. This includes photons and relativistic particles, such as neutrinos in the early Universe. Radiation was the dominant component in the earliest stages of cosmic history and played a major role 1 2 CHAPTER 1. COSMOLOGICAL BACKGROUND Figure 1.1: Possible explanations for the nature of dark matter (Credit: G. Bertone and T. M. P. Tait) [CERN, 2024] in shaping its evolution. ⋆ Dark matter is thought to be a form of matter that does not interact with electromagnetic radiation —meaning it neither emits, absorbs, nor scatters light—, making it detectable only through its gravitational effects on visible matter, the bending of light (gravitational lens- ing), and the dynamics of cosmic structures such as galaxies, galaxy clusters, and the LSS (see Section 1.5.2). It makes up ∼ 27% of the total mass-energy content of the Universe. Despite of thorough studies of diverse characteristics, the true nature of dark matter remains to be one of the most profound questions in modern physics. Although various theoretical models have been proposed to explain its existence, a dark matter particle has not yet been detected. Depending on the energy of the hypothetical particles that make up dark matter, different models describe their behavior. These include cold dark matter (CDM), the most favored model, consisting of non-relativistic particles that lead to hierarchical structure forma- tion—where small structures form first and later merge into larger ones. Another possibility is warm dark matter (WDM), a constrained but not entirely ruled-out model, composed of particles that were relativistic in the early Universe but are slow-moving today, suppressing small-scale structures compared to CDM. Lastly, hot dark matter (HDM) consists of highly relativistic particles that smooth out structure on small scales, preventing early galaxy for- mation. Since this contradicts observed structure formation, HDM is considered ruled out as the primary form of dark matter. There are also several theoretical models developed to ex- plain its nature, like Weakly Interacting Massive Particles (WIMPs), which are stable, weakly interacting particles predicted by supersymmetry; axions, light neutral particles proposed to solve the strong CP problem in QCD and potentially detectable via their interaction with photons; and primordial black holes (PBHs), which could have formed in the early Universe and contribute to dark matter under specific constraints [Arun et al., 2017]. Additionally, more exotic candidates have been proposed. Many of these are shown in Figure 1.1 [CERN, 2024]. ⋆ Dark Energy, a type of energy responsible for the accelerated expansion of the Universe that acts like a negative pressure that counteracts gravity. It does not interact electromagnetically 3 in any way understood nowadays, and its nature is one of the most intriguing questions of modern physics. There are several models that intend to explain it, among them modifications to General Relativity. Numerous efforts and experiments are dedicated to studying and unraveling its nature, such as the Dark Energy Survey (DES; [The Dark Energy Survey Collaboration, 2005], [Abbott et al., 2022]), the Dark Energy Spectroscopic Instrument (DESI; see Chapter 2 [Aghamousa et al., 2016a], [Aghamousa et al., 2016b]), the Vera C. Rubin Observatory Legacy Survey of Space and Time (LSST; [Ivezić et al., 2019]) the Euclid mission ([Laureijs et al., 2011]), and the upcoming Nancy Grace Roman Space Telescope ([Spergel et al., 2015], [Wenzl et al., 2022]). These projects use a variety of observational techniques, including supernovae (see Section 1.5.6), baryon acoustic oscillations (see Sections 1.5.5 and 2.3.1), weak gravitational lensing (see Section 1.5.3), and galaxy clustering (see Section 1.5.7), to probe the expansion history and structure of the Universe. 1.0.1 History of the Universe Since the 1920s, when Edwin Hubble found the first observational evidence of an expanding Uni- verse, several observations and experiments have been focused on the nature of this phenomenon. Together with Georges Lemâıtre, this led to the development of the Hubble-Lemâıtre equation, which describes this expansion mathematically: the velocity v at which a galaxy is moving away from us due to this effect is given by the distance to the galaxy multiplied by the present expansion rate H0 (the Hubble constant), so that v = H0d. This equation has been crucial in understanding the expansion dynamics of the Universe over time starting with a hot, dense early Universe. The measurement of the expansion rate across different epochs has provided valuable insights into the various evolutionary stages of the Universe since the Universe was smaller, hotter, and more ener- getic. These stages are illustrated in Figure 1.2 [NOIRLab, 2021], which highlights key periods in the history of the Universe that are essential for reconstructing its life story. ⋆ Inflation: this initial period of the Universe happened when it was 10−34 sec to 10−32 sec old. During this epoch, it experienced nearly exponential growth, driven by the potential energy of a scalar field known as the inflaton. After this expansion, the inflaton field decayed, transferring its energy into particles and radiation in a process known as reheating. During this transition, quantum fluctuations in the inflaton field were stretched to cosmological scales and became the seeds of primordial density perturbations in the relativistic plasma, which later grew into the large-scale structure we observe today. ⋆ Recombination: the primordial plasma, composed mainly of protons and electrons coupled with photons through Thomson scattering, served as the medium in which temperature fluctu- ations propagated as sound waves producing density fluctuations. As the Universe expanded, its temperature dropped until photons decoupled from protons and electrons, allowing the formation of hydrogen atoms. This caused the patterns made by the density fluctuations to freeze, and leave an imprint in this first emitted light, known as the Cosmic Microwave Background Radiation (CMB), as well as in baryonic matter distribution, which began to clump due to gravitational attraction, falling into dark matter gravity potential wells, even- tually evolving into larger structures and forming the Baryon Acoustic Oscillations (BAO, explained in more detail in Section 2.3.1). These oscillations are observable in the distri- bution pattern of galaxies within the Large Scale Structure (LSS, see Section 1.5.2) of the Universe. 4 CHAPTER 1. COSMOLOGICAL BACKGROUND ⋆ Dark Ages and reionization: after recombination, there is a period called Dark Ages, in which the Universe is not observable in any wavelength, except for some emissions in the Hydrogen 21cm line. During this period, matter collapses gravitationally into overdense regions, gradually giving rise to the first structures. These regions lead to the creation of dark matter halos, the formation of gas clouds (also collapsing gravitationally), and the emergence of the first stars. The radiation from these stars and early Active Galactic Nuclei (AGN) ionized the Intergalactic Medium (IGM). The phase of the Universe when the neutral hydrogen was ionized by the first generation of stars is called reionization. ⋆ Accelerated Expansion of the Universe: after several gigayears (Gyr), the expansion of the Universe began to accelerate, presumably due to the negative pressure attributed to the dominance of dark energy (or the cosmological constant Λ) over radiation and matter, counteracting gravitational attraction. Figure 1.2: Timeline of the history of the Universe. Of particular interest, one can identify the stages of inflation (right after the Big Bang and before the emission of the CMB), recombination (associated with the CMB map), dark ages, reionization, and accelerated expansion of the Universe (labeled implicitly under the term ”modern galaxies”). Credit: N.R.Fuller, National Science Foundation. [Drake, 2018] 1.1 Cosmological Model The observations explained in Section 1.5 support the so called ΛCDM cosmological model, which is the current standard model in cosmology. It is based on General Relativity and the cosmological 1.1. COSMOLOGICAL MODEL 5 principle, that states that the distribution of matter and radiation is homogeneous and isotropic at large scales, and establishes a Universe described by the Friedmann-Lemâitre-Robertson-Walker (FLRW) metric: ds2 = c2dt2 − a(t)2 ( dr2 1− κr2 + r2dθ2 + r2 sin2 θdϕ2 ) . (1.1) The first term of this expression represents the time dimension, while the next three correspond to the spatial components. Here, r, θ, ϕ are the spherical coordinates, and κ = −1, 0, 1 denotes the curvature of the Universe (κ = −1 indicates an open Universe, κ = 0 a flat Universe, and κ = 1 a closed Universe). The term a(t), known as the scale factor, describes the “size” of the Universe relative to the present time, where a = a0 = 1. The starting point for deriving the Friedmann equation is Einstein’s field equations: Gµν + Λgµν = 8πG c4 Tµν , (1.2) where Gµν is the Einstein tensor, Λ is the cosmological constant, gµν is the metric tensor, and Tµν is the energy-momentum tensor. Assuming a perfect fluid form for Tµν and applying the FLRW metric, one solves the (0, 0) component (the time-time component) to obtain the Friedmann equation: H2 = ( ȧ a )2 = 8πGρ 3 − κc2 a2 + Λ 3 , (1.3) with H = ȧ a representing the Hubble parameter, which determines the expansion rate of the Uni- verse, G as the newtonian gravitational constant, and ρ is the total energy density of the Universe. In the rest of this work, we adopt natural units where c = 1. In addition to the Friedmann equation, the Einstein field equations applied to the FLRW metric, together with the covariant conservation of the energy-momentum tensor ∇µT µν = 0, yield the continuity equation: ρ̇ = −3H(ρ+ p), (1.4) which describes the evolution of the energy density ρ of a cosmological fluid with pressure p related by an equation of state of the form p = wρ1, where w is a constant (w = 0 for matter, w = 1/3 for radiation). Using Equations 1.3 and 1.4 we obtain a third one: Ḣ = −4πG(ρ+ p) + κ a2 . (1.5) The ΛCDMmodel describes a universe composed of regular matter, as described by the Standard Model of Elementary Particles, alongside an unidentified form of non-relativistic matter known as Cold Dark Matter (CDM), which interacts solely through gravity. Additionally, the model accounts for accelerated expansion, attributed to a cosmological constant Λ, characterized by an equation of state given by w = p/ρ = −1, where p denotes pressure and ρ represents density. So, considering the equation: 1This relation does not hold in general; it is an approximation valid for ideal fluids with a fixed equation of state. 6 CHAPTER 1. COSMOLOGICAL BACKGROUND ρ = ρi ( a ai )−3(w+1) , (1.6) which is valid for any fluid with a constant w, it follows that a cosmological constant, with w = −1, remains constant with time. In this case, the energy density and pressure are given by ρΛ = Λ 8πG , pΛ = −ρΛ We define the critical density ρcrit as the average matter density needed for the Universe to have a flat geometry (k = 0), assuming the cosmological principle. Following Equation 1.3, the critical density is defined as: ρcrit = 3H2 8πG , (1.7) with H0 the Hubble constant, and G the newtonian gravitational constant. The comparison between the different components of the Universe and the critical density is given by: Ωi = ρi/ρcrit, (1.8) where i refers to ordinary matter, dark matter, dark energy, etc. Ωtotal = ∑ i Ωi = 1 for a flat universe, Ωtotal < 1 for an open universe, and Ωtotal > 1 for a closed one. Observations from the CMB (see Sections 1.4.2 and 1.5) indicate that the Universe has a ge- ometry very close to flat, and that its components are approximately in the proportions shown in Figure 1.3: ⋆ Ωb ≈ 0.05: around 5% is composed of ordinary matter.2 ⋆ ΩDM ≈ 0.25: 25% dark matter, a type of matter that interacts only gravitationally with ordinary matter. ⋆ ΩΛ ≈ 0.7: the remaining 70% is Λ, dark energy, responsible for the accelerated expansion of the Universe at an accelerated rate. Depending on the observations used to obtain these results, the values can vary slightly. 2Also called baryonic matter, and refers to the matter that we know. 1.1. COSMOLOGICAL MODEL 7 Figure 1.3: Pie chart illustrating the components of the universe and their approximate proportions according to the ΛCDM model. Around 5% is composed of ordinary matter, 25% dark matter, and the remaining 70% is dark energy. Depending on the probes used to obtain these results, the values can vary slightly. [DESI Collaboration, 2025a] 1.1.1 Alternative models Aside from the ΛCDM model, there is a wide variety of alternative models that try to explain all the cosmological observations consistently. Some alternative models focus on modifying the properties of dark energy. In the ΛCDM model, dark energy is described by a cosmological constant with a fixed equation of state w = −1. However, dynamical dark energy models, such as quintessence, allow the equation of state to evolve over time, offering a more flexible approach to fitting cosmological data. Another class of alternatives, known as modified gravity models, proposes that the accelerated expansion of the Universe may result from deviations from general relativity, rather than invoking dark energy. Examples of such models include f(R) gravity. These alternative models aim to address the limitations of ΛCDM and provide new perspectives on the fundamental nature of the Universe. Among the popular models, we find the w0wa model, a generalization of the ΛCDM model that allows a more flexible description of dark energy. The w0wa model introduces two parameters, w0 and wa, to describe the equation of state of dark energy, which can evolve over time. The equation of state in the w0wa model is given by: w(z) = w0 + wa z 1 + z (1.9) where z is the redshift, w0 is the present value of the dark energy equation of state, and wa char- acterizes its possible time evolution. The w0wa model allows for dynamical dark energy, meaning that w can evolve as the Universe expands. This flexibility enables the w0wa model to better fit a broader range of cosmological data, especially in cases where observations suggest deviations from the simple cosmological constant hypothesis, providing a more general framework for testing cos- 8 CHAPTER 1. COSMOLOGICAL BACKGROUND mological theories, helping to refine our understanding of the expansion history of the Universe, and the role and nature of dark energy [Chevallier et al., 2001], [Linder, 2003]. However, ΛCDM has been the most popular and, as said before, it is the current standard model in Cosmology. All the methodology and results of this research work assume this model. 1.2 Cosmological Redshift In the 1920s Edwin Hubble observed that nearly all galaxies appear to move away from us, and that their recession velocities increase proportionally to their distance from us. This discovery leads to the conclusion of the expansion of the Universe. The recession velocity can be measured by means of the shift of the light towards the red end of the spectrum. This is called cosmological redshift. Objects that are farther away will have a larger recession velocity and hence a larger redshift. Due to the change in energy levels of electrons, photons are emitted by sources of radiation (or absorbed by the medium between the source and the observer) by different elements of the periodic table at specific and unique wavelengths. These emissions and absorptions have an energy given by the equation E = hν = h/λ, where h is the Planck constant, ν is the frequency of the emission or absorption, and λ is the corresponding wavelength, with c = 1. As the Universe expands, the wavelength of these photons that propagate freely increases, as all physical distances expand. This causes the wavelength of the photons to shift to the red end of the electromagnetic spectrum. This implies that this red-shift of a photon is due to the fact that the Universe was smaller when the photon was emitted. The wavelength of a photon is inversely proportional to the scale factor. Thus, the wavelength at time t0, denoted by λ0, will differ from that at time t1, denoted by λ1, according to λ1 λ0 = a(t1) a(t0) . In observational cosmology, when the source of the emission or absorption moves relative to the observer, the wavelength appears shifted: towards the blue end of the electromagnetic spectrum (blueshifted) if the source is approaching, or toward the red end (redshifted) if the source is receding. Observations of galaxies show that their spectra are redshifted, which is evidence of the expansion of the Universe. To define the redshift z, we consider a light beam emitted by a source at time ti from position ri. This beam is detected by the observer at position r = r0 = 0 at time t0. According to General Relativity, ds2 = 0 by definition, so from the FLRW metric: dt = a(t)dr√ 1− κr2 , (1.10) and ∫ t0 ti 1 a(t) dt = ∫ r0 ri 1√ 1− κr2 dr. (1.11) Let us also suppose that the wave begins to be emitted at time ti (and will be observed at t0), and at time ti + δti, a different “part” of the same wave is emitted (which will be observed at t0 + δt0). Since r0 and ri are comoving, then: ∫ t0+δt0 ti+δti 1 a(t) dt = ∫ r0 ri 1√ 1− κr2 dr, (1.12) hence 1.3. DISTANCES 9 ∫ t0 ti 1 a(t) dt = ∫ t0+δt0 ti+δti 1 a(t) dt. (1.13) If we sum ∫ ti+δti t0 1 a(t) dt to both sides, we obtain: ∫ ti+δti ti 1 a(t) dt = ∫ t0+δt0 t0 1 a(t) dt. (1.14) We note that δt can be as small as needed, then the change in a(t) is negligible in the t, t + δt interval, so it is considered constant. Hence, following the discussion in Section 1.40: δt0 δti = a(t0) a(ti) = λ0 λi . (1.15) From this expression we see that if a(t0) > a(ti), than λ0 > λi. This means that the observed wavelength is longer then it was when emitted (i.e. redshifted). This shift is a consequence of the expansion of the Universe, and objects that present this behavior are tracers of this expansion. This cosmological redshift (or simply redshift) z is defined as: a(t0) a(ti) = 1 + z = λ0 λi (1.16) z = λ0 λi − 1 = λ0 − λi λi (1.17) Due to the expansion of the Universe, objects that are farther away have higher receding ve- locities and, therefore, greater redshifts. Since light from high-redshift galaxies was emitted when the Universe was younger, observing these galaxies allows us to study the evolution of galaxies over cosmic time. Statistically, high-redshift galaxies are the progenitors of present-day galaxies, and any changes in galaxy number density or intrinsic properties with redshift provide a direct window into the formation and evolution of the galaxy population. With modern large telescopes, we can observe galaxies at z > 10, enabling us to explore the galaxy population from a time when the Universe was only about 10% of its current age. 1.3 Distances This section is based on material extracted from [Muñoz-Gutiérrez et al., 2021]. The FLRW metric (Equation 1.1) provides a framework for calculating cosmological distances. Following the discus- sions in [Kolb et al., 1990] and [Mo et al., 2010], dΩ = r2dθ2 + r2 sin2 θ dϕ2 defines the angular line element on a unit sphere. When scaled by r2, it represents angular separations in spherical coordi- nates. Setting dΩ = 0 corresponds to aligning the coordinate axes in a convenient way, restricting the analysis to purely radial motion, simplifying cosmological distance calculations. We then define the following distances: 10 CHAPTER 1. COSMOLOGICAL BACKGROUND Comoving Distance The comoving distance ∆r quantifies the separation between two points r1 and r2 in the Universe, factoring out its expansion. It remains constant with time for objects moving with the Hubble flow (see Section 1.5.4). For a photon traveling radially (i.e., dθ = dϕ = 0) along a null geodesic (ds2 = 0) in the FLRW metric: ds2 = −dt2 + a(t)2 dr2 1− κr2 = 0, (1.18) we obtain the relation: dr√ 1− κr2 = dt a(t) . (1.19) Integrating both sides gives the comoving distance ∆r between times t1 and t2: ∆r = ∫ t2 t1 dt a(t) . (1.20) Alternatively, one can express the comoving distance as a spatial integral over the radial coor- dinate: ∆r = ∫ r2 r1 1√ 1− κr2 dr, (1.21) which yields: ∆r =    r2 − r1 if κ = 0, arcsin(r2)− arcsin(r1) if κ = +1, arcsinh(r2)− arcsinh(r1) if κ = −1. (1.22) This distance represents the coordinate separation between two points, unaffected by cosmic ex- pansion, and serves as a foundational concept in defining other cosmological distances. Physical Distance The physical distance is the real distance between two points with comoving coordinates r1 and r2, and does change with the expansion of the Universe. It is given by: D = ∫ r2 r1 dl, (1.23) with dl = a(t)√ 1−κr2 dr the spatial part of Equation 1.1, hence: D(t) = a(t) ∫ r2 r1 1√ 1− κr2 dr (1.24) D(t) = a(t)∆r. (1.25) For two given times ta and tb, the physical distances are: 1.3. DISTANCES 11 D(ta) = a(ta) ∫ r2 r1 1√ 1− κr2 dr, (1.26) D(tb) = a(tb) ∫ r2 r1 1√ 1− κr2 dr. (1.27) Using these two expressions, and considering that comoving distance does not change with the expansion of the Universe, the physical distance at time ta, D(ta), can be determined if the scale factor a is known at both ta and tb, along with the physical distance at tb, D(tb): D(ta) = a(ta) a(tb) D(tb). (1.28) Proper Distance The proper distance, which refers to the distance a light beam travels, is given by: dp(t) = c ∫ t ti dt = c(t− ti). (1.29) For a light beam, ds2 = 0 (as light travels along null geodesics). If dϕ = 0 and dθ = 0, we have: dt = a(t)dr√ 1− κr2 , (1.30) so: dp(t) = a(t) ∫ r2 r1 1√ 1− κr2 dr = a(t) ∫ t ti 1 a(t′) dt′. (1.31) If we set ti = 0 as the initial time and t0 as the current time, the proper distance becomes the distance to the horizon, which defines the region of the Universe that is causally connected to an observer: dH(t0) = a(t0) ∫ t0 0 1 a(t) dt. (1.32) During different epochs of the Universe, various components such as radiation, matter, and dark energy coexist, each influencing its expansion. The distance to the horizon depends on the relative contributions of these components. Using the Friedmann equation: H2 = 8πGρ 3 − κ a2 , (1.33) where the total density ρ is the sum of the densities of radiation, matter, and dark energy: ρ = ρr + ρm + ρΛ = ρr0 ( a a0 )−4 + ρm0 ( a a0 )−3 + ρΛ0 . (1.34) Multiplying and dividing this expression by the critical density today ρc0 = 3H2 0 8πG , and defining Ωi = ρi ρc , we have: 12 CHAPTER 1. COSMOLOGICAL BACKGROUND ρ = ρc0 ( Ωr0 ( a a0 )−4 +Ωm0 ( a a0 )−3 +ΩΛ0 ) , (1.35) so the Hubble parameter can be expressed as: H = H0 √ Ωr0 ( a a0 )−4 +Ωm0 ( a a0 )−3 + ( a a0 )−2 (1− ΩT0 ) + ΩΛ0 . (1.36) If the scale factor today is a0 = 1 and Ωκ0 = 1−ΩT0 , we can write the equation for the horizon distance as: ∫ dt a(t) = 1 H0 ∫ da a2 1 √ Ωr0a −4 +Ωm0 a−3 +Ωκ0 a−2 +ΩΛ0 (1.37) Luminosity Distance Certain objects in the Universe, such as variable stars and Type Ia supernovae (SNe Ia, see Section 1.5.6), have well-understood intrinsic luminosity L that can be determined from their observed flux F . By establishing a relationship between the flux and the luminosity of these objects, we can estimate their distance from us. This concept is referred to as the luminosity distance dL. The relationship between F and L is given by: F = L 4πd2L , (1.38) where L is: L = E δt . (1.39) In this expression, E represents the energy of the photons involved, and δt denotes a specific time interval. The observed flux changes due to the expansion of the Universe. Thus, for light: δt0 δt1 = λ0 λi = a0 a(t1) = E1 E0 , (1.40) where the quantities with the subscript 0 are measured at the present time, while those with the subscript 1 correspond to the time of emission. If we observe a luminosity on Earth at the present time t0, it would be given by: L0 = E0 δt0 . (1.41) From Equation (1.40), E1 = a0 a(t1) E0, (1.42) and δt1 = a(t1) a0 δt0, (1.43) 1.3. DISTANCES 13 then L0 = E0 δt0 = ( a(t1) ao )2 E1 δt1 = ( a(t1) ao )2 L1. (1.44) As mentioned earlier, the physical distance is related to the comoving distance through the scale factor. For a given time t1 and the present time t0, this relationship can be expressed as: d(t1) = a(t1)∆r, and d0 = a0∆r, (1.45) so for t1: d(t1) = a(t1) a0 d(t0) = a(t1)∆r. (1.46) The flux observed at the present time is: F0 = L0 4πd2o = ( a(t1) ao )2 L1 4πa20(∆r)2 = L1 4πa20(∆r)2(1 + z)2 , (1.47) and the luminosity distance for an expanding universe is defined as: dL = a0∆r(1 + z). (1.48) From the Friedmann equation: H2 = H2 0 ( 8πGρ 3H2 0 − κ a2H2 0 ) = H2 0 ( ∑ i Ω0i ( a a0 )−3(1+wi) − κ a20H0 ) , (1.49) where the sum is over all the components of the Universe (baryonic matter, dark matter, radiation, and dark energy). Substituting κ/a2H0 in terms of ΩT : H2 = H2 0 ( ∑ i Ω0i ( a a0 )−3(1+wi) + (1 + ΩT ) ( a a0 )−2 ) (1.50) For light, ∆r = ∫ t0 t1 dt a(t) . With this, the comoving distance turns into: ∆r = ∫ da a2H0 ( ∑ i Ω0i ( a a0 )−3(1+wi) + (1 + ΩT ) ( a a0 )−2 )(1/2) (1.51) and the luminosity distance into: dL = a0(1 + z) a0 ∫ a a0 da a2 ( ∑ i Ω0i ( a a0 )−3(1+wi) + (1 + ΩT ) ( a a0 )−2 )(1/2) . (1.52) 14 CHAPTER 1. COSMOLOGICAL BACKGROUND Angular Distance Lets consider an object of size D, at a distance d, that subtends an angular size θ. The angular- diameter size dA is given by: dA = D θ . (1.53) To express dA in terms of the FLRW metric, D represents the proper distance between two light signals emitted from two points with the same radial coordinate re at a time te, reaching the observer at t0. Since the physical (proper) size of the object at emission is smaller than its comoving size by a factor of a(te)/a0 = 1/(1 + z), the proper transverse separation between two points separated by angle θ is given by D = aereθ. D will then be the integral of the spatial part of the metric (1.1): D = aereθ = a0re 1 + z θ (1.54) with a0 = a(t0) and ae = a(te). Substituting this result in Equation 1.53, dA = a0re 1 + z = aere (1.55) In a space with no expansion, dA = dL = d, but in the cosmological context these values differ. 1.4 Statistical Tools and Cosmological Parameters There is a set of fundamental quantities that describe the key properties and dynamics of the Universe at large scales. These are known as cosmological parameters. They characterize key aspects, like the expansion rate of the Universe, its composition (contents of matter, energy, dark matter, etc.), geometry, and the evolution of cosmic structures. They set the theoretical basis to interpret observations such as the cosmic microwave background radiation (CMB, see Section 1.0.1), galaxy distribution (see Section 1.5.2), and gravitational lensing (see Section 1.5.3), among others. Measuring these parameters with high precision allows us to understand the current state of the Universe, as well as to reconstruct its history evolution and possible fate. This makes them an essential link between fundamental theories, cosmological models, and cosmological and astronomical observations. The following subsections will introduce the measured cosmological parameters. Before doing so, a couple of statistical tools used to determine them will be defined: the power spectrum and the correlation function. 1.4.1 Power Spectrum P (k) and Correlation Functions (ξ) As explained further in Section 1.5.2, at large scales, matter in the Universe is distributed in different structures, such as clusters of galaxies, filaments, voids, etc., forming the sponge-like- structure called the cosmic web. These structures do not have a random distribution, but follow patterns that can be measured with statistical tools that help us quantify how galaxies and gas group together, and how this changes in different evolutionary stages of the Universe. 1.4. STATISTICAL TOOLS AND COSMOLOGICAL PARAMETERS 15 Among these statistical tools we find the 2-point correlation function ξ, which measures the excess probability to find galaxy pairs separated by a given distance r, with respect to a ran- dom distribution, and the power spectrum P (k), the Fourier transform of ξ, which measures the amplitude of density fluctuations as function of wavenumber k. Correlation Functions ξ(r) As previously mentioned, the correlation function ξ(r) quantifies how much more likely it is to find galaxy pairs separated by a distance r compared to a purely random distribution of galaxies. This quantifies galaxy clustering and provides a direct estimation of their distribution in real space. There are several estimators of the correlation function [Vargas-Magaña et al., 2013]. The most commonly used expression for the correlation function is the Landy & Szalay estimator: ξ(r) = DD(r)− 2DR(r) +RR(r) RR(r) , (1.56) where D means data, and R means random, DD is the number of galaxies separated by a distance r, and DR the number pairs of real galaxies separated at a distance r from one in a random catalog distribution. Power Spectrum P (k) As introduced before, the power spectrum P (k) measures the amplitude of density fluctuations as a function of the wavenumber k, representing the distribution of matter in the Universe on different scales. This is expressed by the relation P (k) = ⟨|δk|2⟩, where δk represents the Fourier modes of the density fluctuation field. This specific form describes the matter power spectrum, which quantifies how matter clusters in Fourier space. In general, however, power spectra can be defined for a variety of cosmological observables, such as temperature anisotropies in the cosmic microwave background or galaxy number counts. Interpretation of P (k) The wavenumber k is associated with spatial scales, where smaller values of k correspond to larger scales, and larger values of k correspond to smaller scales. The shape and amplitude of the power spectrum P (k) are determined by the primordial power spectrum Pprim(k) set by inflation, and its evolution through the transfer function T (k)3, which encodes the physical processes (such as radiation pressure, matter-radiation equality, and baryon acoustic oscillations) that depend on the cosmological parameters defined previously, so that the present-day matter power spectrum is: P (k) = Pprim(k) ∝ T 2(k) (1.57) The shape of the power spectrum features a turnover at small k (large scales, k ≲ 0.1h/Mpc), marking the transition from the linear to the nonlinear regime of structure formation. The slope at intermediate scales (0.1 ≲ k ≲ 1h/Mpc) gives information about the matter content of the Universe, and the shape of P (k) at large k (small scales, k ≳ 1h/Mpc) is affected by non-linear gravitational evolution and galaxy formation. The amplitude of P (k) depends on Ωm and the amplitude of 3T (k) accounts for how perturbations of different scales grow differently due to the expansion history and contents of the Universe (dark matter, baryons, radiation, etc.). 16 CHAPTER 1. COSMOLOGICAL BACKGROUND primordial density fluctuations, and is affected by how galaxies trace (dark) matter distribution, given by the galaxy bias b. Since P (k) is the Fourier transform of ξ(r), both contain the same fundamental information, but each function represents it in a manner tailored to different types of analysis. To transform from Fourier to real space (from ξ(r) to P (k)) we integrate the power spectrum over all wavenumbers: ξ(r⃗) = 1 (2π)3 ∫ P (k)eik⃗·r⃗d3k⃗. (1.58) Conversely, from real to Fourier space (from P (k) to ξ(r)) we integrate the correlation function over all space: P (k⃗) = ∫ ξ(r)e−ik⃗·r⃗d3r⃗. (1.59) The measurement of P (k) and ξ(r) at different redshifts provides valuable information about the expansion of the Universe and the growth of structure. 1.4.2 Measured Cosmological parameters There are several fundamental and derived quantities that describe different aspects of the Universe, such as its geometry, composition, and evolution. These encompass information about the energy density of matter, radiation, dark matter, dark energy, etc, primordial perturbations, the rate of expansion of the Universe, among others, and they define the basis of the cosmological models. This information is measured and constrained by different observations (explained in the next section, 1.5) and, as described before, some are embedded in P (k) and ξ(r). These measurements test theoretical models and predictions. The most important ones, classified by their interpretation are: ★ Density parameters (see Equation 1.8 for the definition of Ωi): ⋆ Ωb: baryonic matter density parameter. ⋆ Ωm: total (baryonic and dark) matter density parameter. ⋆ Ωr: radiation density parameter, which includes Ωγ - photons, and Ων - relativistic neutrinos. ⋆ ΩΛ: dark energy density parameter. ⋆ Ωk: curvature parameter. Represents the spatial curvature of the Universe. Ωk = 0 corresponds to a flat Universe, Ωk > 0 implies a negative curvature, a hyperbolic geometry, and Ωk < 0 represents a positive curvature, which is a spherical geometry. In the ΛCDM model, Ωk ≡ 0. As a result, the total energy density satisfies ΩT = 1. This means that only the density parameters Ωb, Ωm, ΩΛ, and Ωr are needed to describe the energy content of the Universe. However, since ΩT = 1, only three of them are independent. For example, Ωr can be expressed as: Ωr = ΩT − (Ωb +Ωm +ΩΛ) = 1− (Ωb +Ωm +ΩΛ). (1.60) ★ Expansion rate parameters: 1.4. STATISTICAL TOOLS AND COSMOLOGICAL PARAMETERS 17 ⋆ H0: the Hubble constant measured at the current stage (z0 = 0). According to Planck Collaboration, 2020, H0 ≈ 67.9 km s Mpc . ★ Primordial perturbations: ⋆ As 4 is the amplitude of primordial scalar perturbations at a given scale k∗. It describes the overall strength of density fluctuations in the early Universe. ⋆ ns is the spectral index. While As denotes the amplitude of primordial scalar perturba- tions, ns describes how the amplitude changes with scale of the perturbations. ns ≈ 1 corresponds to nearly scale-invariant fluctuations. ⋆ σ8, a derived parameter, quantifies the amplitude of matter fluctuations at scales of 8Mpc/h today. It is a normalization of the matter power spectrum representing the root mean square of such scales. They are derived from As after considering effects of structure formation and cosmic evolution. ★ Dark energy properties: ⋆ w0 is the parameter for dark energy in the equation of state. In the equation w = p/ρ, it represents the cosmological constant when w = w0 = −1. ⋆ wa is a parameter that represents the time dependence of the dark energy equation of state. ★ Reionization: ⋆ τ is the optical depth to reionization, the quantification of the cumulative effect due to scattering of the CMB (see Section 1.0.1) photons due to Thomson scattering along the line of sight (LOS). This measures the ionization of the neutral hydrogen in the intergalactic medium (see Chapter 3) after the first stars and galaxies formed. While τ is not a fundamental cosmological parameter, it is a critical nuisance parameter in CMB studies: its value affects the determination of As, and provides important information about the formation of the first stars and galaxies. ★ Early-Universe Derived Scales: ⋆ rd is the comoving sound horizon at the drag epoch, representing the maximum dis- tance sound waves could travel in the photon–baryon plasma before baryons decoupled from radiation (see Section 1.5.5). It serves as a standard ruler in BAO measurements and depends on early-universe parameters such as Ωb, Ωm, and H(z) before recombina- tion. Though not a fundamental parameter, it is a key derived quantity in constraining cosmological models through LSS observations. According to the results reported by Planck in 2018 [Planck Collaboration, 2020], the values of these parameters are the ones shown in Table 1.1. 4Also ∆2 R(k), dimensionless power spectrum of primordial curvature perturbations, or related to PR(k), primordial power spectrum, to which As is a normalization, as PR(k) = As( k ks )ns−1. 18 CHAPTER 1. COSMOLOGICAL BACKGROUND Parameter Planck (2018) cosmology (TT, TE, EE + lowE + lensing)5 Ωmh2 0.14297 +Ωch 2 0.12 +Ωbh 2 0.02237 +Ωνh 2 0.0006 h 0.6736 ns 0.9649 109As 2.100 Ωm 0.31509 Ωr 7.9638× 10−5 σ8(z = 0) 0.8119 rd [Mpc] 147.09 rd [h−1Mpc] 99.08 Table 1.1: Cosmological parameters from Planck 2018 [Planck Collaboration, 2020], derived using the full-mission temperature and polarization power spectra (TT, TE, EE), low-multipole polariza- tion data (lowE), and measurements of CMB lensing. 1.5 Observational Probes The widespread acceptance of the ΛCDM model is due to its ability to successfully explain nu- merous phenomena. While Λ explains the accelerated expansion of the Universe, CDM explains the observed kinematics in galaxies and clusters, which differ from those expected if only baryonic matter were considered. As explained before, the Universe has been through a number of evolutionary stages. Each of them has been studied by diverse methods, and has provided probes that support the robustness of the ΛCDM model. Following the timeline presented in Section 1.0.1, the most relevant probes are described in the following subsections: 1.5.1 Cosmic Microwave Background As explained in Section 1.0.1, the Cosmic Microwave Background Radiation (CMB) are the photons that decoupled from matter when e− and existing nuclei combined to form the first neutral atoms. At that point, photons ceased to interact significantly with these particles and began to travel freely through space, permeating all the Universe. Nowadays, this first light is detected in the wavelength range of the microwaves, and provides very valuable information of the early Universe. It was predicted by George Gamow, Ralph Alpher, and Robert Herman in 1948, and first detected by Arno Penzias and Robert Wilson in 1964, which earned them the Nobel prize in Physics in 1978. Derived from this event, a variety of experiments were built to analyze this radiation: ground- based experiments, balloons, planes, and satellites. The most relevant are COBE (Cosmic Back- ground Explorer [Smoot et al., 1992]), which obtained the first all-sky map of the CMB, led by George Smoot and John Mather (who also got a Nobel prize for their results), BOOMERanG (Balloon Observations Of Millimetric Extragalactic Radiation and Geophysics [Netterfield et al., 2002]), MAXIMA (Millimeter Anisotropy eXperiment Imaging Array [Rabii et al., 2006]), ACBAR 1.5. OBSERVATIONAL PROBES 19 (Arcminute Cosmology Bolometer Array Receiver [Runyan et al., 2003]), WMAP (Wilkinson Mi- crowave Anisotropy Probe [Bennett et al., 2003]), and Planck [Tauber et al., 2006]. Figure 1.4 shows the full-sky map of the temperature anisotropies in the CMB as measured by the Planck mission, which provides a visual representation of the initial density fluctuations that seeded LSS in the Universe. Figure 1.4: Map of the Cosmic Microwave Background as seen by the Planck mission, as part of the results of Planck 2018 [Planck Collaboration, 2020]. Figure taken from [European Space Agency, 2018a]. 1.5.2 Large Scale Structure The distribution of galaxies in the Universe shows a variety of structures, and contains information about the distribution of matter, the evolutionary history of the Universe and serves as a powerful cosmological probe. Following the Cosmological Principle, the Universe, at large scales, is considered to be homo- geneous and isotropic. This is true at scales larger than 100Mpc/h, which can be seen in the Millenium-II simulation in Figure 1.5 [Max Planck Institute for Astrophysics, 2009]. At smaller scales, we can find galaxies in different types of structures: ⋆ Groups: contain a few to tens of galaxies ⋆ Clusters: contain several hundreds to thousands of galaxies ⋆ Walls or sheets: large, sheet-like structures made of galaxies and groups of galaxies that surround voids. ⋆ Filaments: elongated regions of galaxies in thread-like structures formed by intersections of walls. ⋆ Voids: regions up to 100 Mpc/h in diameter with very few to no galaxies surrounded by walls/sheets and filaments. 20 CHAPTER 1. COSMOLOGICAL BACKGROUND Figure 1.5: Zoom sequence from 100 Mpc/h to 5 Mpch of the Millenium-II Simulation, an N-body simulation of the dark matter in the Large Scale Structure, assuming a ΛCDM model. Figure taken from [Max Planck Institute for Astrophysics, 2009] At z < 2 a large percentage (considered to be ≈ 80− 90% [European Space Agency, 2018b]) of the baryonic matter is found as intergalactic gas. The reconstruction of the Cosmic Web requires mapping it, especially at redshifts above 2. The most abundant element conforming the IGM (see Section 3) is HI, and it helps to make 3D maps of its distribution observing the absorption of light of background quasars. These are called tomographic studies, and there are numerous efforts dedicated to them. Among them, the COSMOS Lyman-Alpha Mapping And Tomography Observations (CLAMATO) survey [Lee et al., 2018], illustrated in Figure 1.6 which aims to map the Cosmic Web at z = 2.3, constructing three-dimensional maps of the IGM through Lyman-α forest tomography, extending beyond traditional H i Lyman-α forest analyses. 1.5. OBSERVATIONAL PROBES 21 Figure 1.6: 3D visualization of the CLAMATO (COSMOS Lyman-Alpha Mapping And To- mography Observations [Hess et al., 2018]) tomographic map of the IGM COSMOS field, at 2.05 < z < 2.55. The box is 20 Mpc/h per side, the shades of blue correspond to densities of δrecF = −0.08 and δrecF = −0.18, and pale yellow dots are galaxy positions. This image is sourced from the interactive figure in Müller, 2017. 1.5.3 Gravitational Lensing The light from distant sources is distorted by massive structures that lie in its path. These distor- tions behave analogously to those produced by optical lenses, and depending on the patterns made by such structures, their mass distribution can be reconstructed. This allows a better understand- ing of the nature of CDM. There are two types of gravitational lenses: Strong lenses: as a result of the deflection of light by a massive foreground object, the shape of a background galaxy can appear significantly distorted. This often results in multiple images of the source, but strong lensing can also occur when only a single highly magnified and distorted image is observed — for example, an arc with no simple linear transformation from the original shape. An extreme case of this is an Einstein ring, in which the background source appears as spread in a circular or ring-like pattern due to perfect alignment between the source, lens, and observer, as in the illustration at the left in Figure 1.7. Weak lenses: the light of the background source presents minor distortions, which are about 1%, as shown in the right illustration of Figure 1.7. These distortions can be measured statistically, examining correlations of shapes of galaxies measured in galaxy surveys, providing a direct mea- surement of the underlying mass distribution. Furthermore, by means of different methods, the intrinsic morphology of the background sources can be reconstructed [Courteau et al., 2014]. 22 CHAPTER 1. COSMOLOGICAL BACKGROUND Figure 1.7: Representation of strong (left) and weak (right) gravitational lensing. This figure illustrates the distortion of the shapes of the field galaxies shown in the center of the image by matter near the line of sight. Credits: ESA, [Euclid Consortium, n.d.]. 1.5.4 Redshift Space Distorsions Cosmic cartography is defined by three dimensions: right ascension (RA), declination (DEC, δ), and the third dimension parametrized by redshift z. As seen in Section 1.2, these quantities are related by the Hubble expansion. The measurement of z has two components: cosmological redshift, caused by the expansion of the Universe, and an additional shift due to the peculiar velocity of the object - its motion relative to the Hubble flow6. These peculiar velocities arise from gravitational interactions with nearby structures and introduce Doppler shifts in the observed redshift. The peculiar velocity v is a tracer of the mass in the LSS [DESI Collaboration, n.d.(b)]. Due to random motions on small scales, particles at the same distance have slightly different redshifts, which is observed as the elongation of structures along the line of sight (LOS), causing the so-called Fingers- of-God (FoG, see Figure 1.8) effect, in which the structures seem to be pointing at the observer. Conversely, on large scales, objects fall towards overdense regions. This makes objects between the overdensity and the observer to look further away, while making objects on the other side of the overdensity look closer. This causes the overdensity to be magnified along the LOS. —a phenomenon known as the Kaiser effect or Kaiser boost [Kaiser, 1987]. These phenomena are collectively referred to as redshift-space distortions [White, n.d.]. The recession velocity cz and distance d to a given galaxy are related by the Hubble-Lemâitre law as: z ∼ Hod, (1.61) where c is the speed of light, and z is the redshift. We define the redshift distance s of a galaxy as s = cz, where z can be measured from its spectrum. The redshift distance s, and the true distance r differ, causing an apparent displacement along the line of sight in redshift space. As a consequence, the pattern in which galaxies cluster presents, in the redshift space, the so-called redshift space distortions, and this effect provides valuable information about large-scale motions of galaxies and structure growth. We define the linear redshift distortion parameter β as the amplitude of coherent large-scale 6The smooth, large-scale recession of galaxies due to the expansion of the Universe, as described by Hubble law. 1.5. OBSERVATIONAL PROBES 23 Figure 1.8: Due to virialized peculiar velocities on small scales, particles at the same distance have slightly different redshifts, which is observed as the elongation of structures along the line of sight, causing the so-called Fingers-of-God (FoG) effect shown in red, in which the structures seem to be pointing at the observer [Peacock, 2003]. distortions. It is related to cosmology by: β = f b , (1.62) where b is the galaxy bias and f is the linear growth rate of structure. The latter is commonly approximated as f = Ωγ m, with γ ≈ 0.55 in General Relativity. Measuring β thus constrains both the matter density Ωm and the bias b, and serves as a key probe of the growth of structure and gravitational dynamics [Hamilton, 1998], [Linder, 2005]. 1.5.5 Baryon Acoustic Oscillations Before the recombination epoch, protons, electrons, and photons comprised a coupled fluid, forming a plasma where the radiation pressure of photons counteracted the attempt of gravity to collapse into dark matter potential wells. This interplay between radiation and matter caused oscillations in the plasma. As the Universe expanded and the temperature decreased, photons ceased to interact with electrons. This led to the formation of neutral hydrogen, making the Universe transparent to photons. These photons freely traveled through the Universe, creating the CMB. In configuration space, the spatial domain in which we analyze correlations between physical objects or fields (see Section 1.4.1) based on their positions in real space, a spherical sound wave propagated through the photon-baryon plasma in the early Universe. This wave froze at the drag epoch. We refer to the drag epoch as the time slightly after recombination,7 when photons dynamically decoupled from 7At a redshift of approximately zd ∼ 1059, whereas recombination occurs at z ∼ 1100. 24 CHAPTER 1. COSMOLOGICAL BACKGROUND Figure 1.9: Left panel: Schematical evolution of CMB anisotropies into the distribution of matter imprinted in the LSS as BAO. Figure taken from [CAS, Swinburne U. of Technology, n.d.] . Right panel: Representation of the Baryon Acoustic Oscillations as tracers of the expansion of the Universe. As the Universe ages, it expands, and so does the BAO scale. This makes it a remarkably useful tracer of the Dark Energy. Sourced from [DESI Collaboration and Lamman, 2020]. Credit: Claire Lamman/DESI collaboration. baryons. At this point, photons began to stream freely (following recombination), and baryons were no longer held back by radiation pressure. It left a characteristic scale, known as the sound horizon, imprinted in the spatial distribution of baryons. Gravitational interactions then drove the formation of structures, evolving into the LSS (see Section 1.5.2) pattern. This frozen oscillations generated the Baryon Acoustic Oscillations (BAO), which kept growing as a consequence of the expansion of the Universe only, and eventually generated patterns in the distribution of objects (galaxies, quasars, IGM clouds, etc.) in the Universe. This imprint is observed as a pronounced peak (the BAO peak) in the two-point correlation function (see Section 1.4.1) and the anisotropy spectrum of the CMB, as schematically shown in the left panel of Figure 1.9 [McDonald and D. J. Eisenstein, 2007]. Figure 1.10 (from [D. Eisenstein et al., 2007]) illustrates the stages of this process, as a function of redshift z, in which BAO were formed. Initially, electrons and protons were coupled in dark matter gravitational potential wells. The second panel depicts the coupled plasma being influenced by the radiation pressure of photons. When conditions allowed it, photons decoupled from the plasma, leaving an imprint of baryons and traveling freely through the Universe. The position of the BAO peak, centered on a comoving distance rd (sound horizon at the drag epoch, rd ≈ 100h−1Mpc [Blomqvist et al., 2018]), determines the ratio DM (z)/rd, where DM (z) = (1+z)DA(z) is the comoving angular distance, and DH(z)/rd (where the Hubble distance is DH(z) = c/H(z) at a given observed redshift). These ratios depend on cosmological parameters, constraining them through BAO peak observations. This frozen scale serves as a standard ruler, first measured by [D. Eisenstein et al., 2005] and [Cole et al., 2005], allowing distance measurements in the LSS clustering at different evolutionary stages of the Universe as a function of z, as shown in the right panel of Figure 1.9 [Bautista et al., 2017], [du Mas des Bourboux et al., 2017]. 1.5. OBSERVATIONAL PROBES 25 1.5.6 Type Ia Supernovae Numerous efforts have been dedicated to understand the nature of the expanding Universe. To achieve this, precise methods for measuring distances are essential. Among these methods, one of the most important is the use of standard candles. Historically, the first standard candles were Cepheid variable stars, used by Edwin Hubble in the 1920s to measure distances to nearby galaxies and demonstrate the expansion of the Universe. Later, Type Ia supernovae (SNe Ia) became key tools for probing cosmological distances at much greater redshifts due to their high luminosity and relative uniformity, ultimately leading to the discovery of the accelerated expansion of the Universe. SNe Ia consist of stellar explosions that follow the same pattern of formation and death, and all have approximately the same luminosity. The results obtained demonstrated that the Universe expands at an accelerated rate. This earned a Nobel prize to Saul Perlmutter, Brian Schmidt and Adam Reiss. SNe Ia arise from binary star systems involving a white dwarf and another accompanying star. When the companion star is attracted to the white dwarf, it begins to accumulate mass until it exceeds a limit (the Chandrasekhar limit, ≈ 1.4M⊙), triggering nuclear processes that enable the burning of heavy elements. This causes the white dwarf to explode, shining as brightly as the center of an average spiral galaxy. Since all SNe Ia are generated in a very similar process, their light curves exhibit similar shapes, and by applying an empirical relation between their color and the stretch of their light curve, type SNe Ia can be standardized and used as precise distance indicators. When we compute their distance modulus using their apparent and absolute magnitudes, we can estimate their luminosity distance. Observations of Type Ia supernovae have revealed that the expansion of the Universe is acceler- ating, providing the first direct evidence for dark energy. They provide information and constraints of cosmological parameters like Ωm, ΩΛ, the equation of state w, and H0. The most recent SNe Ia experiments are the SCP (Supernova Cosmology Project [Goldhaber, 2009]), and Pantheon sample [Scolnic et al., 2018]. The SCP is one of the pioneering collaborations that used SNe Ia to discover the accelerated expansion of the Universe, while The Pantheon sample is a large, combined dataset of 1,048 spectroscopically confirmed SNe Ia in the range of 0.01 < z < 2.26 from various sur- veys, including the Panoramic Survey Telescope and Rapid Response System 1 (Pan-STARRS1), the Sloan Digital Sky Survey (SDSS), the Supernova Legacy Survey (SNLS), low-redshift samples, and the Hubble Space Telescope (HST) observations. This used to tightly constrain cosmological parameters, particularly those related to dark energy [Goldhaber, 2009], [Scolnic et al., 2018]. 1.5.7 Galaxy and Galaxy Cluster Kinematics Rotation Curves of Spiral Galaxies: In spiral galaxies, stars and cold gas in the disk predominantly follow circular orbits within the disk plane. The kinematics of these disks are characterized by the rotational velocity, Vrot(R), as a function of galactocentric radius R, forming what is known as a rotation curve. Observations show that these rotation curves tend to remain flat at large radii, meaning the rotational velocity does not decline as expected from the distribution of visible matter alone. This flat behavior implies the presence of an extended dark matter halo that dominates the galaxy’s mass distribution beyond its baryonic components. 26 CHAPTER 1. COSMOLOGICAL BACKGROUND Galaxy Clusters: Galaxy and galaxy cluster kinematics provide crucial observational support for the ΛCDM model. In spiral galaxies, stars and cold gas in the disk predominantly follow circular orbits within the disk plane. The kinematics of these disks are characterized by the rotational velocity, Vrot(R), as a function of galactocentric radius R, forming what is known as a rotation curve. Observations show that these rotation curves tend to remain flat at large radii, meaning the rotational velocity does not decline as expected from the distribution of visible matter alone. This flat behavior implies the presence of an extended dark matter halo that dominates the galaxy’s mass distribution beyond its baryonic components. Similarly, measurements of velocity dispersion in elliptical galaxies and within galaxy clusters reveal motions that require a significantly larger mass than what is observed in stars and gas, further confirming the existence of dark matter. Using the virial theorem, galaxy velocities in clusters allow for total mass estimates that consistently exceed the luminous mass, reinforcing the dominance of non-baryonic dark matter. Furthermore, gravitational lensing caused by galaxies and galaxy clusters directly measures the mass distribution of both: baryonic and dark matter. Observations of this effect test the ΛCDM model predicted relationships between the kinematics of galaxies and the mass distribution observed through gravitational lensing. Additionally, the redshift evolution of galaxy motions and cluster abundance informs the growth of cosmic structure, providing constraints consistent with the hierarchical formation scenario of the ΛCDM model and the influence of dark energy on the expansion history. When comparing with predictions from ΛCDM simulations, galaxy clusters provide insights into the effect of dark matter, dark energy, and the connection between galaxies and halos in order to use galaxies as cosmological probes [Mo et al., 2010], [Weinberg et al., 2013]. As mentioned in previous sections, galaxy clusters have provided valuable evidence of the ex- istence of dark matter. Today they represent several cosmological tests, like the aforementioned measurement of BAO (Section 2.3.1) RSD (Section 1.5.4), and P (k) (Section 1.4.1) across a range of scales, allowing the estimation of the deviations of a primordial power spectrum. Moreover, deviations from the observations reported by SNe Ia and BAO are also direct evidence of modified gravity as a cause of the accelerated expansion of the Universe, hence galaxy clustering also tests constraints to the cosmological model. While all cosmological probes provide valuable and complementary information, it is important to highlight that BAO constitute the primary observational tool underpinning the development of this research. Their robustness and precision in constraining the expansion history of the Universe make them central to the analysis presented in this work. 1.6 Motivation and Objectives This thesis is driven by a strong scientific curiosity to deepen our understanding of the cosmic web and to explore new ways to trace its structure. While the Lyman-α forest and galaxy surveys have already provided remarkable insights into the LSS, there remains a gap in redshift coverage between these two established tracers. One of the central motivations of this work is to investigate the potential of metal absorption lines, in particular the C iv forest, to help reduce that gap. As 1.6. MOTIVATION AND OBJECTIVES 27 S/N8 continues to improve in current and future surveys, the exploration of additional absorption features, such as the Si iv and Mg ii forests, becomes progressively more viable. These alternative tracers present several challenges. Metals trace the IGM in a different way than neutral hydrogen, and their abundance is significantly lower. Despite these difficulties, they carry valuable astrophysical information. Elements such as C, Si, and Mg are produced in stellar interiors and dispersed into the IGM through feedback processes. Their presence across cosmic time contains clues about stellar evolution, chemical enrichment, and the impact of AGN. Studying their distribution allows us to investigate when different stellar populations formed, how their abundances evolve with redshift, and how they contribute to the overall enrichment of the Universe. Although we cannot directly estimate the Nmetal/NHII 9 in the IGM as is done in studies of the insterstellar medium (ISM), we can measure the clustering of metal absorbers through their bias. Studying how this bias evolves with redshift offers indirect insights into the distribution of metals and their role in the cosmic web. One of the long-term goals of this project is to determine whether the Lyman-β forest, C iv, Si iv and Mg ii forests can be established as reliable tracers of the LSS and eventually be used to detect BAO. As an initial step, this thesis focuses on testing the computational tools and pipelines available for such analyses. Instead of starting with BAO-scale measurements, which occur at ∼ 100 Mpc/h, we begin by measuring correlation functions at smaller scales 40 − 60 Mpc/h. This allows us to validate our methodology and to better understand the effects introduced by the doublet structure of the C iv, Si iv, and Mg ii transitions. Properly modeling these effects is essential for future applications on larger scales and for unlocking the full potential of these absorption features as cosmological tracers. This thesis begins with a general overview of cosmology and the statistical tools used in LSS studies. It then introduces the DESI survey and the quasar absorption sample used in this work. Subsequent chapters detail the methodology, small-scale clustering measurements using the Lyman- β forest, and the first steps toward analyzing alternative absorber (C iv, Si iv, and Mg ii) forests. The final chapter discusses results, current limitations, and future directions. 8Signal-to-noise ratio, a measure of the strength of the desired signal compared to the level of background noise in a dataset or observation 9metal-to-hydrogen density ratio 28 CHAPTER 1. COSMOLOGICAL BACKGROUND Figure 1.10: Snapshots of the process in which the BAO were formed in different epochs (redshifts z). The black lines represent dark matter, the blue ones represent baryons, and the red represent photons. The top left panel shows coupled photons and baryonic matter in the dark matter potential well, the top right panel shows how in the last oscillation of the coupled plasma photons started dragging the baryonic matter through the middle left panel. The middle right panel shows photons decoupling from this plasma as the CMB leaving a fixed BAO scale. The bottom left panel shows fully decoupled photons, and baryons in the fixed BAO scale. The right bottom panel shows how the interaction of dark and baryonic matter caused their distribution, observed by tracers (such as galaxies and IGM) nowadays. Figure taken from [D. Eisenstein et al., 2007] Chapter 2 The Dark Energy Spectroscopic Instrument (DESI) A redshift survey is a census of a section of the celestial vault focused on measuring the redshift of astronomical objects, such as galaxies, galaxy clusters, AGN, quasars, and others. Using a cosmological model, the redshifts are used to estimate the distance to such objects. Combining this information with their angular position, the surveys map the distribution of matter at different epochs of the Universe, providing valuable information of its evolution and thereof validating or falsifying our modeling and gravity theories. The development of this thesis project has temporarily coincided with the transition of two such surveys: in one hand, it has been part of the analysis of the Final Data Release of the Extended Baryon Oscillation Spectroscopic Survey (eBOSS), and in the other hand, the preparation for the Dark Energy Spectroscopic Instrument (DESI), the start of the survey, its first data and public data releases. This chapter briefly describes the evolution and timeline of galaxy surveys, especially the aforementioned, and focuses on the DESI survey. Timeline of Redshift Surveys The LSS of the Universe arises from gravitational interactions and reflects the distribution of both baryonic and dark matter. Observational studies of the LSS provide key insights into cosmic evolu- tion and the accelerated expansion of the Universe. Two fundamental probes derived from galaxy redshift surveys are Baryon Acoustic Oscillations (BAO), used as a standard ruler to measure the expansion history, and Redshift-Space Distortions (RSD), which probe the growth of structure and test gravity on large scales. Resolving these features requires high-precision spectroscopic redshift measurements, prompting the development of numerous redshift surveys with varied strategies and wavelength coverage tailored to specific scientific goals. The first large spectroscopic galaxy sample, the Harvard-Smithsonian Center for Astrophysics (CfA) [Davis et al., n.d.] in the 1970s, consisted of several thousand galaxies for clustering analysis. Since then, and with the development of instruments such as spectrographs and CCDs, several redshift surveys have been designed and conducted, aiming to map a larger amount of objects in a 29 30 CHAPTER 2. THE DARK ENERGY SPECTROSCOPIC INSTRUMENT (DESI) wider and deeper sample of the Universe. Among these: ⋆ CfA2 Redshift Survey [Davis et al., n.d.] (1999): Completed 18,000 redshifts over 17,000 deg2. ⋆ 2dF Galaxy Redshift Survey (2dFGRS) [Cole et al., 2005] (2001, 2003 for final release): Completed 220,000 redshifts over 1500 deg2. ⋆ 6dF Galaxy Survey (6dFGS) [Jones et al., 2004] (2009): Completed 125,000 redshifts over 17,000 deg2. ⋆ Baryon Oscillation Spectroscopic Survey (BOSS) [Dawson et al., 2013](2002-2014): Com- pleted 1,660,000 redshifts over 10000 deg2. ⋆ Extended BOSS (eBOSS) [Dawson et al., 2016] (2014 – 2019): Completed 975,000 redshifts over 6000 deg2. ⋆ Dark Energy Spectroscopic Instrument (DESI) [Aghamousa et al., 2016a][Aghamousa et al., 2016b]: (2019 - ongoing): Aims to measure 40,000,000 spectra over 14,000 deg2. [Wang et al., 2020], [Astrophysics Research Institute, 2024]. Figure 2.1, shows the area (in deg2) covered and the number of spectra per deg2 observed by the main spectroscopic surveys. While this plot does not explicitly show a timeline, it reflects how spectroscopic galaxy surveys have progressively pushed the boundaries of both area and spectral density, culminating in projects like DESI, currently the largest ground-based spectroscopic survey in terms of both sky coverage and redshift depth. This progression is the result of steady technical and scientific improvements over time. 2.1 The DESI Survey Since May 2021, the Dark Energy Spectroscopic Instrument (DESI) is conducting a five-year spec- troscopic survey aimed to be “the most precise measurement of the expansion of the Universe ever obtained” [DESI Collaboration, 2022], [Levi, M. et al., 2013]. DESI is a spectrograph that oper- ates robotically a carefully choreographed arrangement of 5020 optical fibers in the focal plane of the 4m Mayall telescope, capturing up to 5000 spectra simultaneously across a wavelength range extending from 360 nm to 980 nm. The fibers feed into three-arm spectrographs with a resolution R = λ/∆λ ranging from 2000 to 5500. It is designed to be a Stage IV 1 dark energy survey during its five years of operations, obtaining spectra of over 40 million galaxies and quasars, mapping over 14,000 deg2 from nearby stellar objects to redshifts of z > 3.5. The expansion history or distance- redshift relationship of the Universe at this redshift range will be measured using the standard ruler technique with BAO, and several surveys with successively increasing sample sizes have followed 1Stage IV refers to the classification system proposed by the Dark Energy Task Force (DETF)[Albrecht et al., 2006], which ranks surveys according to their scale and precision. Stage IV surveys represent the most advanced generation, aiming to deliver an order-of-magnitude improvement in constraints on dark energy compared to previous efforts. They typically cover large areas of the sky, use multiple cosmological probes (such as BAO, RSD, and weak lensing), and are designed to test the expansion history and structure growth of the Universe with high precision. DESI meets these criteria by targeting an unprecedented volume of the Universe with spectroscopic redshift measurements. 2.1. THE DESI SURVEY 31 Figure 2.1: Area (in deg2) covered and the number of spectra observed per deg2 by the main spectroscopic surveys, comparable with DESI (ELG and BGS surveys within the DESI survey, explained in Section 2.1). Squares represent magnitude-limited surveys, circles are surveys involving color cuts for photometric redshift selection, triangles are highly targeted surveys. The filled symbols represent completed surveys. The color code refers to the wavelength selection of each survey. For more information see [Baldry, n.d.]. 32 CHAPTER 2. THE DARK ENERGY SPECTROSCOPIC INSTRUMENT (DESI) their lead measuring it at different redshifts. DESI will also measure RSD which will give valuable information about the growth of structure from peculiar velocities of galaxies. To achieve these goals, it observes seven different classes of targets, selected based on the Legacy Surveys, a combination of three public imaging surveys2 to map ∼ 14, 000 deg2 of the northern extragalactic sky in optical and infrared bands, producing uniformly deep images and catalogs to support target selection for the DESI experiment [Dey et al., 2019], [Alexander et al., 2023a]: ⋆ Stars: sample of more than ten million stars in galactic latitudes of b±20 of up to a magnitude of 19 visible with exposures of 8-10 min with a S/N = 25 that will provide valuable spec- tra with information to probe the ΛCDM model at small scales by serving as tracers of the gravitational potential, mass distribution, and formation history of galaxies. Radial velocities and proper motions provide the kinematics of stars, which allow us to reconstruct the gravi- tational potential of the galaxy and infer the distribution of dark matter, which is crucial for testing ΛCDM predictions at galactic scales. Effective temperature and surface gravity help determine the evolutionary stage of stars and, when combined with luminosity and metallicity, allow us to estimate stellar ages. The chemical abundance (metallicity) distribution of stars traces the chemical evolution of galaxies, offering clues about star formation histories and the hierarchical assembly of substructures, as predicted by ΛCDM. Stellar ages, when combined with spatial and kinematic information, help reconstruct the formation history of different galactic components such as the halo, thick disk, and thin disk, providing constraints on merger events and dynamical evolution. Additionally, mapping the three-dimensional spatial distribution of stars reveals the morphology of galaxies and their substructures—such as tidal streams, disks, and bulges—offering further tests of the growth of structure through mergers and accretion in the ΛCDM framework. These spectra will enrich the dataset obtained by other surveys like GAIA [Cooper et al., 2023]. ⋆ Bright sample of low redshift galaxies (BGS - Bright Galaxy Sample): r < 19.5 magnitude- limited sample of more than ten million galaxies in a redhsift range between 0 < z < 0.6 (although the best BAO and RSD measurements will be in the range of 0 < z < 0.4), observed during the time when the sky is too bright and hampers the observation of fainter objects. It will contain quiescent and star-forming galaxies [Hahn et al., 2023]. ⋆ Luminous red galaxies (LRG): sample of galaxies that present high luminosities and are red in the rest-frame optical wavelengths. They have high stellar masses and are not forming stars. They also present high clustering, which makes them a good BAO tracer. The sample is constituted by approximately 8 million galaxies in a redshift range of 0.4 < z < 1 [Zhou et al., 2023]. ⋆ Emission Line Galaxies (ELG): sample of approximately 17 million luminous star-forming galaxies, typically late-type spiral and irregular (although ELG refers to galaxies actively forming stars at a sufficiently high rate) at redshifts 0.6 < z < 1.6 (where the tightest constraints are expected to be in the redshift range of 1.1 < z < 1.6) whose spectra present the [O ii] emission doublet (rest-frame wavelengths of 3726 Å and 3729 Å) with S/N ≈ 7, and strong emission lines from ionized H ii regions surrounding short-lived but luminous, massive stars. This survey is the largest sample of DESI targets. Due to the active star formation, they tend to be bluer than galaxies with older stellar populations such as LRGs [Raichoor et al., 2023]. 2The Dark Energy Camera Legacy Survey, the Beijing-Arizona Sky Survey, and the Mayall z-band Legacy Survey 2.1. THE DESI SURVEY 33 ⋆ Quasars (QSO) (also known as clustering quasars, or direct tracers of dark matter): sample of the most luminous extragalactic sources known, reaching 1000 times the brightness of a normal galaxy. Their luminosities are a consequence of the accretion of supermassive black holes in the center of galaxies, and hence associated with Active Galactic Nuclei (Section 3.1.1). These objects are bright enough to outshine their host galaxy. The emitting regions at the centers of even the nearest QSOs are too compact to be resolved, so they appear as point sources in imaging data. As such, they are treated as point tracers of the large-scale matter distribution in the Universe and represent the targets at highest redshift observed by DESI, at 0.9 < z < 2.1 [Yèche et al., 2020], [Chaussidon et al., 2023]. ⋆ Lyman-alpha quasars (or absorption quasars): 8 million quasars at redshifts above 2.1 used to probe the intergalactic medium using the neutral hydrogen absorptions at λ = 1216 Å that builds the Lyman-α forest (described in more detail in Section 3.2.1). These are observed multiple times to improve the measurements of the Lyman-α forest with the required S/N. These objects will be observed in the footprint shown in Figure 2.2, with a redshift depth schemed in Figure 2.3. Figure 2.2: Forecasted footprint of the Dark Energy Spectroscopic Instrument (DESI) survey. The color map represents the time required to reach a specific galaxy depth, relative to the observation at zenith without galactic extinction. The gray dashed line indicates the Galactic plane, while the gray contour is the E(B − V ) = 0.3mag [Schlafly et al., 2024]. According to Alexander et al., 2023b, ≈ 70% of the main DESI survey targets are identified as quasars, ≈ 16% as galaxies, ≈ 8% stars, and ≈ 8% are inconclusive spectra due to low quality and the lack of identifiable reliable features. Achieving the goal of observing this extensive sample of objects within a five-year time-frame requires a significantly high survey speed (number of measure- ments per time unit). The parameters involved in the realization of this objective are the exposure time of each observation, time between consecutive exposures and number of objects observed in each exposure. The design and construction of DESI allows the instrument to [DESI Collaboration, 2022]: 1. Take science exposures of 1000 seconds in nominal conditions (although the longest exposures expected are of 1200 seconds). 34 CHAPTER 2. THE DARK ENERGY SPECTROSCOPIC INSTRUMENT (DESI) Figure 2.3: Tracers observed by DESI and building the cosmic web, closed-up in a circular wedge. Distribution of tracers, from the core (observer) to the outer edges: Bright Galaxy Sample (BGS), Luminous Red Galaxies (LRGs), Emission Line Galaxies (ELGs), Quasars, and absorber quasars (Lya forest quasars). Credit: Claire Lamman/DESI collaboration [DESI Collaboration, 2024b]. 2.2. INSTRUMENT FACILITIES 35 2. Obtain 5000 spectra during each exposure. This is made possible by the 5020 robotic posi- tioners of the focal plane, each equipped with an optical fiber, arranged in ten petals, with 500 positioners allocated to each petal. 3. Change between fields in ≈ 2 min, thanks to the stability of the system and the robotic positioner system that controls the optical fibers. This instrument is installed in the Nicholas U. Mayall 4-meter Telescope, at I’olkam Du’ag (Manzanita Bush Mountain) in the Tohono O’odham Nation, also known as Kitt Peak, Arizona. It is operated by the National Optical-Infrared Astronomy Research Laboratory (NOIRLab) of the National Science Foundation (NSF) of the USA. 2.2 Instrument facilities The Mayall, the largest of the 22 telescopes located at the Kitt Peak National Observatory (KPNO) and commissioned fifty years ago, is a reflector telescope with a 4m diameter primary mirror, po- sitioned on an equatorial mount. It was reconditioned for the installation of the DESI instrument (February 2018) and saw its new first light in October 28, 2019. Figure 2.4 shows a scheme of the facilities of the 4 m Mayall telescope and the instrumental arrangement built for the DESI instru- ment. We can identify six main structures that conform it: corrector, focal plane, fibers system, spectrographs, instrument control system, and data systems. ★ Corrector: its function is to provide a larger field of view to the focal plane, allowing an array of 5000 optical fibers in 8 deg2. It consists of 4 lenses made of fused silica. The two smaller ones are single-surface spherical, while the others are spherical. They were designed to meet the needs of the DESI lenses. Additionally, they have an atmospheric dispersion compensator to calibrate blurry areas away from the zenith. ★ Focal plane: It is located at the top of the telescope and has 5000 robotic positioners, each equipped with an optical fiber. These positioners adjust to direct towards individual sources, ensuring that the fibers collect their respective light, with the additional challenge of avoiding collisions between them. With an approximate diameter of 1 meter, the focal plane is com- posed of ten petals, each with 500 positioners. Additionally, the focal plane includes sensors and lights to ensure the proper alignment of the positioners. ★ Fiber system: The fibers, each 47.5 meters long, carry the light from the 5000 collected targets at the focus to the spectrographs. Each fiber measures 107µm in diameter (roughly the thickness of a human hair) and each is mounted on one of the positioners. Afterwards, they are grouped into sets of 500. ★ Spectrographs: each of the ten spectrographs is fed by a cable of 500 fibers. The light from the fibers is separated into three cameras that are each sensitive to different wavelength ranges and have different resolutions (R = λ/∆λ): ⋆ 360 < λ ≤ 555nm, blue camera;R = 2000− 3200 ⋆ 555 < λ ≤ 656nm, red camera;R = 3200− 4100 ⋆ 656 < λ ≤ 980nm, near infrared (NIR) camera; R = 4100− 5000 36 CHAPTER 2. THE DARK ENERGY SPECTROSCOPIC INSTRUMENT (DESI) Figure 2.4: Scheme of the Dark Energy Spectroscopic Instrument in the Nicholas U. Mayall 4-meter Telescope, at I’olkam Du’ag (Manzanita Bush Mountain) in the Tohono O’odham Nation, also known as Kitt Peak, Arizona, with its main components (described in this chapter: corrector, the structure that supports it, focal plane with positioners, fiber system and cables, and spectrographs) [Abareshi et al., 2022]. 2.3. SCIENTIFIC GOALS OF THE DESI SURVEY 37 The detectors are arrays of 4096 x 4096 pixels of 15µm, each in its own cryostat which is at very low and stable temperatures: 163 K for the blue camera and 140 K for the red and NIR camera, with a precision of ±1 K and a stability of ±0.1 K. ★ Instrument control system: controls and monitors all functions required for the operation of DESI. From the data acquisition, connection to the Mayall Telescope, monitoring infrastruc- ture, maintenance of a database of operations, guiding, coordinate transformations, subsystem coordination and user interfaces. It also has the DESI Online System, that contains the user interfaces for exposure sequences, real-time telemetry, and image previews. ★ Data system: the data are monitored by the observers within the first three minutes after being obtained. Then the raw data are transferred to the National Energy Research Scientific Computing Center (NERSC) at Berkeley Lab for processing. Then the processed spectra are extracted, calibrated, classified, and their redshift measured. 10 TB are expected to be transferred from KPNO to NERSC per year, as well as stored at NOIRLab in Tucson, Arizona. After the data have been processed at NERSC investing millions of CPU hours, ≈ 100 TB will be available per year, containing the contents of the data releases, expected to be within a yearly cadence throughout the five years of DESI operations. 2.3 Scientific Goals of the DESI Survey The scientific goals of DESI are focused on advancing our understanding of dark energy, galaxy formation, and the LSS. DESI aims to map the distribution of millions of galaxies and quasars across a wide redshift range, enabling precise measurements of the expansion history of the Universe. One of its key goals is to constrain the properties of dark energy by studying the expansion rate at different cosmic epochs and the growth of cosmic structures. An example of this is shown in Figure 2.5, which shows the comparison of H(z)/(1+ z) from BOSS/eBOSS and DESI [Aghamousa et al., 2016a]. Section 1.1 mentions that dark energy, represented by Λ, is described by three possible scenarios: a cosmological constant, equivalent to static dark energy with w = −1; a dynamical dark energy with w(a) ̸= −1; or as a manifestation of modified gravity, where deviations from General Relativity on cosmological scales mimic the effects usually attributed to dark energy. DESI is designed to address this fundamental question about the nature of the Universe. By probing the Universe with unprecedented precision, DESI will test cosmological models that seek to explain the nature of dark energy. A primary approach involves measuring BAO (see Sections 2.3.1) and RSD (1.5.4). BAO provides a “standard ruler” for gauging cosmic distances, while RSD provides insight into the growth of cosmic structures. This combined analysis allows DESI to assess the rate of structure formation, which is sensitive to both dark energy and the properties of dark matter. In addition to its primary focus on dark energy, DESI will also contribute to our understanding of galaxy evolution, by providing a vast catalog of galaxy redshifts that will allow researchers to study how galaxies have evolved in various environments over cosmic time. Furthermore, it will offer new insights into the distribution of matter in the Universe, including constraints on neutrino masses and tests of modifications to general relativity. By linking theory with observations, the rich dataset obtained by DESI will serve as a valuable resource for cosmological studies, enhancing our understanding of fundamental physics. 38 CHAPTER 2. THE DARK ENERGY SPECTROSCOPIC INSTRUMENT (DESI) Figure 2.5: Comparison of H(z)/(1+ z) measurements from BOSS/eBOSS (left) and DESI (right). The figure shows the Hubble parameter evolution as a function of redshift, with data points repre- senting BAO measurements from different surveys. The DESI results (right) are expected to provide significantly improved precision, particularly at higher redshifts, highlighting its capability to map the expansion history of the Universe with unprecedented accuracy [Aghamousa et al., 2016a]. 2.3.1 Baryon Acoustic Oscillations One of the primary goals is to refine our understanding of the accelerated expansion of the Universe due to dark energy. By using the BAO signal at different redshifts, DESI can measure the expansion rate at various epochs, providing essential data to model the evolution of dark energy. BAO also allows DESI to independently test the ΛCDM model and investigate new physics, offering a precise way to measure the Hubble constant and compare it with results from other cosmological probes, such as the CMB. Furthermore, these measurements can be used to constrain neutrino masses and the potential effects of modified gravity. Neutrinos, with their unique properties, leave subtle imprints on the LSS of the Universe. By comparing BAO signals across time, DESI can provide new insights into the role neutrinos play in the cosmos. [Aghamousa et al., 2016a] 2.3.2 Redshift Space Distorsions Redshift Space Distortions (RSD) are crucial to understand how galaxies move under the influence of gravity. DESI measures peculiar velocities caused by gravitational attraction, offering direct insights into the growth of cosmic structures. One of the main scientific goals of studying RSD with DESI is to probe gravity on cosmological scales, testing predictions from General Relativity and exploring potential modifications. RSD measurements also help distinguish between different models of dark energy by examining how the large-scale structure of the universe grows over time. Dark energy influences this growth by opposing gravitational collapse, so tracking the rate of structure formation provides an independent test of dark energy models. In addition to probing gravity and dark energy, the RSD analysis of DESI will improve the un- derstanding of galaxy formation and evolution. By disentangling the effects of large-scale clustering from those of small-scale, nonlinear motions, RSD measurements help reveal how galaxies move and 2.4. DESI Y1 39 form in different cosmic environments. RSD data will complement BAO measurements, offering a cross-check on the expansion history and improving precision in measuring the fundamental parameters of the Universe. Together they will provide a powerful tool for understanding the interplay between gravity, dark matter, and dark energy across cosmic time. 2.4 DESI Y1 2.4.1 Data The data used for the cosmological analysis of this research is DESI DR1, the first data release of this survey. In June 2021, DESI officially started mapping the 14,000 deg2 of the celestial vault, in the footprint shown in Figure 2.2, which is planned to be covered in a five year time lapse. The DR1 dataset is composed of other sub-datasets: commissioning and Survey Validation, which are all data taken between December 14, 2020 - June 10, 2021, used for the validation of the survey design and strategy for the DESI Main Survey, that started in May 14, 2021. The first year of the five that conform the DESI operations concluded in June 13, 2022. The data were processed by the data reduction pipeline and the redshift classification algorithms in [Guy et al., 2023]. DR1 is composed of the following amount of spectra: 6,279,965 galaxies conforming the BGS, 3,926,272 ELG, 2,829,517 LRG, 1,340,073 QSO, and 3,641,276 stars of the MWS. They have been observed accross the footprint shown in Figure 2.6, which shows the completeness of the DESI Main Survey by June 13, 2022 for the dark, bright, and backup programs [DESI Collaboration, 2025b]. The second and third data releases of DESI will correspond to the third and fifth years of operations respectively. The DESI collaboration is currently analyzing the data of the third year of operations and extracting cosmological information from them3, while observers at the Mayall are already obtaining the dataset that will conform the third and final data release. Quasar catalog The different Lyman-α analyses require a refined sample of quasars with very precisely determined redshifts. For this, DESI follows a thorough procedure to identify quasar spectra and determine such feature with the requirements met. This procedure is thoroughly detailed in Section 2.1 of [DESI Collaboration, 2025c]. The quasar catalog was made through three classifiers: RedRock [DESI Collaboration, n.d.(a)], a template fitting code that uses Principal Component Analysis using templates for galaxies, stars and quasars in a wide range of redshifts, QuasarNet [Busca et al., n.d.], using a deep convolusional neural network that identifies potential quasar emissions and estimates the redshift, and finally a second iteration of RedRock for targets confirmed as quasars. This catalog is part of the first DESI Data Release (DR1) catalog, released on March 2025. The left panel of Figure 2.7 shows the spatial distribution comparison of the DESI DR1 (green dots) and the SDSS DR16 quasars (red curve) [du Mas des Bourboux et al., 2020]. The right panel of Figure 2.7 shows the redshift distribution of the DESI DR1 (orange) and the SDSS DR16 (green) quasars, together with the DESI Lyman-α pixels. 3Some of these results were published in March, 2025 in [DESI Collaboration, 2025e] and [DESI Collaboration, 2025f] 40 CHAPTER 2. THE DARK ENERGY SPECTROSCOPIC INSTRUMENT (DESI) Figure 2.6: Footprint of the first year of data of the Main Survey obtained with DESI. This sample is composed of 6,279,965 BGS, 3,926,272 ELG, 2,829,517 LRG, 1,340,073 QSO, and 3,641,276 stars of the MWS observed by June 13, 2022. This plot shows the completeness of the dark, bright, and backup programs in shades of green, and the unobserved region in white. The gray dashed line illustrates the Galactic plane, while the gray contour is the E(B−V ) = 0.3mag [DESI Collaboration, 2025b]. 2.4. DESI Y1 41 Figure 2.7: Left panel: footprint of the DESI DR1 quasars (green dots) and expected final footprint (surrounded by the blue curve), compared with the one reported by SDSS DR16 (surrounded by red curve). Right panel: redshift distribution of the DESI DR1 (orange) and the SDSS DR16 (green) quasars, together with the DESI Lyman-α pixels [DESI Collaboration, 2025c]. 2.4.2 Cosmological Results As mentioned previously, one of the primary goals of the DESI experiment is the precise mea- surement of cosmological parameters and test of the ΛCDM model. By measuring BAO and RSD imprinted in the LSS distribution of galaxies and the IGM, DESI provides valuable insights into the expansion history of the universe and the growth of structure. DESI observations of LSS tracers, including galaxies, quasars, and absorbers, provide constraints on fundamental quantities such as H0, Ωm, Ωb, and w. Additionally, the Lyman-α forest, as well as other absorbers (which will be introduced and explained in the following chapters), enables constraints on P (k) and complements galaxy clustering analyses performed at lower redshifts. Alternative Models As explained in Sections 1.1 and 1.1.1, the ΛCDM model has been remarkably successful at ex- plaining numerous observational phenomena. Despite its success, it faces several challenges, such as the nature of dark energy and dark matter, and difficulties in explaining small-scale structures in galaxy formation. These open questions have led to the exploration of alternative cosmological models. Among those, the w0waCDM model extends this framework by allowing a time-varying equation of state for dark energy, given by w(a) = w0+wa(1− a), [Linder, 2003], [Chevallier et al., 2001] offering a more refined understanding of its role in cosmic expansion. The w0wa model appears to be slightly favored by recent DESI data analysis. The precise measurements of DESI of the LSS, including BAO and RSD, enable tighter constraints on the time evolution of dark energy. DESI data, with its extensive redshift coverage and high precision, indicate a slight preference for models where w0 deviates slightly from −1 and wa allows for evolution, suggesting that the assumption of a cosmological constant may not fully describe the nature of dark energy. This evolving dark energy scenario provides a better fit to the data across multiple redshift bins, particularly at higher redshifts, where the influence of dark energy may evolve more significantly than predicted by the ΛCDM model. Nonetheless, the results derived from the Lyman-α forest do not seem to strongly favor either 42 CHAPTER 2. THE DARK ENERGY SPECTROSCOPIC INSTRUMENT (DESI) the w0wa or the ΛCDM model. This can be seen qualitatively in Figure 2.8, (where the results from the Lyman-α forest analysis is shown in violet). This indicates that the Lyman-α forest data alone is insufficient to distinguish between these competing models4, and the whole analysis followed by the DESI Lyman-α working group, and the development of the research presented in this thesis considers a ΛCDM model. Figure 2.8: Hubble diagram incorporating BAO measurements from all tracers. The results show a slight preference for a w0wa model (dashed curve). Measurements from Lyman-α forest, QSO, and LRG+ELG, however, do not favor neither of the models. Credit: Arnaud de Mattia/DESI collaboration [DESI Collaboration, 2024a] . The measurement of Ωm reported by DESI BAO, obtained with the first year of observations [DESI Collaboration, 2025d] is listed in Table 2.9 together with the measurement combined with other external datasets and priors. 4There is no bayesian analysis performed with DESI Lyman-α data alone, since it is made together with other tracers. 2.4. DESI Y1 43 Figure 2.9: Cosmological parameter constraints from DESI DR1 BAO measurements, combined with external datasets and priors, like SNIa experiments (Pantheon, Union3, and Dark Energy Survey Y5), BBN, and CMB, are presented for the baseline flat ΛCDM model as well as for model extensions that include spatial curvature and two parameterizations of the dark energy equation of state. The reported values correspond to marginalized means with 68% credible intervals, including upper limits where applicable. It is important to note that DESI data alone constrain rdh, rather than H0. Table obtained from [DESI Collaboration, 2025d] 44 CHAPTER 2. THE DARK ENERGY SPECTROSCOPIC INSTRUMENT (DESI) Chapter 3 Quasars and IGM as Large Scale Structure Tracers This chapter describes the physics underlying the Lyman-α forest, explaining its role as an impor- tant tracer of the Intergalactic Medium (IGM), Dark Matter, and the Large Scale Structure (see Section 1.5.2). It culminates by exploring the importance of the Lyman-α forest as a tracer of BAO at redshifts above 2.1, as well as its relevance and current state in cosmology. 3.1 Quasars 3.1.1 Active Galactic Nuclei According to Mo et al., 2010, “normal” galaxies are those whose spectra is dominated by the contribution of the black body radiation of all the individual stars conforming them. Their spectra typically span the wavelength range of ∼ 4000 Å and ∼ 20000 Å, which can be extended if it has active star formation (in smaller wavelengths) and heated dust (by these mentioned young stars). These galaxies are distinct from a particular subset of galaxies, known as ‘active galaxies”, char- acterized by a spectral energy distribution (SED) significantly broader than that of typical ones, with wavelengths ranging from X-ray to radio, even accounting for gas and stellar formation processes. These galaxies frequently present intense, broad emission lines in the wavelength ranges of the UV and optical, indicative of energetic activity originating in their central regions. These are known as the Active Galactic Nucleus (AGN), which host a supermassive accreting black hole (SMBH) in their centers. This is the main feature that distinguishes these objects. Despite occupying a very small spatial extent, the luminosity of the AGN can exceed that of their host galaxy by orders of magnitude. To classify an object as an AGN, it should present at least one of the following properties [Mo et al., 2010], [Netzer, 2013]: • A compact nuclear region with a significantly greater luminosity compared to an area of equivalent size within a normal galaxy. • Non-stellar (non-thermal) continuum emission. 45 46 CHAPTER 3. QUASARS AND IGM AS LARGE SCALE STRUCTURE TRACERS • Strong emission lines coming from non-stellar processes. • Variability in continuum emission and/or in emission lines over relatively brief periods. The active galaxies are classified into different types according to their observational properties, although they are commonly referred to as AGNs. Figure 3.1 presents some examples of different AGN spectra, from where quasars are the most important objects for the development of this work. Figure 3.1: Examples of spectra of different types of Active Galactic Nuclei: Seyfert galaxies, quasars, BL Lac (BL Lacertae, named after its prototype) objects, Broad-line radio galaxies (BLRGs), Narrow-line radio galaxies (NLRG), Low-ionization nucelar emission-line regions (LIN- ERS) and OVVs (optically violently variable QSOs), and a normal galaxy. Credit: Bill Keel. Figure taken from [Andika, 2016] 3.1.2 AGN spectra AGN spectra are characterized by two main components: a continuum and prominent emission lines. While all galaxies emit across the electromagnetic spectrum, AGNs stand out due to their exceptionally broad and intense emission, spanning from gamma rays up to radio wavelengths. The overall spectral energy distribution (SED) of an AGN spans a broad range of wavelengths 3.2. INTERGALACTIC MEDIUM 47 and includes both thermal and non-thermal components. In the radio and X-ray bands, the SED can be described by a power law, Fν ∝ ν−α, with 0 ≤ α ≤ 1, characteristic of synchrotron and inverse Compton emission. In the infrared, the emission is dominated by thermal blackbody radiation from dust heated by the AGN, while the optical and ultraviolet bands primarily reflect blackbody emission from the accretion disk around the central supermassive black hole [Harrison, 2016], [Gudiño, 2025]. The SED also presents depressions and bumps, being the blue bump (at ≈ 1015 − 1016Hz) and the broad bump (at ≈ 1020 − 1021Hz) the most important/prominent ones. The wide energy range of the spectrum suggests there are many physical procesess behind their emission, most of them involving relativistic electrons. The relativistic electrons are thought to be generated in the first order Fermi accelerations that occur in shocks from supersonic flows near the central SMBH in the center of the AGN. The Fermi acceleration is the acceleration that charged particles face when they are reflected (passing, scattering and repeated bouncing) by an interstelar magnetic field, resulting in a power-law energy distribution for the accelerated particles. The motion of these relativistic electrons with a power- law energy distribution in a magnetic field produces synchrotron radiation (generated by relativistic electrons spiraling in a magnetic field) with a power-law spectrum that can be of many decades in frequency in the radio band. The X-ray band energies are explained for photons emitted in inverse Compton process (that occurs when a charged particle gives part of its energy to a photon). Emission lines are produced by the transitions of excited atoms. These lines are used to infer the physical properties of the emiting gas (density, temperature, composition, etc.) and the sources. Lines are divided into permitted and forbidden/improbable (there are some semiforbidden lines with intermediate spontaneous transitions) depending on the rate of spontaneous lines between energy levels responsible for the emission. Line ratios are used to distinguish an AGN from a star-forming galaxy, which has emissions due to HII regions generated by young massive stars. The temperature and level of ionization are expected to be higher in an AGN because they have more UV flux, and because the radiation field in an AGN has more high energy photons. 3.1.3 Quasars and QSO The term quasar comes from the term “Quasi-Stellar Radio Source”, and it refers to compact radio sources with a non thermal continuum and strong, broad emission lines, and luminous nuclei that dominate in the blue/UV section of the spectrum and are often variable and sometimes have jets. Their luminosities can reach 1000 times the one of a normal galaxy, typically reaching a bolometric luminosity Lbol = 1045 erg s−1 at the central source [Netzer, 2013], [Gudiño, 2025]. They usually outshine their host galaxy and it is very hard to observe, nonetheless nearby quasars have been detected in both, elliptical or spiral galaxies, and in disturbed interacting systems. 3.2 Intergalactic Medium Galaxies are thought to consist of three primary constituents: dark matter, stars, and gas. The gas is broadly categorized into two components based on its relationship with the galaxy: the interstellar medium (ISM) and the intergalactic medium (IGM). The ISM represents gas directly associated with the galaxy, while the IGM comprises halo gas distributed outside the galaxy but within the host dark matter halo, along with gas unrelated to dark matter halos. Some authors also define the IGM as “anything outside the virial radius of galaxies and clusters (the medium between 48 CHAPTER 3. QUASARS AND IGM AS LARGE SCALE STRUCTURE TRACERS halos rather than the medium between galaxies)” [McQuinn, 2016]. Interactions between the ISM and IGM play crucial roles in the formation and evolution of galaxies through various processes. Consequently, understanding the IGM becomes imperative in comprehending galaxy formation and evolution. Prior to the emergence of stars and galaxies, all baryons were part of the IGM, which later, upon virialization, got accreted by dark matter halos, eventually transforming into stars and the ISM. At present times, more than 50% of the baryonic matter is considered to be in the IGM, less than 10% is found in stars (according to BBN predictions1 and CMB observations), and the rest making the cold molecular and atomic gas associated with galaxies (ISM), and the hot gas in intracluster medium (ICM) [Mo et al., 2010]. The IGM is a very important tool for the study of the evolutionary stages of the Universe. It provides information of cosmological events when analyzed at different redshifts, and origins of structures that turn into galaxies. This allows the potential improvement of constraints of the cosmological initial conditions. All the different characteristics of the IGM, including its density, temperature, chemical composition, and spatial distribution, are studied through its interaction (absorption) with the light of background sources and the radiation field produced by it. The IGM is a valuable tracer of the distribution of baryonic matter and dark matter. When light of a faraway quasar goes through the IGM, electrons changing their orbital levels leave their imprint in the spectrum of the quasar. The density fluctuations of IGM are observed as a continuous absorption field with respect to the unabsorbed emission of quasar spectra. This is illustrated in Figure 3.2. Figure 3.2: Artistic interpretation of the detection of light from background quasars, travelling through and being absorbed by the IGM, until it is detected at the DESI instrument at the Mayall telescope. Credit: KPNO/NOIRLab/NSF/AURA/P. Marenfeld and DESI Collaboration [NOIR- Lab, 2024] 1BBN predicts the total baryon density of the Universe by calculating the primordial abundances of light elements formed in the first minutes after the Big Bang; these abundances, particularly that of deuterium, depend sensitively on the baryon-to-photon ratio η, allowing precise inference of the baryon content when compared with astrophysical observations [Tytler et al., 2000], [Dodelson, 2003]. 3.2. INTERGALACTIC MEDIUM 49 3.2.1 Lyman-α and Lyman-β Forests The Lyman series is a spectral sequence of transitions within the hydrogen atom, manifesting when an electron shifts its energy level from n ≥ 2 to n = 1. These transitions lead to the emission of photons within the ultraviolet segment of the electromagnetic spectrum. Each transition within this series is denoted by a Greek letter, corresponding to the difference between the initial and final energy levels. For instance, the transition from n = 2 to n = 1 is termed Lyman-α, from n = 3 to n = 1 is Lyman-β, from n = 4 to n = 1 is Lyman-γ, and so forth. Among these transitions, Lyman-α is particularly notable. It occurs in hydrogen overdensities within the IGM, where atoms absorb photons emitted by a source, typically a QSO. These absorp- tions leave behind a distinctive print in the form of absorption lines at a rest-frame wavelength of λLyα = 1215.67 Å within the spectra corresponding to different wavelength depending on the red- shift of the IGM overdensity. Figure 3.3 shows a quasar spectrum at a redshift of z = 4.646, where the Lyman-α emission line is shown in a blue dotted line, and the Lyman-α forest is highlighted in a light violet shade. There are other forests formed analogously, named after the transition in the Lyman series that originates them. Among these, the Lyman-β forest, with an absorption wavelength of λLyβ = 1025.18 Å, also shown in a blue dotted line in Figure 3.3 (explained in more detail in Section 5.1) is of peculiar interest as a potential LSS tracer. Figure 3.3: The left panel is the spectrum of a quasar observed by DESI at a redshift of z = 4.646, where the Lyman-α, Lyman-β, and C iv emission lines at rest-frame wavelengths of λLyα = 1215.67 Å, λLyβ = 1025.72 Å, and λCIV = 1549 Å (see 6 for more details about this transition) are shown in blue dotted lines, and the Lyman-α forest is highlighted in a light violet shade. The gray line is the original spectrum, the black one is the smoothed spectrum, and the yellow one the error spectrum. The right panel shows the 18”x18” thumbnail image of the quasar whose spectrum is that of the left panel, observed by the DESI Legacy Imaging Surveys [Alexander et al., 2023b], [Legacy Survey Collaboration, n.d.]. 3.2.2 Absorber Forests The process responsible for generating the Lyman-α forest can also produce other forests with alternative absorption lines. These forests serve as valuable probes, offering complementary insights into various aspects of the history of the Universe at different epochs, including their role as LSS tracers and the evolution of cosmic structure. Of particular relevance to this thesis is the exploration 50 CHAPTER 3. QUASARS AND IGM AS LARGE SCALE STRUCTURE TRACERS of their potential as a source of information and constraints on cosmological parameters through the IGM. Several efforts have been made in this research line studying different regions of the Mg ii forest [du Mas des Bourboux et al., 2019] and proposing the C iv forest as BAO tracer ([Pieri, 2014], followed by a handful of studies that aim to compute several quantities looking to obtain cosmological information from it, e.g. [Blomqvist et al., 2018]] and [Gontcho A. Gontcho et al., 2018]). This research explores different angles of the use of Si iv, C iv, Mg ii, and Lyman-β forests as sources of cosmological information. The phenomena that produce such transitions and forests are thoroughly explained in Chapter 6. These should be distinguished from strong absorption features coming from high column densities (HCD), where logNHI > 17.2 cm−2, also referred to as Lyman Limit Systems (LLS) for column densities within the range 17.2 cm−2 < logNHI < 20.3 cm−2 or Damped Lyman-α Systems (DLAs) when logNHI > 20.3 cm−2. [Herrera-Alcantar et al., 2025][Wolfe et al., 1986] 3.3 Cosmology with the Lyman-α forest One of the most surprising discoveries in the understanding of the Universe has been its accelerated expansion. The existence of new physics resulting in the accelerated expansion of the Universe at late times is a fact. This breakthrough has ignited dedicated efforts, using various techniques, to quantify the properties of the acceleration. The most popular way to address this quantity is the dark energy, a fundamental constituent of the Universe of which we have almost no information except that it has negative pressure and is believed to drive its expansion. The measurements that probe the expansion history of the Universe reveal changes in the energy density of its components as the Universe expands, thereby allowing a deeper understanding of their nature and characteristics. The main probes for this analysis are the type Ia supernovae (SN Ia), relying on the fact that they are standardizable candles, and BAO, using the principal of standard rulers. The most used tracer of BAO are galaxies, and they provide a robust probe at redshifts up to z < 2. However, there is a need to obtain cosmological information of the younger Universe. This is a significant scientific incentive to measure the expansion history of the Universe at higher redshifts. This is provided by means of the Intergalactic Medium (IGM), using mainly the Lyman-α forest at redshifts of z > 1.8. This was first proposed by McDonald (2003) and Weinberg (2003) and first studied in detail by McDonald & Eisenstein (2007) ([McDonald and D. J. Eisenstein, 2007], [Slosar et al., 2013], [McDonald, 2003], [Weinberg, 2003]). Recent analyses using the Lyman-α forest, particularly those from the DESI survey, have demonstrated its power not only as a high-redshift tracer of the LSS but also as a tool to constrain fundamental cosmological parameters. By measuring the position of the BAO peak in the Lyman-α correlation function and cross-correlations with quasars, DESI provides robust estimates of the matter density parameter Ωm, the Hubble constant H0, and the sound horizon at the drag epoch rd, offering complementary constraints to galaxy-based measurements at lower redshifts. This reinforces the Lyman-α forest as a critical probe in the effort to map the expansion history of the Universe across cosmic time. With the start of the millennium, the number of studies using the Lyman-α forest to trace the LSS have grown significantly. Particularly, it has been widely used in the development of cosmology with analyses such as tomographical studies of the IGM (e.g. Lee et al., 2014b; Lee et al., 2014a; Cisewski et al., 2014; Lee et al., 2018; Ravoux et al., 2020; Newman et al., 2020; Horowitz et al., 2022; Kraljic et al., 2022), measurements of the line-of-sight one-dimensional flux power spectrum (P1D) (e.g. Rupert A. C. Croft et al., 1998; Rupert A. C. Croft et al., 2002; McDonald et al., 2006; Kim et al., 2004; Iršič et al., 2017b; Walther et al., 2018; Chabanier et al., 2019; Karaçaylı et al., 3.3. COSMOLOGY WITH THE LYMAN-α FOREST 51 2022; Karaçaylı et al., 2024; Ravoux et al., 2023), constraints of neutrino masses (e.g. Palanque- Delabrouille et al., 2015; Yèche et al., 2017; Palanque-Delabrouille et al., 2020) and dark matter models ((e.g. Viel et al., 2013; Baur et al., 2017; Iršič et al., 2017a; Iršič et al., 2017c; Armengaud et al., 2017; Palanque-Delabrouille et al., 2020)), and the measurement of the BAO scale with correlation functions technique (e.g. Slosar et al., 2011; Busca et al., 2013; Slosar et al., 2013; Font-Ribera et al., 2014; Delubac et al., 2015; Bautista et al., 2017; du Mas des Bourboux et al., 2017; de Sainte Agathe et al., 2019; M. Blomqvist et al., 2019; du Mas des Bourboux et al., 2020), in which this research is centered. The use of this technique has evolved significantly over time. Early studies relied on small quasar samples large enough only to identify Lyman-α absorption and correlate them along individual lines of sight (LOS). Later, with a sample of 10,000 quasars from BOSS, it became possible to probe the full three-dimensional distribution of neutral hydrogen and detect the expected large- range correlations, including redshift-space distortions (RSD) [Slosar et al., 2011]. This progress culminated in successfully achieving the first measurement of the BAO peak in the Lyman-α forest reported in 2013 [Busca et al., 2013] using the ninth Data Release (DR9) of the SDSS, composed of a sample of ≈ 60, 000 QSO at z ∼ 2.3 [Pâris et al., 2012], [Ahn et al., 2012]. The DESI survey seeks to achieve exceptional precision in mapping BAO across a broad spectrum of redshifts during its five-year campaign. During the observations of the first year of operations, it has performed the BAO analysis at z = 2.33 computing auto-correlations of the Lyman-α forest dataset and its cross-correlation with quasar positions using a sample of over 450,000 Lyman-α forest spectra and more than 700,000 quasars over an area of 9500 deg2 from its first data release (DR1). This is the largest dataset ever obtained for this type of analysis, duplicating the sample of the previous eBOSS DR16, which has been the state of the art since [DESI Collaboration, 2025c], [du Mas des Bourboux et al., 2020]. 52 CHAPTER 3. QUASARS AND IGM AS LARGE SCALE STRUCTURE TRACERS Chapter 4 Correlation Function Measurements from Quasars and IGM Tracers The IGM is a powerful probe of the LSS by tracing neutral hydrogen and other abundant elements in the Universe. The statistical properties of the LSS can be studied through the flux correlation function ξF (r), which measures the spatial correlation between absorption features and provides information about the IGM clustering, and the flux power spectrum PF (k), which provides infor- mation about growth of structure and other features from which cosmological parameters and some properties of the IGM can be inferred. Throughout this chapter, these tools will be explained, as well as its use for this research work. 4.1 IGM Absorption Fluctuations In the same way as the correlation function ξ(r) is computed with positions of galaxies, flux cor- relation functions can be derived in a similar manner. However, instead of galaxy positions, these functions rely on the locations of IGM density fluctuations, traced by the flux fluctuations due to the optical depth of the absorption that generates the different forests. Since neutral hydrogen is the most abundant element in the Universe, the best and most used tracer of IGM has been the Lyman-α forest. Nonetheless, since the objective of this thesis work is to explore other transitions as IGM tracers and sources of cosmological parameters, the statistical tools to study absorber forests have been generalized to analyze any transitions whose physics are consistent with the aforementioned. Following the notation used in [DESI Collaboration, 2025c], we define the following spectral regions to develop the analysis of this research: ⋆ Region A: the restframe wavelength range of the QSO spectrum from 1040 Å to 1205 Å. ⋆ Region B: the restframe wavelength range of the QSO spectrum from 920 Å to 1020 Å. These regions are delimited by restframe wavelengths corresponding to specific spectral intervals of the QSO continuum. It is important to distinguish the different absorption transitions (e.g., 53 54 CHAPTER 4. CORRELATION FUNCTIONS FROM QSO AND IGM TRACERS Lyman-α, Lyman-β) that contribute to the absorption features observed in each region, as multiple transitions at different redshifts can overlap in the same observed wavelength range. Defining these regions allows for a more accurate treatment of contamination. For example, metal absorption lines in Region A can be misidentified as Lyman-α by analysis pipelines, biasing the shape of the correlations [Muñoz-Gutierrez, 2019]. Similarly, Region B contains both Lyman-β absorption and contamination from Lyman-α lines at different redshifts. This is illustrated in Figure 4.1. Throughout this work, when referring to the Lyman-β forest, we specifically mean the Lyman-β absorption in Region B, excluding contamination from overlapping Lyman-α lines. Figure 4.1: Quasar spectrum from the first year of the DESI survey, at a redshift of z = 3.14. The dashed lines indicate the positions of prominent emission lines. The colored regions on the blue side of each emission line correspond to the associated absorption forests, named after the respective ions: light green marks the C iii forest, defined by the C iii emission line; light blue corresponds to the C iv forest; indigo highlights the Lyman-α forest and Region A; and purple denotes the Lyman-β forest and Region B [DESI Collaboration, 2025c]. The following paragraphs provide a step-by-step explanation on how to compute the correlation functions taking as a starting point the measurement of the flux of the QSOs in the catalogs. For further reference, this process is thoroughly detailed in [[Gordon et al., 2023], [DESI Collaboration, 2025c]]1[[Bautista et al., 2017], [du Mas des Bourboux et al., 2017], [M. Blomqvist et al., 2019], [de Sainte Agathe et al., 2019], [du Mas des Bourboux et al., 2020]]2 The flux f of an object is the amount of radiation that we observe from it at different wavelengths λ. The observed flux of a quasar is denoted as fq(λ). The unabsorbed quasar continuum, denoted C(λ) (or specifically Cq(λ) for a quasar), corresponds to the intrinsic emission spectrum of the object, representing the photon flux emitted across all wavelengths in the absence of any intervening absorption. F (λ) is the mean transmission of IGM at a given wavelength λ. The transmitted flux fraction F is given by the relation F = e−τ , with τ being the optical depth 1DESI pipeline analysis. 2BOSS and eBOSS pipeline analysis. 4.1. IGM ABSORPTION FLUCTUATIONS 55 (the quantification of how much light from the source is absorbed by the IGM when passing through it). The optical depth τ is connected to local matter overdensities δm. To establish this connection, we rely on two key physical assumptions: first, under adiabatic expansion, the temperature T of the low-density IGM is tightly correlated with its density ρ, following the relation d lnT d ln ρ = γ − 1, (4.1) where γ is the equation-of-state parameter or temperature-density index. Second, assuming pho- toionization equilibrium, the number density of neutral hydrogen atoms scales with the gas density and temperature as nHI ∝ ρ2T−0.7. Since the optical depth is directly proportional to nHI, these assumptions together allow us to express τ as a function of the local matter density. This leads to the formulation known as the aforementioned fluctuating Gunn–Peterson approximation (FGPA) which models the optical depth as τ(z,x) = τ0(z) [1 + δ(z,x)] α(z) , (4.2) where τ0(z) is a redshift-dependent normalization factor encapsulating the photoionization rate and thermal state of the gas, and α(z) = 2− 0.7(γ(z)− 1) reflects the slope of the temperature–density relation. Both τ0(z) and α(z) are treated as free parameters in practice, and the method for choos- ing them is explained in [Farr et al., 2020b]. The fluctuations around the mean transmitted flux fraction are called transmitted flux field, they are given as a function of wavelength δq(λ), and calculated as: δq(λ) = fq(λ) Cq(λ)F (λ) − 1. (4.3) This is the relative ratio of the absorbed flux to the expected flux. The product F (λ)Cq(λ) is the mean expected flux for a quasar q at a wavelength λ. Following [DESI Collaboration, 2025c] and [Gordon et al., 2023], this is estimated with the approximation: F (z)Cq(λ) = C(λRF ) ( aq + bq Λ− Λmin Λmax − Λmin ) ,with Λ = log λ. (4.4) In this expression, C(λRF ) is an estimate of the mean continuum, the rest-frame (λRF ) spectra of all quasars. The pair of parameters (aq, bq) describe the diversity of quasar spectra and the redshift evolution of the mean transmission F (λ). The flux variance (total variance of the data, later used to calculate the pixel weights described in Section 4.3.1) σ2 q in the pixels is given by: σ2 q = (F (λ)Cq(λ)) 2σ2 LSS(λ) + ηpip(λ)σ 2 pip,q(λ). (4.5) In this expression, the first term represents the intrinsic variance of the Lyman-α forest. The second term represents the variance from the instrumental noise, and is conformed by the noise variance estimated by the pipeline σ2 pip,q, normalized with the expected flux, and multiplied by a function that corrects possible mis-calibrations of the instrumental noise, ηpip(λ). The functions η(λ) and σ2 LSS(λ), the intrinsic variance of the Lyman-α forest, are iteratively fitted with the mean quasar continuum C(λRF ) and the parameters (aq, bq). These parameters are typically computed in around 56 CHAPTER 4. CORRELATION FUNCTIONS FROM QSO AND IGM TRACERS five convergent iterations3 by the code PICCA (Package for IGM Cosmological-Correlations Anal- yses), a software package designed to process Lyman-α forest data for LSS analyses taking quasar spectra as input to produce 1D and 3D correlation functions (both auto- and cross-correlations) between quasars and absorption features [IGMHub Collaboration, n.d.]. 4.2 Alcock-Paczyński effect A significant portion of the cosmological information extracted from galaxy surveys does not come directly from the statistical tools explained in Section 1.4.1, but rather from effects that distort the observed signal relative to this ideal case [Percival, 2014]. One such effect is the Alcock–Paczyński (AP) effect [Alcock et al., 1979], which arises from a fundamental challenge in observational cos- mology: distances to cosmological objects must be inferred from redshifts and angular positions, assuming a specific cosmological model. A fiducial cosmology refers to a set of cosmological pa- rameters adopted as a reference model when analyzing data. These parameters are used to convert observables, such as RA, DEC, and z, into physical quantities like distances. However, if the as- sumed fiducial cosmology differs from the true one, this conversion introduces geometric distortions that lead to apparent anisotropies in the spatial distribution of galaxies or other tracers, even if their intrinsic distribution is statistically isotropic, such as a spherical galaxy cluster or a statis- tically isotropic field. The AP effect occurs because converting angular separations and redshift differences into comoving coordinates depends specifically on the Hubble parameter H(z) and the transverse comoving distance DM (z). If the fiducial cosmology adopted for this conversion differs from the true cosmology, the resulting comoving distances will be distorted, leading to apparent anisotropies in the observed clustering of galaxies or the LSS. These geometric distortions manifest as mismatches in radial and transverse directions and provide the basis for the AP test. The AP effect offers a robust and model-independent cosmological probe, as it relies purely on geometry and is largely insensitive to the details of galaxy evolution. By analyzing the anisotropies in statistical features that are expected to be isotropic (such as ξ(r) or P (k) of galaxies or the IGM absorber forests) this test constrains the expansion history and geometry of the Universe. In this way, the AP effect serves as a powerful tool to extract cosmological information from the apparent shapes of structures in redshift surveys. We define α∥ and α⊥ as scaling parameters that quantify deviations between the true and fiducial cosmology in the radial and transverse directions, respectively. These parameters are extracted from the anisotropic clustering of LSS tracers, such as galaxies or the Lyman-α forest, and are a direct outcome of AP measurements. These parameters are interpreted as ratios of cosmological distances normalized by the sound horizon scale rd, with respect to the values in the fiducial cosmology [DESI Collaboration, 2025c]: α∥ = DH(zeff)/rd [DH(zeff)/rd]fid , α ⊥= DM (zeff)/rd [DM (zeff)/rd]fid . (4.6) where DH(z) = c/H(z), and the quantities with subscript “fid” are computed using the fiducial cosmology which, for DESI DR1 analysis (including the one performed in this thesis work) is given by Planck Collaboration, 2020. By comparing observed clustering anisotropies with expectations from the fiducial model, these scaling parameters enable geometric tests of cosmology. In combination with measurements at other 3explained in more detail in [Gordon et al., 2023] 4.3. THE FLUX CORRELATION FUNCTIONS 57 redshifts, they contribute to reconstructing the expansion history H(z) and constraining the energy content of the Universe, including dark energy 4.3 The Flux Correlation Functions In general, there are two types of datasets considered in the analysis of this research. One of them is conformed by the fluctuations δ(λ) of absorber forests, described in Chapters 2 and 4.5. The fluctuations in question are mainly those produced by the Lyman-α, Lyman-β, C iv, Si iv, and Mg ii transitions (detailed in Chapters 5 and 6). The other dataset consists of quasars that make up the sample of absorption quasars in the DESI DR1 catalog, as reported by [DESI Collaboration, 2025c]. This gives a combination of 6 correlations to perform the analysis of these works: • Auto-correlations: Lyman-α, Lyman-β, C iv, Si iv, Mg ii correlating each with fluctuations of the same species. • Cross-correlations: Lyman-α, Lyman-β, C iv, Si iv, Mg ii fluctuations correlated with quasar positions These are measured in bins of comoving separation along and across the line of sight (r∥ and r⊥ respectively), obtained from the angular and redshift separations using the standard cosmological model explained in Chapter 1 with cosmological parameters from Table 1.1 of [Planck Collaboration, 2020]. Their computation is described in the following subsections, following [Gordon et al., 2023] and [DESI Collaboration, 2025c]. The correlation functions between the forests of different absorbers are often represented in different angular bins relative to the LOS. These are called wedges. These are useful for studying different features, such as metal contamination. Since the correlations depend on r∥ and r⊥, instead of analyzing the full correlation function, one can integrate it over specific angular ranges (or wedges) in µ, where µ = r∥/r is the cosine of the angle between the separation vector and the line of sight. By analyzing wedges, one can isolate different physical effects, improving constraints on BAO measurements while studying and mitigating systematics. In the case of cross-correlations, slices refer to fixed-separation cuts through the correlation function. These slices provide a way to visualize the clustering signal and other features more clearly. A fixed r⊥ slice shows how the correlation function varies along the line of sight at a given transverse separation. A fixed r∥ slice reveals how the clustering signal behaves in the transverse direction, highlighting the large-scale structure perpendicular to the line of sight. Wedges and slices are illustrated in the different panels of Figure 4.2. 58 CHAPTER 4. CORRELATION FUNCTIONS FROM QSO AND IGM TRACERS (a) (b) (c) Figure 4.2: Panel (a) displays the angular bin representation (wedges), while panels (b) and (c) illustrate the fixed-separation (vertical and horizontal, respectively) cut representation (slices). 4.3. THE FLUX CORRELATION FUNCTIONS 59 4.3.1 Auto-correlation Figure 4.3: Schematical configuration of the IGM overdensities traced in the auto-correlation func- tion. In order to measure the auto-correlation function, redshift and angular separations (∆z,∆θ respec- tively) are transformed into longitudinal (along the LOS) and transverse comoving separations (r∥ and r⊥). For a pair of measurements (pixel-pixel or pixel-quasar) denoted by (i, j) at redshifts (zi, zj) and separated by an angle ∆θ we calculate r∥ and r⊥ as: r∥ = (DC(zi)−DC(zj)) cos ( ∆θ 2 ) , (4.7) r⊥ = (DM (zi) +DM (zj)) sin ( ∆θ 2 ) . (4.8) For a given pixel i, the redshift is calculated assuming the wavelength of a known absorption, denoted by m such that zi = λobs λm − 1. In Equation 4.7, DC(z) is the comoving distance and DM (z) the angular (or transverse) comoving distance. Assuming a fiducial cosmology with Ωk = 0, DM = DC [DESI Collaboration, 2025c], [Gordon et al., 2023]. For the measurement of this correlation, a weighted covariance estimator is used [Gordon et al., 2023], [Slosar et al., 2011], [du Mas des Bourboux et al., 2020]: ξA = ∑ i,j∈A wiwjδiδj ∑ i,j∈A wiwj , (4.9) with A being a 4Mpc/h width bin (or pixel) in (r∥, r⊥), and wi, wj are the weights4, which are applied to the Lyman-α pixels to account for the diverse signal-to-noise ratios resulting from the varying brightness of quasars and exposure times, ensuring that our measurements of correlations accurately reflect the underlying data (technical details are explained in [Gordon et al., 2023], [DESI 4a modified Lyman-α weights, to be used instead of the inverse of the variance σ2 q , explained in detail in eq. 3.1 in [Gordon et al., 2023] and eq. 3.2 in [DESI Collaboration, 2025c] 60 CHAPTER 4. CORRELATION FUNCTIONS FROM QSO AND IGM TRACERS Figure 4.4: Lyman-α auto-correlation function computed with data from the first year of operations of the DESI survey (points), with the best-fit-models (black solid lines). The different colors represent the four analyzed wedges, where the blue points are the correlation closest to the LOS and the red one corresponds to the farthest. [DESI Collaboration, 2025c] Collaboration, 2025c]). The sum is performed over all pixel pairs i, j across different LOS, excluding the ones in the same one to avoid continuum fitting errors that affect the forest. For each bin A, the model correlation ξmod is evaluated at the weighted mean separation r∥, r⊥ of the Lyman-α pixel pairs in the data. For BAO analysis with the Lyman-α forest, the autocorrelation (schemed in Figure 4.3) is generally measured from [0, 200] Mpc/h in different amount of bins, parallel and perpendicular to the LOS. Figure 4.4 illustrates the Lyman-α autocorrelation function as a function of µ, r, where µ = r∥/r, ranging from µ ∈ [0.95, 1] (closest to the LOS) to µ ∈ [0, 0.5] (furthest from the LOS), and r2 = r2∥ + r2⊥. 4.3. THE FLUX CORRELATION FUNCTIONS 61 4.3.2 Cross-correlation Figure 4.5: Schematical configuration of elements traced in the cross-correlation function between IGM overdensities and quasars. The cross-correlation function measures the relationship between two different tracers (or groups of datasets) as a function of a separation r. In the context of IGM, the correlations are generally IGM fluctuations (often Lyman-α, but for the development of this work any of the absorber forests defined previously) correlated with QSO across different scales, as schemed in Figure 4.5. Similarly to the auto-correlation, the estimator for the cross-correlation is defined as [Gordon et al., 2023], [DESI Collaboration, 2025c]: ξA = ∑ i,j∈A wiwjδi ∑ i,j∈A wiwj , (4.10) for a pixel of a given absorption i and quasar j, and for the weights wj corrected for QSO bias evolution given by: wj = ( 1 + zj 1 + zfid )γq−1 . (4.11) where γq = 1.44 ± 0.08 [du Mas des Bourboux et al., 2019] and zfid = 2.25 [Gordon et al., 2023], and the sum runs over quasar-pixel pairs, excluding pixels from their respective background quasar. The cross-correlation is also computed in terms of the LOS and transverse separation, with r⊥ ∈ [0, 200] Mpc/h and r∥ ∈ [−200, 200] Mpc/h. It is usually defined as negative separations when the IGM pixel is between the observer and the QSO, and positive separations when the quasar is between the observer and the IGM pixel. Figure 4.6 shows the shape of the cross-correlation computed with the Lyman-α forest, as a function of µ, r. It can be seen that the cross-correlation is asymmetric under the permutation of the two tracers due to several reasons: spurious excess correlation due to contamination from other transitions appears at either r∥ > 0 or r∥ < 0, that µ is averaged over µ ∈ [−1, 1], since the cross-correlation has negative values of r∥, systematic redshift errors (see eq. 4.9 of [Gordon et al., 2023]), and the redshift evolution of the bias of each tracer. 62 CHAPTER 4. CORRELATION FUNCTIONS FROM QSO AND IGM TRACERS It can also be seen that the shape of the cross-correlation seems to be reversed with respect to the matter correlation function. This is because the bias of the absorbers is negative (see clarification about bias in Section 4.4), and the one of QSO is positive, resulting in a negative product in general. Figure 4.6: Lyman-α and quasars cross-correlation function computed with data from the first year of operations of the DESI survey (points), with the best-fit-models (black solid lines). The different colors represent the four analyzed wedges, where the blue points are the correlation closest to the LOS and the red ones the farthest. [DESI Collaboration, 2025c] 4.4 Flux Power Spectrum PF (k) According to what was explained in Section 1.4.1, the Fourier transform of a given ξ(r) is a power spectrum P (k). This means that, just as there is a matter P (k) to a ξ(r), there is a flux PF (k) to a flux ξF (r). Since the flux traces the density field, the flux power spectrum PF (k) provides constraints on the total matter power spectrum, considered in the linear regime Plin(k). They are related by the equation: PF (k) = bF (1 + β2 µ) 2Plin(k)D(k) (4.12) where bF is the flux bias, a linear bias parameter that relates fluctuations in the transmitted flux to the underlying linear matter density fluctuations described by Plin(k). This type of bias arises be- cause observable tracers of the LSS (such as galaxies, quasars, and IGM traced by different forests) do not perfectly follow the distribution of dark matter, but instead trace it in a biased way. In the linear regime, where density fluctuations are small, this bias is assumed to be scale-independent and 4.5. PIPELINE VALIDATION AND APPLICATION TO DATA 63 can be modeled as a constant multiplicative factor. Specifically, the flux transmitted through the IGM is anti-correlated with the matter density: higher matter density leads to stronger absorption and thus lower flux. This results in a negative flux bias, meaning that the flux fluctuations trace the matter fluctuations with opposite sign. The accurate measurement of bF is essential, since it allows the inference of the underlying matter power spectrum Plin(k) from IGM statistics, enabling cosmological parameter constraints from Lyman-α forest data. The term (1− β2 µ) 2 originates from linear RSD (Kaiser effect). It enhances clustering along the line of sight due to peculiar velocities of the absorbing gas. While a full treatment of RSD in the ξF is beyond the scope of this work, this factor is included here for completeness as it is a common component in standard flux power spectrum models. Here, β is the RSD parameter, related to the linear growth rate of structure f and bF as β = f/bF 5 and describing the effect of peculiar velocities on the observed clustering, µ is the cosine of the angle between the wavevector k and the LOS. The last term, D(k) introduces the non-linear corrections, which are corrections from small-scale nonlinear effects, including thermal broadening, gas pressure effects, and velocity dispersion of the absorbing gas. Among these terms, it is important to highlight the importance of bF , since it is the quantity that is measured with absorber forests in the development of this research work. 4.5 Pipeline Validation and Application to Simulated and Observational Data Over the years, extensive analyses of the Lyman-α forest have refined our understanding of its role as a high-redshift tracer of LSS. From the measurements of an early BOSS [Dawson et al., 2013] and eBOSS [Dawson et al., 2016], to the latest DESI data, studies performed with the Lyman-α forest have consistently demonstrated their robustness in detecting the BAO signal and constraining cosmological parameters. These efforts have paved the way for high precision galaxy cosmology at redshifts z > 2. This section reviews the observational progress that underpins and validates the methodologies described in the previous sections. By tracing the key developments and results from past surveys, it highlights how successive improvements have consolidated the reliability of the Lyman-α forest as a cosmological probe. As explained in Chapter 2, DESI is a groundbreaking project designed to map the Universe with unprecedented precision by measuring BAO over a range of redshifts. Over the course of its five-year survey, DESI aims to measure the redshift of over 40 million galaxies and quasars, covering 14,000 deg2 of the sky. The project has already achieved significant milestones, including assembling a 5The linear growth rate is defined as f ≡ d lnD+ d ln a , (4.13) where D+(a) is the linear growth function describing the growth of matter overdensities in the linear regime. It is the growing mode solution of the linear perturbation equation δ̈ + 2Hδ̇ − 4πGρmδ = 0, (4.14) with δ the matter overdensity, H the Hubble parameter, and ρm the matter density. The growth factor D+ is normalized to unity today, D+(a = 1) = 1, and its evolution depends on the cosmological parameters, particularly the matter density and dark energy content. For further details see: [Dodelson, 2003], [Amendola et al., 2010], [Aghamousa et al., 2016a]. 64 CHAPTER 4. CORRELATION FUNCTIONS FROM QSO AND IGM TRACERS dataset of its first year of operations with redshifts for approximately 13 million galaxies, 1.5 million quasars, and 4 million stars over more than 9,500 deg2, and is currently performing the first BAO and cosmological analyses assembling a dataset with the data from the third year of operations. A key aspect of DESI is using the Lyman-α forest as a tracer of the LSS of the Universe. These absorption features are sensitive to the distribution of matter on cosmological scales, making them a powerful tool for tracing the underlying dark matter distribution and measuring the expansion history of the Universe at redshifts above z > 2. In its first year of data, DESI has used this dataset to perform seven BAO measurements across different redshifts. Among these, a significant result was achieved using the Lyman-α forest, where DESI performed an auto-correlation analysis at a redshift of z = 2.33. Additionally, a cross- correlation between the Lyman-α forest and the positions of quasars enabled an even more precise measurement of BAO at these high redshifts. While the previous sections of this chapter delves in the methodology followed to make these measurements, this one discusses the results of these analyses, highlighting how DESI uses the Lyman-α forest as an LSS tracer, contributing to our understanding of the expansion of the Universe at redshifts inaccessible galaxy surveying. As mentioned before, the results of the DESI survey presented in [DESI Collaboration, 2025c] are built on the success and constant evolution and improvement of previous cosmological surveys such as the Baryon Oscillation Spectroscopic Survey (BOSS) Dawson et al., 2013 and the extended BOSS (eBOSS) Dawson et al., 2016 have successfully utilized the Lyman-α forest as a powerful probe of LSS. BOSS pioneered the use of the Lyman-α forest to measure the distribution of matter at high redshifts, resulting in the first detection of BAO at z ∼ 2.3 Busca et al., 2013. eBOSS extended these measurements, improving the precision of BAO and providing cross-correlations between the Lyman-α forest and quasar positions. These groundbreaking efforts laid the foundation for current surveys like DESI to refine high-redshift BAO measurements and further explore the expansion history of the Universe. The evolution of these analyses and results is presented in the following paragraphs: Since the first detection of the BAO in 2012, with BOSS DR9 [Busca et al., 2013], the tools for the analysis of cosmo-statistics of the IGM have evolved significantly, converging in the analysis presented in DESI DR1. [Busca et al., 2013], [Bautista et al., 2017], [du Mas des Bourboux et al., 2017], [de Sainte Agathe et al., 2019], [M. Blomqvist et al., 2019], [du Mas des Bourboux et al., 2020], and [DESI Collaboration, 2025c] have collectively traced the evolution of Lyman-α forest cosmology from its pioneering use in BOSS over ten years ago, to the latest developments in DESI. 1. Busca et al [Busca et al., 2013] present the first detection of BAO using the Lyman-α forest from the ninth data release of SDSS-III BOSS [Ahn et al., 2012]. It pioneered the use of the Lyman- α forest for cosmology, demonstrating the potential of this high-redshift probe to map the LSS. The analysis centered on auto-correlation of the Lyman-α forest at redshift z ∼ 2.5, providing the first cosmological constraints on the Hubble parameter and angular diameter distance at such high redshifts. This set the stage for future work by showing that the Lyman-α forest could successfully measure BAO at early cosmic times. 2. Bautista et al. [Bautista et al., 2017] presents a BAO measurement at redshift 2.3 using data from the twelfth data release of SDSS-III BOSS [Alam et al., 2015]. The analysis utilizes the Lyman-α forests of more than 150,000 quasars in the redshift range 2.1 < z < 3.5. It focuses on the Lyman-α auto-correlation, and presents a significant detection of the BAO peak, contributing to the determination of the cosmological parameters associated with the expansion rate of the Universe at high redshift. This high-redshift BAO measurement provides unique constraints on the Hubble parameter H(z) and the angular diameter distance DA(z), enhancing our understanding of dark 4.5. PIPELINE VALIDATION AND APPLICATION TO DATA 65 energy and the large-scale structure of the Universe at earlier cosmic epochs. It also elaborates on the construction of synthetic datasets for the validation of the analyses. 3. Du Mas des Bourboux et al. [du Mas des Bourboux et al., 2017] focuses, complementary, on the cross-correlation between the Lyman-α forest and quasars from the twelfth data release of SDSS-III BOSS [Alam et al., 2015]. This paper refined the methodology for measuring BAO at high redshifts. By incorporating the cross-correlation into the analysis, it achieved even more accurate results for the BAO signal at z = 2.4. Both works were done with significant improvements in data processing, statistical treatment, and the development of synthetic datasets to study, characterize, and mitigate systematic errors. 4. De Sainte Agathe et al. [de Sainte Agathe et al., 2019] provide BAO measurements through the auto-correlation of Lyman-α at z = 2.34 from the twelfth data release of SDSS-IV eBOSS DR14 data [Abolfathi et al., 2018]. Over 170,000 Lyman-α forests were analyzed to determine the parameters of the BAO peak, with results that are consistent with the ΛCDM model. Additionally, Lyman-α absorption in the Lyman-β region (nowadays defined as region B, see Figure 4.1) of 56,154 spectra were used for the first time. 5. Blomqvist et al. [M. Blomqvist et al., 2019] present a parallel measurement of BAO at z = 2.35 by analyzing cross-correlations between Lyman-α absorption and quasar positions from the twelfth data release of SDSS-IV eBOSS DR14 [Abolfathi et al., 2018] data. The study utilizes over 266,000 quasars, expanding the range of Lyman-α forest data to include the Lyman-β region (nowadays defined as region B, see Section 4.1) as well, which increases precision by reducing noise and systematics. 6. Du Mas des Bourboux et al. [du Mas des Bourboux et al., 2020] present BAO measurements from both: the Lyman-α forest auto-correlation and Lyman-α-quasar cross-correlations, utilizing data from the sixteenth data release of SDSS-IV eBOSS Ahumada et al., 2020. This analysis is performed with a sample of 210,005 quasars with redshifts z > 2.1 and 341,468 quasars at z > 1.77. The study is made with an updated version of the code PICCA[IGMHub Collaboration, n.d.] that makes it capable not only of expanding the spectral range to include the Lyman-β region (nowadays defined as region B, see Section 4.1), but also improving the S/N and reducing statistical uncertainties. Combining auto-correlation and cross-correlation yields distances at z = 2.33, consistent with ΛCDM. These high-precision measurements validate Lyman-α forest data as an effective BAO tracer, setting a foundation for further studies like DESI to probe large-scale structure at high redshifts. 7. The DESI Collaboration [DESI Collaboration, 2025c] presents BAO measurements using the sample of 1.5 million quasars described in Chapter 2, as part of the data release from its first year of operations. This was done by applying similar but more robust techniques as in BOSS and eBOSS, performed by the codes PICCA [IGMHub Collaboration, n.d.] and Vega [Cuceu et al., n.d.] (see Section 5.3), but with a dataset that is six times larger. DESI builds on the success of previous surveys by increasing the sample size of quasars and Lyman-α forest spectra, improving the precision of BAO measurements at z = 2.33 using both auto-correlation of the Lyman-α forest and cross-correlation with quasar positions. With this sample, DESI provides an unprecedented level of precision in high-redshift BAO measurements, enabling even tighter constraints on the Hubble parameter and angular diameter distance compared to BOSS and eBOSS (as presented in Table 2.9). To illustrate the comparison between the previous sample to the one conforming the DESI DR1 dataset, the right panel of Figure 2.7 compares the redshift distribution of the quasars that conform the DESI DR1 and SDSS DR16 datasets, together with the Lyman-α pixels. Drawing from the achievements and constant evolution and improvement of the IGM analyses 66 CHAPTER 4. CORRELATION FUNCTIONS FROM QSO AND IGM TRACERS probed by the Lyman-α forest in region A and B and, following the proposal of considering the C iv forest as a possible BAO tracer [Pieri, 2014] together with several analyses of the C iv and Mg ii forests [Blomqvist et al., 2018]], [du Mas des Bourboux et al., 2019], [Gontcho A. Gontcho et al., 2018] the following chapters will use the same techniques, but with using the Lyman-β (in region B), Si iv, C iv, and Mg ii forests as LSS tracers and analyze their potential use as BAO tracers as well with the largest and most precise dataset up-to-date. Chapter 5 The Lyman β Forest as Tracer of the Large Scale Structure While the Lyman-α forest has been extensively used to trace the distribution of matter at high redshifts, the Lyman-β forest remains a less explored but promising alternative. By studying the correlations in forests of multiple LOS, it can be used as an independent tracer of the cosmic web, offering complementary constraints on the matter power spectrum and the growth of structure. One of the key motivations for investigating the Lyman-β forest is its potential to be used as a BAO tracer. Since the restframe wavelengths of the forest is different than the Lyman-α, it allows the study of additional modes of the LSS, potentially reducing statistical uncertainties and systematic biases. Additionally, joint analyses of both forests could improve constraints on cosmological parameters by increasing the effective volume of the intergalactic medium sampled. This chapter explores the theoretical foundations and observational prospects of using the Lyman-β forest as a tracer of LSS, and presents the first cross-correlations between the Lyman-β forest in Region B with quasars from the DESI DR1 sample. The following analysis constitutes orig- inal work developed for this thesis, including the implementation of the cross-correlation estimator and the interpretation of the results. 5.1 The Lyman-β forest Section 3.2.1 describes the phenomenology that generates the Lyman transitions and forests. The Lyman-β forest (from λ = 920 Å to λ = 1020 Å) traces IGM overdensities distributed at a redshift range of 2.91 < z < 8.63. Figure 5.1 shows a quasar spectrum at a redshift of 3.16, observed by DESI, where the Lyman-β emission line and the Lyman-β forest are highlighted in blue, together with the Lyman-α emission line and forest highlighted in red. 67 68 CHAPTER 5. THE LYMAN β FOREST AS TRACER OF THE LSS Figure 5.1: Spectrum of a quasar at a redshift of 3.16, where the Lyman-β emission line at a rest-frame wavelength of λLyβ = 1025.18 Å, and the Lyman-β forest are shown in blue, and the Lyman-α emission line at a rest-frame wavelength of λLyα = 1215.67 Å and the Lyman-α forest are shown in red. 5.2 Measurements of Cross-Correlations of Quasars and the Lyman β Forest Section 3.2.1 introduced the physical origin and characteristics of the Lyman-α and Lyman-β tran- sitions and their corresponding forests. Chapter 4 outlines the methodological framework of this work, detailing the theoretical statistical foundations and computational tools used to compute correlation functions. Building upon this methodology, this section presents the computation of the cross-correlation between Lyman-β transmission fluctuations, δLyβ , and the positions of quasars. Quasar Catalog for the Lyman-β forest analysis Figure 5.2 shows the spatial distribution and redshift of the quasars with Lyman-β forests, used to compute cross-correlations with the aforementioned absorption. Correlation function The cross-correlation function of the Lyman-β forest with quasars at scales of −60 Mpc/h < r < 60 Mpc/h is presented in Figure 5.3. It is displayed in four panels, each representing a different range of r⊥ (see Section 4.3), which cover 0 Mpc/h < r⊥ < 2 Mpc/h, 4 Mpc/h < r⊥ < 6 Mpc/h, 8 Mpc/h < r⊥ < 10 Mpc/h, and 12 Mpc/h < r⊥ < 14 Mpc/h. By convention, positive distances correspond to points where the absorber is located in front of the quasar (i.e., at lower redshift), while negative distances correspond to absorbers located behind the quasar (i.e., at higher redshift). 5.3. BEST FIT PARAMETERS 69 The point r∥ = 0 represents the location where the absorption and the quasar are at the same redshift. This approach allows the study of possible asymmetries in the correlations caused by different astrophysical effects. Across all panels of Figure 5.3, a clear dip in ξ(r∥) can be seen around r∥ = 0. This corresponds to excess absorption in the Lyman-β forest at the redshift of the quasars. This feature indicates that foreground absorbers are statistically more likely to be found near quasars in redshift space, reflecting the correlated large-scale structure that both the quasars and the absorbing gas trace. As r⊥ increases, the amplitude of the absorption feature becomes shallower, consistent with the decrease in clustering strength at larger transverse separations. This behavior confirms the spatial correlation between quasars and the Lyman-β absorption field and is qualitatively similar to what is observed in Lyman-α–quasar cross-correlations, though with lower signal-to-noise due to the weaker Lyman-β transition. 5.3 Best Fit Parameters This section shows the best fit model to the quasar and Lyman-β computed cross-correlation. The results presented below are derived from computations performed specifically for this thesis and rep- resent an original contribution to the study of the first ever measured Lyman-β forest-quasar cross- correlations (hereafter Lyman-β X QSO). This fitting process was carried out using the software Vega, a tool specifically developed for modeling the 3D correlation functions between quasars and intergalactic absorbers, such as the Lyman-α forest. Vega constructs both the theoretical model and the corresponding likelihood function, enabling parameter estimation either through likelihood minimization with iminuit or posterior sampling using PolyChord. It is designed to interface with data products from PICCA, and is primarily used to extract cosmological information (es- pecially BAO signals) from quasar–absorption cross-correlation measurements [Cuceu et al., n.d.], [Cuceu et al., 2023], [du Mas des Bourboux et al., 2020]. We estimate the value of the bias b, letting this as the only free parameter during the fitting process, while all the rest1 remain fixed to avoid degeneracies. Figure 5.4 shows the best fitted model (in red) to the computed cross-correlation (in blue). The correlation is displayed in four wedges that isolate different angular ranges, from nearly parallel to the line of sight (top left, 0.95 < |µ| < 1.0) to nearly transverse (bottom right, 0.0 < |µ| < 0.5). These correlations are commonly plotted as r2ξ(r) to emphasize the BAO feature, expected around r ∼ 100, Mpc/h (which is not shown since the largest scale reached in this analysis is of r = 60 Mpc/h) and to visually enhance the signal at intermediate scales. The overall shape and sign of the correlation confirm that Lyman-β absorption traces the same LSS as quasars, but with a negative bias, as quantified by the computed best-fit value for the bias, given as bη = f/b with b the linear bias and f the growth factor fixed at f = 0.9674, is bηLyβ = −0.0344± 0.0012. When comparing bβ to the Lyman-α × QSO bias reported by [DESI Collaboration, 2025c], bα = −0.099+0.015 −0.013, we obtain: bβ bα ≈ 0.35. (5.1) This ratio reflects the intrinsically weaker nature of the Lyman-β transition compared to Lyman-α, confirming that the Lyman-β forest is a significantly weaker tracer of large-scale structure (LSS). The smaller bias value suggests that Lyman-β absorption features are less strongly correlated with 1like α, a∥, a⊥, β, etc. 70 CHAPTER 5. THE LYMAN β FOREST AS TRACER OF THE LSS the underlying matter density field, potentially originating from regions with lower average over- density and resulting in a lower clustering amplitude. Despite this, the Lyman-β forest remains a valuable cosmological probe. It can complement analyses based on the Lyman-α forest, particularly in multi-tracer studies, by providing independent information and helping to probe redshift ranges where Lyman-α becomes less effective. Figure 5.4: Cross-correlation of the Lyman-β forest with quasars, shown in the wedges, from top left panel to bottom right panel: 0.95 < |µ| < 1.0, 0.8 < |µ| < 0.95, 0.5 < |µ| < 0.8, and 0 < |µ| < 0.5. The blue dots are the computed data, and the red line is the best-fit model found by Vega. The bias for the Lyman-β forest is bηLyβ − 0.0344± 0.0012 5.3. BEST FIT PARAMETERS 71 (a) (b) Figure 5.2: (a) Footprint of the quasar catalog that conforms the sample of Lyman-β forests of DESI DR1 used for this analysis, and (b) histogram of z distribution of such sample. 72 CHAPTER 5. THE LYMAN β FOREST AS TRACER OF THE LSS (a) (b) (c) (d) Figure 5.3: Cross-correlation function of the Lyman-β forest with quasars at a scale of−60 Mpc/h < r < 60 Mpc/h. The four panels represent a different slice: (a) 0 Mpc/h < r⊥ < 2 Mpc/h, (b) 4 Mpc/h < r⊥ < 6 Mpc/h, (c) 8 Mpc/h < r⊥ < 10 Mpc/h, and (d) 12 Mpc/h < r⊥ < 14 Mpc/h. Chapter 6 The SiIV, CIV, and MgII Forests as Large Scale Structure Tracers The BAO scale has been measured with several tracers that probe the history of the expansion of the Universe. At lower redshifts (z < 1.5) it has been measured with galaxies and quasars. At higher redshifts (z > 2) the number of these tracers is much lower, which limits their efficacy for high-precision measurements. As a result, measurements are conducted using the IGM traced by different absorbers. Since hydrogen is the most abundant element conforming the IGM, the main absorber for the BAO measurement is Lyman-α. As a consequence, there is an interval of z (1.5 < z < 2) with less tracers. The analysis of metal forests allows to study structure and its evolution in this interval of z. Figure 6.1 shows how the gas metallicity evolves with redshift (as it goes from z = 4 to z = 0) [Vogelsberger et al., 2014]. This highlights the importance of studying the metallicity of the IGM at different evolutionary stages of the Universe to understand its overall history and the formation of LSS. 6.1 Absorber forests In an analogous way as in the Lyman forests, other atomic transitions can generate their corre- sponding forests. Depending on the intensity of these transitions and the S/N in which they are detected, they can be used as LSS tracers. Here we analyze three alternative forests of particular interest for cosmology: Si iv, C iv, and Mg ii. Table 6.1 shows the wavelength range of each of them, and Figure 6.2 illustrates a spectrum of a quasar at z = 2.63 from DESI DR1 that contains all the aforementioned forests. Figure 6.3 illustrates stacked quasar spectra observed by eBOSS [Yang et al., 2022], where the different transitions studied in this work and highlighting the Si iv, C iv, and Mg ii with their respective doublets. As opposed to the nature of the Lyman-α and Lyman-β forests, that are produced by singlets, the forests explored here are composed by doublets, which are commonly observed in the IGM despite being classified as forbidden or semi-forbidden transitions in laboratory conditions. This is due to the extremely low densities of the IGM, allowing transitions that are highly suppressed or unobservable under laboratory conditions and making such transitions observable in QSO spectra. As a consequence, the measured 3D cross-correlation functions (defined and described in Section 73 74 CHAPTER 6. THE SIIV, CIV, AND MGII FORESTS AS LSS TRACERS Figure 6.1: Redshift evolution of a box slice (21.3 Mpc thickness, 106.5 Mpc width) of the Illustris simulation, from z = 4 to z = 0, showing four projections: dark matter density, gas density, gas temperature, and gas metallicity, shown in green and orange, evolves with redshift (as it goes from z = 4 to z = 0). This highlights the importance of studying the metallicity of the IGM at different evolutionary stages of the Universe to understand its overall history and the formation of LSS. Figure sourced from [Illustris Collaboration, 2014], [Vogelsberger et al., 2014] 6.1. ABSORBER FORESTS 75 Figure 6.2: DESI DR1 quasar at z = 2.63 with the absorber forests in the following colors: Lyman-β in blue, Lyman-α in red, Si iv in green, C iv in yellow, and Mg ii in purple. Forest λmin[ Å] λmax[ Å] λa[ Å] λb[ Å] λeff[ Å] Ly-β°* 920 1020 1025.72 - - Ly-α* 1040 1205 1216 - - Si iv°* 1260 1375 1402.77 1393.76 1396.76 CIV°* 1410 1520 1550.78 1548.20 1549.06 Mg ii* 2600 2760 2803.53 2796.35 2798.75 Table 6.1: Metal absorption forests analyzed in this study. The first column lists the atomic transitions that define each forest. The second and third columns give the minimum and maximum rest-frame wavelengths of the corresponding forest regions. The fourth and fifth columns show the rest-frame wavelengths of the relevant absorption features (either singlets or doublets) associated with each forest. The fifth column is the effective wavelength, an intensity weighted average of the doublet that acts as a singlet for the fitting of the model in the correlations. The definitions of the forests are defined in [Ramirez-Perez et al., 2024] (*), and [du Mas des Bourboux et al., 2019] (°). Forest Scale r [Mpc/h] Ly-β −60 < r < 60 Si iv −60 < r < 60 C iv −40 < r < 40 Mg ii −60 < r < 60 Table 6.2: Scales in Mpc/h in which the correlation functions were computed. 76 CHAPTER 6. THE SIIV, CIV, AND MGII FORESTS AS LSS TRACERS 4.3) are the superposition of two correlations separated by a distance ∆rm, corresponding to the separation of the doublets, where m represents the different forests, and with a different bias. For the best-fit analyses of correlations with doublet nature transitions, an effective wavelength, which is an intensity weighed average λeff is defined. The effective wavelengths λeffSiIV , λeffCIV , and λeffMgII are shown in Table 6.1. 6.1.1 CIV (triply ionized carbon) forest The C iv forest (at rest frame wavelengths λ = 1420 Å to λ = 1520 Å), illustrated in Figure 6.5, is formed by the doublet caused by the electronic transition from the level 2s2S − 2p2p0, where photons are emitted at wavelengths of λa = 1548.2 Å, and λb = 1550.8 Å1, as shown in Figure 6.4 and it is one of the most populated and the strongest of the metal forests observed in quasar spectra. This makes it a good target to analyze the LSS in the Universe at redshift z > 1.4, as it also offers a valuable tool to conduct cosmological analysis at these redshifts with IGM absorption features [Pieri, 2014]. Carbon is considerably less abundant than hydrogen in the Universe. As a consequence, the observation and study of the C iv forest present additional challenges, compared to the Lyman-α forest, such as the need for higher S/N for its detection (which often requires deeper observations achieved through multiple passes or larger exposure times). Nevertheless, since the C iv forest can be observed across a broader redshift range (1.5 < z < 5.4), it makes it a relevant and potentially rich LSS tracer. Given that the presence of metals is a consequence of processes intrinsic to galactic evolution, the C iv not only serves as an effective LSS tracer but also provides insights into evolved structures within the observed forest range. 6.1.2 SiIV (Triply ionized silicon) forest As described in [Blomqvist et al., 2018], the Si iv forest (at rest-frame wavelengths of λ = 1260 Å to λ = 1375 Å) is made by a doublet composed of λSiIVa = 1393.76 Å and λSiIVb = 1402.77 Å, which takes place when electrons transition from 3s2S − 3p2P 02. It can be observed in the redshift range of 1.85 < z < 6.14, making it a potentially valuable source of information, despite the challenges imposed by lower S/N compared to C iv. Figure 6.6 shows a spectrum of a quasar at a redshift of z = 3.16 observed by DESI. The region of the spectrum highlighted in green corresponds to the Si iv forest. 1Also denoted by λCIV(1548) = 1548.20 Å and λCIV(1551) = 1550.78 Å. 2This transition is analogous to Mg ii. The change in energy levels and the structure of Figure 6.7 is valid for Si iv, with the corresponding values found in [Moore et al., 1968]. 6.1. ABSORBER FORESTS 77 Figure 6.3: Stacked quasar spectra observed by eBOSS [Yang et al., 2022]. The highlighted areas show the absorption that cause the different forests: circled in blue is the Lyman-β absorption, in orange the Lyman-α, in green the SIV, in yellow the C iv, and in red Mg ii. The doublet nature of the latter three transitions should be easily noted. 78 CHAPTER 6. THE SIIV, CIV, AND MGII FORESTS AS LSS TRACERS Figure 6.4: Transitions of C iv. The C iv forest analyzed in this work is formed by the doublet caused by the electronic transition from the level 2s2S − 2p2p0, where photons are emitted at wavelengths of 1548.2 Å, and 1550.8 Å, highlighted in orange [Moore et al., 1968]. 6.1. ABSORBER FORESTS 79 Figure 6.5: Spectrum of a quasar at a redshift of z = 3.16 observed by DESI. The C iv emission lines at rest-frame wavelengths of λCIVa = 1548.2 Å and λCIVb = 1550.8 Å and the C iv forest are highlighted in orange, and the λLyα = 1215.67 Å emission line and forest are shown in red. Figure 6.6: Spectrum of a quasar at a redshift of z = 3.16 observed by DESI. The Si iv emisison lines at rest-frame wavelengths of λSiIVa = 1393.76 Å and λSiIVb = 1402.77 Å and the Si iv forest are highlighted in green, while the λLyα = 1215.67 Å emission line and the Lyman-α forest are shown in red. 80 CHAPTER 6. THE SIIV, CIV, AND MGII FORESTS AS LSS TRACERS 6.1.3 MgII (Singly ionized magnesium) forest Figure 6.7: Transitions of Mg ii. The Mg ii forest analyzed in this work is formed by the doublet caused by the electronic transition from the level 3s2S − 3p2p0, where photons are emitted at wavelengths of λMgIIa = 2796.35 Å and λMgIIb = 2803.53 Å, highlighted in violet [Moore et al., 1968]. The Mg ii forest, at rest-frame wavelengths of λ = 2600 Å to λ = 2760 Å, is made by a doublet composed of λMgIIa = 2796.35 Å and λMgIIb = 2803.53 Å, emitted when electrons transition from 3s2S − 3p2p0, as shown in Figure 6.7. It can be observed between redshifts of 0.38 < z < 2.55, making it a complementary tracer at redshifts than cannot be observed by other forests. It is also a region where, as explained in [du Mas des Bourboux et al., 2019], there is no confusion with other doublets (namely C iv and Si iv). Figure 6.8 shows the spectrum of a quasar at a redshift of z = 2.21 observed by DESI. The region of the spectrum highlighted in violet is the Mg ii forest. 6.2. CROSS-CORRELATIONS OF QSO AND ABSORBER FORESTS 81 Figure 6.8: Spectrum of a quasar at a redshift of z = 2.21 observed by DESI. The Mg ii emisison lines at rest-frame wavelengths of λMgIIa = 2796.35 Å and λMgIIb = 2803.53 Å and the Mg ii forest are highlighted in purple, while the λLyα = 1215.67 Å emission line and the Lyman-α forest are shown in red. The Si iv and C iv doublets are shown in green and orange lines respectively. 6.2 Measurements of Cross-Correlations of Quasars and Ab- sorber Forests Following the methodology explained in Section 4.3, the three cross-correlations between the ab- sorber (C iv, Si iv, and Mg ii) forests and quasar samples were computed. The spatial (footprint and z) distribution of the quasars used for each measurement and the respective correlations are shown in the following subsections. The three different correlations are presented in four panels each. These panels present different slices (ranges of r⊥, see Section 4.3). Since the data analysis pipeline transforms wavelengths to redshifts assuming a given absorption (generally Lyman-α, but potentially defined depending on the given absorption for a specific analysis), such line and a second one3 occupying the same physical position are absorbed at different apparent redshifts, resulting in an apparent distance separation given by: r∥ = (1 + z)DH(z)∆λ/λmain, (6.1) where DH is the Hubble distance, ∆λ = λsec−λmain is the wavelength separation of the secondary transition (λsec) with respect to the main one [Herrera-Alcantar et al., 2025]. All the forests analyzed in this chapter are doublets, so the main wavelength corresponds to the most prominent absorption and the secondary one is the following. Given a redshift, the wavelength difference of these doublets ∆λ transforms into a distance r∥ that defines the resolution (binning) in which the 3Which could be a metal contaminating the forest, or a doublet. 82 CHAPTER 6. THE SIIV, CIV, AND MGII FORESTS AS LSS TRACERS correlations are computed. The redshift used to make this conversion is the effective redshift zeff, the weighted average redshift of each of the datasets (catalogs described in Sections 6.2.1, 6.2.2, and 6.2.3). 6.2.1 CIV forest analysis Quasar Catalog sample for the CIV forest analysis Figure 6.9 shows the footprint and redshift distribution of the quasars with C iv forests, used to compute cross-correlations. This sample will hereafter be referred to as C iv quasar4 catalog. Correlation function of CIV forests and quasars Figure 6.10 presents the cross-correlation functions between the C iv forest density fluctuations and the C iv quasar catalog. In order to capture the doublet nature of this transition in the correlation function, which is at r∥ = 4.85 Mpc/h at zeff = 2.09, it had to be computed with a high resolution at scales of −40 Mpc/h < r < 40 Mpc/h. It is presented in four panels, each representing a dif- ferent slice (see Section 4.3), which cover 0 Mpc/h < r⊥ < 1 Mpc/h, 2 Mpc/h < r⊥ < 3 Mpc/h, 4 Mpc/h < r⊥ < 5 Mpc/h, and 6 Mpc/h < r⊥ < 7 Mpc/h. 6.2.2 SiIV forest analysis Quasar Catalog sample for the SiIV forest analysis Figure 6.11 shows the footprint and redshift distribution of the quasars with Si iv forests, used to compute cross-correlations with the aforementioned absorption. Correlation function of SiIV forests and quasars Figure 6.12 presents the cross-correlation functions between the Si iv forest density fluctuations and the Si iv quasar catalog. Since the doublet nature is detected at r∥ = 18.6 Mpc/h at a zeff = 2.23, the correlation was computed at scales of −60 Mpc/h < r < 60 Mpc/h with a standard resolu- tion, and is presented in four panels, each representing a different slice (see Section 4.3) cover- ing 0 Mpc/h < r⊥ < 2 Mpc/h, 4 Mpc/h < r⊥ < 6 Mpc/h, 8 Mpc/h < r⊥ < 10 Mpc/h, and 12 Mpc/h < r⊥ < 14 Mpc/h. 6.2.3 MgII forest analysis Quasar Catalog sample for the MgII forest analysis Figure 6.13 shows the footprint and redshift distribution of the quasars with Mg ii forests, used to compute cross-correlations with the aforementioned absorption. 4Or C iv-QSO 6.3. BEST FIT PARAMETERS 83 Correlation function of MgII forests and quasars Figure 6.14 presents the cross-correlation functions between the Mg ii forest density fluctuations and the Mg ii quasar catalog. Since the doublet nature is detected at r∥ = 7.94 Mpc/h at a zeff = 1.71, the correlation was computed at scales of −60 Mpc/h < r < 60 Mpc/h with a standard resolution, and is presented in four panels each representing a different slice (see Section 4.3) covering 0 Mpc/h < r⊥ < 2 Mpc/h, 4 Mpc/h < r⊥ < 6 Mpc/h, 8 Mpc/h < r⊥ < 10 Mpc/h, and 12 Mpc/h < r⊥ < 14 Mpc/h. 6.3 Best Fit Parameters The obtention of the best-fit model for the cross-correlations between the absorber (Si iv, C iv, Mg ii) forests and quasars is ongoing work. The approach follows the procedure used for the Lyman-β forest in Section 5.3, where the bias parameter bη is computed using the VEGA code. However, adapting this methodology to these absorbers requires further refinements in the fitting process, as well as additional computational and human effort. As mentioned before, the doublet nature of these transitions causes the correlations to behave as the superposition of two signals separated by a fixed distance. The modeling of these features within the fitter involves treating the main line as the absorber that defines the forest, while the secondary one is considered a contaminant. The implementation of this approach is currently in progress, and results are expected in the fall of 2025. 84 CHAPTER 6. THE SIIV, CIV, AND MGII FORESTS AS LSS TRACERS (a) (b) Figure 6.9: (a) Footprint of the quasar catalog that conforms the sample of C iv forests of DESI DR1 used for this analysis, (b) histogram of z distribution of such sample. 6.3. BEST FIT PARAMETERS 85 (a) (b) (c) (d) Figure 6.10: Cross-correlation function of the C iv forest with quasars at a scale of −40 Mpc/h < r < 40 Mpc/h. The four panels represent a different slice: (a) 0 Mpc/h < r⊥ < 1 Mpc/h, (b) 2 Mpc/h < r⊥ < 3 Mpc/h, (c) 4 Mpc/h < r⊥ < 5 Mpc/h, and (d) 6 Mpc/h < r⊥ < 7 Mpc/h. 86 CHAPTER 6. THE SIIV, CIV, AND MGII FORESTS AS LSS TRACERS (a) (b) Figure 6.11: (a) Footprint of the quasar catalog that conforms the sample of Si iv forests of DESI DR1 used for this analysis, (b) histogram of z distribution of such sample. 6.3. BEST FIT PARAMETERS 87 (a) (b) (c) (d) Figure 6.12: Cross-correlation function of the Si iv forest with quasars at a scale of −60 Mpc/h < r < 60 Mpc/h. The four panels represent a different slice: (a) 0 Mpc/h < r⊥ < 2 Mpc/h, (b) 4 Mpc/h < r⊥ < 6 Mpc/h, (c) 8 Mpc/h < r⊥ < 10 Mpc/h, and (d) 12 Mpc/h < r⊥ < 14 Mpc/h. 88 CHAPTER 6. THE SIIV, CIV, AND MGII FORESTS AS LSS TRACERS (a) (b) Figure 6.13: (a) Footprint of the quasar catalog that conforms the sample of Mg ii forests of DESI DR1 used for this analysis, (b) histogram of z distribution of such sample. 6.3. BEST FIT PARAMETERS 89 (a) (b) (c) (d) Figure 6.14: Cross-correlation function of the Mg ii forest with quasars at a scale of −60 Mpc/h < r < 60 Mpc/h. The four panels represent a different slice: (a) 0 Mpc/h < r⊥ < 2 Mpc/h, (b) 4 Mpc/h < r⊥ < 6 Mpc/h, (c) 8 Mpc/h < r⊥ < 10 Mpc/h, and (d) 12 Mpc/h < r⊥ < 14 Mpc/h. 90 CHAPTER 6. THE SIIV, CIV, AND MGII FORESTS AS LSS TRACERS Chapter 7 Discussion and Conclusions Thesis Overview and Future Prospects This thesis investigates the feasibility of using alternative absorption features in the intergalactic medium, specifically the Lyman-β and C iv, Si iv, and Mg ii forests as LSS tracers. Building on the framework developed for the Lyman-α forest, this work adapts and applies a validated pipeline to new spectral features, addressing the methodological challenges introduced by their doublet nature. Using absorption features from the quasar sample observed in the first year of the DESI survey, this thesis presents the first cross-correlation measurements with the Lyman-β forest and explores small- scale clustering in absorber forests. These results establish a foundation for future analyses that aim to compute cosmological parameters and, eventually, to extract BAO signals using these tracers. These results have not been published as they are. They were initially presented within the DESI Lyman-alpha working group in 2023, and since then, the focus has shifted toward analyzing larger scales, including the BAO scale using DR2 data [DESI Collaboration, 2025f]. While these small- scale Lyman-β forest analyses contributed to early discussions, the current collaborative efforts have evolved beyond this initial scope. As data quality improves with upcoming DESI releases, this approach may significantly expand the cosmological utility of the IGM, filling into redshift gaps and offering new insights into both cosmic structure and the astrophysical processes that enrich the IGM over time. Potential of the Lyman-β and CIV forests as BAO tracers Even though the analysis performed in this work covers scales up to 60Mpc/h, the size and resolution of the datasets used for the Lyman-β and C iv forest clustering for DR1 analysis are promising about their potential use for the detection of the BAO features. The Lyman-β cross-correlation functions presented in Chapter 5 are the first ever measured correlations using the Lyman-β forest using it as a tracer of LSS of the IGM. To better understand how the Lyman-α and Lyman-β forests trace the underlying matter distribution, their respective linear biases are compared. The cross-correlation between Lyman-β forest and QSO (Lyman-β X QSO) bias obtained in this work, reported in Figure 5.4, is bβ = −0.0344 ± 0.0012, while the Lyman-α X QSO bias is bα = −0.099+0.015 −0.013 [DESI Collaboration, 2025c]. These values are reported in Table 7.1. The ratio of these two biases is 91 92 CHAPTER 7. DISCUSSION AND CONCLUSIONS Parameter Combined Absorber x Absorber Absorber x QSO bα −0.1078+0.0045 −0.0054 −0.1078± 0.003 −0.099+0.015 −0.013 bβ - - −0.0344± 0.0012 Table 7.1: Comparison of the biases of Lyman-α (ba) and Lyman-β (bβ). The first row are the best-fit values (mean of the posterior) and uncertainties (68% credible intervals) for the Lyman-α bias of DESI DR1 [DESI Collaboration, 2025c]. The second one is the value of the bias of Lyman-β x QSO obtained in this work, reported in Figure 5.4. bβ/bα ≈ 0.35, which indicates that the Lyman-β forest is a significantly weaker LSS tracer than the Lyman-α forest. This is consistent with the strength difference of both transitions, and the generally lower S/N in the B region. Nonetheless, it would be interesting to compare this value using synthetic datasets, where systematics and astrophysical effects influencing the measurement of correlations with both tracers can be better controlled. The smaller absolute value of the bias suggests that Lyman-β absorption features are less strongly correlated with the underlying matter density and may originate from regions with lower average overdensity. A lower bias implies a lower clustering amplitude. Contrary to what one might initially conclude, this does not make the Lyman- β forest less competitive. On the contrary, it highlights its potential as a complementary tracer to Lyman-α in multi-tracer analyses. In addition to offering independent information, the Lyman-β forest can probe redshift ranges where Lyman-α alone becomes less effective. As analyses using this tracer become more detailed, it will be important to consider that the Lyman-β forest is more prone to contamination from Lyman-α absorption of lower-redshift systems. Therefore, dedicated efforts to characterize these contaminants—along with others such as Ovi—will be crucial in enabling its reliable use as a cosmological tracer. Part of the investigation done during the PhD program consisted on testing the C iv forest as a potential BAO tracer in a forecasted synthetic dataset of the final DESI survey, made by the codes CoLoRe and LyaCoLoRe [Alonso et al., 2019], [Ramı́rez-Pérez et al., 2022], [Farr et al., 2020a], [Farr et al., 2020b]. This analysis began during the commisionning stage of the survey, with an early version of PICCA that has required several modifications, making it an ongoing research developed parallelly to the one presented in this thesis. Nonetheless, the amount of data and the resolution used for the binning of DR1 data is starting to present several computational challenges. As a first stage, the strategy was to prioritize the characterization and understanding of the C iv doublet, resolving it in the correlations and modelling it properly in the fitting before stepping into the challenges of the BAO scales found during the work with synthetic datasets. The following steps are to extend the scale of the analysis to cover the BAO scale, as well as modify accordingly PICCA to perform the analysis efficiently with the highest possible resolution. It is important to note that this work faces several interesting challenges moving forward. One key assumption so far is that the relation between optical depth τ and matter overdensity, δm, follows the form given in Equation 4.2. While this relation is well motivated for hydrogen, it does not necessarily hold for other elements. Although we adopt this approximation for the additional transitions analyzed, it should be emphasized that this represents a first approach. Future refinements are both possible and planned, as part of ongoing efforts to improve the physical accuracy of the model. 93 Modeling the doublet As explained in Section 6.1, since Si iv, C iv, and Mg ii forests are doublets and this impacts significantly the shape of the correlation functions, this behavior has to be well understood and characterized. This affects two lines of the analysis: one is in the computation of the correlation function and the second is in the bias fitting. While the doublets do not affect the computation of the correlation directly, but their shape, it is convenient to define a reference wavelength for the analysis. This is to choose the reference absorption feature, which can be the main, the secondary or an effective (weighted average) wavelength. This will only affect the center of the correlations, potentially shifting it slightly toward negative or positive values of r. In the case of the fitting process, the definition of the reference wavelength has a more transcendental impact. The current analysis defines the main absorption of the doublet as the reference, and the secondary one is treated as a contaminant. This will allow the computation of both biases (as free parameters together or separately). Alternative, one could use a different criteria which could be more convenient according to the approach to the analysis (e.g. choosing as the main line the one in the blue side of the spectrum, or the weighted-average wavelength). Improvement of Catalogs The first attempt to build a catalog of each tracer followed the methodology described in [Ramirez- Perez et al., 2024]1. The analysis was made with the datasets obtained filtering by redshift and wavelength the quasars, so that the computations could be performed with the respective forests. Since these catalogs were constructed, the amount of data taken by the DESI instrument has increased significantly. This has led to a more exhaustive discussion among a larger group of col- laborators to re-define the new catalogs for the current and future analyses. While these new criteria do not impact state-of-the-art analyses, their refinement with a larger dataset could introduce some effects. However, these are not expected to significantly alter the results in this line of research. Cosmological model As explained in Section 2.4.2, under Alternative Models, DESI DR1 results tend towards a w0wa cosmology. The results derived from the Lyman-α forest do not strongly favor either the w0wa or the ΛCDM model, since both models fit the Lyman-α forest data similarly well. This is illustrated in Figure 2.8. Most of the forests analyzed in this research meet the same criteria, since the effect of having one model or the other one will affect the observations at much smaller redshifts. Nonetheless, the analysis of Mg ii (particularly, but could be generalized to any forest that could help constraining the cosmological model), where zeffMgII = 1.71, could potentially be affected by the cosmological model, which could make it a very interesting forest to analyze with future datasets. This would require two approaches: observational data analysis with computational tools capable of performing their study, and synthetic datasets created with both, the standard cosmological model, and a different cosmological model (namely w0wa) as priors in order to confront them and have a 1This work presents and validates a catalog of Lyman-α forest flux-transmission fluctuations for 3D analyses, based on 88,511 quasars from the Survey Validation (SV) phase of DESI and the first two months of the main survey (M2), as part of the Early Data Release (EDR). 94 CHAPTER 7. DISCUSSION AND CONCLUSIONS better control of the framework so that it can be re-studied. The same would apply if an alternative model were to gain broader acceptance by the survey, or if the ΛCDM model were to be discarded. Future Undoubtedly, this work is considered to have a lot of potential serving as a door to new research within the area of cosmology with absorbers. There are several lines of research considered to be very interesting that are planned to be followed: ⋆ Perform a thorough analysis on the Lyman-β forest, both as a LSS tracer using correlations at scales of |r| < 60 Mpc/h for clustering purposes and at scales that include the BAO feature. This analysis will most likely include the study of contaminants (namely Lyman-α and Ovi absorption), following the work of [Muñoz-Gutierrez, 2019], [Muñoz-Gutiérrez and Macorra, 2023], [Farr et al., 2020b], [du Mas des Bourboux et al., 2020], and [Herrera-Alcantar et al., 2025]. This will most likely be done with the DESI DR2 dataset currently being analyzed. ⋆ Extend the analysis of C iv, Si iv, and Mg ii forests to scales that include the BAO feature with DESI DR1 and DR2 datasets. ⋆ Once again, following the work of [Muñoz-Gutierrez, 2019], [Muñoz-Gutiérrez and Macorra, 2023], [Farr et al., 2020b], [du Mas des Bourboux et al., 2020], and [Herrera-Alcantar et al., 2025], study the effect of C iv, Si iv, Mg ii, and Ovi in the Lyman-α forest flux correlation function, as well as their biases in BAO and other studies. ⋆ Measure the bias of all the forests mentioned at different redshifts to indirectly infer the abundance of metal elements and gain insights into the evolution of cosmic structure and chemical enrichment across epochs. Such analysis can enhance our understanding of the metal enrichment of the IGM, as well as the physical processes responsible for it - such as AGN feedback, structure formation, star formation, and galaxy formation and evolution in the early Universe. Achieving this will likely require the development of additional tools, including realistic IGM simulations to better capture the dynamics and properties of the enriched IGM. Furthermore, complementary studies with galaxy observations are essential. These include cross-correlations between metal absorbers and galaxy distributions, analy- ses of galaxy properties such as metallicity and star formation rate, circumgalactic medium (CGM) studies using quasar LOS, and investigations of feedback signatures in galaxy spectra. Such approaches can help establish direct connections between galaxy evolution and IGM en- richment, providing a more comprehensive picture of the processes shaping the high-redshift Universe. As explained in Chapter 4, the Gunn–Peterson effect provides an approximation to relate the distribution of dark matter to neutral hydrogen in the IGM. In particular, under the FGPA (Fluctuating Gunn-Peterson Approximation, see Equation 4.2), the optical depth of hydrogen absorption is modeled as a power-law function of the underlying matter density. This framework is valid as a first-order approach on large scales and at redshifts z ≈ 2 − 5, where the hydrogen in the IGM is highly ionized but still produces measurable absorption in quasar spectra. While this approximation has proven effective in linking gas and matter distributions via the Lyman-α forest, its accuracy diminishes at smaller scales and in regions where baryonic physics, temperature fluctuations, and reionization effects become significant. The inclusion of additional absorption lines from metal ions, as C iv, Si iv, and Mg ii, can 95 provide complementary information that could help to constrain deviations from the FGPA and refine our understanding of the matter–gas connection. ⋆ In previous chapters it was established that the model used for this analysis was the ΛCDM framework. Nonetheless, throughout the development of this work the cosmological results of DESI DR1 and DR2 show a preference towards a dynamical dark energy model, specifically the w0wa parameterization. Figure 2.8 was used to illustrate that the Lyman-α forest results, at z < 2, did not seem to favor either of these models. Nonetheless, this may not hold at lower redshifts, where additional metal absorption forests become visible. Employing these forests for cosmological analysis would require more detailed studies, as the growth of fluctu- ations may exhibit different behaviors across redshifts. 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