# Illuminating the Dark Sector

Searching for new interactions between dark matter and dark energy

Elsa Maria Campos Teixeira

[1]∥#1 ∥ [2][#1, #2] School of Mathematics and Statistics The University of
Sheffield cmb short=CMB, long=Cosmic Microwave Background, cdm short=CDM,
long=Cold Dark Matter, eh short=EH, long=Einstein-Hilbert, flrw short=FLRW,
long=Friedman-Lemaître-Robertson-Walker, gr short=GR, long=General Relativity,
hbb short=HBB, long=Hot Big Bang, bao short=BAO, long=Baryon Acoustic
Oscillations, bbn short=BBN, long=Big Bang Nucleosynthesis, dm short=DM,
long=Dark Matter, de short=DE, long=Dark Energy, eos short=EoS, long=Equation
of State, mde short=MDE, long=Matter Dominated Epoch, qft short=QFT,
long=Quantum Field Theory, rde short=RDE, long=Radiation Dominated Epoch, svt
short=SVT, long=Scalar-Vector-Tensor, wimps short=WIMPs, long=Weakly
Interacting Massive Particles, bb short=BB, long=Big Bang, gw short=GW,
long=Gravitational Wave, cl short=CL, long=Confidence Level, dede short=DEDE,
long=Dark Energy Dominated Epoch, shoes short=SH0ES, long=Supernovae H0 for
the Equation of State of Dark Energy, ddb short=DDB, long=Dark D-Brane, ddm
short=DDM, long=Disformal Dark Matter, lss short=LSS, long=Large Scale
Structure, sm short=SM, long=Standard Model, lisa short=LISA, long=Laser
Interferometer Space Antenna, et short=ET, long=Einstein Telescope, sn
short=SN, long=Supernova,

> Tudo no mundo começou com um sim. Uma molécula
> disse sim a outra molécula e nasceu a vida. Mas antes da pré-história
> havia a pré-história da pré-história e havia o nunca e havia o sim.
> Sempre houve. Não sei o quê, mas sei que o Universo jamais começou.
> — Clarice Lispector A Hora da Estrela

> All the world began with a yes. One molecule
> said yes to another molecule and life was born. But before prehistory
> there was the prehistory of prehistory and there was the never and there was
> the yes.
> It was ever so. I don’t know why, but I do know that the Universe never
> began.
> — Clarice Lispector in The Hour of the Star

## Preface

This dissertation is presented to the School of Mathematics and Statistics of
the University of Sheffield for the degree of Doctor of Philosophy. This
project has been supervised by Prof. Carsten van de Bruck, and the contents
report on the results of the research work conducted over the course of the
four years of my PhD, supported by the Fundação para a Ciência e a Tecnologia
(FCT) through the grant SFRH/BD/143231/2019. The Chapters in Part II are based
partially on ongoing work and on articles published in refereed journals, as
the result of collaborations with Carsten van de Bruck, Noemi Frusciante,
Bruno J. Barros, Vasco M.C. Ferreira, Richard Daniel, Gaspard Poulot and
Cameron Thomas. I list the published work below:

  1. 1.

Elsa M. Teixeira, Richard Daniel, Noemi Frusciante, and Carsten van de Bruck
“Forecasts on interacting dark energy with standard sirens”,
Phys.Rev.D 108 (2023) 8, 084070, arXiv:2309.06544 [astro-ph].

  2. 2.

Carsten van de Bruck, Gaspard Poulot, and Elsa M. Teixeira,
“Scalar field dark matter and dark energy: a hybrid model for the dark
sector”,
JCAP 07 (2023), 019, arXiv:2211.13653 [hep-th].

  3. 3.

Elsa M. Teixeira, Bruno J. Barros, Vasco M. C. Ferreira, and Noemi Frusciante,
“Dissecting kinetically coupled quintessence: phenomenology and observational
tests”,
JCAP 11 (2022), 059, arXiv:2207.13682 [gr-qc].

  4. 4.

Carsten van de Bruck, and Elsa M. Teixeira,
“Dark D–Brane Cosmology: from background evolution to cosmological
perturbations”,
Phys.Rev.D 101 (2020) 8, 083506, arXiv:2007.15414 [gr-qc]

This thesis is structured as follows:

  * •

Chapter 1 provides an introduction to the main concepts relevant for modern
cosmology, including the theory of general relativity and the underlying
cosmological principles.

  * •

Chapter 2 is an account of the fundamentals of the standard model of cosmology
- the $\Lambda$CDM model - including its formulation, assumptions, background
and perturbed equations, and a brief history of the Universe.

  * •

Chapter 3 consists of a review of the concepts in observational cosmology that
are relevant for the work included in the thesis. It covers the main
observables and probes pf the dark sector, followed by an account of the
parameters of the $\Lambda$CDM model, and a comment on the current crisis it
faces.

  * •

Chapter 4 provides a summary of the statistical methods necessary for the
numerical analysis and the assessment of the support for the alternative
models considered. It also includes a short introduction to the numerical
tools and data sets for that purpose.

  * •

Chapter 5 introduces the motivation and formulation of models beyond the
standard $\Lambda$CDM model, with particular focus on models with a coupling
in the dark sector.

  * •

Chapter 6 reports on the ongoing work with Cameron Thomas and Carsten van de
Bruck. Cameron Thomas is the author of part of the numerical work conducted
for Section 6.2.

  * •

Chapter 7 draws from published work in the Journal of Cosmology and
Astroparticle Physics (JCAP) Teixeira et al. (2022) in collaboration with
Bruno J. Barros, Vasco M. C. Ferreira and Noemi Frusciante. Bruno J. Barros is
credited with the original formulation of the model.

  * •

Chapter 8 is an account on collaborative work with Richard Daniel, Noemi
Frusciante, and Carsten van de Bruck, which has been published in Physical
Review D and is available at Teixeira et al. (2023). Richard Daniel is the
author of the simulated standard siren catalogue on which we report in
Appendix B.

  * •

Chapter 9 is based partially on published work in Physical Review D van de
Bruck and Teixeira (2020) in collaboration with Carsten van de Bruck and on
ongoing work for the data analysis.

  * •

Chapter 10 reports on work published in the Journal of Cosmology and
Astroparticle Physics (JCAP) van de Bruck et al. (2023) in collaboration with
Gaspard Poulot and Carsten van de Bruck. Gaspard Poulot is credited with part
of the theoretical background work and the fluid approximation.

  * •

Chapter 11 concludes with a summary of the work presented in this
dissertation, with an outlook on coupled and interacting quintessence models,
including some general thoughts on the relevance of the research outlined in
this manuscript and thoughts on the future of the field.

## Acronyms

## Abstract

> Os factos são sonoros mas entre os factos há um sussurro.
> É esse sussurro que me impressiona.00footnotetext: The facts are sonorous
> but between the facts there’s a whispering. It’s the whispering that
> astounds me. — Clarice Lispector in The Hour of the Star
> — Clarice Lispector A Hora da Estrela

The current standard model of cosmology - the $\Lambda$CDM model - is
appropriately named after its controversial foreign ingredients: a
cosmological constant ($\Lambda$) that accounts for the recent accelerated
expansion of the Universe and cold dark matter needed to explain the formation
and dynamics of large scale structures. Together, these form the dark sector,
whose nature remains a mystery. After 25 years of withstanding confirmation
and support for the $\Lambda$CDM model, enough to bypass some of its unclear
theoretical issues, this paradigm is facing its biggest crisis yet. The rapid
advent of technology has brought cosmology to an unprecedented observational
era, with increased technical precision and the emergence of independent
measures, including probes of phenomena that were thought impossible to detect
or even exist, such as the gravitational ripples that propagate in the
spacetime. However, such precision has unveiled cracks in the porcelain of
$\Lambda$CDM, with pieces that seem glued together and difficult to reconcile.
Particularly worrying is the apparent lack of compatibility between
measurements of the Universe’s present expansion rate based on local
measurements and those based on phenomena that occurred far in the early
Universe and that can only be translated into present quantities through
physical propagation under a cosmological model. In this dissertation, we
delve into extensions to the standard model that consider alternatives to the
mysterious nature of the dark sector and any possible new interactions
therein. We analyse these alternative models, hoping to identify measurable
observational signatures of extra degrees of freedom in the dark sector. For
this purpose, we focus on Markov Chain Monte Carlo sampling techniques by
exploiting recent observational data and their combination to constrain and
assess the validity of the model’s parameter space. The extended frameworks
investigated are only found to be marginally or not at all favoured over the
$\Lambda$CDM model, ultimately falling short of resolving the tensions present
in the latter. We conclude with a forecast on the constraining power of
upcoming gravitational wave detectors as standard sirens, independently and
combined with current background data. By generating simulated standard siren
event catalogues following the specifications predicted by the proposed
missions, we discuss how an emerging avenue for independent data may be
decisive in constraining alternative theories of gravity and shed light on the
nature of the dark sector.

##### Units

Unless stated otherwise, in this dissertation I will always adopt natural
units in which the speed of light, Planck’s constant and Boltzmann’s constant
are unity, $c=\hbar=k_{B}=1$. This translates into an equivalence of units for
energy, mass and momentum as an inverse unit of length and time:

$\text{[Energy]}=\text{[Mass]}=\text{[Temperature]}=\text{[Length]}^{-1}=\text{[Time]}^{-1}\,.$
(0.0.1)

Throughout this manuscript we denote derivatives with respect to physical time
$t$ by an upper dot and derivatives with respect to conformal time $\tau$ by a
prime,

$\displaystyle t=\text{cosmic time}\ \ $ $\displaystyle\Longrightarrow\ \
\frac{\mathrm{d}X}{\mathrm{d}t}\equiv\dot{X}\,,$ (0.0.2)
$\displaystyle\tau=\text{conformal time}\ \ $ $\displaystyle\Longrightarrow\ \
\frac{\mathrm{d}X}{\mathrm{d}\tau}\equiv X^{\prime}\,.$ (0.0.3)

Spatial 3-vectors are denoted an upper arrow, such as $\vec{x}$, and four-
dimensional spacetime vectors are denoted in terms of their components as
$x=(x^{\mu})$. We adopt the convention for the signature of the metric as
$(-,+,+,+)$.

###### Contents

  1. Preface
  2. Abstract
  3. Acknowledgements
  4. 1 Introduction
    1. 1.1 Introduction to Cosmology
    2. 1.2 A Comment on Observational Cosmology
    3. 1.3 General Relativity
    4. 1.4 Homogeneity and Isotropy
    5. 1.5 The Hubble-Lemaître Law: Cosmic Expansion
    6. 1.6 The Friedmann-Lemaître-Robertson-Walker Metric
    7. 1.7 Spacetime Geometry
    8. 1.8 The Hot Big Bang Model
  5. 2 The Standard $\Lambda$CDM Model: Successes and Controversies
    1. 2.1 Fundamentals of the Background Cosmology
    2. 2.2 Linear Cosmological Perturbations
    3. 2.3 Inflation
    4. 2.4 A Brief History of the Universe in Four Epochs
    5. 2.5 The Theoretical Problems with $\Lambda$
  6. 3 Cosmological Observational Tests
    1. 3.1 The Matter Correlation Function
    2. 3.2 The Matter Power Spectrum
    3. 3.3 The Cosmic Microwave Background
    4. 3.4 Other Cosmological Probes
    5. 3.5 The Cosmological Parameters of the $\Lambda$CDM Model
    6. 3.6 Cosmological Puzzles and Observational Tensions
  7. 4 Methodology
    1. 4.1 Statistical Methods and Bayesian Inference
    2. 4.2 Einstein-Boltzmann Codes
    3. 4.3 Baseline Data Sets
  8. 5 Beyond the Standard Model: Coupled Dark Sector
    1. 5.1 Beyond $\Lambda$CDM
    2. 5.2 Scalar Field Dark Energy
    3. 5.3 Universal Couplings to the Matter Sector
    4. 5.4 Non-Universal Couplings and Interactions in the Dark Sector
  9. 6 Coupled Quintessence in Flat and Curved Geometries
    1. 6.1 Models and Methodology
    2. 6.2 Update on Flat Coupled Quintessence
    3. 6.3 Coupled Quintessence in Non-Flat Geometries
    4. 6.4 Summary and Discussion
  10. 7 Kinetically Coupled Dark Energy
    1. 7.1 Theoretical Motivation
    2. 7.2 Background Dynamics
    3. 7.3 Evolution of Linear Perturbations
    4. 7.4 Observational Constraints
    5. 7.5 Discussion
  11. 8 Forecasts on Coupled Quintessence with Standard Sirens
    1. 8.1 Standard Sirens
    2. 8.2 Methodology and Data Sets
    3. 8.3 Forecasts Results
    4. 8.4 Discussion
  12. 9 Dark D-Brane Cosmology
    1. 9.1 The Model
    2. 9.2 Background FLRW Cosmology
    3. 9.3 Linear Perturbations
    4. 9.4 Numerical Study
    5. 9.5 Observational Constraints
    6. 9.6 Discussion
  13. 10 A Hybrid Model for the Dark Sector
    1. 10.1 The Hybrid Model
    2. 10.2 Fluid Approximation
    3. 10.3 Observables and Phenomenology
    4. 10.4 Discussion
  14. 11 Conclusions
  15. A Scalar Field Inflation
  16. B Mock Data Set Simulation
  17. C String Theory and the DBI Action

###### List of Figures

  1. 1.1 The Planck 2018 temperature anisotropies map
  2. (a)
  3. 1.2 The SDSS map of galaxies and quasars in the observable Universe
  4. (a)
  5. 2.1 The redshift evolution of the energy densities of the components of the Universe
  6. (a)
  7. 3.1 The Planck 2018 CMB temperature–temperature power spectrum
  8. (a)
  9. 3.2 Contributions to the CMB power spectrum
  10. (a)
  11. 3.3 The Planck 2018 CMB gravitational lensing power spectrum
  12. (a)
  13. 3.4 The distance modulus measurements from Pantheon
  14. (a)
  15. 3.5 BAO distance and expansion rate data from SDSS
  16. (a)
  17. 3.6 The impact of varying each of the six $\Lambda$CDM parameters on the CMB temperature power spectrum
  18. (a)
  19. 3.7 The impact of varying each of the six $\Lambda$CDM parameters on the matter power spectrum
  20. (a)
  21. 3.8 Tension between estimates of $H_{0}$ from different probes
  22. (a)
  23. 3.9 Tension between estimates of $S_{8}$ from different probes
  24. (a)
  25. 6.1 1D marginalised posterior distributions in the flat $\text{M}1$ models
  26. (a)
  27. (b)
  28. 6.2 2D marginalised posterior distributions in the flat $\text{M}1$ models
  29. (a)
  30. (b)
  31. 6.3 1D marginalised posterior distributions in the flat $\text{M}2$ models
  32. (a)
  33. (b)
  34. 6.4 2D marginalised posterior distributions in the flat $\text{M}2$ models
  35. (a)
  36. (b)
  37. 6.5 1D marginalised posterior distributions in the flat $\text{M}3^{-}$ model
  38. (a)
  39. 6.6 2D marginalised posterior distributions in the flat $\text{M}3^{-}$ model
  40. (a)
  41. 6.7 1D marginalised posterior distributions in the curved $\text{M}1$ models
  42. (a)
  43. (b)
  44. 6.8 2D marginalised posterior distributions in the curved $\text{M}1$ models
  45. (a)
  46. (b)
  47. 6.9 1D marginalised posterior distributions in the curved $\text{M}2$ models
  48. (a)
  49. (b)
  50. 6.10 2D marginalised posterior distributions in the curved $\text{M}2$ models
  51. (a)
  52. (b)
  53. 6.11 1D marginalised posterior distributions in the curved $\text{M}3$ models
  54. (a)
  55. (b)
  56. 6.12 2D marginalised posterior distributions in the curved $\text{M}3$ models
  57. (a)
  58. (b)
  59. 6.13 Whisker plot of the $68\%$ CL constraints on $H_{0}$ for the flat and curved models
  60. (a)
  61. (b)
  62. 6.14 Whisker plot of the $68\%$ CL constraints on $S_{8}$ for the flat and curved models
  63. (a)
  64. (b)
  65. 6.15 Whisker plot of the $68\%$ CL constraints on $\Omega_{m}$ for the flat and curved models
  66. (a)
  67. (b)
  68. 6.16 Whisker plot of the $68\%$ CL constraints on $\Omega_{K}$ for the flat and curved models
  69. (a)
  70. (b)
  71. 7.1 Evolution of relative energy densities $\Omega_{i}$, the ratio $\rho_{c}/\rho_{\phi}$ and variations of the Hubble rate $\Delta\mathcal{H}/\mathcal{H}$
  72. (a)
  73. (b)
  74. 7.2 The matter power spectrum and the lensing potential
  75. (a)
  76. (b)
  77. 7.3 The CMB temperature and lensing power spectra
  78. (a)
  79. (b)
  80. 7.4 2D marginalised posterior distributions in the $\Lambda$CDM and Kinetic models
  81. (a)
  82. (b)
  83. 7.5 2D marginalised posterior distributions in the Kinetic model
  84. (a)
  85. 8.1 Triangle plot of 2D marginalised posterior distributions for conformal coupled quintessence with ET
  86. (a)
  87. 8.2 Triangle plot of 2D marginalised posterior distributions for conformal coupled quintessence with LISA
  88. (a)
  89. 8.3 Triangle plot of 2D marginalised posterior distributions for conformal coupled quintessence with ET+LISA
  90. (a)
  91. 8.4 Triangle plot of 2D marginalised posterior distributions for kinetic coupled quintessence with ET
  92. (a)
  93. 8.5 Triangle plot of 2D marginalised posterior distributions for kinetic coupled quintessence with LISA
  94. (a)
  95. 8.6 Triangle plot of 2D marginalised posterior distributions for kinetic coupled quintessence with ET+LISA
  96. (a)
  97. 8.7 Triangle plot of 2D marginalised posterior distributions for disformal coupled quintessence with ET
  98. (a)
  99. 8.8 Triangle plot of 2D marginalised posterior distributions for disformal coupled quintessence with LISA
  100. (a)
  101. 8.9 Triangle plot of 2D marginalised posterior distributions for disformal coupled quintessence with ET+LISA
  102. (a)
  103. 8.10 Triangle plot of 2D marginalised posterior distributions for mixed conformal-disformal coupled quintessence with ET
  104. (a)
  105. 8.11 Triangle plot of 2D marginalised posterior distributions for mixed conformal-disformal coupled quintessence with LISA
  106. (a)
  107. 8.12 Triangle plot of 2D marginalised posterior distributions for mixed conformal-disformal coupled quintessence with ET+LISA
  108. (a)
  109. 9.1 Evolution of the coupling parameter
  110. (a)
  111. (b)
  112. 9.2 Evolution of relative energy densities, the Hubble rate and the effective dark energy EoS
  113. (a)
  114. (b)
  115. (c)
  116. (d)
  117. (e)
  118. (f)
  119. 9.3 Evolution of the effective gravitational constant
  120. (a)
  121. (b)
  122. 9.4 The CMB temperature and lensing angular power spectra and the matter power spectrum
  123. (a)
  124. (b)
  125. (c)
  126. (d)
  127. (e)
  128. (f)
  129. 9.5 Evolution of the lensing potential and its derivative
  130. (a)
  131. (b)
  132. (c)
  133. (d)
  134. 9.6 Marginalised 1D and 2D constraints for the Dark D-Brane model
  135. (a)
  136. (b)
  137. 9.7 2D marginalised posterior distributions in the $\Lambda$CDM and Dark D-Brane models
  138. (a)
  139. (b)
  140. 10.1 Hybrid Inflation Potential
  141. (a)
  142. 10.2 Evolution of the relative energy densities, the ratio $\rho_{\phi}/\rho_{\Lambda}$ and the Hubble rate
  143. (a)
  144. (b)
  145. 10.3 Evolution of relative energy densities and effective gravitational constant
  146. (a)
  147. (b)
  148. 10.4 Evolution of growth factor and matter power spectrum
  149. (a)
  150. (b)
  151. 10.5 Evolution of the lensing potential and its derivative
  152. (a)
  153. (b)
  154. 10.6 The temperature and polarisation CMB angular power spectra
  155. (a)
  156. (b)
  157. B.1 ET and LISA mock data

###### List of Tables

  1. 3.1 Estimates of cosmological parameters in the six-parameter $\Lambda$CDM model derived from Planck
  2. 4.1 The Jeffreys scale
  3. 6.1 Coupled quintessence models
  4. 6.2 Priors on $\Lambda$CDM and coupled quintessence parameters
  5. 6.3 Observational constraints for flat $\Lambda$CDM
  6. 6.4 Observational constraints for the flat M$1^{+}$ model
  7. 6.5 Observational constraints for the flat M$1^{-}$ model
  8. 6.6 Observational constraints for the flat M$2^{+}$ model
  9. 6.7 Observational constraints for the flat M$2^{-}$ model
  10. 6.8 Observational constraints for the flat M$3^{-}$ model
  11. 6.9 Observational constraints for the curved $\Lambda$CDM model
  12. 6.10 Observational constraints for the curved M$1^{+}$ model
  13. 6.11 Observational constraints for the curved M$1^{-}$ model
  14. 6.12 Observational constraints for the curved M$2^{+}$ model
  15. 6.13 Observational constraints for the curved M$2^{-}$ model
  16. 6.14 Observational constraints for the curved M$3^{+}$ model
  17. 6.15 Observational constraints for the curved M$3^{-}$ model
  18. 7.1 Priors on the model parameters
  19. 7.2 Observational constraints for the $\Lambda$CDM model
  20. 7.3 Observational constraints for the Kinetic model
  21. 8.1 Priors on cosmological models
  22. 8.2 Observational constraints for conformal coupled quintessence
  23. 8.3 Observational constraints for kinetic coupled quintessence
  24. 8.4 Observational constraints for disformal coupled quintessence
  25. 8.5 Observational constraints for mixed conformal-disformal coupled quintessence
  26. 9.1 Example solutions of the Dark D-Brane model
  27. 9.2 Priors on the model parameters
  28. 9.3 Observational constraints for the reference $\Lambda$CDM
  29. 9.4 Observational constraints for the Dark D-Brane model

## Acknowledgements

> A culpa é minha ou
> A hora da estrela ou
> Ela que se arranje ou
> O direito ao grito ou
> Quanto ao futuro ou
> Lamento de um blue ou
> Ela não sabe gritar ou
> Uma sensação de perda ou
> Assovio no vento escuro ou
> Eu não posso fazer nada ou
> Registro dos fatos antecedentes ou
> História lacrimogênia de cordel ou
> Saída discreta pela porta dos fundos00footnotetext: It’s all my fault - or -
> The hour of the star - or - Let her deal with it - or - The right to scream
> - or - As for the future - or - Singing the blues - or - She doesn’t know
> how to scream - or - A sense of loss - or - Whistling in the dark wind - or
> - I can’t do anything - or - Account of the preceding facts - or - Cheap
> tearjerker - or - Discreet exit through the back door — Clarice Lispector in
> The Hour of the Star
> — Clarice Lispector A Hora da Estrela

I have to start by expressing my enormous gratitude for having been so lucky
to have Carsten as a supervisor. During these past four years, almost half
spent during a health emergency lockdown and hundreds of miles apart, you were
always available to help me and keep me motivated. Your excitement and
curiosity for the world are definitely not restricted to science and maths and
I’m grateful to have shared that with you. I have enjoyed all the non-academic
times we shared, from pandemic socially distanced walks to insisting on
keeping your word and taking your two PhD ‘siblings’ to Woolsthorpe Manor. I
am also thankful to Sabina for sharing some of these moments with us and for
all the wild berries and mushroom picking.

Despite being affected by the pandemic, I was still lucky to travel abundantly
during my PhD. I am grateful to the School of Mathematics and Statistics of
the Universe of Sheffield for providing me with multiple opportunities to do
so, never denying any of my funding requests, to the European Consortium for
Astroparticle Theory (EuCAPT) and the EU COST Action CosmoVerse, CA2113.
Thanks to the University of Rome’ La Sapienza’ for hosting my STSM,
‘Università di Napoli’ Federico II & INFN Sezione di Napoli for hosting the
EuCAPT exchange grant, and LUPM, Université Montpellier for the introductory
visit. Special thanks to Noemi Frusciante, Alessandro Melchiorri, Giuseppe
Fanizza, and Vivian Poulin for welcoming me.

I am indebted to all the members of the CRAG research group, past and present,
who made it such a fun, friendly, and stimulating environment to work in. I
thank Steffen Gielen, Sam Dolan, Eleonora di Valentino, and Elizabeth
Winstanley for being academic role models and introducing me to various
corners of the Universe. I am incredibly thankful to have shared this
experience with Richard, from the first day in the lift, stealing your desk
and our crazy little matchbox office. I will never forget the times we spent
in that squared metre, whose value is greater than the amount of tea bags
consumed. To Ale for always having the greatest ground-lifting (literally) hug
when it was (and also when it was not) needed. A particular mention has to be
made to Lisa, who has been my absolute role model and best of friends these
last two years, which feel like a decade of companionship. Thank you for
always being there, for your kindness, and the occasional slice of cake. I
cannot thank you enough for reading and scrutinising every line in this
thesis. I am also grateful to have crossed academic paths with some wonderful
people. Thank you Rita, Matilde, Ruchika, William, Jéssica, Kerkyra, Vitor,
Davide, João and Léo, and all the wonderful people that made attending
conferences, workshops and seminars such an enriching experience. I want to
thank my collaborators with whom I learned so much: Noemi Frusciante, Bruno J.
Barros, Vasco M.C. Ferreira, Richard Daniel, Gaspard Poulot and Cameron
Thomas. A special mention should be made to Ana Nunes, Nelson Nunes, Noemi
Frusciante, Eleonora Di Valentino and Danielle Leonard for all the advice and
mentoring, and to Bruno for everything we learned together and for remaining
(in)sane with me during 1.5 years of lockdown.

I cannot overstate the equally great debt I owe to my home friends. I am
especially grateful to Rita and Joana for spreading roots and sharing this
adventure with me, ensuring we never took ourselves too seriously, and always
sending me off with a full bag of laughs and heartfelt understanding. You have
truly shown me the sun behind the English clouds. I thank Inês for being my
companion in this 10-year journey through physics, mathematics and philosophy,
always with amazement and creativity. I cannot wait to keep exploring the
wonders of the Universe with you. Most of all, I owe all of it to my family,
who has always supported me unconditionally with complete freedom to flourish.
To my parents, who have taught me the most important things of all: never take
myself too seriously, always be kind, and, above all, do what makes me happy.
And in particular to my sister and niece for understating my physical absence
in their lives. I dedicate this work to them, who inspire and motivate me
daily. To my pardalinho, in your fairy-tale homonymous words, I can only hope
to inspire you always to remain

> “Curiouser and curiouser!”
> — Lewis Carroll Alice’s Adventures in Wonderland

Part IFoundations of Gravity and Cosmology

##  Chapter 1 Introduction

> Como começar pelo início, se as coisas acontecem antes de acontecer?
> Se antes da pré-pré-história já havia os monstros apocalípticos?
> Se esta história não existe, passará a existir. Pensar é um ato.
> Sentir é um fato. Os dois juntos – sou eu que escrevo o que estou
> escrevendo.00footnotetext: How do you start at the beginning, if things
> happen before they happen? If before the pre-prehistory there were already
> the apocalyptic monsters? If this story doesn’t exist now, it will. Thinking
> is an act. Feeling is a fact. Put the two together — I am the one writing
> what I am writing. — Clarice Lispector in The Hour of the Star
> — Clarice Lispector A Hora da Estrela

###  1.1 Introduction to Cosmology

Cosmology is a unique field of scientific inquiry that seeks to understand the
nature and origins of the Universe. On what might be the only personal note in
this dissertation, I like to think that cosmology parallels the work of
archaeologists piecing together the fragments of ancient civilisations. Both
disciplines aspire to reconstruct a larger, coherent picture from fragmented
pieces - whether it be artefacts buried in the earth or distant sources of
light reaching our telescopes from the vast deepness of the Universe. Each
cosmic signal we analyse is a relic from a long-past event, a fossil of light,
offering clues about the history and structure of the ultimate civilisation -
our Universe. This presents an epistemological challenge unique to cosmology:
working to understand an entire, singular Universe based on isolated data
points, inherently confined by the limits of causality and our observational
reach. The task has an almost philosophical undertone; it grapples with
questions about the very nature of evidence and existence, especially given
that we have only one Universe to study. This singularity not only imposes a
limit on our understanding but also introduces an inherent bias into any laws
or theories we might derive. Thus, cosmology exists at an intersection of
science and philosophy, endeavouring to build a narrative of the Universe that
is both empirical and conceptually profound. It confronts us with the stark
reality that our efforts to understand the cosmic journey are both an
ambitious scientific undertaking and a quest that treads on deeply abstract
grounds.

Metaphysics aside, as an established scientific field, cosmology aspires to
comprehend and articulate the fundamental characteristics and evolution of the
Universe. It devises meticulous and often speculative narratives for the
composition of the Universe, delving further into the journey of those
constituents as they mature into complex structures such as galaxies. It also
examines the dynamics and interplay of those structures within an ever-
changing landscape. These overarching frameworks are referred to as
cosmological models. Unlike the word model might suggest, contemporary
cosmological models consist mainly of abstract mathematical concepts, often
devoid of physical intuition or tangible form.

While commonly referred to as the birth of the Universe, the Big Bang might be
more aptly portrayed as its conception. The period preceding the creation of
the first atoms, culminating in the unimpeded propagation of light, was a
vastly unfamiliar environment, an obscure cosmic fog of elementary particles
and their extremely energetic interactions. When the Universe was
approximately $380\ 000$ years old, it grew more familiar. It gradually
unfolded into the configuration we recognise today, seemingly set on a
perpetual path of expansion, according to the prevailing cosmological model.

At its core, Cosmology stands as a field deeply rooted in ancient human
contemplation. It is impossible to disentangle from the quest to understand
our place in the world, its origins, and ontological meaning. The foundations
of modern cosmology were established in 1917 with Albert Einstein’s seminal
article on his theory of gr, titled Cosmological Considerations in the General
Theory of Gravity Einstein (1917). Building upon Einstein’s work, cosmological
models were further developed by Dutch mathematician Willem de Sitter de
Sitter (1916a, b, 1917), German physicist Karl Schwarzschild Schwarzschild
(1916a, b), and Russian mathematician Alexander Friedmann Friedmann (1924,
1922). These studies demonstrated that general relativity could accommodate
the concept of an expanding Universe, a theoretical hypothesis that came to be
confirmed by the realisation that galaxies stood as rulers to trace
astronomical distances in space and which were, in fact, receding from our own
Milky Way in all directions. In 1929, Edwin Hubble’s landmark discovery, now
known as the Hubble law, revealed a linear relation between the recessional
speed of a galaxy and its distance, culminating in the hypothesis that the
Universe is expanding. This idea was not unexpected and had been largely
predicted from a theoretical perspective. Indeed, it was qualitatively
reinforced by redshift measurements and notoriously explored during the 1920s.

As it stands at the moment of writing this dissertation, the standard
cosmological model - referred to as the $\Lambda$CDM model based on its novel
components - is constructed upon six foundational principles, summarised in
the following equation and which will be discussed below:

$\Lambda\text{CDM Model}\,=\,\text{GR}\,+\,\text{FLRW}\,+\,\text{Standard
Matter}\,+\,\text{Initial Conditions}\,+\,\text{CDM}\,+\,\Lambda\,.$

The GR, FLRW, and Standard Matter triad represent well-established and tested
hypotheses based on local empirical evidence. On the other hand, although
highly supported by cosmological and astrophysical observations, the last
three components infuse new physics into the picture and still lack more
concrete and direct confirmation. In unison, these six hypotheses orchestrate
a comprehensive paradigm for understanding the Universe on large scales, as
listed below:

  1. 1.

gr: The theory of gravity is essential for understanding cosmological
phenomena, as gravity dominates the Universe on large scales and is at the
heart of cosmic evolution.

  2. 2.

flrw metric: GR admits only a few analytical solutions as a metric theory
regulated by non-linear partial differential equations. Therefore, deriving
physical implications requires a simple mathematical description from the
metric, with the FLRW metric being a natural choice assuming maximal spatial
symmetry.

  3. 3.

Standard Matter (from the standard model of particle physics): the directly
detected forms of matter in the Universe - baryons (protons, neutrons,
including the less massive electrons), photons, and neutrinos - whose dynamics
and interactions are based on robust physical principles - must be included in
the cosmological model to recount the expansion history of the Universe
successfully.

  4. 4.

Initial Conditions (from inflation): an early period of rapid cosmic
expansion, inflation is invoked to resolve causality problems and account for
the observed spatial curvature of the Universe. Remarkably, it also provides a
mechanism to explain the Universe’s initial conditions, including generating
perturbations around the background metric and supporting the choice of the
homogeneous and isotropic metric ansatz.

  5. 5.

cdm: a form of pressureless matter that does not interact with light, needed
to accurately describe the dynamics and process of formation of large-scale
structures, such as galaxies and galaxy clusters. The simplest resolution is
to introduce a new, weakly interacting particle species.

  6. 6.

Cosmological Constant ($\Lambda$): an additional degree of freedom that
accounts for the observed accelerated expansion of the Universe, which in the
simplest scenario is a cosmological constant, i.e., some energy density
contribution that remains constant over the expansion.

For completeness of the work, we first provide a general overview of the
relevant concepts of cosmology, focusing mainly on the Universe’s expansion
history and leading towards the physics of the late Universe. The main goal is
to introduce and contextualise the original work presented in Part II of this
manuscript. In that sense, Part I will consist of well-established topics in
the field. Nevertheless, the organisation of the discussion and the
storytelling should be original, connecting the ideas and guiding the reader
through the relevant concepts. We will start by presenting a brief overview of
the dynamics of general relativity. This lays out the basis for the
introduction of cosmology, focusing mainly on the large-scale structure of
spacetime and the dynamics of the expansion history. To understand the full
scope of the changes introduced in each model, a complete account of the
physically relevant epochs of the Universe is needed. The seeds for the
evolution of structures lie in the primordial perturbations produced during an
initial period of inflation. These are imprinted in the cmb radiation and are
a fundamental cosmological probe.

###  1.2 A Comment on Observational Cosmology

The methods of scientific pursuit must be adapted to the limitations of the
field, particularly in the realms of physical cosmology and extragalactic
astronomy, where empirical assessment is limited to distant observations
without direct interaction or manipulation of the objects of study. Moreover,
in the vast expanse of the cosmos, we cannot replicate experiments or
observations. Instead, we must rely on searching for the clues and evidence
left behind in fossils from the ancient past. Some of these relics are found
nearby, such as extragalactic hints on the Earth or the Moon, in our Solar
System or even in stars within our galaxy. Even though these local remnants
give important insight, they are just pieces of a large jigsaw puzzle.
Nevertheless, here, the laws of physics and the principles of general
relativity play in our favour: our vision can reach far beyond, as the light
from distant objects carries information from earlier epochs of the Universe,
based on the fact that observed objects at increasing distances effectively
corresponds to looking back in time, witnessing the earlier stages of cosmic
evolution. Our view of the cosmos is shaped by our past light cone,
representing the collection of all the light that has reached us throughout
history. While the light cone of human existence encompasses an incredibly
thin slice of the Universe’s history, the information travels towards us from
all directions and epochs, spanning a vast range of time and space. Collecting
the remnant pieces of cosmic history and drawing them together is just like
adding the pieces of the jigsaw puzzle, except that there may be infinite
pieces, some inaccessible or lost forever in the vastness of the cosmos,
others waiting to be found (and most likely there are no corner pieces to
start with!).

The debates and discussions surrounding cosmological issues can be met with an
intensity and fervour not matched by the weight of evidence. One reason is
that the observational evidence needed to settle these questions often appears
to be beyond our grasp or even an impossibility. Furthermore, the study of
such fundamental questions is deeply entangled with an overarching interest in
understanding the nature of our own existence and the world we inhabit. It is
inevitable to ponder the Universe’s origins and potential fate, and it is hard
not to develop a philosophical preference for certain scientific outcomes or
perspectives.

The era of precision measurements has drastically changed the scientific
method. Big collaborative teams have developed to address the need to deal
with unprecedentedly large amounts of data, with the natural gap between
theory and observations being transposed with numerical simulations. This
shift has led to heightened scrutiny of research endeavours and the need for
inquiries to be more rigorously justified or supported.

The standard model of cosmology today has reached the status of an empirical
science; that is, it is based on and tested according to what can be observed
or measured. However, scientific advancements are often made on a
theoretically intuitive basis since, while being well-tested and investigated,
a cosmological theory is always incomplete and subject to unpredictability.
The currently accepted $\Lambda$-Cold-Dark-Matter ($\Lambda$CDM) model relies
on the extrapolation of a cosmological history from imprints left in the
observables that resemble those predicted by the model. The theory is then a
useful though incomplete approximate picture of the history of the cosmos,
with remaining cracks and hypotheses on the borderline of established physics.
Nevertheless, the abundance and diversity of empirical tests that the standard
model has undergone maintain outstanding confidence that any alternative
theory that attempts to glue those cracks will portray a Universe that behaves
much like $\Lambda$CDM. For this reason, the first natural step in that
direction is to consider small deviations or extensions to the minimal
$\Lambda$CDM model while remaining conscious that this may not be enough and
the answer may require a new paradigm disconnected from the $\Lambda$CDM
assumptions.

###  1.3 General Relativity

In this section, we provide a summary of general relativity, with no intention
of exhaustively introducing or reviewing its full extent. For this purpose, we
refer to e.g. Wald (1984), Misner et al. (1973).

Albert Einstein’s theory of General Relativity, formulated in 1915 Einstein
(1916), presents an exceptional framework where gravitational interactions are
related to the geometry of spacetime Carroll and Carroll (2004), Mukhanov
(2005), Ellis et al. (2012). The central element is the spacetime
metric111Throughout this manuscript, we adopt Einstein’s summation convention
where repeated indices are summed over. The metric signature is $(-,+,+,+)$
and we use natural units with $c=\hbar=k_{B}=1$. Greek indices range from $0$
to $3$, while Latin indices denote spacetime components only, $1$ to $3$.
degree of freedom $g$, which determines physical distances as

$\odif{s}^{2}=g_{\mu\nu}\odif{x}^{\mu}\odif{x}^{\nu}\,,$ (1.3.1)

and is treated as a fully dynamic variable.

The time coordinate interval between two events, $\Delta t$, is not the same
as perceived by different observers, motivating the introduction of the more
convenient invariant proper-time interval, $\Delta\tau$. Considering the
events $A$ and $B$ which are separated by a time-like interval
($\odif{s}^{2}<0$), the curve which connects the two is denoted by $\gamma$
and parameterised by $\lambda$, such that e.g. $\gamma(\lambda_{A})=A$ and
$\gamma(\lambda_{B})=B$. The proper-time interval $\Delta\tau$ between $A$ and
$B$ is then given by

$\Delta\tau\equiv\int_{\gamma}\sqrt{-\odif{s}^{2}}=\int_{\lambda_{A}}^{\lambda_{B}}\odif{\lambda}\sqrt{-g_{\mu\nu}\odv{x^{\mu}}{\lambda}\odv{x^{\nu}}{\lambda}}\,.$
(1.3.2)

Massive free-falling particles moving in a curved spacetime follow a geodesic
trajectory, $X^{\alpha}(\tau)$, defined as the time-like trajectory that
extremises the proper-time interval $\Delta\tau$ in Eq. 1.3.2. The geodesic
trajectory is the one that minimises the action

$S=m\int\odif{\tau},$ (1.3.3)

where $m$ is the rest-frame mass of such particles. The trajectory of these
particles follows the geodesic equation:

$\odv[order=2]{X^{\alpha}}{\tau}+\Gamma^{\alpha}_{\mu\nu}\odv{X^{\mu}}{\tau}\odv{X^{\nu}}{\tau}=0\,,$
(1.3.4)

where

$\Gamma^{\alpha}_{\mu\nu}\equiv\frac{1}{2}g^{\alpha\lambda}\left[\partial_{\mu}g_{\lambda\nu}+\partial_{\nu}g_{\lambda\mu}-\partial_{\lambda}g_{\mu\nu}\right]\,,$
(1.3.5)

is the torsion-free ($\Gamma^{\alpha}_{\mu\nu}=\Gamma^{\alpha}_{\nu\mu}$)
connection (or Christoffel symbols) that gives a metric-compatible covariant
derivative $\nabla_{\mu}(\cdot)$

$\nabla_{\mu}Y^{\lambda...}_{\nu...}=\partial_{\mu}Y^{\lambda...}_{\nu...}+\Gamma^{\lambda}_{\mu\alpha}Y^{\alpha...}_{\nu...}-\Gamma^{\alpha}_{\mu\nu}Y^{\lambda...}_{\alpha...}+...-...\quad\xrightarrow{\text{\lx@cref{creftype~refnum}{eq:gammadef}}}\quad\nabla_{\alpha}g_{\mu\nu}=0\,,$
(1.3.6)

where $Y^{\lambda...}_{\nu...}$ is some arbitrary-rank tensor.

Einstein’s gravitational theory establishes a connection between the dynamics
of the metric and matter through the curvature of spacetime, with the latter
embodied by the Riemann tensor:

$R^{\alpha}_{\beta\mu\nu}\equiv\partial_{\mu}{\Gamma^{\alpha}_{\nu\beta}}-\partial_{\nu}{\Gamma^{\alpha}_{\mu\beta}}+\Gamma^{\sigma}_{\beta\nu}\Gamma^{\alpha}_{\mu\sigma}-\Gamma^{\sigma}_{\beta\mu}\Gamma^{\alpha}_{\nu\sigma}\,.$
(1.3.7)

An action for the metric is required to endow this formulation with dynamics.
Remarkably, the simplest nontrivial action that can be constructed out of a
scalar from $g_{\mu\nu}$ leads to second-order equations in the effective
field theory and reads:

$S=S_{\text{grav}}+S_{\text{mat}}\,,$ (1.3.8)

where

$S_{\text{grav}}=\int\odif[order=4]{x}\sqrt{-g}\frac{R}{16\pi G},$ (1.3.9)

is the eh action. The action $S_{\text{mat}}$ encodes the matter sector, and
$g$ is the determinant of the metric. The main player in the EH action is the
Ricci (or curvature) scalar, defined in terms of the trace of the Ricci
tensor:

$R\equiv R^{\mu}_{\mu}=R_{\mu\nu}g^{\mu\nu}\quad\text{with}\quad
R_{\mu\nu}\equiv R^{\alpha}_{\mu\alpha\nu}\,.$ (1.3.10)

By employing the principle of least action, the variation of the action in Eq.
1.3.8 with respect to the metric yields the Einstein field equations:

$G_{\mu\nu}\equiv R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=8\pi GT_{\mu\nu}\,,$
(1.3.11)

where $G$ is Newton’s gravitational constant, $8\pi
G=\kappa^{2}=\text{M}_{\text{Pl}}^{-2}$, and $\text{M}_{\text{Pl}}$ ($\kappa$)
is the (reduced) Planck mass. The left-hand side (LHS) is enclosed in the
Einstein tensor $G_{\mu\nu}$. The source term on the right-hand side (RHS),
the energy-momentum tensor, is obtained from the variation of the matter
action with respect to the metric and is covariantly conserved on account of
the Bianchi identities:

$\nabla_{\mu}G^{\mu\nu}=0\quad\xrightarrow{\text{\lx@cref{creftype~refnum}{eq:eineqdef}}}\quad\nabla_{\mu}T^{\mu\nu}=0\quad\text{with}\quad
T^{\mu\nu}\equiv 2\sqrt{-g}\fdv{S_{\text{mat}}}{g_{\mu\nu}}\,.$ (1.3.12)

By definition, $T^{\mu\nu}$ is always symmetric. The $\nu=0$ component of
$\nabla_{\mu}T^{\mu\nu}$ represents energy conservation, while the three
remaining space-like components correspond to momentum conservation. The
Einstein equations epitomise the foundation of general relativity: matter and
geometry are cut from the same fundamental cloth, with the geometry of
spacetime prescribing the dynamics of matter. At the same time, massive bodies
set the curvature of spacetime itself.

The seemingly compact set of Einstein’s field equations, in Eq. 1.3.11, govern
the evolution of the Universe and all gravitating bodies, with the concrete
realisation depending on its matter content and the spacetime metric based on
physical considerations.

To derive the dynamics of the Universe from GR, we can start with the example
of Einstein’s original proposal, based upon two main assumptions, reflecting
the impossibility of filling up an infinite space with matter, which must
itself be finite in nature Janzen (2014):

  * •

The Universe is finite;

  * •

The Universe is static.

The first assumption aims to ensure a metric completely described by the
stress-energy tensor. The second one is more subtle but reasonable in the
scientific context in which Einstein first proposed a cosmological solution to
his equations in 1917 Einstein (1917). By extrapolating the assumption of a
spatially closed Universe to the very largest scales, a uniform distribution
of matter can only be achieved if the curvature of space is constant. This is
in line with Einstein’s intention, which appears to have been to depict a
cosmological model wherein a finite density of matter shaped the structure of
the world. During that period, only stars with minor peculiar velocities had
been detected, and it was not until 1922 that nebulae were acknowledged as
independent stellar systems beyond our galaxy Eddington (1923), Lemaître
(1927), Friedmann (1922), supporting the existence of a reference frame in
which matter is at rest O’Raifeartaigh et al. (2017). However, the two
postulates above are not compatible with the Einstein field equations, given
in Eq. 1.3.11, according to which a finite density of matter collapses on
itself due to the gravitational attraction, invalidating the static Universe
assumption. Einstein resolved this issue by simply introducing a constant
term, $\Lambda$, in his equations to counterbalance the gravitational pull of
matter, consistent with the principles of General Relativity and with the
Bianchi identities $\nabla^{\mu}T_{\mu\nu}=0$, leading to a new set of field
equations Straumann (2012):

$G_{\mu\nu}=8\pi GT_{\mu\nu}+\Lambda g_{\mu\nu}\,.$ (1.3.13)

With this relation, the $\Lambda$ term stands as a negative effect against
gravity, so both effects compensate. In the context of Einstein’s static
Universe, and with the solution proposed by Friedmann-Lemaître-Robertson
Walker (which we assume for now but will be discussed in detail in Section
1.6), these equations imply

$8\pi G\rho=\frac{1}{a^{2}R_{E}^{2}}=3\Lambda\,,$ (1.3.14)

where $\rho$ stands for the total mass density of the Universe, and the
parameter $R_{E}$ is its curvature radius222The radius of curvature of space
in Einstein’s static Universe is also referred to as Einstein’s radius. In the
context of the FLRW solution (see Section 1.6), the definition of $R_{E}$
emerges from setting both the first and second-time derivatives of the scale
factor, $\dot{a}$ and $\ddot{a}$, to zero in the Friedmann equations, arriving
at $aR_{\text{E}}=1/\sqrt{4\pi G\rho}$, where $G$ is Newton’s gravitational
constant and $\rho$ represents the spatial density of this Universe., for a
closed spherical geometry. Hence, the cosmological constant $\Lambda$ plays
the role of some repulsive force counteracting the gravitational collapse of
the Universe. It could be included in the theory as a fundamental constant
that determines the density of matter and the radius of cosmic space. However,
the solution for a steady Universe is unstable because a change in the value
of $\Lambda$ in Eq. 1.3.14 (or equivalently a local rearrangement of the mean
mass density) would cause the Universe to either expand or collapse, like a
ball that is carefully balanced on the top of a mountain, thus invalidating
the static assumption. Remarkably, in 1922, Friedmann proposed a new
cosmological model that abolished the static hypothesis and replaced
Einstein’s assumptions with Friedmann (1922):

  * •

The Universe is homogeneous;

  * •

The Universe is isotropic;

  * •

The Universe is expanding.

It was only in 1929 that evidence for the expansion of the Universe came to
light with Hubble’s discovery of the recession of galaxies. This led to the
cosmological constant being cast aside, only to be revived at the end of the
millennium. Einstein himself was determined to dismiss the cosmological
constant, as the goal for its introduction had been to represent a finite
density of matter in the Universe. When Friedmann showed that this could be
achieved, either with or without the $\Lambda$-term - provided that the cosmic
distance scale was time-dependent, which was further substantiated by Hubble’s
empirical confirmation that cosmic distances indeed change with time -
Einstein found himself able to reject the term. He maintained that this
constant undermined the theoretical elegance and logical simplicity of his
original formulation. In his own words Schilpp (1949):

The introduction of such a constant implies considerable renunciation of the
logical simplicity of theory, a renunciation which appeared to me unavoidable
only so long as one had no reason to doubt the essentially static nature of
space. After Hubble’s discovery of the "expansion" of the stellar system and
since Friedmann’s discovery that the unsupplemented equations involve the
possibility of the existence of an average (positive) density of matter in an
expanding Universe, the introduction of such a constant appears to me, from
the theoretical standpoint, at present unjustified.

Perhaps one of the most vocal critics of Einstein’s rejection of the
cosmological constant was Arthur Eddington. He fundamentally disagreed with
Einstein’s view, especially regarding the interpretation of cosmic expansion,
seeing it as a logical consequence of the physical theory Eddington (1932).
Eddington maintained that the general relativistic field equations expressed
that length measurements should be made relative to a finite constant radius
of curvature. As he famously wrote Eddington (1923),

From this point of view it is inevitable that the constant $\Lambda$ cannot be
zero; so that empty space has a finite radius of curvature relative to
familiar standards. An electron could never decide how large it ought to be
unless there existed some length independent of itself for it to compare
itself with.

This disagreement between Einstein and Eddington was not rooted in scientific
differences but rather philosophical ones, as both relied on the same
empirical evidence to support their views. Einstein believed that the time-
varying spatial scale introduced by the expansion in Hubble’s law rendered the
$\Lambda$-term superfluous and that his theory would reduce to a simpler
statement without it. On the other hand, Eddington advocated that not only did
the cosmological constant simplify the theory’s formulation, but this reformed
version also aligned with the observed expansion, confirming the need for a
significant cosmic repulsion on large scales.

Eddington’s view was exalted by the Nobel Prize-winning observation of the
acceleration of the cosmic expansion in 1998 Riess et al. (1998), Perlmutter
et al. (1999) (which will be discussed in Chapter 2), which validated the
cosmological constant through its predictive power. Indeed, in retrospect, the
evidence gathered since Einstein first proposed $\Lambda$ seems more aligned
with Eddington’s interpretation, who believed that the natural gravitational
repulsion embodied by $\Lambda$ could set the stage for the understanding of
the Universe’s evolution, referring to the cosmic expansion as Einstein’s
almost inadvertent prediction Eddington (1932).

###  1.4 Homogeneity and Isotropy

Modern cosmology emerged from Albert Einstein’s investigation of how his
theory of general relativity could be employed to understand the large-scale
nature of the Universe. Einstein believed that a philosophically reasonable
Universe should have the same properties everywhere and in all directions
(therefore justifying the proposal for a uniform distribution of matter at the
boundary), with only minor inhomogeneities, such as the agglomerates of matter
that make up planets or stars Einstein (1917). This opposed the standard
scientific approach of examining structures separately and at specific
hierarchical levels based on a progressive narrative, such as the study of
molecules, atoms in molecules, nuclei in atoms and so on Peebles (2020). The
same premise can be applied on larger scales: the study of the nature of
planets around stars is followed by stars in galaxies, galaxies in clusters,
and so the story proceeds.

Einstein’s vision was that the hierarchical structure of the Universe
culminates in a new ideal for modern science: large-scale homogeneity.
According to this argument, the Universe is assumed to be homogeneous and
isotropic on large scales, meaning it is the same on average regardless of the
observer’s position or orientation. Einstein’s large-scale homogeneity opened
the possibility of considering and testing a theory that describes the
Universe as a whole rather than focusing on theories for specific levels of
structure. The power of this hypothesis is that observations from our position
in the Universe can be taken to represent what the Universe looks like from
any other place. Nevertheless, just like any hypothesis, it requires empirical
evidence. The concept of large-scale homogeneity grew from philosophical
reasoning, intuition, collaboration, and perhaps even some wishful thinking.
It was not initially rooted in empirical evidence but in the pursuit of a
unified, coherent framework to describe the Universe.

In more precise terms, the cosmological principle is the backbone of the
standard cosmological model. It states that on large scales (greater than
$\sim 100\,\text{Mpc}$), the Universe is isotropic (rotational invariance) and
homogeneous (translational invariance). This assumption implies no privileged
positions or directions in the Universe, limiting the vast array of possible
cosmological models. Isotropy333A general theorem of geometry ensures that
isotropy around every point implies homogeneity Kolb and Turner (1990). has
been supported on intermediate and large scales through various observations
Aluri et al. (2023), discussed in Chapter 3, including:

  * •

The distribution of galaxy clusters and super-clusters;

  * •

Observations of large scale structure and of the interstellar medium in radio
and X-ray wavelengths;

  * •

The uniformity of the cmb radiation temperature, with $\Delta T/T\sim
10^{-5}$, indicating a strongly isotropic Universe at the time of radiation
emission (approximately 300 thousand years after the Big Bang).

The cosmological principle is also a modern formalisation of the old
Copernican principle, in the sense that not only do we not hold a particular
position in the Solar System, but also the position of the Milky Way in the
Universe should not be statistically distinguishable from the spatial
distribution of other galaxies. While the Universe contains inhomogeneous
structures like galaxies and galaxy clusters on smaller scales, at early
times, it looked roughly the same at every point and in all directions. This
is best illustrated through Fig. 1.1, where the impressively detailed map of
the temperature distribution of the CMB as measured by the Planck satellite is
shown. The CMB is the monument marking the largest (and hence most ancient)
scales that can be directly observed, depicting remarkable homogeneity.

We have just seen the picture of a homogeneous, isotropic and expanding
Universe on large scales. However, which scales can be classified as
sufficiently large? A standard distance unit in Cosmology is the Megaparsec,
$1\,\text{Mpc}=3.2615\times 10^{6}\,\text{light-years}=3.0856\times
10^{24}\,\text{cm}$. This is the typical distance between nearby galaxies,
e.g. the distance between Andromeda and the Milky Way is around
$0.7\,\text{Mpc}$. Determining the exact scale for which the distribution of
galaxies can be considered to be homogeneous is not a trivial endeavour. Even
though large-scale structure observations Alam et al. (2021) show that the
galaxy density inhomogeneities are still of the per cent order on scales of
$\sim 150\,\text{Mpc}$, the geometry of the Universe hints that these
deviations must be small already on the Mpc scale. The geometry can be tested
through gravitational lensing, the peculiar motion of galaxies and, more
importantly, the CMB Aghanim et al. (2020a).

Ultimately, the cosmological principle ensures that the laws of physics
governing the Universe at large scales are identical for all observers, making
it comprehensible to us.

(a) Figure 1.1:  The sky map of temperature anisotropies in the Cosmic
Microwave Background as measured by the Planck Collaboration (taken from the
Planck Legacy Archive) Aghanim et al. (2020b). (a) Figure 1.2:  A map of
galaxies and quasars in the observable Universe found by the Sloan Digital Sky
Survey (SDSS) from 2000 to 2020 Alam et al. (2021). This particular wedge of
the full survey covers around 10 degrees. It encompasses about $200\,000$
galaxies and quasars reaching from our observable position (not special
according to the cosmological principle) up to a look-back time of 12 billion
years and cosmological redshift $5$, with redness indicating increasing
redshift and distance. Almost every dot in the nearby lower part of the
illustration represents a galaxy, and the upper part represents a distant
quasar. The cosmic web structure is clearly illustrated, with the
gravitational attraction between the nearby galaxies leading to a local
Universe more condensed and filamentary than the distant Universe. The edge is
the CMB temperature map as measured by the Planck Collaboration Aghanim et al.
(2020b). Visualisation created at John Hopkins University:
https://mapoftheUniverse.net.

Combined with the Einstein equations and a specified energy-matter
distribution, the Cosmological Principle provides a framework for
understanding how the Universe evolves. However, this simplistic view of the
homogeneous model has inherent limitations. Specifically, it cannot account
for the formation and filament-like distribution of galaxies and galaxy
clusters, such as depicted in Fig. 1.2, nor for general variations in the
matter distribution (a perfectly homogeneous and isotropic Universe would not
even allow for our existence to discuss it in the first place!). Likewise, the
distribution of these inhomogeneities and the variations in the temperature
and polarisation of the CMB photons are intimately connected. One can employ
an approach based on linear perturbation theory to address these limitations.
If one first solves for the evolution of the background quantities (i.e. non-
perturbative), that information can be incorporated into the equations that
govern the changes in these small-scale variations. The details of this method
and how it connects with observations will be the focus of Chapters 2 and 3.

Nevertheless, this perturbative approach is not without its limitations.
During the period when matter dominates, structure formation starts taking
place, invalidating the assumption of small density perturbations. Over this
regime non-linear perturbation theory must be employed, a topic which falls
outside the scope of this dissertation. Nevertheless, the linear analysis of
matter fluctuations and CMB anisotropies remains valuable. Its strength
resides exactly in the fact that various methodologies can be complementary
and mutually reinforcing.

###  1.5 The Hubble-Lemaître Law: Cosmic Expansion

It was Edward Arthur Milne who recognised the power of homogeneity in
formulating a cosmological scenario and named it Einstein’s cosmological
principle Milne (1933). He showed that independent of general relativity, this
principle explains the relation between the recession velocity $v$ of a galaxy
and its distance $r$, known as Hubble’s law:

$v=H_{0}r\,.$ (1.5.1)

The proportionality constant, $H_{0}$, is known as Hubble’s constant, and the
subscript is meant to indicate that it is a measure of the present rate of
expansion of the Universe. In an evolving Universe, the expansion rate is a
function of time, $H(t)$. This proposal was based on the observation that
light from distant galaxies is shifted towards longer wavelengths
(redshifted), indicating their recession from us. A simple way to picture this
in Euclidean geometry is to imagine two galaxies at arbitrary points and write
their velocities as the vector relation $\vec{v}=H_{0}\vec{r}$. Then an
observer on galaxy $a$ sees galaxy $b$ moving away at velocity

$\vec{v}_{b}-\vec{v}_{a}=H_{0}(\vec{r}_{b}-\vec{r}_{a})\,.$ (1.5.2)

This demonstrates that regardless of their position, all observers witness a
consistent pattern of galaxies receding from one another, as expected in a
homogeneous Universe. For nonrelativistic recession velocities, the
cosmological redshift is defined as the ratio $z=v/c$, akin to a Doppler
shift, where $c$ is the speed of light.

This redshift-distance relation has recently been coined with a new name, the
Hubble-Lemaître law, in acknowledgement of Lemaître’s theoretical prediction
of the expanding Universe. However, the recognition could be extended to the
significant contribution of other players. Vesto Melvin Slipher’s redshift
measurements444In fact Slipher measured the shift of spectral lines of
galaxies, from which it was possible to infer their velocities using the
Doppler shift. This led him to the groundbreaking conclusion that the nebulae
being measured had to be outside of the Milky Way. Slipher (1913) and
Henrietta Leavitt’s discovery of the Cepheid period-luminosity relation from
variable stars in the Magellanic Clouds Leavitt (1908) were indispensable to
the production of Hubble’s famous redshift-distance plot Hubble and Humason
(1931), while Milton Humason’s redshift measurements Hubble and Humason
(1931), Adams et al. (1935) and Milne’s cosmological considerationsMilne
(1933) in the 1930s were instrumental in providing compelling and precise
evidence for the nature of this result.

###  1.6 The Friedmann-Lemaître-Robertson-Walker Metric

To extract cosmological implications from general relativity, we need to
specify the components of the spacetime metric in Eq. 1.3.1. There are no
general analytical solutions to the Einstein field equations, Eq. 1.3.11, but,
amongst others, Einstein realised that a coherent physical description of the
Universe could only be achieved under specific assumptions. The fundamental
underlying assumption is that the Universe is expanding and encapsulates the
cosmological principle. That is to say that the metric needs to be such that
it describes a time-varying Universe which is, at each time, spatially
homogeneous and isotropic. Assembling all the geometric and symmetry
arguments, the most generic expanding spacetime for a homogeneous and
isotropic Universe with matter uniformly distributed as a perfect fluid is
given by the Friedmann-Lemaître-Robertson-Walker (FLRW) metric Walker (1937),
Robertson (1936a, b, 1935), Lemaître (1933, 1927), Friedmann (1924, 1922) and
has the following form:

$\odif{s}^{2}=-\odif{t}^{2}+a^{2}(t)\gamma_{ij}\odif{x}^{i}\odif{x}^{j}=-\odif{t}^{2}+a^{2}(t)\odif{l}^{2}\,,$
(1.6.1)

where $\gamma_{ij}$ are the spatial components of the metric. The spatial part
of the line element, $\odif{l}^{2}$, can be expressed in traditional Cartesian
coordinates or, more informatively, in spherical polar coordinates
$(x^{1},x^{2},x^{3})=(r,\theta,\varphi)$ as

$\odif{l}^{2}=\frac{\odif{r}^{2}}{1-Kr^{2}/R_{0}^{2}}+r^{2}\odif{\theta}^{2}+r^{2}\sin^{2}(\theta)\odif{\varphi}^{2}\,,$
(1.6.2)

in which case

$\gamma_{11}=\left(1-Kr^{2}/R_{0}^{2}\right)^{-1}\,,\quad\gamma_{22}=r^{2}\,,\quad\gamma_{33}=r^{2}\sin^{2}(\theta)\,.$
(1.6.3)

The scale factor $a(t)$ describes the time evolution of the expansion scale of
the Universe. It is dimensionless and typically normalised at present as
$a(t_{0})=1$. $r$ represents the radial comoving coordinate, while $\theta$
and $\varphi$ are the comoving angular coordinates. The whole term in Eq.
1.6.2 corresponds to the line element of a maximally symmetric 3D space with a
constant curvature parameter $K$ and curvature scale $R_{0}$. The derivation
of the FLRW solution can be found in Weinberg (2008), Baumann (2022). Under
the assumption of homogeneity and isotropy, the intrinsic curvature must be
constant, and its character is encoded in $K$:

  * •

Flat space ($K=0$): the simplest case corresponding to a three-dimensional
Euclidean space. In such a space, parallel lines never intersect.

  * •

Positively curved space ($K=+1$): known as spherical space as it can be
thought of as a three-sphere embedded in a four-dimensional Euclidean space.
This is the space in which all parallel lines will eventually meet.

  * •

Negatively curved space ($K=-1$): called the hyperbolic space as it can be
represented by a hyperboloid embedded in a four-dimensional Lorentzian space.
In this space, parallel lines will always diverge.

It is readily apparent that the time and space coordinates are not scaled in
the same way in the FLRW metric, Eq. 1.6.1, since the scale factor $a(t)$ only
multiplies the spatial sector. For that purpose, the cosmic time $t$ may be
replaced by the conformal time $\tau$, which is particularly useful for the
study of the propagation of photons (like in the CMB), which is defined as

$\odif{\tau}\equiv\frac{\odif{t}}{a(t)}\,.$ (1.6.4)

Moreover, without assuming a particular curvature parameter, it is convenient
to redefine the radial component as
$\odif{\chi}\equiv\odif{r}/\sqrt{1-Kr^{2}/R_{0}^{2}}$ and the metric on the
unitary two-sphere,

$\odif{\Omega}^{2}=\odif{\theta}^{2}+\sin^{2}(\theta)\odif{\varphi}^{2}\,,$
(1.6.5)

such that the FLRW metric in Eq. 1.6.1 becomes

$\odif{s}^{2}=a^{2}(\tau)\left[-\odif[order=2]{\tau}+\left(\odif{\chi}^{2}+S_{k}^{2}(\chi)\odif{\Omega}^{2}\right)\right]\,,$
(1.6.6)

where $a(\tau)$ is now the conformal scale factor and

$S_{k}(\chi)\equiv R_{0}\begin{cases}\chi/R_{0}\quad&K=0\,,\\\
\sin(\chi/R_{0})\quad&K=+1\,,\\\ \sinh(\chi/R_{0})\quad&K=-1\,,\end{cases}$
(1.6.7)

expresses the difference between $r$ and $\chi$ for non-flat geometries. In
general we will adopt the conventions in which $r$ is normalised to $R_{0}$,
which vanishes from the equations. Finally, it is worth noting that the FLRW
metric encapsulates all the degrees of freedom of the spacetime through a
scaling function of time, $a(\tau)$, and the constant $K$.

###  1.7 Spacetime Geometry

Having introduced the FLRW metric, we can now compute the Einstein tensor by
evaluating the left-hand side of Eq. 1.3.11.

####  1.7.1 Connections

Substituting $g_{\mu\nu}=\diag(-1,a^{2}\gamma_{ij})$ from Eq. 1.6.1 into the
definition in Eq. 1.6.1, it is straightforward to compute the Christoffel
symbols. All the components with two time indices vanish, that is,
$\Gamma^{\mu}_{00}=\Gamma^{0}_{0\nu}=0$. The non-vanishing components are:

$\displaystyle\Gamma^{0}_{ij}=a\dot{a}\gamma_{ij}\,,\quad\Gamma^{i}_{0j}=\frac{\dot{a}}{a}\delta^{i}_{j}\,,\quad\Gamma^{i}_{jk}=\frac{1}{2}\gamma^{il}\left(\partial_{j}\gamma_{kl}+\partial_{k}\gamma_{jl}-\partial_{l}\gamma_{jk}\right)\,.$
(1.7.1)

according to Eq. 1.3.5 and the Christoffel symbols are symmetric under the
exchange of the lower indices. In spherical polar coordinates, under the
normalisation $R_{0}=1$ over $r$ and remembering that $H=\dot{a}/a$, this
becomes

$\displaystyle\Gamma^{0}_{ij}$
$\displaystyle=a^{2}H\gamma_{ij}\,,\quad\Gamma^{i}_{0j}=H\delta^{i}_{j}\,,\quad\Gamma^{1}_{11}=\frac{Kr}{1-Kr^{2}}\,,\quad\Gamma^{1}_{22}=-r(1-Kr^{2})\,,$
(1.7.2) $\displaystyle\Gamma^{1}_{33}$
$\displaystyle=-r(1-Kr^{2})\sin^{2}\theta\,,\quad\Gamma^{2}_{12}=\frac{1}{r}\,,\quad\Gamma^{2}_{33}=-\sin\theta\cos\theta\,,\quad\Gamma^{3}_{13}=\frac{1}{r}\,,\quad\Gamma^{3}_{23}=\cot\theta\,.$

####  1.7.2 Ricci Tensor

The time-space components vanish according to the isotropy of the spacetime:
$R_{0i}=R_{i0}=0$. The non-zero components are:

$\displaystyle R_{00}$
$\displaystyle=-3\frac{\ddot{a}}{a}=-3\left(H^{2}+\dot{H}\right)\,,$ (1.7.3)
$\displaystyle R_{ij}$
$\displaystyle=a^{2}\left[\frac{\ddot{a}}{a}+2\left(\frac{\dot{a}}{a}\right)^{2}+2\frac{K}{a^{2}}\right]\gamma_{ij}=a^{2}\left[3H^{2}+\dot{H}+2\frac{K}{a^{2}}\right]\gamma_{ij}\,.$
(1.7.4)

####  1.7.3 Ricci Scalar

The Ricci scalar is computed from the components of the Ricci tensor:

$R=6\left[\frac{\ddot{a}}{a}+\left(\frac{\dot{a}}{a}\right)^{2}+\frac{K}{a^{2}}\right]=6\left[\dot{H}+2H^{2}\right]\,.$
(1.7.5)

####  1.7.4 Einstein Tensor

The non-zero components of the Einstein tensor, Eq. 1.3.11, in the form
$G\indices{{}^{\mu}_{\nu}}\equiv g^{\mu\lambda}G_{\nu\lambda}$ are:

$\displaystyle G\indices{{}^{0}_{0}}$
$\displaystyle=-3\left[\left(\frac{\dot{a}}{a}\right)^{2}+\frac{K}{a^{2}}\right]=-3\left[H^{2}+\frac{K}{a^{2}}\right]\,,$
(1.7.6) $\displaystyle G\indices{{}^{i}_{j}}$
$\displaystyle=-\left[2\frac{\ddot{a}}{a}+\left(\frac{\dot{a}}{a}\right)^{2}+\frac{K}{a^{2}}\right]\delta\indices{{}^{i}_{j}}=-\left[2\dot{H}+3H^{2}+\frac{K}{a^{2}}\right]\delta\indices{{}^{i}_{j}}\,,$
(1.7.7)

where again isotropy ensures that the mixed components vanish.

###  1.8 The Hot Big Bang Model

We close with a note on the remarkable observational evidence for what exactly
is meant by the Big Bang mentioned at the beginning of this chapter. The
advent of General Relativity (GR) sparked a paradigm shift, revolutionising
our understanding of the Universe’s evolution. Georges Lemaître was the first
to propose that Einstein’s equations yielded a solution describing an
expanding Universe that could be followed back in time to a single point
Lemaître (1931), since $\rho\to\infty$ as $a\to 0$. Subsequently, Edwin
Hubble’s experimental discovery of the Universe’s expansion Hubble and Humason
(1931) culminated in the development of the successful hbb model for the
observable Universe (for a more detailed description and historical context
about the HBB model, we refer the reader to the review in Ref. Liddle and Lyth
(2000)).

The standard HBB model has provided a theoretical basis for explaining various
experimental observations Kolb and Turner (1990), namely:

  1. 1.

Predicting the existence and the shape of the spectrum of the cosmic microwave
background;

  2. 2.

Explaining the mechanism behind the primordial abundances of light elements
through nucleosynthesis;

  3. 3.

Describing the expansion of the Universe and providing a framework to
understand the gravitational collapse of matter, leading to the formation of
galaxies and other large-scale structures observed today.

The Hubble parameter $H(t)=\dot{a}/a$ plays a crucial role as a physical
observable, describing the expansion rate of an FLRW Universe and setting its
characteristic time and distance scales. The Hubble time $t_{H}=H^{-1}$ and
the Hubble radius $d_{H}=cH^{-1}$ (in units where $c=1$ we have $d_{H}=t_{H}$)
define the time and length scales of the FLRW spacetime, respectively, and are
strictly local quantities (i.e. they are defined by the instantaneous
expansion rate at a given time $t$).

##  Chapter 2 The Standard $\Lambda$CDM Model: Successes and Controversies

> E – e não esquecer que a estrutura do átomo não é vista mas sabe-se dela.
> Sei de muita coisa que não vi. E vós também. Não se pode dar
> uma prova da existência do que é mais verdadeiro, o jeito é
> acreditar.00footnotetext: And — and don’t forget that the structure of the
> atom cannot be seen but is nonetheless known. I know about lots of things
> I’ve never seen. And so do you. You can’t show proof of the truest thing of
> all, all you can do is believe. — Clarice Lispector in The Hour of the Star
> — Clarice Lispector A Hora da Estrela

In this chapter, we build upon the gravitational framework introduced in
Chapter 1 to introduce the current standard model of cosmology, the
$\Lambda$CDM model, by setting forth its underlying theoretical assumptions
and predictions. Assuming the Friedmann-Lemaître-Robertson-Walker background
metric as the starting point, we will explore the equations governing the
Universe’s expansion history in Section 2.1. Moreover, we examine the linear
perturbations produced in the early Universe in Section 2.2, which serve as
the initial inhomogeneities that are the seeds of structure formation.
Bringing everything together, we summarise the most relevant epochs in the
Universe’s evolution and the physics involved at the relevant scales in
Sections 2.3 and 2.4. We conclude by contemplating the need for the
cosmological constant, the simplest form of dark energy, while also discussing
the conceptual challenges that arise from its inclusion in Section 2.5.

###  2.1 Fundamentals of the Background Cosmology

In the framework of the Hot Big Bang model for the early Universe, tracing
back the evolution until the beginning of time leads to a problematic point
known as a singularity, a fundamental limitation of standard general
relativity. However, it is widely expected that there exists a more
comprehensive theory that resolves this singularity issue, with the concept of
cosmological inflation being a prominent avenue of investigation. While
inflation is a significant topic, this chapter primarily focuses on the
subsequent cosmic evolution following inflation and the events in the very
early Universe, deliberately setting aside the open question of what preceded
(or more precisely, what was) the bb.

####  2.1.1 Observable Universe

Let us start by clarifying what is meant by the observable Universe. According
to GR, the speed of light $c$ (and therefore the speed of any detectable
signal carrying information) is finite. It follows that not every spacetime
region is within our reach. Adopting the Big Bang scenario further implies the
existence of an intrinsic limit set by the age of the Universe on the maximum
distance from which a point in space could have received information. This
limit is known as the particle horizon (defined as the time integral of
$1/a(t)$), and its size keeps evolving. This means that whenever we make
considerations about the Universe, these are based on the portion of the
Universe that is causally connected to us and which lies within our particle
horizon. Anything beyond this region is inaccessible and, more importantly,
will not directly impact our observations. Nevertheless, two different
spacetime regions can be accessible and causally connected to some observer
while remaining disconnected from each other. All it takes is for these two
regions to be contained inside the observer’s particle horizon while their
individual particle horizons do not overlap. It is intuitively understandable
that any causally connected regions can exhibit homogeneity (without an
extremely high level of fine-tuning) if they have had the chance to interact
and reach some thermal equilibrium. On the other hand, if both regions are not
causally connected, then local interactions cannot justify the observed
homogeneity, which must only be related to some initial homogeneity of the
Universe itself.

Our observed Universe demonstrates a remarkable level of uniformity on scales
larger than anticipated, given its age and the problems posed above. This
challenge is known as the horizon problem Guth (1981) and was first formalised
in the 1960s. One possible resolution would be to consider that the Universe
is significantly older. However, this would imply substantial deviations from
what is estimated by observations by several orders of magnitude Baumann
(2022). Consequently, under the current cosmological model, this uniformity is
better explained by the inherent particularities of its initial conditions -
in a brief moment at the beginning of the Universe which cosmologists believe
happened and refer to as inflation, and which will be further discussed in
Section 2.3.

####  2.1.2 Invisible Universe

One of the most revolutionary proposals of the standard model was the
assertion that the majority of matter and energy in the Universe is in some
form invisible to our detectors (non-luminous), meaning that it can only be
observed through gravitational interactions or other unknown indirect signals.
More precisely, the standard matter we can see and interact with amounts to
around $5\%$ of the total share. The remaining $95\%$ are in the form of the
mysterious dark sector, split into two components with rather distinct nature
and purpose: dark matter (DM) and dark energy (DE).

The term dark matter was coined by Fritz Zwicky in 1933 when he observed that
galaxies in the Coma cluster moved faster than anticipated in order to remain
bounded together Zwicky (1933) given the measure of visible mass within them.
This implied that either the clusters were flying apart, or they had to be
held together by the gravitational attraction of mass outside the luminous
parts of the cluster, or otherwise, something had to be wrong with our
understanding of physics. To explain the stability of the cluster, he proposed
the existence of an additional form of invisible matter. Later in the 1970s,
Vera Rubin and colleagues measured the rotation speed of gas in the outer
regions of galaxies Rubin and Ford (1970), also reporting exceptionally high
speeds, with nearly flat rotation curves. These and subsequent observations,
based on X-rays, radio emission, bbn or the developing studies on lss, could
be explained by hypothesising halos of some excess non-luminous matter
surrounding the galaxies and in clusters of galaxies, further supporting the
need for this weakly interacting matter. Nevertheless, the most robust
indirect evidence for dark matter so far comes from the gravitational lensing
of the CMB. As the CMB photons travel across the Universe, they are
gravitationally deflected by the nearby large-scale structure, creating a
pattern of distortions in the hot and cold spots of the CMB. This phenomenon
has been mapped in great detail by the Planck satellite Aghanim et al.
(2020c), which confirmed that its magnitude relates directly to the total
amount of matter in the Universe. Observations of the abundance of light
elements suggest that ordinary baryonic matter exists in relatively less
abundance. The exact amount of dark matter is also necessary to explain the
measured rate of gravitational clustering since the density fluctuations
generated in the early Universe rely on dark matter to grow at such a rapid
and efficient pace de Swart et al. (2017), Bertone and Hooper (2018).

In the 1980s, even after the proposal of DM, cosmology was still at a turning
point. The extrapolated age for a Universe populated only by matter appeared
to be shorter than the ages of the oldest stars within it. Moreover,
observations of the large-scale structure were incompatible with the
predictions from inflationary cosmology of a quasi-flat Universe, suggesting
that only $30\%$ of the critical density (the total energy-density
contribution in the Universe to ensure spatial flatness) could be accounted
for by the total matter density. As discussed in Section 1.3, it was once
suggested that Einstein’s cosmological constant, introduced to create a static
Universe solution, or some other form of dark energy (a term that appears to
have first been coined by Huterer and Turner Huterer and Turner (1999)), might
provide a solution to these issues. However, lacking empirical support, this
idea remained a theoretical hypothesis. Later came the simultaneous proposal
by Ratra and Peebles Ratra and Peebles (1988) and Wetterich Wetterich (1988)
in 1988, positing that the cosmological constant may have been gradually
approaching its natural value - zero - over an extended period.

However, this question was ultimately settled by the unanticipated discovery
of the accelerating character of the Universe by two separate teams: the
Supernova Cosmology Project Perlmutter et al. (1999), and the High-z Supernova
Search Team Riess et al. (1998), by studying standard distant supernova events
and measuring the distances and recession speeds of galaxies, analogous to
Hubble’s analysis using local sources. These studies revealed that the
expansion rate was decelerating in the early stages when the Universe was
dominated by matter and transitioned to an accelerating phase quite recently
(in cosmological terms). It turns out that the simplest way to account for the
accelerated expansion within the framework of Einstein’s theory of gravity is
if the Universe has just entered a phase dominated by Einstein’s cosmological
constant $\Lambda$; more precisely, a source with an energy density that does
not dilute with the expansion and which exerts a negative pressure symmetric
to its energy density. Remarkably, the cosmological constant hypothesis
accommodates the oldest stars in the estimated age of the Universe and fulfils
the critical density as required in the inflation paradigm, with detailed
studies of the CMB anisotropies map placing the dark energy fractional density
to the missing $70\%$ of the total energy in the present-day Universe Aghanim
et al. (2020a).

The addition of these two dark sector components is the genesis of the
standard model of cosmology, the Lambda-Cold-Dark-Matter ($\Lambda$CDM) model.
However, and as will be one of the main focuses of this dissertation, despite
addressing the problems above and fitting the data with outstanding accuracy,
the $\Lambda$CDM model now faces a new crisis. On the fundamental side, the
cosmological constant cannot be reconciled with predictions for the vacuum
energy from quantum field theory, with several orders of magnitude of
difference between the predicted and observed values. On the other hand, the
increase in independent precision probes has brought to light observational
incompatibilities between model-dependent measurements from the early Universe
and studies within the local Universe. The origin of these inconsistencies
remain one of the most significant unresolved problems in cosmology and
fundamental physics Abdalla et al. (2022).

####  2.1.3 Expanding Universe

The Hubble expansion rate is one of the protagonists of modern cosmology and
is defined as:

$H\equiv\frac{\dot{a}}{a}\,,$ (2.1.1)

with the fact that it provides information about the Universe’s expansion rate
now made explicit. Generally, $H$ is a function of time, and its value
measured at present, denoted as $H_{0}$, is referred to as the Hubble constant
or parameter. As was the case for defining the conformal time, Eq. 1.6.4, it
is crucial to differentiate between the comoving distance $r$ \- which is
fixed to the expansion and thus oblivious to its effect - and the physical
distance $r_{\text{phys}}$, which increases proportionally to the expansion.
An observer studying an object at a fixed comoving distance $r$ will measure a
physical distance to it that depends on time. Therefore, the physical velocity
of a point must also account for the added contribution of the expansion and
is given by:

$r_{\text{phys}}\equiv a(t)r\quad\xrightarrow{\mathrm{d}/\odif{t}}\quad
v_{\text{phys}}\equiv\frac{\odif{r_{\text{phys}}}}{\odif{t}}=a\dot{r}+r\dot{a}\equiv
v_{\text{pec}}+Hr_{\text{phys}}\,.$ (2.1.2)

The first term, $v_{\text{pec}}$, is the peculiar velocity, that is, the
velocity of the object relative to the coordinate grid. The second term,
$Hr_{\text{phys}}$, is the Hubble flow resulting from the expansion of the
coordinate grid itself. For galaxies moving in the Hubble flow, sufficiently
distant for their peculiar velocities to be insignificant in comparison to the
velocity arising from the expansion, Eq. 2.1.2 simplifies to:

$v_{\text{phys}}\simeq Hr_{\text{phys}}\,,$ (2.1.3)

which we have encountered before as the Hubble-Lemaître law, valid for local
distances (in comparison to $H$), in which case $H\approx H_{0}$. In other
words, the recession of galaxies away from us happens at speeds proportional
to their physical distance and can be solely attributed to the expansion of
the Universe. The proportionality constant measures the present rate of
expansion, $H_{0}$. Edwin Hubble’s reported observation of his homonym law in
1929 Hubble and Humason (1931) by plotting $v$ against $r$ leading to an
estimated $H_{0}=500\,\text{km/s/Mpc}$, was the onset of evidence of the
Universe’s expansion and provided strong support for the hot Big Bang model.
During the past century, the endeavour of measuring and independently
confirming the value of the Hubble constant progressed. At the moment, $H_{0}$
is typically inferred from Cosmic Microwave Background (CMB) measurements
assuming a specific cosmological model or estimated directly in the local
Universe, such as the luminosity patterns of Type Ia supernovae, in a model-
independent fashion.

As is the case for the $H_{0}$ estimates, most observational probes in
cosmology rely on detecting light emitted by astrophysical and cosmological
sources. In an expanding Universe, the momentum of particles that transverse
it, as perceived by comoving observers, decays with the expansion at the rate
$1/a$, resulting in an intrinsic effect in the light we observe. If the light
is emitted with a specific wavelength $\lambda$, that wavelength will be
stretched by the expansion proportionally to the scale factor $a$.
Consequently, a photon emitted at time $t$ is detected on Earth at time
$t_{0}$ with a wavelength given by:

$\lambda(t_{0})=\frac{a(t_{0})}{a(t)}\lambda(t)\,.$ (2.1.4)

In an expanding Universe, the scale factor increases with time, meaning that
$a(t_{0})>a(t)$, implying that the wavelength at detection is larger than the
wavelength at emission. This leads to the introduction of the redshift
parameter111We had previously defined the redshift as $z=v/c$. This is valid
as long as the recessional velocity of an object is much smaller than $c$ and
$\lambda_{0}\approx(1+v/c)\lambda$ from the Doppler effect., defined as the
fractional change in the wavelength of a photon emitted at time $t$ and
detected today on Earth:

$z\equiv\frac{\lambda(t_{0})-\lambda(t)}{\lambda(t)}\,,$ (2.1.5)

and which quantifies how much the wavelength has been stretched. Employing Eq.
2.1.4, we find $1+z=a(t_{0})/a(t)$, which together with the conventional
normalisation $a(t_{0})=1$, yields:

$1+z\equiv\frac{1}{a(t)}\,.$ (2.1.6)

Given its direct relation with the time-dependent scale factor, the redshift
parameter $z$ is frequently preferred for the time-keeping of specific events
during the history of the Universe.

The Hubble parameter is often expressed in terms of the dimensionless quantity
$h$:

$H_{0}=100h\,\text{km/s/Mpc}=2.1332h\times 10^{-42}\,\text{GeV}\,,$ (2.1.7)

and can be used to define the average cosmological density in the Universe,
also called the critical density:

$\rho_{\text{crit},0}\equiv\frac{3H_{0}^{2}}{8\pi G}=1.88h^{2}\times
10^{-29}\,\text{g}\,\text{cm}^{3}\,.$ (2.1.8)

Lastly, the Hubble rate defines a Hubble time

$t_{H}\equiv\frac{1}{H_{0}}=9.78\times 10^{9}\,h^{-1}\,\text{years}\,,$
(2.1.9)

which is a crude measure of the age of the Universe, and the present Hubble
radius

$D_{H}\equiv\frac{c}{H_{0}}=2998\,h^{-1}\,\text{Mpc}\,,$ (2.1.10)

a fair approximation to the largest scales that can currently be observed. The
following two sections will address other observationally meaningful
parametrisations of time and distance.

####  2.1.4 Time Parametrisations

Before moving on to the details of the standard model, it is essential to
summarise the various clocks that refer to the notion of time in different
epochs and contexts depending on the physical system under consideration. The
relation between the time $t$ (and equivalently $\tau$) and the redshift $z$
(and equivalently $a$) encoded in Eq. 2.1.6 is especially relevant when
dealing with relativistic components due to the direct redshift-dependent
correspondence between energy at emission and detection, encoding a valuable
measure of energy loss over time which would not be evident for the proper
time. Thus, $t$, $\tau$, $a$, and $z$ are all relevant and distinctively
informative definitions of time.

However, in the very early Universe, when particle physics processes play a
crucial role, the concept of temperature $T$ constitutes a trigger for
different events. For photons with energy $E$, there is a direct
correspondence between temperature and redshift since $1+z\propto E\propto T$
(normalised to the temperature of the Universe today:
$T_{0}=2.7255\,\text{K}\simeq 2.348\times 10^{-4}\,\text{eV}$).

Finally, and as will be extensively discussed in the following sections, the
tiny energy fluctuations in the early-Universe evolve according to a wave-like
behaviour and can be decomposed into their Fourier modes $k$, related to their
wavelength $\lambda$ as $k=2\pi a/\lambda$. Assuming the separate Universe
approximation, i.e. that each mode evolves independently, $k$ becomes a proxy
for the scales of a given event. At early times, when the Universe was
smaller, only modes with small wavelengths contributed significantly to the
evolution, as those with much larger wavelengths were minimally affected by
processes occurring at small scales. The Fourier modes convey an idea of the
size of the Universe, which, in turn, is related to the time elapsed since the
beginning of its expansion.

Therefore, all these quantities will be used interchangeably throughout the
text depending on the context and the cosmological epoch.

####  2.1.5 Cosmological Distances

In the same way, as for the time parametrisations considered above, different
rulers can be used to measure distances, depending on which quantity is being
used as a proxy of distance. Much of the evidence for dark energy comes from
measurements of cosmological distances. In order to understand how present
observations relate to other processes happening in the Universe, it is
essential to understand photon propagation. Since photons have no mass, they
traverse null geodesics:

$\odif{s}^{2}=-\odif{t}^{2}+a^{2}(t)\odif{\chi}^{2}=0\ \ \Leftrightarrow\ \
\odif{\chi}^{2}=a^{-2}(t)\odif{t}^{2}\,,$ (2.1.11)

with the radial component $\chi$ as the direction of the photons’ propagation.
The last expression can be integrated to find the line-of-sight comoving
distance $d_{c}$ travelled by the photon in some time interval
$[t_{i},t_{f}]$:

$d_{c}(t)=\chi(t)=\int_{t_{f}}^{t_{i}}\frac{c\odif{t}}{a(t)}\quad\xrightarrow{\odif{t}=\odif{a}/(aH)=-\odif{z}a/H}\quad
d_{c}(z)=\int_{z_{i}}^{z_{f}}\frac{c\odif{z}}{H(z)}\,.$ (2.1.12)

Another important observable in cosmology is the absolute luminosity of
astrophysical objects $L$, defined in terms of the observed flux $F$, over a
sphere of radius $\chi$ in an expanding spacetime. This sphere has an area

$4\pi a^{2}(t_{0})d_{M}^{2}\quad\text{where}\quad d_{M}\equiv S_{k}(\chi)\,,$
(2.1.13)

and is called the metric or transverse comoving distance, with $S_{k}(\chi)$
as defined in Eq. 1.6.6, which reflects the fact that in a curved spacetime
$d_{M}\neq\chi$. $d_{c}$ in Eq. 2.1.12 is often used when referring to the
spatially flat geometry, in which case both quantities are equivalent. The
luminosity distance is then expressed in terms of $F$ as,

$d_{L}^{2}=\frac{L}{4\pi F}\quad\xrightarrow{\text{expanding space}}\quad
d_{L}(z)=(1+z)d_{M}(z)\,,$ (2.1.14)

where the multiplicative redshift factor accounts for the energy loss
experienced by the photons and the reduction in their number due to cosmic
expansion since

$\frac{L_{i}}{L_{f}}=\frac{\Delta E_{i}}{\Delta E_{f}}\frac{\Delta
t_{f}}{\Delta t_{i}}=(1+z)^{2}\,.$ (2.1.15)

This will be important in Chapter 3 where we will discuss how distance
measurements can be used to infer the value of cosmological parameters such as
$H_{0}$.

For objects of some physical size $\Delta x$ that are observed to subtend an
angle in the sky $\Delta\theta$ (perpendicular to the line of sight and
$\Delta\theta\ll 1$ in radians for cosmological distances), it is more
convenient to define the angular diameter distance:

$d_{A}=\frac{\Delta x}{\Delta\theta}\quad\text{and}\quad\Delta
x=a(t)S_{k}(\chi)\Delta\theta\quad\xrightarrow{\text{expanding space}}\quad
d_{A}(z)=d_{M}(z)/(1+z)\,,$ (2.1.16)

where we have used geometrical considerations assuming $\Delta\theta\ll 1$,
and the dividing redshift factor is now included to reflect the Universe’s
expansion since the light was emitted until it is observed in the sky. In
summary, we have defined three different distance observables which are
related as:

$d_{M}(z)=(1+z)d_{A}(z),\quad\text{and}\ \quad d_{L}(z)=(1+z)^{2}d_{A}(z)\,.$
(2.1.17)

The last expression is the distance-duality or Etherington relation
Etherington (2007), which holds for general metrics under photon number
conservation.

####  2.1.6 Friedmann Equations

Assuming perfect fluids coupled to Einstein’s field equations, Eq. 1.3.11, we
can gather all the ingredients and derive the Friedmann equations that
describe the evolution of an FLRW Universe:

$\displaystyle\left(\frac{\dot{a}}{a}\right)^{2}=H^{2}$
$\displaystyle=\frac{8\pi G}{3}\rho-\frac{K}{a^{2}}\,,$ (2.1.18)
$\displaystyle\frac{\ddot{a}}{a}=\dot{H}+H^{2}$ $\displaystyle=-\frac{4\pi
G}{3}(\rho+3p)\,,$ (2.1.19)

where $\rho$ and $p$ are the generic contributions of all the fluids in the
Universe, remembering that the constant $K$ represents the spatial curvature
of the Universe. It is worth remarking that Eq. 2.1.19 is also known as the
Raychaudhuri equation and is a curvature-independent measure of the expansion,
as it does not depend directly on $K$.

####  2.1.7 Cosmic Ingredients

The cosmological principle dictates that the background energy-momentum tensor
resembles that of a perfect fluid Baumann (2022), expressed as

$T_{\mu\nu}=(\rho+p)u_{\mu}u_{\nu}+pg_{\mu\nu}\,,$ (2.1.20)

where $\rho$ and $p$ represent the energy density and pressure in the rest
frame of the fluid, respectively, and $u^{\mu}$ is the four-velocity relative
to a comoving observer, under the constraint $u^{\mu}u_{\mu}=-1$, which we
take to be

$u_{\mu}=(a,\vec{0})\quad\text{and}\quad u^{\mu}=(a^{-1},\vec{0})\,,$ (2.1.21)

for all cosmological components. The Bianchi identities, Eq. 1.3.12, imply
conservation of energy from the time component, leading to the conservation or
continuity equation:

$\nabla^{\mu}T_{\mu\nu}=0\quad\xrightarrow{\nu=0}\quad\dot{\rho}+3H(\rho+p)=0\,.$
(2.1.22)

The second Friedmann equation, introduced in Eq. 2.1.19, is equivalent to
taking the first Friedmann equation, Eq. 2.1.18, together with the continuity
equation, Eq. 2.1.22.

Assuming a Universe dominated by a single species with an equation of state
$p=w\rho$ with constant $w$ (which will no longer hold, for instance, for
dynamical dark energy), Eq. 2.1.22 can be integrated to obtain

$a\propto(t-t_{i})^{\frac{2}{3(1+w)}}\quad\text{and}\quad\rho\propto
a^{-3(1+w)}\,,$ (2.1.23)

where the subscript denotes the initial time. This result embodies how
different cosmological fluids are diluted by cosmic expansion, which we list
below for the three ingredients that play essential roles in the evolution of
the Universe. It also shows how spatial curvature contributes to the first
Friedmann equation, Eq. 2.1.18, as a fluid with $\rho\propto a^{-2}$.

The critical energy density for a flat spatial geometry ($K=0$), defined in
Eq. 2.1.8 at present, can be expressed as a time-evolving function as

$\rho_{\text{crit}}=\frac{3H^{2}}{8\pi
G}\quad\text{with}\quad\rho_{\text{crit},0}\approx 5\times
10^{-6}\,\text{GeV}\,\text{cm}^{-3}\,,$ (2.1.24)

where the label $0$ denotes present time, which we often omit when discussing
the current matter distributions, and it is implied from the context. This can
be used to express the relative energy density of the fluid components $f$ in
a dimensionless manner:

$\Omega_{f}=\frac{\rho_{f}}{\rho_{crit}}\quad\text{for}\quad
f=m,r,\Lambda,...\,,$ (2.1.25)

where the subscripts $(m,r)$ denote the collective matter and radiation
components, whose evolution will be detailed below, and $\Lambda$ gives the
contribution from the cosmological constant. Defining
$\Omega_{K}\equiv-K/(aH_{0})^{2}$ (in which case $\Omega_{K}<0$ actually
corresponds to $K>0$ and vice versa), the Friedmann equation, Eq. 2.1.18, can
be rewritten as the Friedmann constraint for the curvature density. This can
be expressed according to the measured relative energy densities at present
time:

$\Omega_{0}\equiv\Omega_{m}^{0}+\Omega_{r}^{0}+\Omega_{\Lambda}^{0}\quad\xrightarrow{a(t_{0})\equiv
1}\quad\Omega^{0}_{K}=1-\Omega_{0}\,.$ (2.1.26)

If the Universe’s expansion is decelerating (i.e., $\ddot{a}<0$), then the
absolute value of the curvature term $|\Omega_{K}|$ will continue to rise.
This is except in the particular case where the Universe has been precisely
flat ($K=0$) from its early stages. According to the Planck 2018 data Aghanim
et al. (2020a), this quantity is contrained to be
$\Omega_{K}^{0}=-0.044^{+0.018}_{-0.015}$ at present at $68\%$ cl. For
$|\Omega_{K}|$ to decrease during the Universe’s evolution, a phase of cosmic
acceleration ($\ddot{a}>0$) is necessary, or otherwise, the Universe must have
emerged from an initial state extremely close to flat. To account for the
current observed level of the Universe’s flatness, a phase of cosmic inflation
is needed before the radiation-dominated era, during which the scale factor
must have increased by a factor greater than $\mathup{e}^{60}$ Liddle and Lyth
(2000).

Next, according to Eq. 2.1.23, we list the Universe’s constituents by their
properties, deriving the corresponding evolution for the approximated single-
component case. The evolution of each comic fluid with redshift is depicted in
Fig. 2.1.

(a) Figure 2.1:  Evolution of the energy densities $\rho$ of radiation $r$
(grey dotted line), baryons $b$ (green dashed line), cold dark matter cdm (red
filled line), and the cosmological constant $\Lambda$ (blue dot-dashed line),
as a function of redshift $z$ in the $\Lambda$CDM model.

##### Matter

The broad heading matter refers to fluids with negligible pressure compared to
the energy density (non-relativistic particles), such that $p\ll\rho\
\Rightarrow\ w\approx 0$, resulting in the dilution rate

$a\propto(t-t_{i})^{2/3}\quad\text{and}\quad\rho_{m}\propto a^{-3}\propto
t^{-2}\quad\xrightarrow{H=\dot{a}/a}\quad H=\frac{2}{3t}\,.$ (2.1.27)

One way to interpret this result is to imagine that the volume of a given
region of three-dimensional space increases as $V\propto a^{3}$. In contrast,
the number of particles (and therefore the energy in that region) remains
constant. The relevant non-relativistic matter components for the Universe’s
evolution are:

  * •

Baryons: refers to ordinary matter, including the less massive electrons
(fermions), thoroughly comprehended by physicists as an integral part of the
standard model of particle physics.

  * •

Dark Matter: approximately five times more abundant than baryons, the exact
characteristics of dark matter remain a mystery, as was discussed in Section
2.1.2. In the standard model, it is cold dark matter (cdm). Other potential
candidates include wimps, which have yet to be detected.

##### Radiation

Radiation is a gas composed of relativistic particles characterised by
$p=\rho/3\ \Rightarrow\ w=1/3$ according to statistical mechanics. As derived
from Eq. 2.1.23, this results in

$a\propto(t-t_{i})^{1/2}\quad\text{and}\quad\rho_{r}\propto a^{-4}\propto
t^{-2}\quad\xrightarrow{H=\dot{a}/a}\quad H=\frac{1}{2t}\,.$ (2.1.28)

The energy redshift of relativistic particles, expressed as $E\propto a^{-1}$,
adds to the dilution due to the expansion.

The non-relativistic components are

  * •

Photons: relativistic massless particles. Photons were the dominant
contribution during nucleosynthesis (formation of the first atoms) and are now
detected as the cosmic microwave background.

  * •

Light particles: In the early Universe, all the particles in the standard
model had a radiation character due to the high kinetic energy relative to
their mass. They behaved like matter once the temperature dropped enough for
their masses to play a role.

  * •

Neutrinos: contributed to the early Universe radiation domination, and only
recently have their small masses become relevant and started to act like
matter.

At present, both photons and neutrinos only exist in minimal quantities.

##### Dark Energy

In order to have cosmic acceleration, The Friedmann equations require
$\ddot{a}>0$ in Eq. 2.1.19, which implies $p<-\frac{\rho}{3}$, or
equivalently, $w<-\frac{1}{3}$. When $w=-1$ and $p=-\rho$, the relation in Eq.
2.1.23 only holds if $\rho$ is a constant. This is the case for the
cosmological constant, which (in a flat Universe) implies that $H$ is also
constant, in which case the scale factor evolves exponentially:

$a\propto\mathup{e}^{H_{\Lambda}t}\quad\text{and}\quad\rho_{\Lambda}=\frac{\Lambda}{4\pi
G}\propto a^{0}\quad\xrightarrow{H=\dot{a}/a}\quad H=\frac{\sqrt{8\pi
G}}{3}\rho_{\Lambda}\,.$ (2.1.29)

The cosmological constant cannot be responsible for inflation in the early
Universe, as that would mean the accelerated expansion would never cease,
contrary to the current cosmic acceleration, which may continue indefinitely.

The cosmological constant proposal can be generalised under the broader
category of dark energy, referring to any fluid either with a constant
$w_{\text{DE}}\approx-1$, or a time-varying equation of state parameter,
$w_{\text{DE}}(t)$, in which case Eq. 2.1.23 no longer holds.

Bringing all the ingredients together for the $\Lambda$CDM model, the first
Friedmann equation, Eq. 2.1.18, can be decomposed into

$E^{2}(z)=\frac{H^{2}}{H_{0}^{2}}=\Omega_{r}^{0}a^{-4}+\Omega_{m}^{0}a^{-3}+\Omega_{K}^{0}a^{-2}+\Omega_{\Lambda}^{0}\,,$
(2.1.30)

where $E(z)$ is the normalised Hubble rate.

###  2.2 Linear Cosmological Perturbations

Up to this point, our treatment of the Universe assumed perfect homogeneity.
However, to comprehend the Universe’s large-scale structure, it becomes
necessary to introduce inhomogeneities and study their evolution. As long as
these perturbations remain relatively small, we can analyse them using linear
perturbation theory.

In practice, the quantities that can be determined observationally for a given
model are not typically the exact values of the perturbation variables,
subject to quantum fluctuations, but rather realisations of expectation
values. However, this methodology has an inherent problem since we cannot
precisely measure expectation values. The usual statistical considerations
fail because we only have access to one Universe, one sample of the stochastic
process that generates these fluctuations. Therefore, when attempting to
determine the mean square fluctuation on a given scale $\lambda$, the best
approach is to average over multiple disjoint patches of size $\lambda$ and
take this spatial averaging as an ensemble averaging based on an ergodic
hypothesis analogue. If the scale $\lambda$ is much smaller than the Hubble
horizon $1/H$ (our observational limit), then we can compute averages over
many independent volumes, and this approximation works well. However, as we
approach scales of the order of the horizon,
$\lambda\approx\mathcal{O}(1/H_{0})$, the statistical approximation breaks
down due to cosmic variance.

The perturbations are included in the Einstein equations by taking slight
deviations - denoted by $\delta$ \- to the background quantities - denoted by
an upper bar - in the metric (the Einstein tensor) and the matter distribution
(the energy-momentum tensor):

$\displaystyle g_{\mu\nu}(t,\vec{x})$ $\displaystyle=\bar{g}_{\mu\nu}+\delta
g_{\mu\nu}(t,\vec{x})\,,$ (2.2.1) $\displaystyle T_{\mu\nu}(t,\vec{x})$
$\displaystyle=\bar{T}_{\mu\nu}+\delta T_{\mu\nu}(t,\vec{x})\,.$ (2.2.2)

We can then expand the Einstein and conservation equations to linear order in
the perturbations. For simplicity, we will focus on flat spacetimes only. The
same derivation, including the terms dependent on the curvature scalar, can be
found e.g. in Liddle and Lyth (2000), Kolb and Turner (1990), Peacock (1998).

For the study of the dynamics of the linear perturbations, it is convenient to
define each relevant quantity in terms of its Fourier transform, where
$k=2\pi/\lambda$ are the Fourier modes. Taking any perturbed quantity
$f(\tau,\vec{x})$, the Fourier transform is

$f(\tau,\vec{k})=\int\mathup{e}^{-i\vec{k}\cdot\vec{x}}f(\vec{x})\odif{\vec{x}}^{3}\,,$
(2.2.3)

while the inverse reads

$f(\tau,\vec{x})=\frac{1}{(2\pi)^{3/2}}\int\mathup{e}^{-i\vec{k}\cdot\vec{x}}f(\vec{k})\odif{\vec{k}}^{3}\,,$
(2.2.4)

where $f(\vec{k})$ are the Fourier components of $f(\tau,\vec{x})$ and cosmic
time has been replaced by the conformal time $\tau$, related by
$\odif{t}=a\odif{\tau}$ and $\dot{(\cdot)}\mapsto(\cdot)^{\prime}/a$. This
implies the mapping $\nabla^{2}\rightarrow-k^{2}$ when the spatial derivatives
are applied to the relevant quantities.

The general perturbed metric containing only scalar perturbations222In
general, one can perform a scalar-vector-tensor decomposition of the metric
perturbations, such that the Einstein equations for scalars, vectors and
tensors are independent at linear order and can be treated separately. We will
only deal with scalar fluctuations and the corresponding density
perturbations, so vectors and tensors may be disregarded at linear order.
Vector perturbations are not produced by standard inflation but would, in any
case, decay quickly over the expansion. Tensor perturbations are an important
test of inflation but not significant for the scales considered in this work.
is subject to a non-unique choice of coordinates or gauge choice. Hence,
different choices of coordinates can introduce significant changes to the
values of perturbed variables. Some non-physical perturbations may arise
associated with that choice, the so-called gauge modes, which can be
eliminated with more convenient coordinates. Equivalently, a physical
perturbation can be dropped, for instance, in comoving coordinates, but these
will never truly vanish; they will instead be incorporated into additional
perturbations in the metric. Therefore, a more physical approach to
perturbation theory is one which remains invariant under changes in
coordinates, allowing for a more robust and reliable study of the true
perturbed cosmological degrees of freedom in cosmology. This can be achieved
via the so-called Bardeen variables Bardeen (1980), which encode the true
spacetime perturbations that cannot be erased by a gauge transformation. More
details on gauge invariance and transformations can be found, for example, in
Mukhanov et al. (1992), Malik and Wands (2009), Riotto (2003), Ma and
Bertschinger (1995). It is often convenient to fix the gauge depending on the
scenario in question, at the cost of carefully keeping track of all the
perturbations in the metric and the matter (there are no free lunches!). For
the cosmological scenarios this dissertation considers, we will focus on two
common gauge choices, primarily for consistency and cross-checking reasons:
the conformal Newtonian and synchronous gauges.

####  2.2.1 Newtonian Gauge

For this dissertation, we will only be interested in cosmological scenarios
well-described by the FLRW background metric $\bar{g}_{\mu\nu}$, as defined in
Eq. 1.6.6. In the context of dealing with the Einstein and Friedmann
equations, the scalar metric perturbations are most simply expressed in the
Newtonian or longitudinal gauge:

$\odif{s}^{2}=a^{2}(\tau)\left[-(1+2\Psi)\odif[order=2]{\tau}+(1-2\Phi)\delta_{ij}\odif{x}^{i}\odif{x}^{j}\right]\,.$
(2.2.5)

The metric perturbations $\Phi$ and $\Psi$ are also called the Bardeen
potentials Bardeen (1980), as they are gauge invariant and hence of physical
interest. The Newtonian gauge is usually preferred for studying structure
formation exactly because of its closest resemblance to the Newtonian
analysis, with $\Phi$ playing the role of the gravitational potential.

##### Metric Perturbations

This lays out the metric of the spacetime as

$g_{\mu\nu}=a^{2}\begin{pmatrix}-(1+2\Psi)&0\\\
0&(1-2\Phi)\delta_{ij}\end{pmatrix}\,,$ (2.2.6)

from which we can compute the evolution equations for the metric perturbations
encoded in the perturbed Einstein equations. Namely, starting from the
perturbed line element, Eq. 2.2.5, we can compute the inverse metric and,
through the definition of the connections, further calculate the perturbations
of more complex objects, such as the Ricci tensor, $\delta R_{\mu\nu}$, and
the Ricci scalar, $\delta R$. These are needed to write the linear order
Einstein equations in terms of the perturbed Einstein tensor $\delta
G_{\mu\nu}$:

$\delta G_{\mu\nu}=\delta R_{\mu\nu}-\frac{1}{2}\left(\delta
Rg_{\mu\nu}+R\delta g_{\mu\nu}\right)=8\pi G\delta T_{\mu\nu}\,.$ (2.2.7)

For the sake of completion, in what follows, we list the main results for the
perturbed quantities relevant to the Einstein equations, considering only the
linear terms. Further details and a thorough derivation can be found e.g. in
Refs. Baumann (2022), Gorbunov and Rubakov (2011), Riotto (2003).

##### Inverse Metric

$g^{\mu\nu}=a^{-2}\begin{pmatrix}-1+2\Psi&0\\\
0&(1+2\Phi)\delta^{ij}\end{pmatrix}\,,$ (2.2.8)

##### Connections

$\displaystyle\delta\Gamma^{0}_{00}$
$\displaystyle=\Psi^{\prime}\,,\quad\delta\Gamma^{0}_{i0}=\partial_{i}\Psi\,,\quad\delta\Gamma^{0}_{ij}=-\left[\Phi^{\prime}+2\mathcal{H}(\Phi+\Psi)\right]\delta_{ij}\,,$
(2.2.9) $\displaystyle\delta\Gamma^{i}_{00}$
$\displaystyle=\partial^{i}\Psi\,,\quad\delta\Gamma^{i}_{j0}=-\Phi^{\prime}\delta^{i}_{j}\,,\quad\delta\Gamma^{i}_{jk}=-2\delta^{i}_{(j}\partial_{k)}\Phi+\delta_{jk}\partial^{i}\Phi\,.$
(2.2.10)

##### Ricci Tensor and Ricci Scalar

The perturbed Ricci tensor is calculated from the connections as:

$\delta
R_{\mu\nu}=\partial_{\alpha}\delta\Gamma^{\alpha}_{\mu\nu}-\partial_{\nu}\delta\Gamma^{\alpha}_{\mu\alpha}+\delta\Gamma^{\alpha}_{\mu\nu}\Gamma^{\beta}_{\alpha\beta}+\Gamma^{\alpha}_{\mu\nu}\delta\Gamma^{\beta}_{\alpha\beta}-\delta\Gamma^{\alpha}_{\mu\beta}\Gamma^{\beta}_{\alpha\nu}-\Gamma^{\alpha}_{\mu\beta}\delta\Gamma^{\beta}_{\alpha\nu}\,.$
(2.2.11)

The Ricci Scalar follows from contractions of the Ricci tensor and the metric,
yielding:

$\delta R=\delta g^{\mu\alpha}R_{\alpha\mu}+g^{\mu\alpha}\delta
R_{\alpha\mu}\,.$ (2.2.12)

##### Einstein Tensor

We now have all the ingredients to write explicitly the different components
of the perturbed Einstein tensor in Eq. 2.2.7:

$\delta G_{\mu\nu}=\delta R_{\mu\nu}-\frac{1}{2}\delta
g_{\mu\nu}R-\frac{1}{2}g_{\mu\nu}\delta R\,,$ (2.2.13)

or, equivalently, with one of the indices raised to facilitate the comparison
with the matter sector:

$\delta G\indices{{}^{\mu}_{\nu}}=\delta
g^{\mu\alpha}G_{\alpha\nu}+g^{\mu\alpha}\delta G_{\alpha\nu}\,.$ (2.2.14)

Writing the perturbed terms explicitly, and according to the perturbed metric
in Eq. 2.2.5, we arrive at the different components of the perturbed Einstein
tensor:

$\displaystyle\delta G\indices{{}^{0}_{0}}$
$\displaystyle=2a^{-2}\left[3\mathcal{H}(\mathcal{H}\Psi+\Phi^{\prime})-\nabla^{2}\Phi\right]\,,$
(2.2.15) $\displaystyle\delta G\indices{{}^{0}_{i}}$
$\displaystyle=-2a^{-2}\partial_{i}\left(\Phi^{\prime}+\mathcal{H}\Psi\right)\,,$
(2.2.16) $\displaystyle\delta G\indices{{}^{i}_{j}}$
$\displaystyle=2a^{-2}\left[(\mathcal{H}^{2}+2\mathcal{H}^{\prime})\Psi+\mathcal{H}\Psi^{\prime}+\Phi^{\prime\prime}+2\mathcal{H}\Phi^{\prime}\right]\delta^{i}_{j}+a^{-2}\left[\nabla^{2}(\Psi-\Phi)\delta^{i}_{j}+\partial^{i}\partial_{j}(\Phi-\Psi)\right]\,.$
(2.2.17)

##### Matter Perturbations

For the purpose of this study, we assume that the background perfect fluids
remain perfect fluids when perturbed, in which case there is no anisotropic
stress. Matter perturbations, that is, perturbations to the energy-momentum
tensor of each species, can first be defined in terms of a perfect fluid:

$\displaystyle T\indices{{}^{0}_{0}}$ $\displaystyle=-\delta\rho\,,$ (2.2.18)
$\displaystyle T\indices{{}^{0}_{i}}$ $\displaystyle=(1+w)\bar{\rho}v^{i}\,,$
(2.2.19) $\displaystyle T\indices{{}^{i}_{j}}$ $\displaystyle=\delta
p\delta^{i}_{j}\,,$ (2.2.20)

where $v^{i}$ is the so-called scalar bulk velocity, the matter peculiar
velocity with respect to the expansion, $v^{i}=\odv{x^{i}}{\tau}$. The
different terms in the energy-momentum tensor from each species $f$ need to be
added together to make up the perturbation of the total contribution
$T_{\mu\nu}=\sum_{f}T_{\mu\nu}^{f}$:

$\delta\rho=\sum_{f}\delta\rho_{f}\,,\ \ \delta p=\sum_{f}\delta\rho_{f}\,.$
(2.2.21)

The density and velocity perturbations are often expressed in terms of the
dimensionless density contrast and the velocity divergence

$\displaystyle\delta$ $\displaystyle\equiv\delta\rho/\bar{\rho}\,,$ (2.2.22)
$\displaystyle\theta$ $\displaystyle\equiv\partial_{i}v^{i}\,.$ (2.2.23)

Once $\delta$ becomes of the order of unity, linear perturbation theory no
longer applies.

The perturbed four-velocity takes the form

$u^{\mu}\equiv\bar{u}^{\mu}+\delta u^{\mu}=a^{-1}(1-\Psi,v^{i})\,,\ \
u_{\mu}\equiv\bar{u}_{\mu}+\delta u_{\mu}=a(-(1+\Psi),v_{i})\,.$ (2.2.24)

This leads to the perturbed energy-momentum tensor of a perfect fluid:

$\delta
T\indices{{}^{\mu}_{\nu}}=\rho\left[\delta(1+c_{s}^{2})\bar{u}^{\mu}\bar{u}_{\nu}+(1+w)(\delta
u^{\mu}\bar{u}_{\nu}+\bar{u}^{\mu}\delta u_{\nu})+\delta
c_{s}^{2}\delta^{\mu}_{\nu}\right]\,.$ (2.2.25)

This equation introduces the sound velocity, $c_{s}^{2}\equiv\delta
p/\delta\rho$. For the barotropic fluids, $p$ is only a function of $\rho$,
even after being perturbed, implying

$c_{s}^{2}\equiv\frac{\delta
p}{\delta\rho}=\odv{p}{\rho}=\frac{\dot{p}}{\dot{\rho}}\,,$ (2.2.26)

where the last step only holds for the FLRW metric and represents the
adiabatic sound speed, discarding any other possible internal degrees of
freedom of the fluid.

##### Einstein Equations

Bringing together the expressions for the perturbed Einstein and energy-
momentum tensors gives rise to the perturbed Einstein equations.

The time-time component is

$\delta
G\indices{{}^{0}_{0}}=-2a^{-2}\left[\nabla^{2}\Phi-3\mathcal{H}(\Phi^{\prime}+\mathcal{H}\Psi)\right]\,,$
(2.2.27)

from which it follows that

$\nabla^{2}\Phi-3\mathcal{H}(\Phi^{\prime}+\mathcal{H}\Psi)=4\pi
Ga^{2}\delta\rho\ \ \xrightarrow{\text{F.T.}}\ \
k^{2}\Phi+3\mathcal{H}\left(\Phi^{\prime}+\mathcal{H}\Phi\right)=-4\pi
Ga^{2}\delta\rho\,,$ (2.2.28)

in Fourier space. The second derivative of the potential hints at the fact
that this is a relativistic generalisation of the Poisson equation. Inside the
Hubble radius ($k\gg\mathcal{H}$), the second term of the left-hand side
becomes negligible, and Eq. 2.2.28 reduces to the Poisson equation in the
Newtonian limit: $\nabla^{2}\Phi\approx 4\pi Ga^{2}\delta\rho$. Only on scales
comparable to the Hubble radius do the GR corrections start to play an
important role.

For the trace-free part of the spatial components, we get the following:

$\nabla^{2}\left(\Phi-\Psi\right)=0\ \ \xrightarrow{\text{F.T.}}\ \
k^{2}\left(\Phi-\Psi\right)=0\,,$ (2.2.29)

and the approximation $\Phi\approx\Psi$ holds under the assumption of
vanishing anisotropic stress considered for the scope of this work, such that:

$\partial_{i}(2\Phi^{\prime}+\mathcal{H}\Psi)=4a^{2}\pi
G(\bar{\rho}+\bar{p})v^{i}\ \ \xrightarrow{\text{F.T.}}\ \
k^{2}\left(\Phi^{\prime}+\mathcal{H}\Psi\right)=4\pi
Ga^{2}(\bar{\rho}+\bar{p})\theta\,,$ (2.2.30)

where
$(\bar{\rho}+\bar{p})\theta\equiv\sum_{f}\rho_{f}\left(1+w_{f}\right)\theta_{f}$.

Finally, the trace of the space-space Einstein equation translates into the
following evolution equation for the metric potential

$\Phi^{\prime\prime}+\frac{1}{3}\nabla^{2}(\Psi-\Phi)+\mathcal{H}(\Psi^{\prime}+2\Phi^{\prime})+\Psi\left(\mathcal{H}^{2}+2\mathcal{H}^{\prime}\right)=4\pi
Ga^{2}\delta p\,,$ (2.2.31)

and $\delta p\equiv\sum_{f}\rho_{f}\delta_{f}c_{s,f}^{2}$ is the total
pressure perturbation. Assuming the approximation $\Psi\approx\Phi$, Eq.
2.2.29 becomes:

$\text{\lx@cref{creftype~refnum}{eq:pein3}}\ \ \xrightarrow{\text{F.T.}}\ \
\Phi^{\prime\prime}+3\mathcal{H}\Phi^{\prime}+\Phi\left(\mathcal{H}^{2}+2\mathcal{H}^{\prime}\right)=4\pi
Ga^{2}\delta p\,.$ (2.2.32)

This gives a closed equation for the evolution of $\Phi$ in terms of $\delta
p$, and the relation between $\delta\rho$ and $\Phi$ is encoded in the
Poisson-like equation, Eq. 2.2.28. Combined with the conservation equations we
derive next, this leads to a closed system of linearised equations.

Altogether, the general linearly perturbed Einstein equations in Fourier space
are

$\displaystyle
k^{2}\Phi+3\mathcal{H}\left(\Phi^{\prime}+\mathcal{H}\Psi\right)$
$\displaystyle=-4\pi Ga^{2}\rho\,,$ (2.2.33) $\displaystyle
k^{2}\left(\Phi^{\prime}+\mathcal{H}\Psi\right)$ $\displaystyle=4\pi
Ga^{2}\rho\left(1+w\right)\theta\,,$ (2.2.34)
$\displaystyle\Phi^{\prime\prime}+2\mathcal{H}\Phi^{\prime}+\mathcal{H}\Psi^{\prime}+\left(\mathcal{H}^{2}+2\mathcal{H}^{\prime}\right)\Psi$
$\displaystyle=4\pi Ga^{2}\delta\rho c_{s}^{2}\,,$ (2.2.35)
$\displaystyle\Phi$ $\displaystyle\approx\Phi\,,$ (2.2.36)

relating the metric perturbation $\Phi$ to the perturbed matter content. For
simplicity, we have omitted the sums over each fluid $f$, running over the
matter species for a generic equation of state,
$w_{f}=\bar{p}_{f}/\bar{\rho}_{f}$, and sound speed, $c_{s,f}^{2}=\delta
p_{f}/\delta\rho_{f}$.

##### Conservation Equations

Additionally, the perturbed evolution equations for each component are derived
from the corresponding conservation relations for the energy-momentum tensor:

$\nabla_{\mu}T\indices{{}^{\mu}_{\nu}}=\partial_{\mu}T\indices{{}^{\mu}_{\nu}}+\Gamma^{\mu}_{\mu\sigma}T\indices{{}^{\sigma}_{\nu}}-\Gamma^{\sigma}_{\mu\nu}T\indices{{}^{\mu}_{\sigma}}=0\,.$
(2.2.37)

From the expression above, employing the definitions for matter perturbations
from Section 2.2.1 and the Christoffel symbols in the Newtonian gauge, Eqs.
2.2.9 and 2.2.10, we can derive the following general perturbed conservation
equations, before particularising for each fluid.

  * •

Perturbed continuity equation: Taking the $\nu=0$ component of Eq. 2.2.37 we
find:

$\delta\rho_{f}^{\prime}=-3\mathcal{H}(\delta\rho_{f}+\delta
p_{f})-(\bar{\rho}_{f}+\bar{p}_{f})(\theta_{f}-3\Phi^{\prime})\,,$ (2.2.38)

for the evolution of the density perturbation of the fluid $f$. The different
effects can be identified on the right-hand side of the equation: firstly, we
have the dilution with the background expansion; secondly, the contributions
from the local fluid flow; and lastly, a purely relativistic effect due to
perturbations to the local expansion rate itself, encoded in $\Phi$. Employing
the definition of the density contrast, $\delta\equiv\delta\rho/\bar{\rho}$,
and transforming to Fourier space:

$\delta_{f}^{\prime}+3\mathcal{H}\left(c_{s,f}^{2}-w_{f}\right)\delta_{f}=\left(1+w_{f}\right)\left(3\Phi^{\prime}-\theta_{f}\right)\,,$
(2.2.39)

recalling that $c_{s,f}^{2}=\delta p_{f}/\delta\rho_{f}$ and
$w_{f}=\bar{p}_{f}/\bar{\rho}_{f}$.

  * •

Euler equation: If, on the other hand, we take the spatial components $\nu=i$,
we obtain:

${v_{i}^{f}}^{\prime}=-\left(\mathcal{H}+\frac{\bar{p}_{f}^{\prime}}{\bar{\rho}_{f}+\bar{p}_{f}}\right)v^{f}_{i}-\frac{1}{\bar{\rho}_{f}+\bar{p}_{f}}\partial_{i}\delta
p_{f}-\partial_{i}\Psi\,.$ (2.2.40)

On the right-hand side, we have a correction to the momentum components from
the velocity flow and contributions from pressure and gravity, respectively.
Once again, using the relation $\theta=\partial_{i}v^{i}$ and taking the
derivative of Eq. 2.2.40, leads to an equivalent equation for the velocity
divergence in Fourier space:

$\theta_{f}^{\prime}+\left[\mathcal{H}\left(1-3w_{f}\right)+\frac{w_{f}^{\prime}}{1+w_{f}}\right]\theta_{f}=k^{2}\left[\frac{c_{s,f}^{2}}{1+w_{f}}\delta_{f}+\Psi\right]\,.$
(2.2.41)

The conservation equations depend on the individual characteristics of the
fluid $f$ whose evolution we wish to track. We remark below the special cases
of matter and radiation at sub-horizon scales ($k\gg\mathcal{H}$), for which
the equations greatly simplify:

  * •

Matter: Taking $p_{m}=0$ the continuity and Euler equations reduce to:

$\displaystyle\delta_{m}^{\prime}$
$\displaystyle=-\theta_{m}+3\Phi^{\prime}\,,$ (2.2.42)
$\displaystyle\theta_{m}^{\prime}$
$\displaystyle=-\mathcal{H}\theta_{m}+k^{2}\Psi\,.$ (2.2.43)

Combining derivatives of these two equations, it is possible to write a single
clustering evolution equation for the matter perturbations:

$\delta_{m}^{\prime\prime}+\mathcal{H}\delta_{m}^{\prime}=-k^{2}\Psi+3\left(\Phi^{\prime\prime}+\mathcal{H}\Phi^{\prime}\right)\,.$
(2.2.44)

The second term on the left-hand side accounts for the expansion and, for this
reason, is often called the Hubble friction. The $-k^{2}\Psi$ accounts for the
scale-dependent time-varying gravitational potential that the matter particles
feel.

  * •

Radiation: For $p_{r}=\rho_{r}/3$ and no viscosity, the continuity and Euler
equations become:

$\displaystyle\delta_{r}^{\prime}$
$\displaystyle=-\frac{4}{3}\theta_{r}+4\Phi^{\prime}\,,$ (2.2.45)
$\displaystyle\theta_{r}^{\prime}$
$\displaystyle=k^{2}\frac{\delta_{r}}{4}+k^{2}\Psi\,.$ (2.2.46)

Likewise, we can compute the second-order evolution equation for $\delta_{r}$:

$\delta_{r}^{\prime\prime}+k^{2}\frac{\delta_{r}}{3}=-k^{2}\frac{4}{3}\Psi+4\Phi^{\prime\prime}\,.$
(2.2.47)

A crucial difference from the relativistic matter case is that the radiation
perturbations are not subject to the Hubble friction but, instead, feel an
induced pressure contribution from the second term on the left-hand side. The
interplay between the effects in both these equations, Eqs. 2.2.44 and 2.2.47,
is the genesis of the acoustic oscillation in the primordial plasma, which
will be discussed in more detail in Chapter 3.

Since the equations are linear, the decomposition in Fourier modes ensures
that each plane wave follows the same equations but with a different comoving
wavenumber $k$. During the phase of linear evolution, the physical scale
$\lambda_{p}=\frac{2\pi}{k}a$ of the perturbation expands along with the
cosmic expansion. However, this linear treatment becomes inadequate if the
perturbation enters the non-linear regime. At that point, the perturbation
decouples from the Hubble expansion and starts to collapse gravitationally.

Lastly, it should be noted that this treatment of the conservation equations
supposes no additional couplings between the individual fluids. For example,
suppose interactions of dark matter with other components are allowed. In that
case, its density perturbations will evolve according to a modified version of
Eq. 2.2.44, including particular terms to account for such non-standard
interactions, e.g. for interacting dark energy models which we will introduce
in Chapter 5. The gravitational interaction of the different fluids is
implicitly accounted for by the terms proportional to the expansion
$\mathcal{H}$ and the metric potentials $\Phi$ and $\Psi$.

####  2.2.2 A Few Words on the Synchronous Gauge

The synchronous gauge is a historically significant gauge, first introduced by
Lifshitz during his groundbreaking work on cosmological perturbation theory
Lifshitz (1946). It played a vital role in the development of early CMB codes,
as will be introduced in Section 4.2. However, a drawback of the synchronous
gauge is the emergence of spurious gauge degrees of freedom, which were at the
origin of considerable confusion in the initial stages of cosmological
perturbation theory. This was one of the driving forces behind Bardeen’s
development of a gauge-invariant approach Bardeen (1980).

In the previous section, we worked on the Newtonian gauge to simplify the
analysis. However, the synchronous gauge is also useful to study the evolution
of perturbations. The transformation between both gauges can be achieved
trivially Ma and Bertschinger (1995), so it also serves as a cross-check of
results. Therefore, for completeness, in this section, we provide the
analogous perturbation equations in the synchronous gauge, in which the line
element is written as

$\mathrm{d}s^{2}=a^{2}\left(\tau\right)\left[-\mathrm{d}\tau^{2}+\left(\delta_{ij}+\mathpzc{h}_{ij}\right)\mathrm{d}x^{i}\mathrm{d}x^{j}\right]\,,$
(2.2.48)

where $\mathpzc{h}_{ij}$ represents the metric perturbation. In what follows,
we will adopt Ma $\&$ Bertschinger’s Ma and Bertschinger (1995) notation.
However, we use $\mathpzc{h}$ for the scalar metric perturbation instead of
the original $h$ to avoid confusion with other quantities. Once more, the
metric perturbation can be decomposed into scalar, vector and tensor modes:

$\mathpzc{h}_{ij}=\mathpzc{h}\delta_{ij}+\mathpzc{h}^{\parallel}_{ij}+\mathpzc{h}^{\perp}_{ij}+\mathpzc{h}^{T}_{ij}\,.$
(2.2.49)

The parallel and perpendicular components, $\mathpzc{h}^{\parallel}_{ij}$ and
$\mathpzc{h}^{\perp}_{ij}$, can still be expressed in terms of a scalar $\eta$
and a divergence-free vector $\vec{A}$,

$\displaystyle\mathpzc{h}^{\parallel}_{ij}$
$\displaystyle=\left(\partial_{i}\partial_{j}-\frac{1}{2}\delta_{ij}\nabla^{2}\right)\eta\,,$
(2.2.50) $\displaystyle\mathpzc{h}^{\perp}_{ij}$
$\displaystyle=\partial_{i}A_{j}+\partial_{j}A_{i}\,.$ (2.2.51)

Therefore, the scalar mode of the metric perturbations is fully characterised
by $\mathpzc{h}$ and $\eta$, while the vector and tensor modes $A_{i}$ and
$\mathpzc{h}^{T}_{ij}$ can be discarded for the purpose of this study. In what
follows, we write down the relevant equations in Fourier space in terms of the
Fourier transforms $\mathpzc{h}(k,\tau)$ and $\eta(k,\tau)$.

Analogously to the treatment for the Newtonian gauge, we derive the perturbed
Einstein equations:

$k^{2}\eta-\frac{1}{2}\mathcal{H}\mathpzc{h}^{\prime}=-4\pi
Ga^{2}\sum\delta\rho_{f}\,,$ (2.2.52) $k^{2}\eta^{\prime}=4\pi
Ga^{2}\sum\rho_{f}\left(1+w_{f}\right)\theta_{f}\,,$ (2.2.53)
$\mathpzc{h}^{\prime\prime}+2\mathcal{H}\mathpzc{h}^{\prime}-2k^{2}\eta=-24\pi
Ga^{2}\sum\delta p_{f}\,,$ (2.2.54)
$\mathpzc{h}^{\prime\prime}+6\eta^{\prime\prime}+2\mathcal{H}\left(\mathpzc{h}^{\prime}+6\eta^{\prime}\right)-2k^{2}\eta=0\,.$
(2.2.55)

The perturbed continuity and Euler equations for the uncoupled baryonic and
radiation fluids are derived directly from the perturbation of the
conservation relations $\nabla_{\mu}T^{\mu}_{\nu}=0$ and are given by