$\delta_{f}^{\prime}+3\mathcal{H}\left(c_{s,f}^{2}-w_{f}\right)\delta_{f}=-\left(1+w_{f}\right)\left(\frac{\mathpzc{h}^{\prime}}{2}+\theta_{f}\right)\,,$
(2.2.56)
$\theta_{f}^{\prime}+\left[\mathcal{H}\left(1-3w_{f}\right)+\frac{w_{f}^{\prime}}{1+w_{f}}\right]\theta_{f}=k^{2}\frac{c_{s,f}^{2}}{1+w_{f}}\delta_{f}\,.$
(2.2.57)

The advantages of using the synchronous gauge become evident from when
$f=\text{cdm}$. In this case, because for cold dark matter $w=p=0$, Eq.
2.2.57, implies that the fluid velocity remains zero as long as it has a
vanishing initial condition ($\theta_{\text{cdm}}(\tau_{i})=0$), removing the
gauge freedom without loss of generality. Nevertheless, this remarkable
simplification is lost by introducing a coupling between cold dark matter and
the other matter components (we will focus on the dark energy case), and the
fluid velocity becomes non-vanishing in general.

###  2.3 Inflation

The first cosmologically relevant epoch in the expansion history of the
Universe is the initial inflation period, which came to a halt at around
$a=10^{-28}$ (or equivalently about $10^{-32}$ seconds). The mechanism of
inflation was first proposed by Guth in 1980 Guth (1981) as an explanation for
the absence of magnetic monopoles and further developed by Linde Linde (1982)
and Albrecht and Steinhardt Albrecht and Steinhardt (1982). In short, it is a
phase in the very early Universe in which the comoving Hubble radius
$(aH)^{-1}$ decreased dramatically through accelerated expansion, which drives
$\Omega_{0}$ in Eq. 2.1.26 to unity, ensuring spatial flatness, which is
preserved up to the present.

Although there are some theoretical gaps and a lack of experimental proof for
this inflation mechanism, it is currently accepted in the community as it
addresses several fundamental issues. Namely, it provides a simple and
effective solution to the horizon problem concerning the origin of the
observed homogeneity and isotropy of the Universe, enabling a full causal
connection of photons in the early Universe that leads to the thermal
equilibrium needed to explain the CMB measurements. Remarkably, this setting
also predicts the quantum fluctuations needed to seed the small
inhomogeneities that grow into the cosmic structure distribution that we
observe at present.

Essentially, inflation addresses the origin of the isotropy in the
distribution of photons at large angular distances. Indeed, we observe that
all the photons emitted from the last scattering surface, i.e. when photons
began to freely traverse the Universe without interacting with other matter,
possess the same temperature up to minor variations. If inflation had not
occurred and the Universe had expanded gravitationally from an epoch dominated
by radiation, then most of these photons would have been separated by
distances greater than their comoving horizon at the time of their last
scattering, meaning that prior to their detection, they could never have been
in causal contact and thereby still exhibit the same temperature today.

The problems presented above (further discussed in Appendix A) are related to
causality and, therefore, can be effectively addressed by postulating a period
of extremely fast inflation before the hot Big Bang to generate the
homogeneity state of the Universe and the correlation of the CMB fluctuations.

The reason why this inflation solves the causality problems is that, keeping
in mind the importance of the Hubble radius during the standard Big Bang
period, this would be equivalent to considering an initial phase of the early
Universe during which the comoving Hubble radius is decreasing:

$\odv{}{t}(aH)^{-1}<0\Leftrightarrow\ddot{a}>0\,,$ (2.3.1)

given that $H\equiv\dot{a}/a$ and the notion of an accelerated expanding
period becomes evident in the last expression. The Hubble radius (a present
particle communication limit) becomes much smaller than the particle horizon
(an absolute causality and particle communication limit), then this means that
even though particles (or patches of the sky) are disconnected today, they
could still have been in causal contact early on. Of course, this is simply an
intuitive picture of how inflation can address the causality problems, and
detailed calculations of how this is achieved for particular frameworks can be
found in Baumann (2022).

A possible approach to realise inflation lies in the introduction of one or
more scalar fields known as the inflaton(s) $\varphi$, which dominate at the
earliest stages and possess a property denoted as $w_{\varphi}<-1/3$, thereby
triggering a phase of accelerated expansion in the early Universe. The entire
model’s degrees of freedom are determined by the shape of its potential
$V(\varphi)$. As the field moves along its potential, it releases significant
kinetic energy. This energy may lead to the generation of sm particles through
couplings with the inflaton, akin to Yukawa couplings, potentially giving rise
to all the particles found in the standard model, a process referred to as
reheating. For inflation to occur and persist for an adequate duration, the
inflaton field must exhibit slow-roll behaviour. To achieve this, its
potential energy must dominate over kinetic energy ($\dot{\varphi}^{3}\ll V$),
and the acceleration must be slow, expressed as as $\ddot{\varphi}\ll
3H\dot{\varphi}$ during inflation. These conditions ensure that the energy
density of the Universe remains nearly constant with $\rho_{\varphi}\simeq V$
during inflation, resulting in exponential growth of the scale factor, similar
to the $\Lambda$-dominated case. However, the balance between kinetic and
potential energy must eventually change, causing inflation to cease and
reheating to commence. When treating the inflaton as a quantum field, it
becomes possible to predict the perturbations generated by its quantum
fluctuations. These primordial quantum fluctuations, stretched to macroscopic
scales by the Universe’s expansion, constitute the initial seeds for the
formation of large-scale structures. Employing a perturbative approach, we can
anticipate the evolution of deviations $\delta\varphi$ of the field from
homogeneity. These deviations evolve as a harmonic oscillator with a time-
varying mass, enabling a quantum treatment resembling that of a quantum
oscillator. For a more detailed introduction to slow-roll inflation see
Appendix A.

The remarkable consistency with observations and the support for its role in
the early Universe has granted inflation a place as an ingredient in the
standard cosmological model, even without consensus on a particular
inflationary model.

In summary, the physical predictions of inflation include:

  * •

Negligible Curvature: Inflation requires that the spatial curvature of the
Universe be extremely close to zero. Observations of the CMB, combined with
lensing effects and bao measurements, put tight bounds on the curvature,
$\Omega_{k}=0.0007\pm 0.0019$ Aghanim et al. (2020a), in agreement with the
inflationary predictions.

  * •

Adiabatic, Gaussian, and Nearly Scale Invariant Initial Perturbations: while
the specific details may vary depending on the inflationary model,
observations of the CMB provide strong constraints on departures from these
assumptions Akrami et al. (2020a).

In Appendix A a more detailed account of the problems addressed by inflation
is provided, along with the illustrative example of a scalar field model of
slow-roll inflation.

###  2.4 A Brief History of the Universe in Four Epochs

Bringing together the information in the previous sections, we can reconstruct
the historical epochs of the Universe by tracing its expansion backwards in
time Cicoli et al. (2023). Since the Universe is expanding, the scale factor
would approach zero at earlier times. Under such circumstances, the Universe
would have been in a drastically different physical state, significantly
hotter, with its baryonic matter ionised and photons unable to propagate
freely in this primordial plasma due to the extreme density of the
environment.

Beginning with this early time, the Universe has undergone various epochs as
part of its cooling and expansion process, in a hierarchy of gradually
occuring phase transitions. We will also explore the subsequent era marked by
structure formation and followed by the more recent phase of accelerated
expansion. Nevertheless, all the processes described are continuous, meaning
there is no absolute division between each epoch. A concise overview of the
evolution of each cosmic fluid during these periods, as described in Section
2.1.7, is illustrated in Fig. 2.1, for the $\Lambda$CDM model.

####  2.4.1 Early Universe: The Hot Big Bang Epoch

Indeed, according to the Hot Big Bang model, the early Universe was a much
denser and high-temperature environment filled with elementary particles. As
we venture back in time, we encounter periods beyond the reach of particle
physics and reproduction with our present-day particle colliders. Rather than
relying on these tools, we can look for the Universe’s thermal evolution
relics, providing an unparalleled laboratory for particle physics. Assuming
conservation of entropy from thermodynamic, and as discussed briefly in
Section 2.1.4, the events in this epoch can also be pinpointed according to
the cooling down temperatures:

$T(t)=T_{0}\frac{a_{0}}{a(t)}\,.$ (2.4.1)

A more detailed overview can be found in Gorbunov and Rubakov (2011).

  * •

The Planck Epoch ($t\sim 10^{-43}\,s$ or $T\sim 10^{19}\,\text{GeV}$): At the
earliest conceivable moment after the Big Bang, the nature of the Universe was
profoundly different, dominated by extreme energy conditions
($E_{\text{Pl}}\sim 10^{19}\,\text{GeV}$ is the Planck energy). The Universe
was hot, dense and governed by quantum gravity effects. The fundamental forces
of electromagnetism, gravitation, weak nuclear interaction, and strong nuclear
interaction are thought to have been unified, although a complete theory
remains elusive. Conventional Big Bang cosmology predicts a gravitational
singularity prior to this time, reflecting our inherent ignorance on the
"origins" of the Universe. General relativity is assumed to break down at such
scales due to the quantum effects, requiring a coherent theory of quantum
gravity ori (2009), Rovelli (2004), arm (2021) or at least some quantum
compatible classical treatment.

  * •

The Grand Unification Epoch ($10^{-43}\,s\lesssim t\lesssim 10^{-36}\,s$ or
$10^{19}\,\text{GeV}\gtrsim T\gtrsim 10^{15}\,\text{GeV}$): The gravitational
force separated from the other three fundamental forces. The Universe was
still too hot and dense for particles as we know them to exist. This is the
era where Grand Unified Theories (GUTs) aim to explain unfamiliar physics. The
transition to the inflationary epoch is predicted to generate a vast number of
magnetic monopoles Preskill (1979), which are conspicuously absent in current
observations. This discrepancy was one of the problems addressed and resolved
by the introduction of the theory of cosmic inflation Guth (1981), Linde
(1982).

  * •

Inflationary Epoch ($10^{-36}\,s\lesssim t\lesssim 10^{-32}\,s$ or
$10^{15}\,\text{GeV}\gtrsim T\gtrsim 10^{13}\,\text{GeV}$): Described in
Section 2.3 and in Appendix A, during this brief moment the Universe underwent
rapid exponential expansion, smoothing itself out. This explains the observed
large-scale uniformity of the CMB radiation. Cosmologists generally believe
that inflation happened and could be explained through a scalar field slowly
rolling down a potential, called the inflaton. Inflation led to a vast
thinning of the elementary particles, which were regenerated through the
symmetry breaking as the inflaton field dropped to the minimum of its
potential energy, refilling the Universe with a hot and dense plasma of
elementary particles in a process called reheating.

  * •

The Electroweak Epoch ($10^{-32}\,s\lesssim t\lesssim 10^{-12}\,s$ or
$10^{13}\,\text{GeV}\gtrsim T\gtrsim 10^{3}\,\text{GeV}$): The Universe cooled
to approximately $10^{15}\,\text{GeV}$ and the strong nuclear force separated
from the electroweak force, leading to distinct electromagnetic and weak
nuclear forces. The Higgs field also gave particles mass in this period Aad et
al. (2012). Around this period, a process known as baryogenesis occurred
($T\sim 10^{12}\text{GeV}$), accounting for the observed predominance of
matter over anti-matter in the present-day Universe.

  * •

Quark Epoch ($10^{-12}\,s\lesssim t\lesssim 10^{-6}\,s$ or
$10^{3}\,\text{GeV}\gtrsim T\gtrsim 1\,\text{GeV}$): The Universe was still
too hot for quarks to combine into protons and neutrons. It was filled with a
quark-gluon plasma, a hot soup of quarks and gluons.

####  2.4.2 Radiation Dominated Epoch

The rde is the period extending from $10^{-10}\,s\lesssim t\lesssim
50\,\text{kyrs}$ and encompasses significant sub-phases, each characterised by
cooling temperatures, ultimately reaching an energy level of approximately
$10^{2}\,\text{GeV}$. The RDE is when light elements form. Since radiation
decays faster with the expansion than non-relativistic matter, this epoch will
eventually cease and give way to a matter-dominated regime. The radiation-
dominated epoch is depicted in grey in Fig. 2.1.

  * •

Electroweak Symmetry Breaking ($t\lesssim 10^{-6}\,s$ or $1\,\text{GeV}\gtrsim
T$): For temperature around $100\,\text{GeV}$ the forces have solidified into
their current forms, and particles acquire masses through the Higgs mechanism.

  * •

Neutrino Decoupling and Hadron Epoch ($10^{-6}\,s\lesssim t\lesssim 1\,s$ or
$1\,\text{GeV}\gtrsim T\gtrsim 1\,\text{MeV}$): The temperature dropped enough
for quarks to bound to create hadron and anti-hadron pairs which annihilate as
the Universe cools, leaving a residue of protons and neutrons. Dark matter is
expected to decouple from the primordial plasma by the end of the hadron epoch
($t\sim 10^{-5}\,\text{s}$ or $T\sim 200\,\text{MeV}$ for WIMPs), while
neutrinos start to decouple around this time ($t\sim 0.2\,\text{s}$ or $T\sim
1\,\text{MeV}$).

  * •

Lepton Epoch ($1\,s<t<200\,s$ or $1\,\text{MeV}\gtrsim T\gtrsim
0.1\,\text{MeV}$): the temperatures are still high enough for positron-
electron pairs to form. For $T\approx m_{e}\sim 0.5\,\text{MeV}$, the photons
are no longer energetic enough to keep this equilibrium and the annihilation
process results in energy transfer to the thermal bath.

  * •

bbn Epoch ($200\,s\lesssim t\lesssim 300\,\text{s}$ or $0.1\,\text{MeV}\gtrsim
T\gtrsim 0.05\,\text{MeV}$): nuclear fusion starts and protons and neutrons
combine into atomic nuclei. Free neutrons start fusing with protons at
$T\approx 0.1\,\text{MeV}$. Around three minutes after the Big Bang, the light
elements were formed, starting with deuterium, which rapidly transformed into
${}^{4}\text{He}$. As the Universe’s temperature and density quickly decrease
to the point where nuclear fusion is no longer possible, we enter the mde. All
the neutrons have been incorporated into helium nuclei, leaving a mass ratio
of roughly three times more hydrogen than ${}^{4}\text{He}$, with only trace
quantities of other nuclei.

  * •

Photon Epoch ($300\,\text{s}\lesssim t\lesssim t_{\text{eq}}$ or
$0.05\,\text{MeV}\gtrsim T\gtrsim 1\,\text{eV}$): For the rest of the
radiation epoch, the Universe consisted of a dense hot plasma filled with
nuclei, electrons, and photons. This epoch extends through the initial stages
of the mde when $t>t_{\text{eq}}$.

##### Matter-Radiation Equality

At some point, the matter density dominates over radiation, which dilutes
faster. In particular, the RDE ended when the densities equalled:

$\frac{\rho_{m}}{\rho_{r}}=\frac{\rho_{m,0}}{\rho_{r,0}}\frac{a^{4}}{a^{3}}=\frac{\rho_{m,0}}{\rho_{r,0}}(1+z_{\text{eq}})^{-1}=1\,,$
(2.4.2)

and thus, the redshift at the equivalence between matter and radiation was
$z_{\text{eq}}\sim 3400$ or $t_{\text{eq}}\sim 50\,000$ years (for fiducial
values as measured by Planck Aghanim et al. (2020a)). This is represented by
the line dividing the grey and red sectors in Fig. 2.1.

####  2.4.3 Matter Dominated Epoch

At the beginning of the MDE, the Universe was still dominated by photons.
Although neutral atoms had not formed yet, the Universe was becoming
increasingly transparent. The MDE is depicted in red in Fig. 2.1.

Shortly after equality, the Universe’s temperature dropped to approximately
$1\,\text{eV}$, equalling the binding energy of the hydrogen atom. For the
first time since the Big Bang, the Universe started emancipation from the
dense primordial plasma of photons, electrons, and protons. Nevertheless, it
remained virtually opaque to electromagnetic radiation since photons could
travel only short distances before colliding with a charged particle through
Compton scattering. In this epoch it becomes relevant to also keep track of
time in terms of the redshift, which for matter domination, evolves
approximately as:

$z-z_{0}\approx(t-t_{0})^{-2/3}\,.$ (2.4.3)

  * •

Recombination and Decoupling Epoch ($18\,\text{kyrs}\lesssim t\lesssim
380\,\text{kyrs}$ or $6000\gtrsim z\gtrsim 1100$): As the Universe continued
to expand and cool down, the formation of neutral hydrogen became
energetically favoured and ionised hydrogen and helium atoms captured the free
electrons. Shortly after this transitional phase, Compton scattering became
inefficient, and photons decoupled from matter, leading to the Cosmic
Microwave Background Radiation we can detect today. This event defines the
last scattering surface. Until this point, photons and baryons had been
tightly coupled, oscillating in acoustic waves generated by the
compression/rarefaction cycle due to the balance between the baryons’
gravitational attraction and the photons’ radiation pressure. Once the two
species decouple, the baryons are effectively released from the drag of the
photons. This decoupling marks the end of the drag epoch and allowed photons
to travel unimpeded through the Universe without interacting with matter,
marking the earliest epoch observable today - the last scattering surface. On
the other hand, the baryons are frozen in the acoustic wave pattern at a scale
determined by the size of the horizon at the drag epoch. It should be
emphasised that photon decoupling and recombination are distinct events,
albeit closely related. The photons present at the time of decoupling have
since freely streamed through the Universe and are the same ones that we
observe today in the CMB radiation, albeit significantly cooled by the
expansion. By the time this period ended, the Universe’s composition had
shifted to a dark fog containing roughly $75\%$ of hydrogen and $25\%$ of
helium, with only small amounts of lithium.

  * •

Dark Ages ($380\,\text{kyrs}<t<150\,\text{Myrs}$ or $1100\gtrsim z\gtrsim
60$): With no stars, the Universe was in a dark state, as photons were not
effectively being produced yet and hence there was no visible light. Matter
consisted of a gas of atoms that kept being dragged into denser regions by the
pull of gravity alone, but the first stars had not yet ignited. Slowly, the
gravitational attraction pulled matter together, and overdensities started
growing into clumps as dark matter started to dominate the total energy
density of the Universe.

  * •

Cosmic Dawn ($150\,\text{Myrs}<t<200\,\text{Myrs}$ or $60\gtrsim z\gtrsim
20$): As matter started to clump subjected to gravity’s rule, the Jeans length
$\lambda_{J}=\sqrt{\frac{\pi c_{s}^{2}}{10G\rho_{0}}}$, which determines the
smallest structures that can form, began to decrease and the density
perturbations started to grow in amplitude. Cold dark matter became the
dominating component, setting the stage for gravitational collapse to amplify
the slight inhomogeneities left by cosmic inflation. This process made dense
areas denser and sparse regions more rarefied. Once the overdensities overcome
the Jeans mass dictated by the Jeans length, they begin to spontaneously
collapse and the first stars form

  * •

Star Formation and Reionisation ($200\,\text{Myrs}<t<1\,\text{Gyrs}$ or
$20\gtrsim z\gtrsim 6$): The process of formation of the first starts heated
the surrounding medium, once again ionising the hydrogen in the majority of
the Universe. This moment is known as reionisation and happened approximately
one billion years after the Big Bang, marking the last phase transition in the
Universe and the end of the dark ages since the star-light across much of the
electromagnetic spectrum was finally able to travel unimpeded through the
cosmos, eventually revealing the Universe as we see it today.

  * •

Large Scale Structure Formation Epoch ($1\,\text{Gyrs}<t<5\,\text{Gyrs}$ or
$6\gtrsim z\gtrsim 1$): Subsequently, stars start assembling into galaxies,
galaxy clusters, super-clusters and all the large-scale structures. The
process of structure formation is dictated by non-linear gravitational
effects, which thread the galactic cosmic web, leaving behind large empty
regions called cosmic voids. This epoch extends to the present day, even after
dark energy took over the evolution at the dark energy-matter equality.

##### Dark Energy-Matter Equality

Likewise, we can estimate the redshift value at which the transition from a
mde, decelerating expansion during which large-scale structures form, to a new
regime of accelerating expansion occurs. Neglecting radiation and for $K=0$ we
have that $\ddot{a}=0$ when

$\frac{\rho_{m}}{2\rho_{\Lambda}}=\frac{\rho_{m,0}}{2\rho_{\Lambda,0}}\frac{1}{a^{3}}=\frac{\Omega_{m,0}}{2\Omega_{\Lambda,0}}(1+z_{\text{acc}})^{3}=1\,,$
(2.4.4)

which corresponds to a redshift $z_{\text{acc}}\sim 0.7$, assuming fiducial
Planck values for $\Lambda$CDM. This equality epoch is depicted as the line
dividing the red and blue regions in Fig. 2.1.

####  2.4.4 Dark Energy Dominated Epoch

In the $\Lambda$CDM model, due to the cosmological constant’s premise of non-
dilution with the expansion, $\Lambda$ will eventually take over the
Universe’s evolution from matter, whose density decays with $a^{-3}$, giving
rise to the current dede. The Universe will keep expanding faster and faster,
leading to a potential Big Freeze, where stars burn out and the Universe
becomes dark and cold.

  * •

Late Universe ($t\gtrsim 5\,\text{Gyrs}$ or $z\lesssim 1$): The Sun forms in
the Milky Ways and the Solar System emerges in an orchestrated dance around
$t\sim 9\,\text{Gyrs}$. Around the $10\,\text{Gyrs}$ mark, the primordial
forms of life on Earth emerge, while our modern human ancestors came to be
only about $20\,0000$ years ago.

  * •

Present time ($t\sim 13.8\,\text{Gyrs}$ and $z=0$): After billions of years of
cosmic evolution, the Universe is as we observe it today, filled with
galaxies, stars, and a variety of complex structures. We start studying the
vastness of the cosmos, which now spans a radius of $46.5$ billion light-years
$\sim$ $14$ Gpc $\sim$ $10^{26}$ m, larger than the naive guess from the age
of the Universe since the light is carried along with the expansion of the
Universe.

###  2.5 The Theoretical Problems with $\Lambda$

Throughout its twenty-year lifespan, the $\Lambda$CDM model described above
has successfully withstood numerous tests under its inherent simplicity. Based
on our current understanding of fundamental physics, this framework provides a
detailed account of most of the Universe’s expansion history with only 6 free
parameters to be fixed by the observational data. More intricate models rarely
manage to achieve statistical preference from the data, as any extra model
parameters must provide a sufficiently better fit to the data to justify their
inclusion (see Section 4.1). However, the cosmological constant hypothesis
still faces longstanding challenges, from motivation in quantum field theory
to its role in an expanding cosmology. Moreover, more recently, the
observational tensions in the parameters of $\Lambda$CDM, discussed in detail
in Section 3.6, have granted extended models an extra layer of encouragement
and support.

For the moment, and for what concerns the core of the standard model, we will
discuss the most prominent challenges: the cosmological constant and the
cosmic coincidence problems.

####  2.5.1 The Cosmological Constant Fine Tuning Problem

In order to realise the cosmic acceleration today, according to Eq. 2.1.29, we
require that the cosmological constant $\Lambda$ is of the order of the square
of the present Hubble parameter $H_{0}$, whose value was estimated in Eq.
2.1.7. This corresponds to the following approximate value for the energy
density (with $\text{M}_{\text{Pl}}\approx 10^{19}\,\text{GeV}$):

$\rho_{\Lambda}\approx 10^{-47}\,\text{GeV}^{4}\approx
10^{-123}\,\text{M}_{\text{Pl}}^{4}\,.$ (2.5.1)

This is incompatible with the vacuum energy scales predicted for qft, as we
will illustrate next. This issue was identified long before the discovery of
the late-time acceleration or the cosmological constant proposal. At that
time, the prevailing belief was that the cosmological constant was precisely
zero, and much effort was devoted to explaining why this should be the case.
The vanishing of a constant in physics often hints at some underlying
symmetry.

Initially referred to as the vacuum catastrophe Adler et al. (1995), the issue
lies in the qft predictions of an inexplicably large value for the vacuum
energy, arising from the sum of the zero-point energies of each degree of
freedom in all quantum fields within the Universe, up to a particular cut-off
scale. Such a large value should manifest itself via significant gravitational
effects which have never been observed. Observational constraints on the upper
limit of this vacuum energy completely contradict the QFT result, with
differences reported up to an outrageous level of $120$ orders of magnitude
Martin (2012). For example, the vacuum energy estimated for the extreme cut-
off scales of the order of $\text{M}_{\text{Pl}}$ or of the scales predicted
in particle physics (such as $\approx 0.1\,\text{GeV}$ for quantum
chromodynamics (QCD)), the vacuum energy density can be estimated in limit
values as:

$\rho_{\text{vac}}^{\text{M}_{\text{Pl}}}\approx
10^{74}\,\text{GeV}^{4}\quad\text{and}\quad\rho_{\text{vac}}^{QCD}\approx
10^{-3}\,\text{GeV}^{4}\,.$ (2.5.2)

This shows how, even while selecting a different cut-off scale for the
cumulative zero-point energy may somewhat alleviate this problem, reducing the
discrepancy to around forty or sixty orders of magnitude Bernardo et al.
(2023), such a solution remains highly insufficient. The observed value for
$\Lambda$ is much smaller than all particle physics scales, endowing this
problem with a fundamental character. The ultimate resolution of the
cosmological constant problem remains uncertain, and whether it will emerge
from QFT or cosmological arguments is unclear. The former could reflect our
incomplete understanding of how to estimate the zero-point energy of quantum
fields in curved spacetimes, while the latter could introduce some bare
cosmological constant to cancel out the contributions from all fields except
the observed cosmological constant. Another alternative is to stick to the
pattern of explaining physical mysteries by postulating a scalar field (like
the Higgs field Aad et al. (2012) or the inflaton Dimopoulos (2020)). In this
context, a potential approach, as we will further dissect, is to discard the
cosmological constant altogether and promote it to a scalar field (or some
similar effect from modified gravity) to account for the late-time
acceleration Copeland et al. (2006), Tsujikawa (2013).

####  2.5.2 The Cosmic Coincidence Problem

The coincidence problem embodies the improbable fact that the observed values
of the dark matter and dark energy densities are of the same order of
magnitude today, given that these species evolve at entirely different rates
throughout the history of the Universe, as discussed in Section 2.4. This
issue is generally less consensual than the cosmological constant problem
Velten et al. (2014), often attributed to more philosophical or metaphysical
considerations, out of the scope of cosmological physical theories cha (2017).
This debate reflects the lack of consensus in the community over the relevance
of anthropic arguments Carter (1974). In essence, the anthropic principle
suggests that the values of physical quantities in the current state of the
Universe are what they are because such conditions are necessary for the
existence of observers in our condition, and it is not up to us, inevitable
inhabitants of the Universe, to decide on the statistical likelihood of the
present design of the Universe cha (2017). This doctrine can be endorsed with
the idea that such statistics implied by the terms coincidence or likelihood
do not make sense, to begin with, since cosmic variance implies that we have
only one realisation of the Universe at our disposal Smeenk and Ellis (2017).

However, the anthropic principle does not easily align with the cosmological
principle, on whose pillars observational cosmology stands, and which implies
that we do not occupy any preferential position in the Universe. Therefore, if
we accept this incompatibility, the coincidence problem can serve as
additional motivation to dismiss the cosmological constant and devise
alternative dark energy models. In particular, dynamical dark energy models,
where the energy density of dark energy varies over time or there is some
interaction in the dark sector, stand out due to their ability to explain the
seemingly coincidental similarities between the matter and dark energy
densities simply through their natural dynamical character. Further discussion
on this type of models will be the focus of Chapter 5.

##  Chapter 3 Cosmological Observational Tests

> Estou me interessando terrivelmente por fatos: fatos são pedras duras.
> Não há como fugir. Fatos são palavras ditas pelo mundo.00footnotetext: I’m
> becoming terribly interested in facts: facts are hard rocks. You can’t
> escape. Facts are words spoken by the world. — Clarice Lispector in The Hour
> of the Star
> — Clarice Lispector A Hora da Estrela

Cosmology differs fundamentally from most branches of physics by being an
observational science rather than an experimental one. Given that the early
Universe events played out without an audience (as much as we are aware!), and
the dawn and subsequent evolution of the Universe cannot be replicated (we
have only one sample!), our understanding is primarily based on measuring the
spatial correlations between cosmological structures at late times. The
fundamental endeavour of modern cosmology is no longer to map the night sky
into constellations that track the seasons but rather to formulate a coherent
narrative for the history of the Universe that provides motivated explanations
and is compatible with these observed correlations. In this sense, cosmology
is a wide-ranged effort to reconcile various observational data with
predictions from theoretical models based on coherent physical principles.

The study of high-redshift Type Ia supernovae (SNe) provided the first
compelling evidence for the late-time accelerated expansion of the Universe
Riess et al. (1998), Perlmutter et al. (1999). These supernovae, which serve
as standardisable candles based on the fixed luminosity at which they occur,
appeared fainter than expected in a spatially flat, matter-dominated Universe.
Subsequent observations from other probes further supported this picture, with
the additional assumptions that make up the $\Lambda$CDM model described in
Chapter 2.

After the success of previous seminal probes and ground-based experiments
Durrer (2020), the Planck satellite has given the most precise measurement of
the anisotropies in the Cosmic Microwave Background (CMB) Aghanim et al.
(2020b). This is one of the most valuable and comprehensive sources of
cosmological information from the initial perturbations that led to the
formation of cosmic structures via gravitational collapse. The CMB has also
been instrumental in constraining cosmological parameters within the standard
cosmological model and its extensions. Simultaneously, galaxy surveys such as
the Sloan Digital Sky Survey Tegmark et al. (2004) have extensively mapped the
distribution of the Large Scale Structure across the Universe, identifying the
distinct pattern known as the Baryon Acoustic Oscillation (BAO) scales within
the matter correlation function. By leveraging these findings, a standard
ruler is established, whose length can be quantified across various
cosmological epochs, enabling scientists to monitor and trace the expansion of
the Universe.

The precision and compatibility level between these observations have put the
standard cosmological model under increasingly rigorous scrutiny. Verifying
the standard model’s hypotheses through independent means solidifies its
validity, and observational support can be established from multiple sources.
Upcoming missions like the space-based interferometer LISA Belgacem et al.
(2019) and the Euclid satellite mission Amendola et al. (2013) will offer
highly detailed maps of the cosmic microwave background and large-scale
structures. These missions promise even finer measurements of the BAO scales
across different cosmological epochs. At the same time, additional insight
will be extracted from phenomena such as supernovae, both weak and strong
gravitational lensing, galaxy clustering, and gravitational wave events, among
others. These advancements will enrich our understanding of the large-scale
homogeneity and any deviations therein, providing extended tests for the
validity of general relativity.

This chapter is structured as follows. In Section 3.1 we introduce the matter
correlation function, needed to define the matter power spectrum in Section
3.2. These two introductory sections are based on the Chapter 3 of Amendola
and Tsujikawa (2015). Section 3.3 provides a description of the main
properties of the cosmic microwave background and how to relate its features
to the cosmological framework. In Section 3.4 we report on other relevant
probes of the physics of the Universe, in particular for dark energy
signatures. Section 3.5 provides a complete account of the impact of the six-
parameters of the $\Lambda$CDM model on the matter power spectrum and the CMB
spectrum of temperature anisotropies. Finally, in Section 3.6 we list the
cosmological tensions and anomalies that have brought the standard model to a
crisis and which motivate the exploration of alternative models.

###  3.1 The Matter Correlation Function

As elaborated in Chapter 2, the Universe’s large-scale structure is not
randomly distributed. Instead, cosmic objects are arranged into a tangled
cosmic web with galaxies anchored along dense filaments and clustering at
their intersections, surrounded by vast empty regions known as cosmic voids.
The mapping of these structures encodes information about the initial state of
the hot Big Bang when the small overdensities that grew into clusters are
thought to have formed and the subsequent governing physical processes.
Different cosmological models predict different patterns of structure
formation, allowing us to infer the best-fitting values for specific
cosmological parameters from the statistical characteristics of the Universe’s
large-scale structure.

We rely on indispensable statistics tools to assess the validity of any
theoretical model and its connection to the real world. Because we have only
one realisation of the observable Universe, spanning a wide range of physical
scales, the statistical properties of the distribution of events in the sky
are as vital to cosmologists as flour to a baker. Statistics methods in
cosmology are generally categorised into two main branches: descriptive
statistics, which aims to cast the data into a compact and physically
understandable way, and estimation or inferential statistics, deals with the
problem of extracting information about the model and its parameters. This
chapter will focus on the former and how observational insight can be derived
from the background and perturbation theory introduced in Chapter 2. The
latter will be deferred to Chapter 4.

The initial step is to consider a random particle distribution, which may
correspond to various astrophysical objects such as galaxies or quasars. For
$N$ points contained within a volume $V$, the primary statistical descriptor
is the average numerical density $\rho_{0}=N/V$. However, this is insufficient
to distinguish between, for example, $N$ points clustered in one location and
$N$ points uniformly scattered throughout the volume $V$. If, instead, we
consider an infinitesimal volume $\odif{V}$ randomly selected within $V$, then
the term $\rho_{0}\odif{V}$ represents the mean number of particles in this
infinitesimal volume. If $\odif{N}_{ab}=\langle n_{a}n_{b}\rangle$ is the
average number of pairs in the infinitesimal volumes $\odif{V}_{a}$ and
$\odif{V}_{b}$, separated by a distance $r_{ab}$, the next significant
statistical descriptor is the two-point correlation function $\xi(r_{ab})$,
defined as:

$\odif{N}_{ab}=\langle
n_{a}n_{b}\rangle=\rho_{0}^{2}\odif{V}_{a}\odif{V}_{b}\left[1+\xi(r_{ab})\right]\,,\quad\text{with}\quad
r_{ab}>0\,.$ (3.1.1)

In our preceding discussion, the implicit assumption is that $r_{ab}>0$,
meaning the two infinitesimal volumes do not overlap. In this context, the
notion of average, denoted by $\langle\cdot\rangle$, conveys the volume or
sample average111Although we have previously introduced this notation as the
average over random initial conditions (like the ones generated from
inflation), the large-scale cosmic structure we observe in practice is
essentially one realisation of this random process. To reconcile this with
theoretical models, we rely on the principle of ergodicity, according to
which, as volume approaches infinity, ensemble averages converge to spatial
averages. In this context, several different patches of the Universe with
various sizes can be considered separate realisations of the same random
process. Thus, averaging measurements over a large volume mimics averaging
over various realisations of the Universe. However, our observations are
inherently limited to a finite volume, introducing what is known as sample
variance. In the case of the CMB, where this finite volume is the single whole
observable Universe, this limitation is appropriately called cosmic variance.
and considers pairs at different points within a single realisation, separated
by the same distance $r_{ab}$. If the points are sufficiently far apart to be
uncorrelated, the results would be similar to taking an average over many
realisations of the distribution. Despite the limitation of being confined to
observations from a single Universe, the correlation function remains a
practical and meaningful descriptor. For a completely random distribution of
$N$ particles, $\odif{N}_{ab}$ should be independent of specific locations,
leading to a vanishing correlation function $\xi$. In contrast, a non-
vanishing $\xi$ indicates spatial correlation among the particles. The
correlation function can then be expressed as a spatial average of the product
of density contrasts at two distinct points:

$\xi(r_{ab})=\frac{\odif{N}_{ab}}{\rho_{0}^{2}\odif{V}_{a}\odif{V}_{b}}-1=\langle\delta(r_{a})\delta(r_{b})\rangle\quad\text{with}\quad\delta(r_{a})=n_{a}/(\rho_{0}\odif{V}_{a})-1\,,$
(3.1.2)

where we used the fact that
$\langle\delta(r_{a})\rangle=\langle\delta(r_{b})\rangle=0$. For a sample
average, $\xi$ can be computed over all possible positions as:

$\xi(\vec{r})=\frac{1}{V}\int\delta(\vec{y})\delta(\vec{y}+\vec{r})\odif{V}_{y}\,.$
(3.1.3)

In practical terms, calculating the correlation function is often simplified
by selecting a volume $\odif{V}_{a}$ such that $\rho_{0}\odif{V}_{a}=1$ is the
average particle at a distance $r$ from a test particle. In that case, the
count of particle pairs reduces to the number of particles contained within
the volume $\odif{V}_{b}$:

$\odif{N}_{b}=\rho_{0}\odif{V}_{b}[1+\xi(r_{b})]\,.$ (3.1.4)

Thus, the correlation function is typically computed as:

$\xi(r)=\frac{\odif{N}(r)}{\rho_{0}\odif{V}}-1=\frac{\langle\rho_{c}\rangle}{\rho_{0}}-1\,,$
(3.1.5)

yielding the average number of particles at a distance $r$ from any given
particle, normalised by the expected number in a uniformly distributed system.
This is sometimes termed the conditional density contrast. Moreover,
considering a bounded volume with $N=\rho_{0}V$ particles, imposes an integral
constraint on $\xi(r)$:

$\int\xi(r)\odif{V}=\frac{1}{\rho_{0}}\int\frac{\odif{N}}{\odif{V}}\odif{V}-V=\frac{N}{\rho_{0}}-V=0\,.$
(3.1.6)

A positive correlation $\xi(r)$ signals more particles than expected in a
uniform distribution. In such instances, the distribution is termed as
positively clustered. Generally, one is more concerned with the dependency on
the magnitude $r$, leading to the selection of a shell of distance $r$
surrounding each particle.

As introduced in Chapter 2, the decomposition in Fourier modes of perturbation
variables is useful for practical purposes because these modes evolve
independently at the linear level. Given that the average of a perturbation
variable is, by definition, zero, the next statistically meaning quantity is
the average of the pairs. When expressed in Fourier space, ($k=2\pi/\lambda$)
these yield the power spectrum:

$P_{\delta}(k)=V|\delta_{k}|^{2}\,,$ (3.1.7)

where $\delta_{k}$ are the Fourier coefficients corresponding to the density
contrast, and $V$ is the power spectrum’s normalisation factor. In cases where
the quadratic form involves two distinct variables, such as
$|\delta_{k}\theta_{k}|$, $P(k)$ is said to be a cross-correlation power
spectrum.

The power spectrum is the most widely used measure for describing clustering
in linear and mildly non-linear regimes and is one of the most important
observables in the field. It serves as the Fourier-space counterpart of the
correlation function, expressed as:

$P(\vec{k})=\frac{1}{V}\int\xi(\vec{r})\mathup{e}^{-i\vec{k}\cdot\vec{r}}\mathrm{d}V\quad\text{and}\quad\xi(\vec{r})=\frac{V}{(2\pi)^{3}}\int
P(\vec{k})\mathup{e}^{i\vec{k}\cdot\vec{r}}\mathrm{d}k\,.$ (3.1.8)

The Fourier volume factors of $1/V$ and $V$ in $P(\vec{k})$ and
$\xi(\vec{r})$, respectively, are conventionally added to the Fourier
transformation to guarantee that both functions have the same dimensions. When
assuming spatial isotropy, that is, when the correlation function depends
solely on the magnitude $r=|\vec{r}|$, the spectrum likewise depends only on
$k=|\vec{k}|$.

In the following sections, we will see in more detail how the power spectrum
decomposition can be used to study density fluctuations in the matter
distributions.

###  3.2 The Matter Power Spectrum

As mentioned in Section 2.3, the theory of inflation is currently the leading
explanation for the first generating moments of the Universe through a rapidly
expanding phase in its early stages. During this period, the initial
anisotropies are believed to have been generated through the quantum
fluctuations of a scalar field called the inflaton. This exponential expansion
is essential as it amplifies the density perturbations responsible for the
observed anisotropies in the CMB and the formation of large-scale structures
such as galaxies and galaxy clusters in the Universe. Therefore, it is helpful
to focus on what is happening at different scales to study the patterns in
this matter distribution resulting from inflation. This is achieved exactly
through the power spectrum decomposition of the density perturbations. The
density contrast $\delta(k)$, which characterises the deviation of the density
of a Fourier scale $k$ from the background density, is the Fourier counterpart
of Eq. 2.2.22.

The large-scale structure in the Universe is not distributed randomly but has
interesting correlations between spatially separated points. By definition,
the mean value of the density perturbations is zero, $\langle\delta\rangle=0$.
Therefore, the first non-trivial statistical measure of the density field (at
a fixed time) is exactly the two-point correlation function introduced in
Section 3.1.

Treating these density contrasts as variations arising from primordial quantum
fluctuations, we can now formally write the power spectrum $P(k)$ in $k$-space
originating from the matter distribution seeds as follows:

$\langle\delta(\vec{k})\delta^{*}(\vec{k^{\prime}})\rangle=\frac{2\pi^{2}}{k^{3}}P(k)\delta^{3}(\vec{k}-\vec{k^{\prime}})\,,$
(3.2.1)

now following the convention of defining the power spectrum with the
normalisation factor $2\pi^{2}/k^{3}$, rendering $P(k)$ dimensionless. The
modes are uncorrelated via the Dirac delta function $\delta^{3}(\cdot)$. When
considering scalar perturbations, we can derive the matter power spectrum for
both the pre-and post-radiation-matter equality epochs, approximately
resulting in Baumann (2022):

$P(k)\propto\begin{cases}k^{n_{s}},\ \ k<k_{\text{eq}}\,,\\\ k^{n_{s}-4},\ \
k>k_{\text{eq}}\,,\end{cases}$ (3.2.2)

where $n_{s}$ is the spectral index of the power spectrum produced during
inflation and $k_{\text{eq}}=a_{\text{eq}}H(a_{\text{eq}})$ is the scale
parametrising the distinction between large and small scales, corresponding to
the mode that entered the Hubble radius exactly at the radiation–matter
equality. On large scales, $P(k)$ grows for increasing $k$, while it is a
decreasing function of $k$ on small scales. This implies a peak in the power
spectrum, directly related to the scale $k_{\text{eq}}$, which is sensitive to
the background and perturbed expansion history. The shape of the power
spectrum changes with parameter variations is depicted for the six
$\Lambda$CDM parameters in Fig. 3.7. Furthermore, there is a subtle
oscillatory signature in the tail of the spectrum at small scales (large $k$),
which results from the baryonic acoustic oscillations (BAO) imprinted by
cosmic evolution.

The comparison between the theoretical power spectrum of dark matter and the
observed power spectrum of the galaxy distribution poses a significant
challenge. Assessing how these two spectra should be related is not trivial,
an issue commonly referred to as biasing. It is commonly assumed that both
spectra differ only by a constant factor, based on the premise that, on
sufficiently large scales, both have their evolution dominated by the
gravitational effects. This hypothesis is not necessarily universally valid as
it can vary depending on the details of galaxy formation and evolution, and is
assessed ultimately through N-body simulations and observations Desjacques et
al. (2018).

This chapter introduces the observational probes relevant to this
dissertation, categorises them and discusses the cosmological information they
offer. In particular, we focus on characteristic signatures and effects that
can be decisive in distinguishing between different models, considering the
underlying assumptions and the potential for model-independent analysis. While
this discussion provides a tailored overview, more comprehensive information
is available in recent reviews such as Moresco et al. (2022) and the sources
cited within each subsection.

###  3.3 The Cosmic Microwave Background

More than any other cosmological probe, observations of the Cosmic Microwave
Background (CMB) spectrum, mapping both intensity and polarisation, have
converted cosmology into a precision science, providing a comprehensive
understanding of the Universe’s geometry, evolution and composition.
Measurements of the temperature anisotropies over the last three decades have
played a crucial role in establishing the standard cosmological model, as
described in Chapter 2. These observations have confirmed the existence of
primordial fluctuations, which are the seeds of structure formation consistent
with the inflationary period. The power of the CMB as a cosmological probe
lies in the fact that it captures these fluctuations at an early stage when
they were still small and aptly described using linear perturbation theory,
unlike the complex and non-linear structure formation process. The evolution
of small fluctuations in the primordial plasma can be understood from
fundamental physics enclosed in well-defined equations, allowing for precise
theoretical predictions of the expected temperature anisotropies’ distribution
in the CMB.

As discussed in Section 2.4, the pivotal moment in which light started to
propagate freely and the Universe became transparent to light occurred at
redshift $z\approx 1090$, when the Universe was around $380000$ years old and
the effective temperature of the free streaming photons was $\sim
3000\,\text{K}$, is termed photon decoupling. Since then, the expansion of the
Universe has redshifted and diluted these photons, according to the relation
$T\propto(1+z)$ discussed in Section 2.1.4. They are detected today as the CMB
radiation, with a characteristic average temperature of $T_{0}\approx
2.725\,\text{K}$. These photons carry imprints of their cosmic journey across
the expansion history, from the early Universe until today, including the
influence of gravitational wells near massive clusters. This information is
encoded at different scales as deviations to this average temperature
distribution. Because these spatial fluctuations break the isotropy of the
CMB, they are referred to as the CMB anisotropies. Being one of the most
robust and complete cosmological probes, numerous experiments have been
conducted to measure the temperature variations in the CMB radiation.
Additionally, the anisotropies can be modelled for different theories
rigorously and systematically using modern Einstein-Boltzmann codes, which
will be the focus of Section 4.2. These solve the Boltzmann equation, which
expresses how the distribution function of some particle species evolves,
assuming Gaussian random fields.

Fig. 1.1 presents the map of the CMB anisotropies as measured by the Planck
collaboration. This is a captivating image of the so-called last scattering
surface of the Universe, with the colour scale encoding variations in the
temperature of the CMB photons, which, in turn, correspond to density
fluctuations at the time of photon decoupling. The study of the CMB
temperature fluctuations distribution follows precise statistical techniques
to find the correlations between hot and cold spots as a function of their
angular separation. This results in an angular power spectrum decomposed in
multipoles $\ell$, as depicted in Fig. 3.1. The figure also includes the
theoretical prediction of the CMB spectrum in the $\Lambda$CDM model when
fitted to the observed data. The striking agreement between the theoretical
prediction and the observed data highlights the remarkable success of the
standard cosmological model. For a more detailed overview of the physics of
the CMB, we refer the reader to Ref. Dodelson (2003), Durrer (2020).

(a) Figure 3.1:  Figure of the power spectrum of the CMB anisotropies
(temperature). The measured spectrum
$\mathcal{D}_{\ell}^{TT}=\ell(\ell+1)C_{\ell}^{TT}/(2\pi)$ has been scaled to
make the smaller scale features visible. The large-scale data ($\ell>30$) is
binned up by multipole intervals. The blue curve is the $\Lambda$CDM
prediction for the Planck 2018 bestfit values, produced with the CLASS code
Lesgourgues and Tram (2011), Blas et al. (2011). The black markers represent
the Planck data Aghanim et al. (2020a) and corresponding error bars. The data
is collected from the Planck legacy archive http://pla.esac.esa.int/pla/.

####  3.3.1 The TT-Spectrum: Measuring the Anisotropies

As observers, we are limited to a realisation of the CMB map at a fixed point
in space and time ($\vec{x}=\vec{x}_{0}$ and $\tau=\tau_{0}$), leading to the
evolution equation for the temperature:

$T_{\text{CMB}}(\vec{x},\vec{p},\tau)=T_{\text{CMB},0}(\tau)\left[1+\Theta(\vec{x},\vec{p},\tau)\right]\,,$
(3.3.1)

with $T_{\text{CMB}}\propto(1+z)$ and $\vec{p}$ is the direction of
propagation of the photons. Next, we will discuss how to compare the observed
temperature distribution function against the theoretical predictions.

Being a sky map, the distribution of CMB anisotropies is better expressed in
the projection of the photon direction in the celestial sphere, i.e. in polar
coordinates $(\theta,\phi)$. Since our fields are assumed to be statistically
homogeneous, the averages over an ensemble will be independent of the
position. Together with the assumption of isotropy of the initial conditions
of the fluctuations, this implies that the distribution of the temperature
anisotropies will be the same for all directions. To compare the theoretical
predictions for the temperature anisotropies with CMB observations, the
temperature perturbation, $\Theta^{\text{obs}}=\Delta T/T$, is expanded in
terms of spherical harmonic functions, $Y_{\ell m}$, which form an orthonormal
basis for the points on a sphere, particularly suited for describing the pixel
patches of the sky map:

$\Theta^{\text{obs}}(\theta,\phi)\equiv\frac{\Delta
T}{T}(\theta,\phi)=\sum_{\ell=1}^{\infty}\sum_{m=-\ell}^{+\ell}a_{\ell
m}^{T}Y_{\ell m}(\theta,\phi)\,,$ (3.3.2)

where $\ell$ and $m$ are conjugate to a unit vector in real space,
representing the direction of propagation of the incoming photons. In
particular, the relation $\theta=\pi/\ell$ in radians holds, meaning that
larger scales correspond to lower multipoles. The corresponding coefficients
$a_{\ell m}$ encode the dependence on the direction of observation and hence
are essential for parametrising the shape and overall distribution of the
anisotropies in the CMB map in Fig. 1.1. The monopole temperature is $a_{00}$
and is typically omitted from the spectra along with the dipole terms 222The
monopole term essentially represents a uniform temperature across the sky. It
encoded the average temperature of the CMB, but it does not provide any
information about fluctuations or anisotropies. The dipole anisotropy is
primarily due to our motion with respect to the CMB rest frame. It has been
extensively mapped but is not particularly interesting for cosmological
purposes. The dipole does not contain information about the early Universe but
about the present motion. On this basis, it is usually discarded to unveil
intrinsic anisotropies of cosmological interest.. Being statistically
independent, the average value of the single harmonic coefficients vanishes,
i.e. $\langle a_{\ell m}\rangle=0$. For the two-point correlator, similarly to
the Fourier decomposition of random fields in space, the off-diagonal
correlators of the expansion coefficients $a_{\ell m}$ of the expansion
vanish, ensuring orthogonality. The intrinsic variance of the coefficients is
defined as:

$\langle a_{\ell
m}a^{*}_{\ell^{\prime}m^{\prime}}\rangle=\delta_{\ell\ell^{\prime}}\delta_{mm^{\prime}}C_{\ell}^{\text{obs}}\,.$
(3.3.3)

The temperature-temperature (TT) spectrum is conventionally scaled in terms of
the multipoles, resulting in

$\mathcal{D}_{\ell}^{TT}=\frac{\ell(\ell+1)}{2\pi}C_{\ell}^{TT}\,,$ (3.3.4)

such as the example depicted in Fig. 3.1. It is also possible to compute the
three-point non-Gaussian correlations, a piece of information directly encoded
by the primordial non-Gaussianity induced by some models of inflation, which
CMB experiments have constrained Bartolo et al. (2004), Akrami et al. (2020b)
but goes beyond the scope of this work.

This method is so robustly studied according to fundamental physics that most
of the statistical features had been predicted long before they were
effectively observed. In particular, the TT-spectrum has been accurately
mapped up to $\ell\sim 4000$ and it is largely believed that most of the
available cosmological information has already been derived from it. In fact,
CMB observations encode potential access to six different power spectra:
temperature-temperature (TT), E-mode-E-mode polarisation (EE) and B-mode-B-
mode polarisation (BB) autocorrelations and temperature-E-mode (TE),
temperature-B-mode (TB), E-mode-B-mode (EB) cross-correlations. The BB spectra
have much lower power compared to TT, and are still quite suppressed in
comparison to the EE polarisation case (around 2 orders of magnitude below).
So, for simplicity reasons, the work reported in Part II of this dissertation
only considers CMB data from TT, EE, and TE spectra. Nevertheless, the B-mode
spectra carry unique information, and their detection and imprints are an
independent research line Kamionkowski and Kovetz (2016). This is because
while scalar perturbations can only generate E-mode polarisations, primordial
B-modes would unequivocally imply the existence of tensor perturbations, most
likely generated during inflation. Even though the observational prospect for
primordial B-modes still faces technical limitations due to secondary B-mode
signals, there is a lot of potential for sensitivity improvements and their
detection would significantly enrich our understanding of the early Universe.

This work will focus on scalar perturbations only, uncorrelated with the
vector and tensor modes. The vector perturbations decay, meaning their
contribution is negligible (assuming that the initial conditions were set very
early, as is the case in inflation), and the tensor perturbations are constant
on super-horizon scales, subject to damped oscillations after entering the
horizon, becoming suppressed compared to scalar perturbations. In a broader
context, scalar perturbations can be compared to sound waves propagating in
the primordial plasma, leading to areas of compression and rarefaction. These
variations produce the pattern of peaks and troughs in the CMB power spectrum,
as seen in Fig. 3.1, where the spectrum has been rescaled by
$\ell(\ell+1)/(2\pi)$. Hence, the specific positions of these peaks depend
non-trivially on the composition and evolution of the Universe and represent a
valuable probe for testing signatures of alternative models to $\Lambda$CDM.

####  3.3.2 Modelling the Anisotropies

The line of sight integration method provides an informative, straightforward
way to study the physical effects behind the CMB anisotropies Seljak and
Zaldarriaga (1996). For a spatially flat Universe and adopting the conformal
Newtonian gauge Eq. 2.2.5, the linear temperature anisotropy
$\Theta\equiv\delta T/T$ due to the Fourier mode $k$ and projected on the
multipole $\ell$ today can be written as Durrer (2020)

$\Theta_{\ell}^{(s)}(k,\tau_{0})=\int_{0}^{\tau_{0}}\odif{\tau}S_{T}^{(s)}(k,\tau)j_{\ell}(k(\tau_{0}-\tau))\,.$
(3.3.5)

The integrand in Eq. 3.3.5 is a convolution between the $\ell$-th spherical
Bessel function, denoted by $j_{\ell}$, which gives the projection from
Fourier to harmonic space. The $S_{T}^{(s)}$ terms in the integrand are the
scalar sources:

$S_{T}^{(s)}(k,\tau)\equiv
g\left(\Theta_{0}+\Psi\right)+\left(gk^{-2}\theta_{b}\right)^{\prime}+\mathup{e}^{-\tau_{\text{op}}}\left(\Phi^{\prime}+\Psi^{\prime}\right)\
+\ \text{polarisation terms}\,.$ (3.3.6)

The term

$\tau_{\text{op}}(\tau)=\int_{\tau}^{\tau_{0}}n_{e}\sigma_{T}a\odif{\tau^{\prime}}\,,$
(3.3.7)

represents the optical depth (not to be confused with conformal time),
determined by the free electron fraction $n_{e}$ and the Thompson scattering
cross section $\sigma_{T}$. An optically thin (thick) medium is characterised
by $\tau_{\text{op}}\ll 1$ ($\tau_{\text{op}}\gg 1$), with
$\tau_{\text{op}}=1$ defining the optical transition. Furthermore,
$g\equiv-\tau_{\text{op}}^{\prime}\mathup{e}^{-\tau_{\text{op}}}$ corresponds
to the visibility function, which quantifies the probability distribution that
a photon last scatters at a given redshift333For simplicity, we omit the terms
from the polarisation spectrum from different sources since the class of
models considered does not typically affect the CMB polarisation, unless there
is some specific coupling to photons.. In addition to the gravitational
potentials $\Phi$ and $\Psi$, the source terms include the temperature
monopole, the baryon velocity $\theta_{b}$ and subdominant polarisation modes,
all dynamical functions of $\tau$ and $k$. The line-of-sight approach
dramatically simplifies the computation while isolating the different
contributions to the total anisotropy. The terms in the first and second
brackets in Eq. 3.3.6 are proportional to the visibility function, which peaks
around the recombination epoch and are known as primary anisotropies since
their effect is mainly imprinted at $z\sim 1090$ when the photons decouple. In
more detail, the contributions are the following:

  * •

Primary Monopole $\Theta_{0}$ (density perturbations): anisotropy proportional
to the local photon energy density fluctuation at the last scattering surface.
Overdensities within the early primordial plasma are anticipated to exhibit
higher temperatures, whereas underdensities are expected to be colder. These
variations in over- and under-densities inherited from the inflationary phase,
will undergo an evolutionary process governed by the perturbation equations
described in Section 2.2. This evolution contimues until the epoch of
decoupling, ultimately leaving a distinctive imprint in the CMB.

  * •

Sachs-Wolfe (SW) Effect $\Psi$ (gravitational perturbations): redshift of the
photons as they climbed up the initial potential wells when emitted. Regions
linked to overdensities are correspondingly linked to more robust
gravitational fields. Photons traversing within these perturbed gravitational
fields will undergo a gravitational redshift, as elucidated by Sachs and Wolfe
in Sachs and Wolfe (1967). Unlike adiabatic perturbations, this effect is
anticipated to generate cooler over-densities and warmer under-densities.

  * •

Doppler Shift (velocity perturbations): effects induced by the baryon velocity
$\theta_{b}$ and $\theta_{b}^{\prime}$.

The third bracket terms are proportional to $\mathup{e}^{-\tau_{\text{op}}}$,
which is active since the last scattering. This gives rise to secondary
anisotropies:

  * •

Integrated Sachs-Wolfe (ISW) Effect $\Phi^{\prime}+\Psi^{\prime}$: accounts
for how the photons travelling from the last scattering surface are affected
by the potential wells they encounter. Comparable to the SW effect, the ISW
occurs on a larger scale when photons traverse through the vast structures in
the Universe. This results in an overall shift in the photon’s frequency,
arising from the cumulative influence of the mass encountered along its path.
If the potential wells were static, the blueshift gained when entering would
be compensated by the redshift necessary to escape. Nevertheless, if the
potentials evolve in time, there is a net contribution to the temperature
anisotropy.

The polarisation terms can be classified as tertiary contributions and
discarded for simplicity. Gravitational lensing of the CMB provides an
additional secondary contribution to the temperature fluctuation in Eq. 3.3.6.
However, lensing effects require bending the photon trajectories due to
inhomogeneities, which are already linear effects. Hence, they constitute a
higher order correction to Eq. 3.3.6.

The contribution of each effect to the temperature power spectrum defined by
Eq. 3.3.5 is depicted in Fig. 3.2.

(a) Figure 3.2:  The individual main contributions to the CMB temperature
power spectrum as detailed in Section 3.3.2, namely the total lensed and
unlensed spectra, the Sachs-Wolfe, Integrated Sachs Wolfe, Doppler effect and
the contributions from polarisation. This plot was generated through the
Boltzmann code CLASS Lesgourgues (2011a, b) and has no multipole scaling
compared to Fig. 3.1.

The measured value $C_{\ell}^{\text{obs}}$ in Eq. 3.3.3 can be directly
compared to the the theoretical angular power spectrum $C^{(s)}_{\ell}$, in
terms of the temperature field $\Theta^{(s)}_{\ell}(k)$, as the multipole
$\ell$ cross-correlation function in Fourier space $k$:

$C^{(s)}_{\ell}=\frac{2}{\pi}\int_{0}^{\infty}\odif{k}k^{2}\left|\Theta^{(s)}_{\ell}(k)\right|^{2}\,,$
(3.3.8)

with

$\Theta_{l}\equiv\frac{1}{2(-i)^{\ell}}\int_{-1}^{1}\odif{\mu}P_{\ell}(\mu)\Theta(\mu)\,,$
(3.3.9)

in terms of the Legendre polynomials $P_{\ell}(\mu)$, where
$\mu=\hat{k}\cdot\hat{p}$ gives the direction between the direction in which
the photon is propagating $\hat{p}$ and its wavenumber $\hat{k}$.

####  3.3.3 Scale Dependence of the CMB

Scalar perturbations can be envisioned as sound waves echoing through the
primordial plasma. These waves correspond to regions of rarefaction and
compression, resulting in the pattern of peaks and valleys in the CMB power
spectrum, which are the dominant contribution at small scales, $\ell\gtrsim
100$. The exact location of these peaks depends on the Universe’s composition
and evolution, which are an important probe of dark energy. Initial
perturbations within the baryon-photon fluid can only travel a finite distance
until recombination444To be more precise, during recombination, most electrons
combined with the atomic nuclei formed during Big Bang nucleosynthesis.
However, photons kept being scattered by the remaining free electrons. This
phase should not be confused with the drag epoch, where photons fully
decoupled from the remaining electrons and could finally free stream
throughout the Universe, a process which occurred about $z=80$ later.. The
extent of this propagation distance experienced by an acoustic wave depends on
its speed of sound:

$c_{s}=\frac{1}{\sqrt{3}}\left(1-\frac{3\rho_{b}}{4\rho_{\gamma}}\right)^{-1/2}\,,$
(3.3.10)

for the oscillations imprinted in the CMB anisotropies with frequency
$\omega_{k}(\tau)=kr_{s}(\tau)$, where $r_{s}$ is this limit distance, known
as the sound horizon,

$r_{s}(z_{\text{dec}})=\int_{0}^{(1+z_{\text{dec}})^{-1}}\frac{c_{s}(a)}{a^{2}H(a)}\odif{a}\,.$
(3.3.11)

The redshift at decoupling $z_{\text{dec}}$ can be deduced by intricate
formulae Hu and Sugiyama (1996) or by taking the entire background history
through fitting numerical codes, which is particularly important for non-
standard scenarios with changes to the early Universe dynamics.

As detailed in Amendola and Tsujikawa (2015), an approximate relation for the
comoving wavelength of the acoustic peaks is $\lambda_{c}=2\pi/k=2r_{s}/n$,
where $n$ are integer numbers. The first peak is found at an angular scale of
$\theta=1$ degree which corresponds to $\ell=180$ (according to
$\theta=\pi/\ell$). It is helpful to define a characteristic angular location
for these peaks’ positions in terms of observable quantities:

$\theta_{\text{A}}=\frac{r_{s}(z_{\text{dec}})}{d_{A}^{c}(z_{\text{dec}})}\,.$
(3.3.12)

This acoustic scale is a quasi-universal observable, as it can easily be
defined across any statistically isotropic or spherically symmetric cosmology.
$d_{A}^{c}$ is the comoving angular diameter distance, defined from Eq. 2.1.14
as

$d_{A}^{c}(z)=(1+z)d_{A}(z)\,.$ (3.3.13)

The multipole associated with the angle $\theta_{\text{A}}$, is expressed as

$\ell_{A}=\frac{\pi}{\theta_{\text{A}}}=\frac{\pi
d_{A}^{c}(z_{\text{dec}})}{r_{s}(z_{\text{dec}})}\,.$ (3.3.14)

Furthermore, the comoving angular diameter distance can be parametrised by

$d_{A}^{c}(z_{\text{dec}})=\frac{c}{H_{0}}\frac{\mathcal{R}}{\sqrt{\Omega_{m,0}}}\,,$
(3.3.15)

where $\mathcal{R}$ stands for the CMB shift parameter related to the sound
horizon and given by

$\mathcal{R}=H_{0}\sqrt{\Omega_{m,0}}\int_{0}^{z_{\text{rec}}}\frac{\odif{z}}{H(z)}\,,$
(3.3.16)

implying $\ell_{A}\propto\mathcal{R}$, which can be bounded by CMB
measurements, which also correspond to estimates of the sound horizon
$r_{s}(z_{\text{dec}})$. The CMB shift parameter depends directly on the
Universe’s expansion history from recombination to the current epoch, defining
the acoustic peaks’ position. This can be appreciated in Fig. 3.1 depicting a
comparison between the theoretical prediction of the CMB temperature power
spectrum for $\Lambda$CDM against the 2018 observations from Planck. The
predictions of peak locations and magnitudes in a Universe with dark energy,
fully defined by a cosmological constant, align closely with the observed
data.

The TT spectrum’s highest amplitudes are attributed to two main components:
the $\ell=0$ component, representing the monopole (the mean value of the
blackbody signal), and the $\ell=1$ component, corresponding to the dipole as
discussed in Section 3.3.2. Moving beyond these components, the predicted and
observed higher modes ($\ell\geq 2$) are displayed in Fig. 3.1, exhibiting a
characteristic oscillatory pattern. Further insight into the shape of the
spectrum is provided in Fig. 3.2, which decomposes the contributions from the
sources of anisotropies introduced in Section 3.3.2. The behaviour of this
spectrum at different scales can be summarised as follows:

  * •

For $\ell<100$, the Sachs-Wolf plateau emerges. These angular scales exceed
the size of the causal horizon at the time of the last scattering surface.
Consequently, regions separated by such scales had limited interaction and co-
evolution since the end of inflation. The Sachs-Wolf plateau quantifies the
primordial perturbations, primarily arising from mechanisms outlined in
Section 3.3.2, particularly the Sachs-Wolf effect. These perturbations exhibit
a distinct scale-invariant behaviour, manifesting as a flat
$\mathcal{D}_{\ell}^{TT}=\ell(\ell+1)C_{\ell}^{TT}/(2\pi)$, consistent with
inflationary predictions.

  * •

In the range of $100\leq\ell\leq 1000$, we enter the realm of the acoustic
peaks. Scales smaller than the horizon at the last scattering surface capture
causal interactions within the primordial plasma prior to decoupling. The
intricate wavelike structure in this region results from the oscillations of
the coupled photon/baryon fluid within the gravitational field, primarily
influenced by the massive and non-interacting dark matter. Here, gravitational
forces compete with fluid motion driven by photon radiation pressure. The
acoustic peaks are characterised by a prominent peak around $\ell\simeq 200$,
followed by two smaller peaks. These visible perturbations in the plasma
resemble sound waves, justifying the term acoustic.

  * •

For $\ell\leq 1000$, we encounter the damping tail. In this region, the
spectrum exhibits rapid oscillations with a progressively diminishing
amplitude. This behaviour arises due to the thickness of the last scattering
surface, averaging over multiple small-scale behaviours that tend to cancel
each other out. This averaging process gradually erases structures at scales
smaller than the thickness of the last scattering surface.

####  3.3.4 CMB Lensing

In the context of general relativity, massive objects serve as gravitational
lenses that bend the paths of photons. The large-scale structure between the
last scattering surface and the observer gravitationally lenses both the
temperature and polarisation anisotropies of the CMB. These imprints can be
reconstructed as a map of the lensing potential, whose gradient determines the
lensing deflections. For example, in propagating through a large, over-dense
clump of matter in the line of sight, angular structures in the CMB get
magnified, appearing more extensive in the sky. Essentially, by looking at how
the typical size of hot and cold spots in the CMB temperature map vary across
the sky, we can reconstruct the lensing deflections and, hence, the integrated
distribution of dark matter. This lensing map provides a new cosmological
observable, similar to maps of cosmic shear estimated from the shapes of
galaxies555Cosmic shear is induced by weak lensing as a consequence of the
overall mass distribution, including non-luminous matter. Compared to galaxy
counting, the advantage is that it is not affected by bias uncertainty.. Its
power spectrum, as depicted in Fig. 3.3, provides access to cosmological
parameters from the CMB alone that affect the late-time expansion and geometry
of the Universe and the growth of structure - parameters that have only
degenerate effects in the primary CMB anisotropies.

This lensing effect offers a unique window into the Universe’s structure,
including its dark matter distribution. It affects various cosmological
observables, such as blurring the acoustic peaks in the CMB and altering
polarisation modes.

Strong lensing phenomena, often involving quasars lensed by single galaxies,
offer direct ways to study cosmological parameters like the rate of the
Universe’s expansion. On the other hand, weak gravitational lensing, usually
featuring distant galaxies lensed by closer galaxy clusters, offers insights
into the growth and density of cosmic structures. Both strong and weak lensing
serve as powerful tools for cosmological study, each providing complementary
data that can help resolve current tensions in our understanding of parameters
like the matter density and structure growth, as we will discuss in more depth
in Section 3.6.2.

(a) Figure 3.3:  The power spectrum of the CMB gravitational lensing
$\mathcal{D}^{\phi\phi}_{\ell}=10^{7}\left[\ell(\ell+1)\right]^{2}C_{\ell}^{\phi\phi}/(2\pi)$
factor. The blue curve is the $\Lambda$CDM prediction for the Planck 2018
bestfit values, produced with the CLASS code Lesgourgues and Tram (2011), Blas
et al. (2011). The black markers represent, as an example, data from the
Planck collaboration Aghanim et al. (2020c), POLARBEAR 2019 Aguilar Faúndez et
al. (2020), SPTpol 2019 Wu et al. (2019), SPT/Planck 2017 Simard et al. (2018)
and ACTpol 2016 Sherwin et al. (2017), and corresponding error bars. The data
is part of a compilation from
https://lambda.gsfc.nasa.gov/education/lambda$\\_$graphics/more/lensing$\\_$power$\\_$source.html.

###  3.4 Other Cosmological Probes

Despite the fact that the CMB provides a powerful and robust measure of the
cosmological parameters of the $\Lambda$CDM model, these can be reinforced and
independently confirmed by combining multiple surveys, ideally probing the
Universe at different temporal and spatial scales. In fact, this is what
allows for the breaking degeneracies in the model parameters inferred from the
CMB, as will be discussed in Section 3.5. Below we detail the various
observational probes relevant for the context of this dissertation which
correspond mostly to probes of dark energy. Nevertheless, a multitude of other
emerging independent cosmological probes could be added to this picture, such
as the $21$ cm and Lymann-$\alpha$ lines, quasars, cosmic chronometers,
measurements of redshift drift, surface brightness fluctuations, to name only
a few. A complete review can be found in Moresco et al. (2022).

####  3.4.1 Standard Candles: Cepheid Variables and Type IA Supernovae

As we have briefly mentioned in the context of the seminal measurements of the
accelerated expansion of the Universe Riess et al. (1998), Perlmutter et al.
(1999), an alternative method of estimating the expansion rate draws on the
knowledge associated with two particular astrophysical objects: a distance
calibrator and some objected with well-known standard luminosity. The most
conventional pair corresponds to Cepheid variables as calibrators for Type Ia
supernovae666The explosion of supernovae results in an intense burst of
radiation, making them exceedingly luminous events. These supernovae can be
categorised based on the presence of chemical element absorption lines in
their spectra. Specifically, if the spectrum of a supernova displays a
hydrogen absorption line, it is labelled as a Type II supernova. In the
absence of such a line, it is called Type I. Further sub-classifications exist
within Type I supernovae: Type Ia contains an absorption line of singly
ionised silicon, Type Ib features a helium line, and Type Ic lacks silicon and
helium absorption lines. The explosion characteristic of Type Ia supernovae is
triggered when a white dwarf in a binary system crosses the Chandrasekhar mass
limit Chandrasekhar (1931) due to the accretion of gas from its companion
star. One remarkable feature of Type Ia supernovae is the near-constancy of
their peak absolute luminosity. Consequently, these supernovae serve as a form
of standardisable candles, enabling the observational measurement of
luminosity distance by assessing their apparent brightness..

Cepheid variables are a vastly studied type of pulsating stars characterised
by a well-established correlation between their luminosity and the period of
their pulses. We can infer their intrinsic luminosity from the observed
pulsation period by categorising Cepheids as Population I or Population II
stars. By measuring the differences in the magnitude (a measure of a luminous
object’s brightness, related to the photon flux), it is possible to establish
a connection with the luminosity distance, defined in Eq. 2.1.14.

More precisely, defining the distance modulus, $\mu=m-M$, simply as the
difference between the apparent $m$ (measure of an object’s brightness when
observed from the Earth) and absolute $M$ magnitudes, we find the key relation

$\frac{\mu}{5}+1=\log\left(\frac{c(1+z)}{H_{0}}\int_{0}^{z}\frac{\odif{z^{\prime}}}{E(z^{\prime})}\right)\,,$
(3.4.1)

where the expansion rate $E(z)$ is defined as $H^{2}(z)=H_{0}^{2}E^{2}(z)$,
scaled by the Hubble constant $H_{0}$ in
$\text{km}\,\text{s}^{-1}\,\text{Mpc}^{-1}$, and assuming a flat geometry
($\Omega_{K}=0$ and $d_{M}(z)=d_{c}(z)$). Since the corrected peak magnitude
$M$ is the same for all the supernovae ($M\approx-19$ for type Ia), the
luminosity distance can be directly obtained from $m$. This methodology allows
us to establish a connection between the apparent magnitudes of Cepheid
variables and supernovae and the Hubble constant. The redshift associated with
the supernova can be determined independently by analysing the wavelength of
the light emitted and the shifts in its characteristic spectral absorption
lines. With a significant sample size of supernova observations, one can
establish the relationship between the observed luminosity and redshift. Large
catalogues like Pantheon (1048 supernovae Scolnic et al. (2018)) and its
recent update Pantheon+ (1550 supernovae Brout et al. (2022)) compare these
observations against calibrated distance-redshift measurements from the
wavelength of the observed light based on the spatially flat $\Lambda$CDM
model. From this comparison, key cosmological parameters such as $\Omega_{m}$
and $\Omega_{\Lambda}$ can be inferred, leaving only one free parameter: the
Hubble constant. The SH0ES collaboration estimates $H_{0}$ employing Cepheid
measurements as calibrators analysed using the Hubble Space Telescope (HST)
and the Gaia mission Riess et al. (2022a). Planned surveys such as the Rubin
Legacy Survey of Space and Time (Rubin LSST) are expected to provide
catalogues of various transient calibrators, increasing the number of known sn
Ia by a factor of $\sim 10$ Hambleton et al. (2023).

However, the measured values of the Hubble constant obtained according to this
method keep showing increasing tension with estimations derived from the
cosmic microwave background angular distance as the data accumulates and the
results become more precise. This tension has been highlighted in the latest
paper from the Pantheon+ catalogue analysis by the SH0ES collaboration Riess
et al. (2022a). Section 3.6 will investigate this issue more thoroughly.

(a) Figure 3.4:  The distance modulus $\mu(z)$ as a function of redshift $z$
for the Pantheon dataset (black dots), along with the expected $\mu(z)$
predicted values according to the $\Lambda$CDM model (blue curve). The error
bars show the uncertainties in the $1048$ data points used to construct the
Pantheon dataset Scolnic et al. (2018).

####  3.4.2 Standard Rulers: Baryonic Acoustic Oscillations

Before the recombination epoch, photons and baryons were tightly coupled,
meaning that the sound waves that resulted in the temperature anisotropies in
the CMB also left signatures in the baryon perturbations. The drag epoch
refers to the era when photons were liberated from the baryons’ drag due to
Compton scattering. The oscillations produced by the acoustic waves in the
CMB, as they interact with baryonic matter, also leave discernible signatures
within the matter power spectrum, as it plays an instrumental role in
determining the distribution of galaxies. The associated large-scale redshift-
space correlation function displays a peak at the Baryon Acoustic Oscillations
(BAO) scale at the sound horizon size $r_{s}$. The BAO peak was initially
observed at $\sim 100\,h^{-1}$ Mpc from a spectroscopic sample of luminous red
galaxies from the Sloan Digital Sky Survey (SDSS) Eisenstein et al. (2005).
The precise location of this peak provided further evidence supporting the
existence of a dominant dark energy component in the late Universe. The sound
horizon at $z=z_{\text{drag}}$, which marks the baryons’ release from the
Compron drag of photons, can be defined as:

$r_{s}(\tau_{\text{drag}})=\int_{0}^{\tau_{\text{drag}}}c_{s}(\tau)\odif{\tau}\quad\xrightarrow{\odif{\tau}=\odif{a}/(a^{2}H(a))}\quad
r_{s}(z_{\text{drag}})=\int_{0}^{(1+z_{\text{drag}})^{-1}}\frac{c_{s}(a)}{a^{2}H(a)}\odif{a}\,,$
(3.4.2)

where $c_{s}(\tau)$ denotes the sound speed of the photon–baryon plasma, given
by:

$c_{s}^{2}=\frac{\delta p_{\gamma}}{\delta\rho_{\gamma}+\delta\rho_{b}}\,,$
(3.4.3)

with the subscripts $(\gamma,b)$ denoting photon and baryon quantities,
respectively, and $\tau$ is the conformal time.

The power spectrum in redshift space, decomposed into wavenumbers parallel and
perpendicular to the line of sight, provides the observed redshift and angular
distributions of galaxies within redshift space. From this power spectrum, the
following quantities can be obtained Amendola and Tsujikawa (2015), in analogy
to Eq. 3.3.12 for the acoustic peak in the CMB:

$\displaystyle\theta_{s}$
$\displaystyle=\frac{r_{s}(z_{\text{drag}})}{(1+z)d_{A}(z)}\,,$ (3.4.4)
$\displaystyle\delta z_{s}$
$\displaystyle=\frac{r_{s}(z_{\text{drag}})H(z)}{c}\,,$ (3.4.5)

with $d_{A}(z)$ as previously defined in Eq. 2.1.17. The angle $\theta_{s}$
corresponds to observations perpendicular to the line of sight, and $\delta
z_{s}$ corresponds to observations made along the line of sight.

Although current BAO data is still not sufficiently abundant to provide
independent measurements of these two distance estimators, a combined distance
scale ratio can be obtained from the spherically averaged power spectrum
Amendola and Tsujikawa (2015):

$\left[\theta^{2}_{s}\delta
z_{s}\right]^{1/3}\equiv\frac{r_{s}(z_{\text{drag}})}{\left[(1+z)^{2}\,d_{A}^{2}(z)c/H(z)\right]^{1/3}}\,,$
(3.4.6)

or the related effective distance Eisenstein et al. (2005):

$d_{V}(z)\equiv\left[(1+z)^{2}d^{2}_{A}(z)\frac{cz}{H(z)}\right]^{1/3}\,.$
(3.4.7)

Thus, predictions for the distance scale $d_{V}(z)$ can be compared to
observational data. These observations indeed reveal an amplified number of
galaxies separated by a distance of roughly $150\,\text{Mpc}$ or simply
$100h^{-1}\,\text{Mpc}$, which, in turn, provides a method for calculating the
expansion rate $H_{0}$ or its dimensionless counterpart $h$
($H_{0}=100h\,\text{km}\,\text{s}^{-1}\,\text{Mpc}^{-1}$).

This probe focuses on the BAO peak in galaxy clustering, which differs from
the acoustic oscillations observed in the CMB, as it primarily reflects the
impact of low-redshift phenomena. The data collected from the Sloan Digital
Sky Survey (SDSS) Alam et al. (2017), Ata et al. (2018), Beutler et al.
(2017), Ross et al. (2015), Beutler et al. (2011), amongst other sources, was
instrumental in measuring this BAO peak. A recent collection of BAO
measurements is depicted in Fig. 3.5, along with the corresponding distance
and expansion rate theoretical $\Lambda$CDM curves. It is an important probe
for breaking degeneracies in the CMB data. The large scale structures of the
Universe can also be retrieved from the reciprocal weak lensing effect, which
measures the response of photons to a gravitational potential, thereby mapping
the mass distribution in the Universe through its effect on deflecting the
light of background galaxies. For this purpose, large photometric and
spectroscopic surveys are under development, including the 18th data release
(DR18) from the fifth generation of the SDSS survey (SDSS-V) Almeida et al.
(2023) and the Dark Energy Spectroscopic Instrument (DESI) Levi et al. (2019).
During the process of writing this thesis (July 2023), ESA’s Euclid satellite
was launched. This cosmological survey mission was designed to map the
extragalactic sky, exploring the expansion history and the formation of
structures, with the main aim of providing new insight on the nature and
properties of dark energy and dark matter on universal scales Amendola et al.
(2013). I am an active member of the consortium since 2022, as part of
modified gravity and dark energy theory working groups.

(a) Figure 3.5:  BAO distance and expansion rate data, derived from the SDSS
collaboration’s final results that span eight different redshift intervals,
improved over two decades Alam et al. (2021). The data includes isotropic BAO
measurements denoted by $D_{V}(z)/r_{s}$, where $D_{V}(z)$ is the spherically
averaged volume distance and $r_{s}$ is the sound horizon at baryon drag, both
defined in Eqs. 3.4.7 and 3.4.2. It also comprises anisotropic BAO
measurements, expressed as $D_{M}(z)/r_{s}$ and $D_{H}(z)/r_{s}$, where
$D_{M}(z)$ is the comoving angular diameter distance and $D_{H}(z)=c/H(z)$ is
the Hubble distance, both elaborated in Eqs. 2.1.17 and 2.1.10. These
measurements are detailed in Table III of Alam et al. (2021) (see references
therein). The compilation of data to produce this plot was provided by William
Giarè.

####  3.4.3 Standard Sirens: Gravitational Wave Detection

In the context of GR, and assuming propagation in a vacuum, gravitational
waves follow the wave equation

$h_{P}^{\prime\prime}(\tau,k)+2\mathcal{H}h_{P}^{\prime}(\tau,k)+k^{2}h_{P}(\tau,k)=0\,,$
(3.4.8)

in which $h_{P}(\tau,k)$ are the Fourier modes of the gravitational wave
amplitude, and the index $P=+,\,\times$ runs over the two possible
polarisation states of the gw Schutz (1986), Zhao et al. (2011).

The groundbreaking detection of gws sourced by the merger of two black holes
was first achieved in 2015 by the LIGO-Virgo collaboration Abbott et al.
(2016). This provided compelling evidence for the existence of gravitational
waves, as remarkably predicted by GR. However, it can be argued that an
equally noteworthy event for cosmology occurred two years later with the
observation of a signal pointing to a binary neutron star merger along with a
counterpart detection in the electromagnetic spectrum Abbott et al. (2017a,
b). This immediately allowed the establishment of notably stringent
constraints on the speed of the tensor modes, thereby ruling out theories of
modified gravity that predict a speed of gws different from the speed of light
Belgacem et al. (2019).

Namely, these signals can serve as standard sirens, which means they provide a
method of constructing a Hubble diagram and derive estimations of the value of
$H_{0}$, independent of the traditional distance ladder method Schutz (1986).
This complementary approach is possible because the amplitude of gravitational
waves is practically inversely proportional to the luminosity distance
$d_{L}(z)$ to the source event from which it originates Holz and Hughes
(2005). The electromagnetic counterpart signal provides redshift measurements,
meaning that $H_{0}$ can be estimated using the relation between the comoving
distance and the luminosity distance in Eq. 2.1.14.

Standard sirens cannot yet compete with other methods of estimating $H_{0}$,
with the current constrained value subjected to significant errors:
$H_{0}=70.0^{+12.0}_{-8.0}\,\text{km/s/Mpc}$ Di Valentino et al. (2018).
However, the prospects for future observations are promising, as next-
generation surveys can yield dozens or even hundreds of additional
measurements of $H_{0}$ using standard sirens.

Presently, we have second-generation (2G) ground-based operational detectors,
which include Virgo Acernese et al. (2015), LIGO Aasi et al. (2015), and the
Kamioka Gravitational Wave Detector (KAGRA) Somiya (2012), which will be
joined by the Indian Initiative in Gravitational-wave Observations (IndIGO)
Iyer (2015), planed for around 2030. The investment in maintaining and
developing more detectors augments the potential of GW astronomy, increasing
the number of detected events (covering a larger cosmic volume) and enhancing
sky localisation (better triangulation to the source), which, in turn,
facilitates counterpart searches. However, their third-generation (3G)
successors aim for increased sensitivity, precision, and a broader frequency
range. The Einstein Telescope (ET) warrants particular attention, as it is
predicted to boost the present sensitivity tenfold Punturo et al. (2010).
Moreover, ET is projected to widen the redshift horizon, for instance, $z\sim
5$ for binary black holes (BBHs) in contrast to the $z\sim 0.5$ threshold of
2G instruments Sathyaprakash et al. (2012). Forecasts suggest that the number
of traceable multi-messenger events might touch the count of tens of thousands
of standard sirens Maggiore et al. (2020). While ground-based detectors might
span a frequency spectrum of $1\lesssim f\lesssim 10^{3}$ Hz Cai and Yang
(2017), forthcoming space-based 3G detectors, with particular focus on the
Laser Interferometer Space Antenna (LISA) Amaro-Seoane et al. (2017), could
reach peak sensitivities close to $10^{-3}$ Hz and identify GW events beyond
$z=20$. Other 3G GW detectors have been planned, such as the DECi-hertz
Interferometer Gravitational Wave Observatory (DECIGO) Kawamura et al. (2011).
In Chapter 8, we will explore a related application of standard sirens centred
on ET and LISA for comparison and joint analysis of ground and space-based
experiments. More precisely, we will forecast their constraining power for
different models in direct comparison with current data from standard candles
and rulers.

####  3.4.4 Redshift Space Distortions

As we have seen in the previous sections, the measurements of distances to
galaxies rely on measuring their redshift. If the Universe were perfectly
homogeneous, there would be a straightforward mapping from the radial distance
(real space) to the redshift space Kaiser (1987). However, we have just seen
that the Universe is not homogeneous777We have assumed that there is a
homogeneous distribution on cosmological scales, which is a different
statement.. The growth of inhomogeneous structures leads to peculiar
velocities in galaxies moving in clusters that add to the expansion redshift
and manifest through distortions in the mapping to redshift space. These
distortions result in an apparent enhancement of large-scale clustering in the
radial direction compared to the transverse direction, and the redshift
distribution of galaxies appears disfigured. These redshift space distortions
(RSD) are a probe for the linear growth of structures and have been
extensively reviewed in the literature, e.g. in Hamilton (1997), Saito (2016).

The linear growth rate $f(z,k)$ of structures is a differential measure of the
evolution of the matter fluctuations:

$f(z,k)=\frac{\odif{\text{ln}}\delta_{\text{m}}(z,k)}{\odif{\text{ln}\,a}}=\frac{1}{\mathcal{H}}\frac{\delta_{\text{m}}^{\prime}(z,k)}{\delta_{\text{m}}(z,k)}\,,$
(3.4.9)

where the subscript $m$ stands for the collective matter component
(baryons+CDM) Hamilton (2001). As the name indicates, $f(z,k)$ is a measure of
the rate at which the matter perturbations grow, highly correlated with the
anisotropic clustering observed in redshift space distortions. Therefore, RSDs
serve as a probe for the combined quantity $f\sigma_{8}$, which is the product
of the growth rate and the root mean square mass fluctuation amplitude for
spheres of size $8h^{-1}\,\text{Mpc}$, and which can be used as a
normalisation factor in the matter power spectrum at that scale Alam et al.
(2021). More precisely, $\sigma_{8}$ is defined as

$\sigma_{8}^{2}=\int_{0}^{\infty}W^{2}(kR)\frac{P(k)}{k}\odif{k}\,,$ (3.4.10)

where $W$ is a top hat filter function in Fourier space $k$

$W(kR)=3\left[\frac{\sin(kR)}{(kR)^{3}}-\frac{\cos(kR)}{(kR)^{2}}\right]\,.$
(3.4.11)

Here, $P(k)$ denotes the matter power spectrum as defined in Eq. 3.2.1, and
$R$ represents the $\sigma_{8}$ scale radius $8h^{-1}\,\text{Mpc}$ Liddle and
Lyth (2000). The combination $f\sigma_{8}$ becomes then

$f\sigma_{8}(z,k_{\sigma_{8}})=\frac{\sigma_{8}(0,k_{\sigma_{8}})}{\mathcal{H}}\frac{\delta_{\text{m}}^{\prime}(z,k_{\sigma_{8}})}{\delta_{\text{m}}(0,k_{\sigma_{8}})}\,,$
(3.4.12)

with $k_{\sigma_{8}}=0.125h$ Mpc-1, and $\sigma_{8}$ is defined as

$\sigma_{8}(z,k_{\sigma_{8}})=\sigma_{8}(0,k_{\sigma_{8}})\frac{\delta_{\text{m}}(z,k_{\sigma_{8}})}{\delta_{\text{m}}(0,k_{\sigma_{8}})}\,.$
(3.4.13)

For observational and degeneracy-breaking purposes, a related parameter is
defined as $S_{8}=\sigma_{8}\sqrt{\Omega_{\text{m}}/3}$. $S_{8}$ is
conventionally employed to express the tension in measurements of $\sigma_{8}$
for weak lensing and CMB probes, and their correlation with measurements of
$\Omega_{\text{m}}$, and will be the focus of Section 3.6.

###  3.5 The Cosmological Parameters of the $\Lambda$CDM Model

The anisotropies of the CMB are dependent on the values of various parameters.
The standard approach typically relies on six free and independent parameters
chosen to avoid degeneracies and to improve the convergence of the model fit
to the data. These parameters are
$\\{\omega_{b},\omega_{c},h,\tau_{\text{reio}},A_{s},n_{s}\\}$, which play a
significant role in describing the primordial power spectrum of fluctuations,
the reionisation epoch, the expansion rate, and the constituents of the
Universe. In this section, we delve into the specifications of these
parameters, exploring the implications of their individual variation on the
power spectrum of the CMB for the $\Lambda$CDM model by keeping the remaining
parameters fixed to Planck 2018 fiducial values Aghanim et al. (2020a), given
in Table 3.1. The corresponding CMB temperature power spectrum change for
variation of the six $\Lambda$CDM parameters is shown in Fig. 3.6 and detailed
below.

Parameters of $\Lambda$CDM from Planck
---
Parameter | Estimated value
Baryon density | $\omega_{\text{b}}$ | $0.02236\pm 0.00015$
CDM density | $\omega_{\text{c}}$ | $0.1202\pm 0.0014$
Dimensionless Hubble constant | $h$ | $0.6727\pm 0.0060$
Optical depth | $\tau_{\text{reio}}$ | $0.0544^{+0.0070}_{-0.0081}$
Scalar tilt | $n_{s}$ | $0.9649\pm 0.0044$
Scalar amplitude | $\ln\left(10^{10}A_{s}\right)$ | $3.045\pm 0.016$
Matter density | $\Omega_{m}$ | $0.3166\pm 0.0084$
$\Lambda$ density | $\Omega_{\Lambda}$ | $0.6834\pm 0.0084$
Proxy angular scale of the sound horizon | $100\theta_{\text{MC}}$ | $1.04090\pm 0.00031$
Normalisation of matter power spectrum | $\sigma_{8}$ | $0.8120\pm 0.0073$
$\sigma_{8}(\Omega_{m}/0.3)^{1/2}$ | $S_{8}$ | $0.834\pm 0.016$
Angular scale of the sound horizon | $100\theta_{*}$ | $1.04109\pm 0.00030$
Sound horizon at recombination |  $r_{*}$ [Mpc] | $144.39\pm 0.30$
Damping scale |  $k_{D}$ [Mpc-1] | $0.14090\pm 0.00032$
Reionisation redshift | $z_{\text{reio}}$ | $7.68\pm 0.79$
Helium mass fraction | $Y_{P}^{BBN}$ | $0.246716^{+0.000062}_{-0.000055}$
Age of the Universe |  $t_{0}$ [Gyr] | $13.800\pm 0.024$
Table 3.1: Estimates of the $\Lambda$CDM model cosmological parameters (upper
section). The remaining parameters are either compatible with zero or can be
derived (lower section) from the standard six parameters above Aghanim et al.
(2020a). The results are obtained from the combination of all the Planck
spectra. The bounds on the parameters correspond to $1\sigma$ errors ($68\%$
CL). (a) Figure 3.6:  Figure with CMB temperature power spectrum variations
from variations $\delta p$ of the standard $\Lambda$CDM cosmological
parameters $p$. The $\Lambda$CDM fiducial case is depicted as a dashed line
for reference. Variations in $\omega_{b}$ and $\omega_{c}$ implicitly hold $h$
fixed, while variations in $h$ hold the cold dark matter and baryon densities
fixed. Figure produced using CLASS and inspired by Hart (2020). (a) Figure
3.7:  Figure with variations in the matter power spectrum from variations
$\delta p$ of the standard $\Lambda$CDM cosmological parameters $p$. The
$\Lambda$CDM fiducial case is depicted as a dashed line for reference.
Variations in $\omega_{b}$ and $\omega_{c}$ implicitly hold $h$ fixed, while
variations in $h$ hold the cold dark matter and baryon densities fixed. Figure
produced using CLASS.

####  3.5.1 Baryon and Cold Dark Matter Densities: $\omega_{\rm b}$ and
$\omega_{\rm c}$

Having fundamentally different properties, each present matter density
influences the anisotropies’ imprint in distinct ways.

The baryon density, denoted as $\omega_{\text{b}}=\Omega_{\text{b}}h^{2}$,
alters the drag experienced by baryons within the tightly coupled baryon-
photon fluid. Increasing the baryon abundance increases the drag, reducing the
sound speed of the acoustic oscillations described in Section 3.4.2, as
depicted in the upper left panel of Fig. 3.6. Consequently, larger
overdensities will be generated by the cold dark matter potential wells,
enhancing the odd-numbered peaks of the acoustic oscillations relative to the
even-numbered ones associated with compression due to the gravitational
potential. Moreover, while more sensitive to the total matter density
$\omega_{\text{b}}+\omega_{\text{c}}$, baryons induce changes in the sound
speed, which impact the size of the sound horizon, shifting all the peaks
towards larger multipoles. While also influencing the free electron fraction
$X_{e}$888$X_{e}=n_{e}/n_{H}$ represents the redshift dependence of the
ionization/free electron fraction., higher values of $\omega_{\text{b}}$ are
associated with an increase in the thickness of the last scattering surface,
leading to a minor suppression of the diffusion damping at small multipoles
(related to time variations in the optical depth, $\dot{\tau}_{\text{reio}}$).

In the upper left panel of Fig. 3.7, we also identify an overall suppression
(enhancement) of the matter power spectrum for higher (lower) values of the
baryon density. Higher baryon densities can amplify the large-scale BAO
features in the matter power spectrum. Because baryons are more affected by
pressure effects from structure formation or radiation compared to dark
matter, it does not cluster as efficiently on small scales and higher values
of $\omega_{\text{b}}$ induce a suppression of the amplitude of the spectrum.
Finally, a higher baryon density effectively increases the Jeans length (below
which they do not collapse to form structure) and thus further suppresses
structure formation on small scales.

Even though dark matter does not interact electromagnetically, it is
responsible for the gravitational wells in which baryons and photons
oscillate. This is illustrated in the upper right panel of Fig. 3.6, by an
enhancement of the BAO features at large multipoles with the increase of
$\omega_{\text{c}}=\Omega_{\text{c}}h^{2}$. CDM has an opposite effect on
variations of $\tau_{\text{reio}}$ compared to baryons. An increase in
$\omega_{\text{c}}$ shifts the epoch of matter-radiation equality to larger
redshifts. Since the cold dark matter’s influence surpasses the photon-baryon
fluid contribution to the gravitational potential after equality, this implies
that the radiation pressure plays a less pivotal role after equality
($\tau>\tau_{\text{eq}}$). Being the motor of the baryon acoustic
oscillations, the attenuation of the radiation’s influence weakens the CMB
anisotropies, particularly in the first peak, as the relative abundance of
dark matter is increased Dodelson (2003), a phenomenon known as radiation
driving. Dark matter is the primary driver of structure formation, with the
variations in dark matter density becoming imprinted on the CMB through the
ISW effect at low multipoles.

This is also reflected in the matter power spectrum depicted in the upper
right panel of Fig. 3.7, as a higher density of CDM will lead to more
structure on all scales, effectively increasing its amplitude. Consequently, a
higher $\omega_{\text{c}}$ is associated to larger values of $\sigma_{8}$.
Higher CDM density results in faster structure growth, leading to more power
on smaller scales and the peak of the matter power spectrum is shifted to
smaller scales

####  3.5.2 The Hubble Constant: $H_{0}$

The Universe’s expansion rate can be directly extrapolated from the peaks of
the CMB TT-spectrum. The angular scale of the first peak, denoted as
$\theta_{\text{rec}}$, sets the horizon size at the time of recombination and
is parametrised as

$\theta_{*}=\frac{s_{*}}{r_{s,*}}\approx\frac{\tau_{\text{rec}}}{\tau_{0}}\,,$
(3.5.1)

where $s_{*}$ is the sound horizon at recombination, approximately given by

$s_{*}=\int_{0}^{\tau_{\text{rec}}}c_{s}(\tau)\odif{\tau}\approx\frac{\tau_{\text{rec}}}{\sqrt{3}}\,.$
(3.5.2)

The following peaks are located at an angular scale following the same
pattern, that is, $\theta_{*,n}\approx n\theta_{*}$, placing the higher
harmonics in the decomposition. A higher value of $H_{0}$ (and equivalently
$h$) makes the angular diameter distance smaller, which means that the same
physical scale corresponds to a larger angle on the sky. This shifts the
acoustic peaks to larger angular scales (i.e., smaller multipoles), as
illustrated in the central left panel of Fig. 3.6. The value of $H_{0}$
indirectly affects early-Universe physics through its relation to other
cosmological parameters. The ISW effect, relevant at large angular scales (low
$\ell$), is sensitive to the rate of expansion related to the critical density
of the Universe, which in turn determines the redshift at which matter and
radiation densities become equal, with higher values of $H_{0}$ leading to an
enhancement of the low-$\ell$ tail of the spectrum.

Analogously, in the central left panel of Fig. 3.7, we identify an enhancement
of the amplitude of the matter power spectrum for higher values of $h$,
associated with a slight shift of the peak towards lower $k$-modes. There are
different effects at play, but, more importantly, a faster expansion can
suppress the growth of structure, leading to less clustering on large scales.
At the same time, it also changes the maximum size of the cosmic structures by
modifying the Hubble horizon, shifting the turnover peak to larger scales.

####  3.5.3 Reionisation Optical Depth: $\tau_{\text{reio}}$

Parameters like the optical depth to reionisation $\tau_{\text{reio}}$ capture
the degree and timing of reionisation. A larger optical depth damps the CMB
anisotropies on all scales because some fraction of the CMB photons have last
scattered with free electrons not at the last scattering surface but more
recently. This overall damping is a multiplicative effect across all
multipoles $\ell$ and is depicted in the central right panel of Fig. 3.6. More
precisely, increasing the optical depth leads to an overall suppression of the
CMB power spectrum. This effect is highly degenerate with the one generated by
the scalar power spectrum parameter $A_{s}$, as will be discussed next. This
degeneracy is only broken by considering measurements of CMB polarisation
Aghanim et al. (2020a).

In the central right panel of Fig. 3.7, we see that $\tau_{\text{reio}}$ has a
negligible impact on the matter power spectrum for the scales considered. This
is because the matter power spectrum reflects the distribution of matter at
later times, long after reionisation has ceased. By this time, most of the
imprints of this epoch have been washed out or are secondary to other physical
processes.

####  3.5.4 Amplitude and Tilt of the Primordial Power Spectrum: $A_{s}$ and
$n_{s}$

The matter power spectrum, introduced in Section 3.2, is sourced by the
primordial (scalar) power spectrum, which in turn is characterised by the
parameters $A_{s}$ and $n_{s}$, defined at a pivot scale of
$k_{*}=0.05\,\text{Mpc}^{-1}$ for Planck observations. These determine the
strength and the scale dependence of the initial density perturbations,
respectively. A higher amplitude ($A_{s}$) means more pronounced density
fluctuations in the matter distribution, leading to stronger clustering of
matter and thus a more peaked matter power spectrum, as depicted in the lower
left panel of Fig. 3.6. The amplitude is directly related to $\sigma_{8}$ and
directly scales the entire matter power spectrum, as shown in the lower left
panel of Fig. 3.7.

The parameter $n_{s}$ accounts for the tilt of the power spectrum: for
$n_{s}>1$, there is more power on smaller scales; for $n_{s}<1$, there is more
power on larger scales, as depicted in the lower right panel of Fig. 3.6.
Changing the tilt alters the relative height of the peaks and the angular
scales at which they appear. Since the tilt $n_{s}$ determines directly how
the clustering power is distributed across different length scales, in the
lower right panel of Fig. 3.7, we see that larger values of $n_{s}$ lead to
more power on smaller scales, making the matter distribution more clumpy at
small scales, greatly suppressing the power on larger scales.

####  3.5.5 Derived Parameters

In addition to the standard six parameters mentioned above, other parameters
embody more complex changes to the physics of the CMB anisotropies. These can
be parameters of the standard model, which are usually kept fixed - such as
the number of relativistic degrees of freedom $N_{\text{eff}}$, the fraction
of primordial helium $Y_{H}$ or the mass of massive neutrinos $m_{\nu}$ \- or
parameters that account for the introduction of new physical degrees of
freedom in alternative cosmological models. The effect of some examples of
both classes of modifications in the CMB anisotropies will be the focus of
Part II. The estimated constraints on some of the relevant cosmological and
derived parameters of $\Lambda$CDM for the Planck 2018 analysis are reported
in Table 3.1.

###  3.6 Cosmological Puzzles and Observational Tensions

Albeit initially yielding great consistency confirmation in favour of
$\Lambda$CDM, the recent advent of observational precision and techniques has
revealed unexpected irreconcilable predictions from different probes, bringing
to light a significant crisis for the standard model of cosmology Weinberg
(1989, 2000), Martin (2012), Di Valentino et al. (2021a), Abdalla et al.
(2022). These tensions arise from discrepancies between observations of the
early and late stages of the Universe, such as those of the model-dependent
CMB data analysis Aghanim et al. (2020b), Aiola et al. (2020), Balkenhol et
al. (2021) and varied distance-ladder surveys Riess et al. (2019), Wong et al.
(2020), Pesce et al. (2020), Freedman (2021), Uddin et al. (2023),
respectively. Although some of these inconsistencies could be attributed to
data errors and systematics, the statistical significance of roughly a
$5\sigma$ tension suggests flaws in the standard model itself, with many
extensions being proposed in the literature to ease this problem. Exploring
alternative models may reveal new insights into the enigmatic nature of the
dark sector, the physics of the early Universe, or even the fundamental
assumptions of $\Lambda$CDM Saridakis et al. (2021).

The level of concordance or discordance quantifies the tension $T_{P}$ (if
any) on a parameter $P$ between the data sets $i$ and $j$. Considering the
tension in the estimate of a single parameter with a posterior distribution
that is approximately Gaussian, the difference between the mean values in the
posterior $P_{i}$ and $P_{j}$ divided by the respective quadrature sum of the
standard deviation of the respective data sets, $\sigma_{P_{i}}$ and
$\sigma_{P_{j}}$, represents a robust tension metric:

$T_{P}\equiv\frac{P_{i}-P_{j}}{\sqrt{\sigma^{2}_{P_{i}}+\sigma^{2}_{P_{j}}}}\,.$
(3.6.1)

But if we want to assess a tension in two or more parameters it can become
inadequate, as not all of the information can be captured with one-dimensional
projections. There is no unique and widely accepted metric for quantifying the
tensions and for simplicity reasons we will always use the metric defined in
Eq. 3.6.1 as a first approximation. Nevertheless we refer to Lemos et al.
(2021) for a more detailed account of this problem, including alternative
metrics and illustrative examples.

####  3.6.1 The Hubble Tension

In broader terms, the Hubble tension often refers to the $\approx 5.0\,\sigma$
disagreement between the value of the Hubble constant predicted by thePlanck
collaboration Aghanim et al. (2020a), $H_{0}^{\textit{Planck}}=\left(67.27\pm
0.60\right)\,\text{km/s/Mpc}$ (red squared data point in the whisker plot of
Fig. 3.8) at $68\%$ confidence level (CL), and the latest constraint from the
shoes collaboration R22 Riess et al. (2022a), $H_{0}^{\text{R22}}=(73.04\pm
1.04)\,\text{km/s/Mpc}$ at 68% CL (blue crossed data point in the whisker plot
of Fig. 3.8), based on the supernovae calibrated by Cepheids. A CMB analysis
by the Atacama Cosmology Telescope (ACT), combining data from the WMAP
satellite, finds a value consistent with and independent from Planck:
$H_{0}^{\text{ACT+WMAP}}=(67.26\pm 1.1)\,\text{km/s/Mpc}$ Aiola et al. (2020).
Nevertheless, it is crucial to recognise that there are multiple sets of
measurements and estimates for $H_{0}$. These can be divided into two
categories that are in agreement within their groups, reflecting the
persistence of the $H_{0}$ tension: $(i)$ indirect model-dependent estimates
at early times (such as CMB and Baryon Acoustic Oscillation experiments) that
assume a $\Lambda$CDM cosmology, and $(ii)$ direct late-time model-independent
measurements (such as distance ladder and strong lensing). That is why the
tension is said to arise between the CMB data and the direct local
measurements. This is depicted in the whisker plot of Fig. 3.8 for a
collection of recent $H_{0}$ estimates, showing the consistent trend between
early and late time probes.

(a) Figure 3.8:  Various recent estimates of $H_{0}$ from different probes, as
detailed in Section 3.6.1. The red and blue vertical bars illustrate the
$68\%$ CL regions for the values measured by the Planck Aghanim et al. (2020a)
and SH0ES Riess et al. (2022a) collaborations. The dashed line separates
measurements from probes based on the early and late Universe. The colours
group the type of estimates: red for CMB with Planck, dark red for CMB without
Planck, blue for SNIa-Cepheid, navy for SNIa-TRGB, sea green for SNIa-Miras,
dark green for Masers, green for Tully Fisher, charcoal for surface brightness
fluctuation, yellow for SNII, and orange for GW related measurements. Plot
adapted from https://github.com/lucavisinelli/H0TensionRealm and based on Di
Valentino et al. (2021b), Abdalla et al. (2022).

####  3.6.2 The $S_{8}$ Tension

Recent observations of the large-scale structure in the Universe have allowed
us to place constraints on the clustering strength of matter. However, these
constraints also differ from those reported by probes of the early Universe.
More precisely, there exists a tension in the matter clustering power between
the primary anisotropies of the CMB, as measured by the Planck satellite, and
lower redshift probes such as weak gravitational lensing and galaxy
clustering. Asgari et al. (2020, 2021), Joudaki et al. (2020), Abbott et al.
(2022), Amon et al. (2022), Secco et al. (2022), Loureiro et al. (2022),
Joudaki et al. (2017a), Heymans et al. (2021), Hildebrandt et al. (2020),
Abbott et al. (2020), Macaulay et al. (2013), Skara and Perivolaropoulos
(2020), Kazantzidis and Perivolaropoulos (2019), Joudaki et al. (2017b), Bull
et al. (2016), Kazantzidis and Perivolaropoulos (2018), Nesseris et al.
(2017), Philcox and Ivanov (2022) at the level of $2-3\sigma$. This tension is
typically quantified in terms of the parameter $S_{8}$, defined as
$S_{8}\equiv\sigma_{8}\sqrt{\Omega_{\rm m}/0.3}$, which regulates the
magnitude of weak lensing measurements Li et al. (2016), Gil-Marín et al.
(2017). The $S_{8}$ parameter is closely related to $f\sigma_{8}(z=0)$
measured by redshift space distortions (see Section 3.4.4) and lensing probes,
with $f=[\Omega_{\rm m}(z)]^{0.55}$ being a good proxy for the growth rate in
GR, based simply on the matter density parameter $\Omega_{\rm m}$ at a given
redshift $z$. In particular, lower redshift probes tend to favour a lower
value of $S_{8}$ compared to the high redshift CMB estimates (red in the
whisker plot of Fig. 3.9). As an example, the latest cosmic shear analysis of
KiDS (KiDS-1000) and DES (DES-Y3) set
$S_{8}^{\text{KiDS-1000}}=0.759^{+0.024}_{-0.021}$ Asgari et al. (2021) and
$S_{8}^{\text{DES-Y3}}=0.759^{+0.025}_{-0.023}$ Amon et al. (2022), Secco et
al. (2022) (blue crossed data point in the whisker plot of Fig. 3.9),
respectively, consistent with the $2-3\sigma$ tension with the fiducial Planck
analysis, which gives $S_{8}^{\textit{Planck}}=0.834\pm 0.016$. The addition
of secondary anisotropies from CMB lensing narrows the constraint to
$S_{8}^{\textit{Planck}+\text{lens}}=0.832\pm 0.013$ (red squared data point
in the whisker plot of Fig. 3.9). The agreement with high and other low
redshift measurements is recovered from the CMB lensing data alone. On the
other hand, a combination of the high-$\ell$ data from ACT and the low-$\ell$
measurements of WMAP Aiola et al. (2020) is compatible with the Planck results
$S_{8}^{\text{ACT+WMAP}}=0.840\pm 0.030$, even though the error bars are more
accommodating.

It is important to note that unlike locally measured quantities such as the
Hubble constant, $S_{8}$ relies on a specific cosmological model for its
interpretation. In the cases reported in Fig. 3.9, the underlying model is
always the standard flat $\Lambda$CDM model, which, despite providing a
reasonable fit to the different data sets, yields a lower rate of structure
formation compared to what observations seem to suggest. Therefore, to arrive
at a fair and accurate comparison of $S_{8}$ values, it becomes imperative to
reassess the analysis of weak lensing data while assuming a cosmological model
beyond $\Lambda$CDM (as will be the focus of Part II), before quantifying any
tension. This holds particularly true for cosmological models that exhibit
variations in the growth of structure compared to the standard scenario. While
not necessarily a trivial task, by conducting this reanalysis under different
cosmological frameworks, we can gain a more comprehensive understanding of how
$S_{8}$ is affected and whether the observed tension persists across various
models. This approach promotes a more refined assessment of the discrepancies
in $S_{8}$ values, disentangling the contributions of different cosmological
parameters to the overall tension.

(a) Figure 3.9:  Various recent estimates of $S_{8}$ from different probes, as
detailed in Section 3.6.2. The red and blue vertical bars illustrate the
$68\%$ CL regions for the values measured by the Planck Aghanim et al. (2020a)
and DES-Y3 Amon et al. (2022), Secco et al. (2022) collaborations. The dashed
line separates measurements from probes based on the early and late Universe.
The colours group the type of estimates: red for CMB with Planck, dark red for
CMB without Planck, blue for cosmic shear, navy for galaxy weak lensing, sea
green for CMB weak lensing, green for peculiar velocities, charcoal for RSD,
and orange for cluster abundance measurements. Plot adapted from
https://github.com/lucavisinelli/H0TensionRealm and based on Di Valentino et
al. (2021b), Abdalla et al. (2022).

####  3.6.3 Lensing Excess

There is a potential indication of a systematic error in the Planck data,
known as the $A_{\rm lens}$ anomaly. The parameter $A_{\rm lens}$ was
introduced in Ref. Calabrese et al. (2008) as a rescaling factor for the
effects of gravitational lensing on the CMB angular power spectra. The term
anomaly is employed since it is an unphysical parameter computed through
smoothing the peaks in the CMB damping tail. When $A_{\text{lens}}=0$, there
is no lensing effect, while $A_{\rm lens}=1$ corresponds to the value expected
in general relativity. In Ref. Aghanim et al. (2020a), an analysis of
$\Lambda$CDM+$A_{\text{lens}}$ finds $A_{\text{lens}}=1.180\pm 0.065$ at
$68\%$ CL. In fact, the Planck CMB power spectra show a preference for $A_{\rm
lens}>1$ at more than $95\%$ CL for both the standard Plik Aghanim et al.
(2020d) and the alternative CamSpec Efstathiou and Gratton (2019, 2020)
likelihoods, resulting in a significant improvement of the statistical fit.
When BAO data is included, the evidence for $A_{\rm lens}>1$ strengthens to
more than and approximately $99\%$ CL for Plik and for CamSpec, respectively.

Remarkably, the inclusion of the $A_{\rm lens}$ parameter in the analysis
introduces a slight alleviation of the cosmological tensions; namely, it
reduces the Hubble tension from $5\sigma$ to $3.9\sigma$ and mitigates the
$S_{8}$ tension with weak lensing experiments to less than $2\sigma$, but is
still insufficient to resolve the cosmic tensions fully. If $A_{\rm lens}$ is
attributed to systematic errors in the Planck data analysis, it is still
valuable to understand how such systematics affect the constraints on $H_{0}$,
$S_{8}$, and ultimately the cosmological tensions. Nevertheless, it should be
noted that ground-based CMB experiments such as ACT and SPT, which are
unaffected by the lensing excess, still find disagreement in the values of
$H_{0}$ and $S_{8}$.

The evidence for $A_{\rm lens}>1$ can also be interpreted as evidence for
missing physics in the theoretical frameworks. It may require considering a
closed Universe, which poses challenges to several observational measurements
and to the simplest inflationary models Di Valentino et al. (2019), Handley
(2021), Di Valentino et al. (2021c), or hint at alternative cosmological
theories that modify GR Ade et al. (2016a), Di Valentino et al. (2016),
Aghanim et al. (2020a). If the $A_{\rm lens}$ anomaly is otherwise associated
with some subtle and unobserved systematic error persistent through all
releases of the Planck data Ade et al. (2014), Di Valentino et al. (2013), Ade
et al. (2016b), Aghanim et al. (2020a) then its impact on the cosmological
tensions needs to be comprehended. Recent measurements from ACT, SPT-3G, and
the combination of Planck TT and SPTPol data all exhibit consistency with the
standard lensing effect predicted in the $\Lambda$CDM model, e.g. $A_{\rm
lens}^{\text{ACT}}=1.01\pm 0.11$ Aiola et al. (2020). This suggests that the
lensing anomaly is unique to the Planck data at small scales and not present
in other CMB experiments.

####  3.6.4 Hints for a Closed Universe from Planck Data

The enhanced lensing amplitude signal ($A_{\rm lens}$ anomaly) observed in the
CMB power spectrum Calabrese et al. (2008), Aghanim et al. (2020a), as
indicated by various data releases from the Planck collaboration, defies the
standard $\Lambda$CDM model paradigm. One of the explanations that have been
put forward is the possibility of having a closed Universe999Or something that
mimics the same signal and is being expressed through the variation of the
spatial curvature parameter., which is favoured over the flat scenario at
$3.4\sigma$ level Di Valentino et al. (2019), Aghanim et al. (2020a), Handley
(2021). This preference for a closed Universe is ascribed to the strong
degeneracy between the $A_{\rm lens}$ parameter and the spatial curvature
parameter $\Omega_{K}$. Moreover, a closed Universe could potentially address
the tension between the low and high multipoles of the angular CMB power
spectrum Addison et al. (2016), Aghanim et al. (2017), Di Valentino et al.
(2019). Nevertheless, both effects could be manifestations of hidden
systematics in the Planck data or even just statistical fluctuations.

However, while Planck 2018 Aghanim et al. (2020a) reports
$\Omega_{K}=-0.044^{+0.018}_{-0.015}$, implying evidence for a closed Universe
at about $3.4\sigma$, this evidence is reduced by replacing the baseline Plik
likelihood Aghanim et al. (2020d) with the alternative CamSpec Efstathiou and
Gratton (2019, 2020). That being said, the estimated value
$\Omega_{K}=-0.035^{+0.018}_{-0.013}$ still shows evidence above the $99\%$
CL. Other studies have also shown that taking a closed Universe in the context
of $\Lambda$CDM leads to an increase in the dark matter abundance, which in
turn exacerbates both the Hubble and $S_{8}$ tensions, leading to a
drastically small prediction of $H_{0}\sim 55\text{km/s/Mpc}$ Aghanim et al.
(2020a), Di Valentino et al. (2019), Handley (2021) from Planck.

However, alternative data combinations have been explored in the literature,
leading to varying degrees of preference (or lack thereof) for a closed
Universe. For example, the recent results from the ground-based experiment
ACT, in an analysis including data from the WMAP experiment, are fully
compatible with a flat Universe,
$\Omega_{K}^{\text{ACT}}=-0.001^{+0.014}_{-0.010}$ Aiola et al. (2020).
Bringing together the ACT data with a fraction of the Planck data recovers
some slight preference for the closed Universe scenario Aiola et al. (2020).
On the other hand, the combination of Planck and BAO Beutler et al. (2011),
Ross et al. (2015), Alam et al. (2017), Benisty and Staicova (2021), Planck
and CMB lensing Aghanim et al. (2020c) or Planck and Pantheon data Scolnic et
al. (2018), favour a flat Universe in all cases. However, the robustness of
this analysis is questionable since the combined data sets are in disagreement
at more than $3\sigma$ when the curvature is left to vary Di Valentino et al.
(2019), Handley (2021), Vagnozzi et al. (2021).

Moreover, while the spatial curvature prediction appears to be unaffected by
local measurements of the Hubble constant ($H_{0}$) from the Cepheid distance
ladder Zuckerman and Anchordoqui (2022), its variation exacerbates both the
$H_{0}$ and $S_{8}$ tensions Di Valentino et al. (2019). Therefore, adding the
curvature degree of freedom to the standard model cannot effectively account
for the observed tensions and anomalies. Surveys such as the recently launched
Euclid Amendola et al. (2013) have the potential to provide tighter
constraints on $\Omega_{K}$ \- optimistically of the order of $\sim 10^{-3}$
at 95% CL - and help break parameter degeneracies Di Dio et al. (2016),
Vardanyan et al. (2009) and solve the anomalies in the Planck data Leonard et
al. (2016).

##  Chapter 4 Methodology

> Vivemos exclusivamente no presente pois sempre e eternamente é o dia de hoje
> e o dia de amanhã será um hoje, a eternidade é o estado das coisas neste
> momento.00footnotetext: We live exclusively in the present because it is
> always eternally today and tomorrow will be a today, eternity is the state
> of things at this very moment. — Clarice Lispector in The Hour of the Star
> — Clarice Lispector A Hora da Estrela

This section outlines the relevant statistical methods and tools used in
cosmology for analysing observational data, focusing on estimating model
parameters. Any particular model can only make statistical predictions about
the Universe’s properties, meaning that we need to optimise the amount of
information that can be extracted from the available data to evaluate the
validity and predictions of these models effectively. For a more detailed
overview of statistical methods in cosmology, we refer to Liddle (2009),
Trotta (2008), Hobson et al. (2009).

In Section 4.1 we provide a brief summary of the theory and methodology for
statistical analysis and inference of parameters, including tests to assess
the goodness of fit of a particular model to the data and its evidence
compared with the standard $\Lambda$CDM case. In Section 4.2 we present a
brief review of the Einstein-Boltzmann codes in cosmology and their connection
with MCMC codes for model sampling. We close with a detailed account of the
data sets considered for the parameter estimates in this work.

###  4.1 Statistical Methods and Bayesian Inference

Cosmology is a unique science that fosters a strong synergy between
theoretical model development, numerical analysis, and the planning and
treatment of observational data. As new theories are put forth daily, data
analysis in cosmology has become essential to test and possibly falsify their
statistical support. This is part of the quest to find a framework that can
handle vast amounts of data and address existing incompatibilities while
navigating complexities such as theoretical biases, model independence,
overfitting due to excessive parameters, and parameter space degeneracies.
Ultimately, developing and improving current statistical methods linking
precision data with accurate theoretical predictions is paramount. Due to its
nature, analysis problems are inherently inverse problems, i.e. from collected
data, we aim to deduce something about the underlying physical processes that
generated the data. For instance, we may use Planck CMB data to fit the
$\Lambda$CDM model and then infer e.g. the relative amount of dark matter or
baryons from these observations.

Hence, we may categorise the data analysis process into three classes:

  1. 1.

Bayesian inference is the tool used to transform observations into constraints
on theoretical models, which should balance complexity for data explanation
and simplicity for concrete prediction.

  2. 2.

Parameter estimation aims to constrain the values of the model’s parameters
and their uncertainties, considering errors from experimental and inherent
statistical uncertainty from physical process randomness. A prominent example
is the limited predictability of the CMB pattern due to quantum processes
during inflation, which can nonetheless still be statistically modelled and
taken into account along with measurement uncertainties.

  3. 3.

Model selection involves selecting among multiple models proposed to explain
the observational data, each representing different physical processes and
offering a unique set of parameters to be varied for data fitting.

####  4.1.1 Bayesian Inference

Bayesian inference is the dish of the day in the statistical analysis methods
menu. Other subjects have traditionally favoured the frequentist approach,
which relies on the repeatability of the experiment. The essence of Bayesian
analysis lies in assigning probabilities to all variables, handling them based
on a set of rules that embody Bayes’ theorem. This theory aims to keep
updating our understanding as new data comes to light. This means that while
all subsequent steps follow some algorithm, a significant aspect of this
approach is the need to quantify our base knowledge before gathering new data.
This is called the prior probability, which can be subject to different views
among researchers.

The posterior probability, the probability that the model’s varied parameter
will adopt specific values after the data is collected, is represented as
$p(\theta|d)$. Here, $\theta$ represents the model’s unknown parameter, while
$d$ represents the observed data. Using this formulation, one can determine
the expected values of the parameters and their associated uncertainties.

Usually, it is easier to calculate the reverse probability, $p(d|\theta)$. For
example, we consider a Gaussian model with mean $\mu$ and variance
$\sigma^{2}$. This model includes two parameters, $\theta=(\mu,\sigma)$, and