Cosmic voids as cosmological laboratories Carlos Mauricio Correa FaMAF - UNC

2021

This work is under a Creative Commons Attribution 4.0 International License.

Dr. Dante Javier Paz Facultad de Matemática, Astronomía, Física y Computación
Universidad Nacional de Córdoba - República Argentina Doctorate Thesis

Authorship declaration

Thesis presented at the Faculty of Mathematics, Astronomy, Physics and
Computer Science111In Spanish, Facultad de Matemática, Astronomía, Física y
Computación. (FaMAF) of the National University of Córdoba222In Spanish,
Universidad Nacional de Córdoba. (UNC), in the Argentine Republic, as a
requirement to obtain the PhD degree in Astronomy.

I declare that I am the author of this thesis. Any source used directly or
indirectly is appropriately cited in the References section. I also declare
that I have not presented this thesis in any other institution in order to
obtain another degree.

This is a faithful English translation of the official version of this thesis,
which was written in Spanish. It is archived in the Daniel Oscar Sonzini
Library of FaMAF. Also available in digital format:
<https://rdu.unc.edu.ar/handle/11086/21041>.

This thesis is dedicated to my family. To my mom, Susana, who has always been
my emotional support, and taught me through her example how a good person
should be. To my dad, Carlos, who instilled in me a love for science, the
importance of working hard and honestly, and fundamentally, to never give up.
To my brother, José, who is also my best friend. To my grandparents, Sarita,
Marta, Carlos and Mauricio, who have been my second parents. To my girlfriend,
Daniela, the love of my life. Thank you all for your unconditional support.

My PhD research was carried out at the Institute for Theoretical and
Experimental Astronomy333In Spanish, Instituto de Astronomía Teórica y
Experimental. (IATE - CONICET - UNC) in the city of Córdoba, Argentina, and
was financially supported by the Argentine National Scientific and Technical
Research Council444In Spanish, Consejo Nacional de Investigaciones Científicas
y Técnicas. (CONICET). This project has also received a partial financial
support from the European Union’s Horizon 2020 Research and Innovation
programme under the Marie Sklodowska-Curie grant agreement number 734374 -
project acronym: LACEGAL, and from the Munich Institute for Astro- and
Particle Physics, which is funded by the Deutsche Forschungsgemeinschaft
(German Research Foundation) under Germany’s Excellence Strategy – EXC-2094 –
390783311 and the Technical University of Munich. Numerical calculations were
performed on the computer clusters from the UNC - High Performance Computing
Centre (<https://ccad.unc.edu.ar/>). Plots were made with the ggplot2 package
(??) of the R software (??). Figures 1, 1 and 2 were designed with the help of
Daniela Taborda, a special thanks to her. This thesis was structured from a
template developed by the abnTeX2 team (<http://www.abntex.net.br/>). I want
to specially acknowledge the hospitality of the Max Planck Institute for
Extraterrestrial Physics (MPE) and the Pontifical Catholic University of Chile
(PUC), where part of this work has been done.

A special appreciation to my PhD examiners: Dr. María Laura Ceccarelli (IATE -
CONICET - UNC), Dr. Yan-Chuan Cai (Institute for Astronomy - University of
Edinburgh) and Dr. Emanuel Gallo (Enrique Gaviola Institute for Physics -
CONICET - UNC), who have reviewed my work thoroughly and have made valuable
contributions to improve it. It has been an honour for me to have been
evaluated by such prestigious professionals.

I would like to thank my collaborators and also coauthors of my papers: Dr.
Dante Paz (IATE - CONICET - UNC), Dr. Nelson Padilla (PUC), Dr. Ariel Sánchez
(MPE), Dr. Andrés Ruiz (IATE - CONICET - UNC) and Dr. Raúl Angulo (Donostia
International Physics Center), for their invaluable contribution, help and
fruitful discussions. A special thanks to Dante, Ariel and Nelson for giving
me the opportunity to travel abroad to visit the MPE and PUC academically, to
present my work at various conferences and workshops, and to contact other
colleagues working on the same topic.

Finally, I would like to thank Dante for being an excellent supervisor, not
only academically, but also as a person.

No te des por vencido, ni aun vencido, no te sientas esclavo, ni aun esclavo;
trémulo de pavor, piénsate bravo, y arremete feroz, ya mal herido.

Almafuerte

[Abstract]

One of the major challenges of modern Cosmology is to understand the nature of
dark energy, which drives cosmic acceleration. In this sense, cosmic voids are
promising cosmological probes. Voids are vast underdense regions that encode
key information about the expansion history and geometry of the Universe. They
are receiving special attention nowadays in view of the new generation of
galaxy spectroscopic surveys, which will cover a volume and redshift range
without precedents. Furthermore, voids offer two advantages over traditional
clustering studies: their dynamics is less affected by non-linearities, and
Modified Gravity theories predict that potential deviations from the
predictions of General Relativity must be more pronounced in this type of
environments.

There are two primary statistics in void studies: the void size function,
which characterises their abundance, and the void-galaxy cross-correlation
function, which contains information about the density and velocity fields in
their surrounding regions. Both statistics are affected by distortions in the
observed spatial distribution of galaxies, which are originated by two main
sources: the dynamics of galaxies (redshift-space distortions, RSD), and the
fiducial cosmology assumed to establish a physical distance scale (Alcock-
Paczyński effect, AP). However, both of them are of high dynamical and
cosmological content.

We present a new cosmological test based on two perpendicular projections of
the correlation function with respect to the line-of-sight direction. We treat
correlations directly in terms of angular distances and redshift differences
between void-galaxy pairs, without the need to assume a fiducial cosmology.
The method is based on a physical model that reproduces the coupled RSD and AP
distortions on the correlation function, and also the mixture of scales due to
the projection ranges. The model primarily depends on two cosmological
parameters: the present-day matter fraction of the Universe and the growth
rate of cosmic structures. The covariance matrices associated with the method
are relatively small and their inversion is numerically more stable, which
reduces the noise in the likelihood analysis. This aspect allows us to
significantly reduce the number of mock catalogues needed to estimate them. We
tested the method with the Millennium XXL simulation for different redshifts
and projection ranges.

There is, however, an obstacle that has prevented the use of voids as reliable
tools. Our standard picture of distortions around voids is incomplete.
Traditionally, we have focused only on the spatial distribution of galaxies.
It turns out that the RSD and AP effects also affect global properties of
voids, such as their number, size and spatial distribution. This generates
additional distortion patterns on observations that lead to biased
cosmological constraints if they are not taken into account appropriately.
Moreover, given the precision achievable nowadays with modern surveys, it is
imperative a full description of these systematicities. Using a spherical void
finder and the Millennium XXL simulation, we addressed this problematic by
describing physically the connection between the identification in real space
(unaffected by distortions) and in redshift space (affected by them). We found
three important results. First, redshift-space voids above the shot-noise
level have a unique real-space counterpart spanning the same region of space,
hence void number conservation can be assumed. Second, the volume of voids is
affected by three sources: (i) the RSD induced by tracer dynamics at scales
around the void radius (t-RSD), which manifests as a systematic expansion,
(ii) the AP effect, which manifests as an overall expansion or contraction
depending on the fiducial cosmology, and (iii) the intrinsic ellipticity of
voids (e-RSD). Third, void centres are systematically shifted along the line-
of-sight direction, a by-product of a different class of RSD induced by the
global dynamics of the whole region containing the void (v-RSD). We also
present an evaluation of the impact that these redshift-space effects in voids
have on the void size function and the void-galaxy cross-correlation function.
In this way, we lay the foundations for improvements in current models for
these statistics in order to obtain unbiased cosmological constraints.

Finally, we present preliminary results from observational data, namely, the
Baryon Oscillation Spectroscopic Survey Data Release 12. The analysis is
focused on understanding the origin of the anisotropic patterns observed on
the correlation function by applying the methodology developed throughout the
work. We conclude that, in addition to its practical cosmological importance,
the analysis carried out in this thesis also has profound implications on
studies of the structure and dynamics of the Universe at the largest cosmic
scales.

Key words: large-scale structure of the Universe – cosmic voids – dark energy
– cosmological parameters – distance scale – galaxies: distances, redshifts
and velocities

lof.0lof.0List of FiguresList of Figureslof.0

###### List of Figures

  1. 1  Maps of the SDSS, BOSS and eBOSS spectroscopic surveys.
  2. 1  Redshift evolution of the different notions of distance used in Cosmology.
  3. 2  Fundamental cosmological experiments for studying dark energy: SNe Ia, CMB and BAO.
  4. 3  Evolution of the Universe.
  5. 1  Dynamical distortions in redshift space.
  6. 2  Visual and comparative impression of the RSD and AP effects.
  7. 1  Models for the abundance of voids: linear, SvdW and Vdn.
  8. 2  Velocity and integrated density contrast profiles for samples of voids selected from the SDSS survey.
  9. 3  Void-galaxy cross-correlation function for samples of voids selected from the SDSS survey.
  10. 1  Spatial distribution of dark matter in the Millennium XXL simulation.
  11. 2  Radius distribution of voids from the MXXL simulation (Table 1).
  12. 3  Slices of the MXXL simulation showing the spatial distribution of their dark-matter haloes and voids.
  13. 4  Radius distribution of voids from the BOSS DR12 survey.
  14. 1  Binning scheme used to measure the void-galaxy cross-correlation function in observable space.
  15. 2  POS and LOS projections of the void-halo cross-correlation function for a sample of voids selected from the MXXL simulation.
  16. 3  Density contrast and velocity profiles for a sample of voids selected from the MXXL simulation.
  17. 4  Normalised covariance matrices associated with the projected correlations method, measured with the MXXL simulation.
  18. 5  Evaluation of the cosmological test: marginalised likelihood distributions over the cosmological parameters $\Omega_{m}$ and $\beta$.
  19. 6  Evaluation of the cosmological test: marginalised likelihood distributions onto the plane $\Omega_{m}-\beta$.
  20. 7  Evaluation of the cosmological test: analysis of the full parameter space.
  21. 1  Void size functions of the TC void catalogues (Table 2).
  22. 2  Redshift-space effects in voids: expansion (t-RSD), AP change-of-volume and off-centring (v-RSD).
  23. 3  Statistical demonstration of the expansion effect.
  24. 4  Characteristics of the global movement of voids.
  25. 5  Isotropy in the global movement of voids.
  26. 6  Statistical demonstration of the off-centring effect: displacement.
  27. 7  Statistical demonstration of the off-centring effect: displacement and velocity.
  28. 8  Statistical demonstration of the independence between the expansion and off-centring effects.
  29. 1  Void size functions of the FC void catalogues (Table 2). Alcock-Paczyński correction.
  30. 2  Expansion-effect correction of void abundances.
  31. 3  Expansion-effect correction: comparative analysis between the full and bijective catalogues, and between the factors $q^{s}_{\mathrm{RSD}}$ and $q^{l}_{\mathrm{RSD}}$.
  32. 1  Samples of voids used to study the impact of the AP, t-RSD, v-RSD and e-RSD effects on the correlation function. Manifestations in real and redshift space.
  33. 2  POS and LOS projections of the correlation function measured in the $z\times z$\- and $r\times z$-space configurations. Comparison with model predictions.
  34. 3  Density contrast and velocity profiles: analysis of the $r\times r$-space configuration.
  35. 4  Impact of the impurity of a sample on the measurement of the correlation function due to non-bijective voids.
  36. 5  Impact of the off-centring effect on the measurement of the correlation function.
  37. 6  Density contrast maps measured in the $r\times r$-space configuration: anisotropies in real space.
  38. 7  Velocity maps measured in the $r\times r$-space configuration: over-expanding and under-expanding voids.
  39. 8  Impact of the ellipticity of voids on the measurement of the correlation function (e-RSD).
  40. 9  Testing the validity of the Gaussian streaming model by incorporating information about the ellipticity of voids.
  41. 1  POS and LOS projections of the void-galaxy cross-correlation function for a sample of voids selected from the BOSS DR12 survey. Comparison with the Patchy mocks.
  42. 2  Normalised covariance matrix measured with the Patchy mocks used to analyse BOSS data.
  43. 3  Marginalised likelihood distribution onto the plane $\Omega_{m}-\beta$ from the analysis of BOSS data.
  44. 4  Complementary analysis of Figure 1. Comparison between measurements and model predictions. Manifestation of the v-RSD and e-RSD effects in observations.

*

lot.0lot.0List of TablesList of Tableslot.0

###### List of Tables

  1. 1  Main characteristics of the dark-matter halo and void catalogues used to develop the cosmological test of Chapter 5.
  2. 2  Main characteristics of the void catalogues used to study the AP, t-RSD, v-RSD and e-RSD effects in Chapters 6, 7 and 8.

*

Two Degree Field Galaxy Redshift Survey

Alcock-Paczyński effect/test

Baryon acoustic oscillations

Big Bang Nucleosynthesis

Baryon Oscillation Spectroscopic Survey

BOSS Data Release 12

Code for Anisotropies in the Microwave Background

Cold dark matter

Cosmic microwave background

Dark energy

Dark Energy Spectroscopic Instrument survey

BOSS extension

Einstein - de Sitter cosmological model

Equation or formula

Ellipticity effect

Fiducial-cosmology catalogue of voids: $\Omega^{l}_{m}=0.2$-cosmology version

Fiducial-cosmology catalogue of voids: $\Omega^{u}_{m}=0.3$-cosmology version

Standard cosmological model

Friedmann-Robertson-Walker

Gaussian streaming model

Hobby-Eberly Telescope Dark Energy Experiment survey

Interquartile range

Line of sight

Markov Chain Monte Carlo

Millennium XXL cosmological simulation

Plane of the sky

Projection range

Configuration: $r$-space voids, $r$-space tracers

Redshift-space distortions

Real space

Configuration: $r$-space voids, $z$-space tracers

Sloan Digital Sky Survey

Supernovae Ia

Sheth $\&$ van de Weygaert model for the abundance of voids

True-cosmology catalogue of voids: MXXL-cosmology, $r$-space, bijective
version

True-cosmology catalogue of voids: MXXL-cosmology, $r$-space, full version

True-cosmology catalogue of voids: MXXL-cosmology, $z$-space, bijective
version

True-cosmology catalogue of voids: MXXL-cosmology, $z$-space, full version

Expansion effect

Volume-conserving model for the abundance of voids

Off-centring effect

Void size function

ZOnes Bordering On Voidness (void finder)

Redshift space

Configuration: $z$-space voids, $z$-space tracers

toc.0toc.0ContentsContentstoc.0

###### Contents

  1. 0 Introduction
  2. 1 The standard model of Cosmology
    1. 1 The cosmological principle
    2. 2 Dynamics of Friedmann-Robertson-Walker universes
    3. 3 Matter and radiation
    4. 4 Dark energy
    5. 5 Distances in Cosmology
      1. 1 The comoving distance
      2. 2 The cosmological redshift
      3. 3 The angular-diameter distance
      4. 4 The luminosity distance
    6. 6 The redshift-distance relation
      1. 1 Standard candles
      2. 2 Standard rulers: the Alcock-Paczyński test
    7. 7 Our Universe: flat-$\Lambda$CDM
  3. 2 Structures in the Universe
    1. 1 The density field and the power spectrum
    2. 2 The two-point correlation function
      1. 1 Estimators
    3. 3 The bias parameter
    4. 4 The evolution of density fluctuations
      1. 1 The evolution equations
      2. 2 The growth factor
      3. 3 The matter power spectrum
      4. 4 The peculiar velocity field
      5. 5 Non-linear evolution of density fluctuations
    5. 5 Redshift-space distortions
      1. 1 The observed redshift
      2. 2 The impact of peculiar velocities
      3. 3 The RSD parameter
    6. 6 The fiducial cosmology
  4. 3 Cosmic voids
    1. 1 Void evolution
    2. 2 The abundance of voids
    3. 3 The density field in voids
    4. 4 The velocity field in voids
    5. 5 The void-galaxy cross-correlation function
    6. 6 Redshift-space distortions around voids
      1. 1 Linear model
      2. 2 Gaussian streaming model
    7. 7 Cosmological relevance of voids
  5. 4 Data set
    1. 1 Millennium XXL simulation
    2. 2 Spherical void finder
    3. 3 Void catalogues I
    4. 4 Void catalogues II
    5. 5 The Baryon Oscillation Spectroscopic Survey
    6. 6 The MultiDark Patchy mock surveys
    7. 7 Void catalogues III
  6. 5 Designing a new cosmological test
    1. 1 Foundations of the test
      1. 1 Geometrical and dynamical distortions
      2. 2 Fiducial-free test and mixture of scales
    2. 2 The projected correlation functions
    3. 3 Model
      1. 1 Density profile
      2. 2 Velocity profile
      3. 3 Integrated density profile
    4. 4 Testing the method
      1. 1 Likelihood analysis and covariance matrices
      2. 2 Cosmological constraints
  7. 6 Redshift-space effects in voids
    1. 1 The standard picture of RSD around voids
    2. 2 Bijective mapping
    3. 3 Theoretical description
      1. 1 Void number conservation
      2. 2 Expansion effect
      3. 3 Alcock-Paczyński change-of-volume effect
      4. 4 AP and RSD combined contributions
      5. 5 Off-centring effect
    4. 4 Statistical analysis
      1. 1 Correlations between $r$-space and $z$-space void radii
      2. 2 Correlations between displacement of centres and net velocity
      3. 3 Cross correlations
  8. 7 Impact on the void size function
    1. 1 A comment about the VSF modelling
    2. 2 Alcock-Paczyński correction
    3. 3 Expansion-effect correction
    4. 4 Free of off-centring effect and cosmological relevance
  9. 8 Impact on the correlation function
    1. 1 Void sample
    2. 2 The projected correlation functions revisited
      1. 1 Different configurations
      2. 2 Simplification of the model
    3. 3 Impact of the impurity of the sample
    4. 4 Expansion-effect correction
    5. 5 Alcock-Paczyński correction
    6. 6 Off-centring-effect correction
    7. 7 Void ellipticity
    8. 8 Towards an improved model
  10. 9 Analysis of BOSS DR12 data
    1. 1 The projected correlation functions in BOSS
    2. 2 Analysis of distortions
  11. 10 Conclusions
    1. 1 Designing a new cosmological test
    2. 2 Redshift-space effects in voids
    3. 3 Impact on the void size function
    4. 4 Impact on the correlation function
    5. 5 Analysis of BOSS DR12 data

*

##  Chapter 0 Introduction

Modern Cosmology rests on six observational pillars.

  1. 1.

The expansion of the Universe. The radial velocity of galaxies, measured by
means of the shift of spectral lines, is positive for nearly all of them
(redshift), i.e. they appear to be moving away from us. Moreover, this
velocity increases linearly with the distance. This is Hubble’s law,
interpreted such that the Universe is expanding. We are able to trace back the
cosmic expansion under the assumption that the known laws of Physics were also
valid in the past. From that we get the Big Bang model of the Universe,
according to which our Universe has evolved out of a very dense and hot state.

  2. 2.

The cosmic microwave background (CMB). This is a background electromagnetic
radiation coming from all directions with an almost perfect black-body
spectrum, confirming the smooth, hot and dense past of the Universe. The
temperature of this radiation is (??)

$T_{\mathrm{CMB}}=2.72548\pm 0.00057\leavevmode\nobreak\ \mathrm{K}.$ (1)

  3. 3.

The abundance of light elements. The observed abundance of the chemical
elements shows that the Universe is composed of about $75\%$ of hydrogen-1 and
$25\%$ of helium-4, with small traces of deuterium, helium-3 and lithium, and
negligible heavier elements, in good agreement with the predictions of the Big
Bang Nucleosynthesis theory (BBN). Moreover, BBN indicates that ordinary
matter represents only a $15\%$ of total matter, and hence provides a
compelling argument for the existence of non-baryonic dark matter (??).

  4. 4.

The supernova experiment. Hubble’s law is inaccurate to explain the redshift-
distance relation of Supernovae Ia (SNe Ia), the farthest standard candles
that can be observed, for which the notion of distance must be redefined
according to the cosmological model used to explain the evolution of the
Universe. The surprising result is that data are not compatible with a flat
matter-dominated universe, nor are they with an open universe. Instead, data
favour a flat Universe with $30\%$ matter and $70\%$ of an enigmatic component
known as dark energy. The existence of this component means that the Universe
is currently in a phase of accelerated expansion, a completely unexpected
result (????). This astonishing result means that dark matter and dark energy
make up $95\%$ of the total energy budget of the Universe, whose true nature
we ignore.

  5. 5.

The large-scale spatial distribution of galaxies. The last decades saw a
number of large spectroscopic surveys of galaxies designed to measure
structure in the Universe. These culminated in two large surveys, the Two
Degree Field Galaxy Redshift Survey (??, 2dFGRS) and the Sloan Digital Sky
Survey (??, SDSS), which compiled the redshift of a million galaxies. The
left-hand panel of Figure 1 shows the spatial distribution of the galaxies in
the SDSS. Each dot is a galaxy, the colour gradient shows the local density.
This map shows that the galaxies are clearly not distributed randomly, the
Universe has structure on large scales. For instance, by the action of
gravity, galaxies form groups, clusters and even larger structures like
filaments and walls. Complementarily, they leave on their way vast underdense
regions in this process, the so-called cosmic voids (darkest zones in the
figure). However, no evidence of structures with dimensions larger than
$150\leavevmode\nobreak\ h^{-1}\mathrm{Mpc}$ have been found. Hence, the
Universe seems to be basically homogeneous if averaged over these scales. The
existence of dark matter and dark energy plays a fundamental role when trying
to explain this large-scale distribution from a theoretical point of view.

  6. 6.

Anisotropies in the CMB. Although the CMB is almost homogeneous and isotropic,
there are small anisotropies in the temperature distribution at the level
between $10^{-4}$ and $10^{-5}$, the seeds of structure. Present-day
measurements of the anisotropies in the CMB are the best example of the
accuracy attainable by current cosmological experiments. For this reason, it
is said that we have entered an era of high-precision Cosmology. These
observations provide tight constraints on the basic set of parameters of
cosmological models, and have been decisive in establishing the current
standard model of Cosmology (??).

Figure 1:  Left-hand panel. The SDSS map of the Universe. Each dot is a
galaxy; the colour bar shows the local density. The darkest zones are the
underdense regions of the Universe, and hence, potential cosmic voids. Right-
hand panel. BOSS and eBOSS coverage of the Universe. Figure credit: M. Blanton
$\&$ SDSS <https://www.sdss.org/>.

In this scenario, one of the major challenges of modern Cosmology is to
understand the nature of dark energy. The standard model rests on two
fundamental hypotheses. One is the cosmological principle, which states that
the Universe is spatially homogeneous and isotropic on large scales. The other
is that gravitational interactions are described by Einstein’s General Theory
of Relativity. One plausible explanation for the dark-energy phenomenon is
that it is a manifestation of the cosmological constant present in Einstein’s
field equations, which can be also associated with the vacuum energy.
Alternatively, this could be an indication that General Relativity must be
reviewed, leading to the field of Modified Gravity theories.

Cosmic voids are the largest observable structures. Therefore, they encode key
information about the expansion history and geometry of the Universe, emerging
in this way as natural cosmological probes to test the standard model.
Moreover, they do not only constitute a complementary view of the large-scale
structure, but they offer two main advantages over galaxy clustering studies.
On the one hand, void dynamics can be treated linearly, hence it is easier to
model systematicities such as redshift-space distortions. On the other hand,
Modified Gravity theories predict that deviations from General Relativity
should be more pronounced in these low density environments. In view of this,
cosmic voids are ideal for testing different dark-energy models. We will
tackle the study of voids in Chapter 3.

The cosmological potential of voids has increased recently with the
development of the new generation of spectroscopic surveys. Some of them are
the SDSS Baryon Oscillation Spectroscopic Survey (??, BOSS), its extension
eBOSS (??), the Hobby-Eberly Telescope Dark Energy Experiment (??, HETDEX),
the Dark Energy Spectroscopic Instrument (??, DESI) and the Euclid mission
(??), which will probe our Universe covering a volume and redshift range
without precedents. This will allow us to obtain rich samples of voids at
different redshifts, and in this way, to test the evolution of the Universe
with high precision. The right-hand panel of Figure 1 shows the coverage
planned for BOSS and eBOSS.

There are two primary statistics in void studies: the void size function and
the void-galaxy cross-correlation function. We will develop these tools in
Chapter 3. The void size function, on the one hand, describes the abundance of
voids. It is analogous to the mass function of dark-matter haloes, hence it
can be modelled using the excursion set theory combined with the spherical
evolution of matter fluctuations derived from the cosmological perturbation
theory. The void-galaxy cross-correlation function, on the other hand,
characterises the void density field when considered at small to intermediate
scales. Both statistics are affected by distortions in the observed spatial
distribution of galaxies, which translate into anisotropic patterns in the
measurements. There are two main sources of distortions: the redshift-space
distortions (??, hereinafter RSD), a dynamical effect caused by the peculiar
velocities of the galaxies, and the Alcock-Paczyński effect (??, hereinafter
AP), a geometrical effect caused by the selection of a fiducial cosmology
necessary to transform the angles and redshifts provided by a survey into
distances expressed in physical units. These systematicities can be modelled
from physical grounds, hence they encode valuable information about the
dynamics of the large-scale structure and the cosmology in which it is
embedded.

The present thesis can be divided into five parts. In the first part, we will
present the theoretical background and necessary tools to develop the work,
and also to discuss the main conclusions. This part will be developed in
Chapters 1, 2, 3 and 4. In Chapter 4, in particular, we will present the data
set used throughout the work. Specifically, we will describe the numerical
simulations and mock catalogues used to calibrate the different tests and
models, the void finding method, and finally, the spectroscopic survey used to
compare observational results with the corresponding theoretical predictions.

Next, in the second part, we will present a new cosmological test based on the
void-galaxy cross-correlation function. This will be developed in Chapter 5.
Our method provides two novel aspects. In the first place, we treat
correlations directly in terms of void-centric angular distances and redshift
differences between void-galaxy pairs, hence it is not necessary to assume a
fiducial cosmology. In this way, the AP effect is taken into account
naturally. Secondly, we model a third type of systematicity besides the RSD
and AP effects that also affects the comparison between predictions and
observations. This is the mixture of scales, which is due to the binning
scheme used in the measurement process. In particular, this allows us to work
with two perpendicular projections of the correlation function with respect to
the line-of-sight direction. This variant of measuring correlations offers
some advantages over the traditional case. For instance, it represents two
simple one-dimensional profiles affected by distortions in a different way,
with associated covariance matrices easier to manipulate. The first aspect
implies that any degeneration in the parameter space due to the coupled RSD
and AP distortions can be easily broken, whereas the last aspect means that
the covariance matrices are smaller and numerically more stable, which reduces
the noise in the likelihood analysis and allows us to use a smaller number of
mock catalogues to estimate them.

The statistical properties of voids depend on the method used to identify them
from the spatial distribution of galaxies. There are different classes of void
finders. Despite the intrinsic differences between them, though, there is a
consensus on their basic statistical properties. However, the RSD and AP
distortions have a direct impact on the void identification process itself,
affecting global properties of voids, such as their number, size and the
position of their centres. This is true for all methods, and translates into
additional anisotropic patterns in the measurements of the void size function
and the correlation function. Given the precision achievable nowadays with
modern surveys, it is extremely important to detect and model all these
systematicities in order to obtain unbiased cosmological constraints. One
solution is to use the reconstruction technique, an algorithm to approximately
recover the real position of the galaxies based on the Zel’dovich
approximation (????). This technique has been recently applied to data (????),
showing a robust power in constraining the cosmological parameters. However,
it also presents some disadvantages, particularly, it loses the physical
information about the structure and dynamics of voids encoded in these
additional anisotropic patterns, which could otherwise be extracted when voids
are directly identified from the observed galaxies. In the present work, we
propose an alternative approach. With the help of a numerical simulation, we
will analyse the void finding method in order to find a physical connection
between the identification with and without distortions, and in this way, be
able to detect the physical effects responsible for these additional
systematic effects. This is the third part of this work, developed in Chapter
6.

In the fourth part, we will analyse the impact that these additional void
systematicities have on the measurements of the void size function and the
void-galaxy cross-correlation function. This will be developed in Chapters 7
and 8. We will focus the analysis on the theoretical framework developed in
Chapter 6 to model them. The importance of this framework resides in the fact
that it depends strongly on cosmological considerations, hence it must be
incorporated in current models for the abundance of voids and the correlation
function. In this way, we lay the foundations for a full treatment that will
allow us to obtain unbiased cosmological constraints from modern spectroscopic
surveys. But besides this practical cosmological importance, this also has
profound implications in our understanding of the large-scale structure of the
Universe, since these effects encode key information about the structure and
dynamics of voids.

Finally, in Chapter 9, we will present a preliminary analysis of the void-
galaxy cross-correlation function measured from observational data taken from
the BOSS DR12 survey. The main goal will be to understand the origin of the
observed anisotropic patterns in the context of the dynamical and cosmological
effects that affect voids studied in the previous chapters.

##  Chapter 1 The standard model of Cosmology

Before we start111This chapter and the next one are mainly based on the
lectures on Formation and evolution of cosmic structures given by Dr. ??) at
the Ludwig Maximilians University in Munich, and on the books of ??) and ??).,
a comment about a few important conventions to be followed. In the first
place, the word universe will refer to a generic space-time, whereas our
particular Universe will be denoted with a capital letter. We will use units
in which the speed of light is $c=1$, although we will include it explicitly
when relevant to compare with results from the literature. We will also use
the convention in which Latin indices represent spatial coordinates, running
from $1$ to $3$ (e.g. $i=1,2,3$), whereas Greek letters represent space-time
coordinates, running from $0$ to $3$ (e.g. $\mu=0,1,2,3$). Moreover, we will
use the summation convention in which identical lower and upper indices are
summed over all their possible values. A dot over a variable will represent a
derivative with respect to the time coordinate $t$, whereas an apostrophe next
to a variable, a derivative with respect to conformal time $\tau$. Finally,
the covariant derivative will be denoted by $\nabla_{\mu}$, and the brackets
$\langle\rangle$ will denote an average over an assembly of universes with the
same statistical properties.

###  1 The cosmological principle

The standard model of Cosmology rests on the following two fundamental
hypotheses.

  1. 1.

Validity of General Relativity. Gravitational interactions, which shape the
global evolution of the Universe, are described by Einstein’s General Theory
of Relativity.

  2. 2.

Cosmological principle. This principle assumes that the Universe is spatially
homogeneous and isotropic on large scales. This means that there are no
preferred locations nor directions. Of course, this hypothesis must be
understood in a statistical sense. For instance, the Universe exhibits large
density fluctuations, such as galaxies, galaxy clusters, filaments and voids.

Under these hypotheses, the most general form that the metric tensor
$g_{\mu\nu}$ can adopt is the Friedmann-Robertson-Walker (FRW) metric:

$ds^{2}=-dt^{2}+a^{2}(t)[\gamma_{ij}(x^{k})dx^{i}dx^{j}],$ (1)

where $\gamma_{ij}$ is the spatial part of the metric, and $x^{i}$ denotes
comoving coordinates. In these coordinates, the change in position of a given
point of space as the universe evolves is absorbed into the scale factor
$a(t)$. In terms of spherical coordinates, the FRW metric can be written as

$ds^{2}=-dt^{2}+a^{2}(t)[d\chi^{2}+S^{2}_{K}(\chi)(d\theta^{2}+\mathrm{sin}^{2}\theta
d\phi^{2})],$ (2)

where we used the convention for the radial coordinate given by

$d\chi:=\frac{dr}{\sqrt{1-Kr^{2}}},$ (3)

and

$S_{K}(\chi):=\left\\{\begin{array}[]{lll}\frac{1}{\sqrt{K}}\mathrm{sin}(\sqrt{K}\chi)&\mathrm{if}\leavevmode\nobreak\
K>0\\\ \chi&\mathrm{if}\leavevmode\nobreak\ K=0\\\
\frac{1}{\sqrt{|K|}}\mathrm{sinh}(\sqrt{|K|}\chi)&\mathrm{if}\leavevmode\nobreak\
K<0.\end{array}\right.$ (4)

This metric is completely determined by two quantities: $a(t)$, which
determines the kinematics of the universe, and $K$, which characterises its
spatial curvature. Depending on the value of $K$, FRW universes can be
classified as open $(K<0)$, flat $(K=0)$ or closed $(K>0)$. These terms refer
to the topological properties of the spatial hypersurfaces of the universe.

In some cases, it will be convenient to express the metric in terms of
conformal time $\tau$, defined by the relation

$dt=a(\tau)d\tau,$ (5)

so that

$ds^{2}=a^{2}(\tau)[-d\tau^{2}+\gamma_{ij}(x^{k})dx^{i}dx^{j}].$ (6)

In this case, the FRW metric can be factorised into a static metric, for which
$K=0$ corresponds to the Minkowski metric, multiplied by a time-dependent
factor.

###  2 Dynamics of Friedmann-Robertson-Walker universes

The relation between the geometry of space-time, characterised by the metric
tensor $g_{\mu\nu}$, and the energetic components that it contains,
encompassed by the energy-momentum tensor $T_{\mu\nu}$, is governed by
Einstein’s field equations:

$G_{\mu\nu}=8\pi GT_{\mu\nu},$ (7)

where $G$ is the Newtonian constant of gravitation, and $G_{\mu\nu}$ is the
Einstein tensor, defined as

$G_{\mu\nu}:=R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R.$ (8)

Here, $R_{\mu\nu}$ is the Ricci tensor, and $R:=g^{\mu\nu}R_{\mu\nu}$, the
scalar curvature.

When the FRW metric is introduced into Einstein’s field equations, we obtain
equations for $a(t)$ and $K$. To do this, we need to first assume a form for
the energy-momentum tensor. The most general form of $T_{\mu\nu}$ that is
compatible with the cosmological principle is that of a perfect fluid:

$T^{\mu\nu}=(p+\rho)U^{\mu}U^{\nu}+pg^{\mu\nu},$ (9)

where $U^{\mu}$ is the four-velocity of the fluid, and $\rho$ and $p$
correspond to its energy density and pressure, respectively. In this way, the
field equations are reduced to two independent equations, the so-called
Friedmann’s equations:

$\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8\pi G}{3}\rho-\frac{K}{a^{2}}$ (10)

and

$\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}(\rho+3p).$ (11)

Einstein’s field equations implicitly contain the energy conservation
equation: $\nabla_{\nu}T^{\mu\nu}=0$. This equation can also be obtained by
differentiating Eq. (10) and combining it with Eq. (11), which leads to the
following relation:

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=0.$ (12)

To derive explicit solutions for $a(t)$ and $K$, we can use Eq. (10) in
combination with Eq. (12). To do so, it is also necessary to assume a relation
between the pressure and density of the ideal fluid in the form of an equation
of state: $p=p(\rho)$. Using this relation it is possible to integrate Eq.
(12) to obtain $\rho=\rho(a)$. In Cosmology, we are mostly interested in
barotropic fluids, for which the pressure is proportional to the density:
$p=w\rho$, and are then characterised by a constant sound speed:
$c^{2}_{s}=dp/d\rho$. The parameter $w$ is known as the equation-of-state
parameter. For barotropic fluids, the energy conservation equation implies
that

$\rho\propto a^{-3(w+1)}.$ (13)

If the universe contains $N$ different components with equations of state
given by $w_{i}$, this result will hold for each species separately as long as
they do not interact. If we denote as $\rho_{i,0}$ the present-day density of
a given species $i$, then the total energy density of the universe at a time
$t$ is given by

$\rho=\sum_{i=1}^{N}\rho_{i,0}a^{-3(w_{i}+1)},$ (14)

where we have set $a_{0}:=a(t_{0})=1$ for simplicity, and $t_{0}$ denotes the
present time. In this way, Eq. (10) can be written as

$\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8\pi
G}{3}\sum_{i=1}^{N}\rho_{i,0}a^{-3(w_{i}+1)}-\frac{K}{a^{2}}.$ (15)

This is the desired equation for $a(t)$.

We define now the Hubble parameter:

$H:=\frac{\dot{a}}{a}.$ (16)

Note that $H>0$, since we are interested in expanding universes ($\dot{a}>0$).
Therefore, this parameter quantifies the expansion rate of the universe at a
time $t$. The present-day value of $H$ is denoted as $H_{0}$, and referred to
as the Hubble constant. This constant is usually parametrised in terms of a
dimensionless quantity $h$ in the following way:

$H_{0}=100\leavevmode\nobreak\ h\leavevmode\nobreak\
\mathrm{km}\leavevmode\nobreak\ \mathrm{s}^{-1}\leavevmode\nobreak\
\mathrm{Mpc}^{-1}.$ (17)

Using this definition, and evaluating Eq. (15) at present time, we get the
following expression:

$K=H_{0}\left(\frac{8\pi G}{3H_{0}^{2}}\sum_{i=1}^{N}\rho_{i,0}-1\right).$
(18)

This is the desired equation for $K$.

In Eqs. (15) and (18), there are $N+1$ unknowns: $H_{0}$ and the densities
$\rho_{i,0}$. They are not variables. They are measurable quantities that must
be obtained from observational experiments. In this way, it is clear that the
geometry of space-time and its evolution are determined by the energy content
of the universe. In particular, it will be flat (K = 0) if the total energy
density is equal to a critical density, $\rho_{c}$, given by

$\rho_{c}=\frac{3H_{0}^{2}}{8\pi G}=2.773\times
10^{11}h^{-1}\mathrm{M}_{\odot}(h^{-1}\mathrm{Mpc})^{-3}=11.26\leavevmode\nobreak\
h^{2}\leavevmode\nobreak\ \frac{\mathrm{protons}}{\mathrm{m}^{3}}.$ (19)

Instead of working with the quantities $\rho_{i,0}$, it is a common practice
to define a dimensionless density parameter for each species by expressing
their energy density in units of the critical density:

$\Omega_{i}:=\frac{\rho_{i,0}}{\rho_{c}}=\frac{8\pi G}{3H_{0}^{2}}\rho_{i,0}.$
(20)

A similar expression is defined for the curvature:

$\Omega_{K}:=-\frac{K}{H_{0}^{2}}.$ (21)

In this way, it is possible to express the equation for the curvature (Eq. 18)
in a more compact form:

$\sum_{i=1}^{N}\Omega_{i}+\Omega_{K}=1.$ (22)

Also, with this notation, the equation governing $a(t)$ (Eq. 15) can be
written in the following way:

$\dot{a}^{2}=H_{0}^{2}\left[\sum_{i=1}^{N}\Omega_{i}a^{-(3w_{i}+1)}+\Omega_{K}\right].$
(23)

As an example, consider a flat universe with a unique energy component
characterised by an equation of state with $w\neq-1$. Then, the time evolution
of the scale factor is given by

$a(t)\propto t^{\frac{2}{3(1+w)}}.$ (24)

###  3 Matter and radiation

In this section, we analyse two main components of the universe: matter and
radiation, reviewing the consequences that each of them has on the evolution
of the scale factor.

Matter (m) behaves as a fluid with vanishing pressure, i.e. $w_{m}=0$, which
according to Eq. (13), this implies that

$\rho_{m}\propto a^{-3}.$ (25)

This simply states that the density of matter decreases in proportion to the
volume of the universe as it expands. From Eq. (24), we see that the evolution
of the scale factor in a flat matter-dominated universe ($\Omega_{m}=1$ and
$\Omega_{K}=0$) is given by

$a(t)\propto t^{2/3}.$ (26)

This solution is also known as the Einstein-de Sitter (EdS) universe.

Radiation ($\gamma$), on the other hand, behaves as a relativistic fluid with
an equation of state given by $w_{\gamma}=1/3$, which according to Eq. (13),
this implies that

$\rho_{\gamma}\propto a^{-4}.$ (27)

This dependence can be understood in the following way: as the universe
expands, a factor $a^{-3}$ can be associated with a change in the volume
element, whereas an additional factor $a^{-1}$ can be associated with the
stretching of the radiation wavelength. As in the case of matter, we can study
the behaviour of a flat radiation-dominated universe by inspecting Eq. (24).
This leads to

$a(t)\propto t^{1/2}.$ (28)

Analysing the dependences with the scale factor in Eqs. (25) and (27), it is
straightforward to realise that matter dominates the evolution and energy
budget of the universe at later times, but radiation is the dominant component
at early times. Furthermore, a comparison between Eqs. (26) and (28) shows
that the universe expands more slowly in the radiation-dominated era than in
the matter-dominated era.

###  4 Dark energy

In recent years, a variety of precise cosmological observations have shown
that our Universe is currently undergoing a phase of accelerated expansion
($\ddot{a}>0$). According to Eq. (11), this requires the presence of a new and
strange component with an equation of state $w_{\mathrm{DE}}<-1/3$, known as
dark energy (DE). Fluids for which $\rho+3p\geq 0$ are said to satisfy the
strong energy condition. This implies that dark energy must violate this
condition. There are several models to explain this phenomenon. Here, we will
only focus on the cosmological constant and its association with the vacuum
energy.

Given the fact that the field equations lead to dynamic universes, Einstein
originally modified them in order to find static solutions by introducing an
additional term proportional to the metric tensor, as

$G_{\mu\nu}-\Lambda g_{\mu\nu}=8\pi GT_{\mu\nu}.$ (29)

With the introduction of the constant $\Lambda$, known as the cosmological
constant, the Friedmann’s equations (Eqs. 10 and 11) are modified as

$\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8\pi
G}{3}\rho-\frac{K}{a^{2}}+\frac{\Lambda}{3}$ (30)

and

$\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}(\rho+3p)+\frac{\Lambda}{3}.$ (31)

Later, due to the solid evidence about the expansion of the Universe, the
cosmological constant was left aside. However, models with $\Lambda>0$ can
also lead to solutions with $\ddot{a}>0$, hence the $\Lambda$-term appeared
again as a possible explanation for this phenomenon.

It is common to find an association between the cosmological constant and the
vacuum energy. Zero-point fluctuations must satisfy Lorentz invariance, and
therefore the relation $\langle
T_{\mu\nu}\rangle=\rho_{\mathrm{vac}}g_{\mu\nu}$, which corresponds to a
barotropic fluid with an equation of state given by $w=-1$. In this way, the
contribution from the vacuum must be included in the right-hand side of
Einstein’s field equations:

$G_{\mu\nu}-\Lambda g_{\mu\nu}=8\pi
G(T_{\mu\nu}+\rho_{\mathrm{vac}}g_{\mu\nu}).$ (32)

This clearly shows that the effect of the vacuum energy cannot be
distinguished from that of a cosmological constant. The net effect is that of
an effective cosmological constant given by

$\Lambda_{\mathrm{eff}}:=\Lambda+8\pi G\rho_{\mathrm{vac}},$ (33)

with an effective energy density given by

$\rho_{\mathrm{eff}}:=\frac{\Lambda}{8\pi G}+\rho_{\mathrm{vac}}.$ (34)

Even when both phenomena produce the same effect, they are very different in
nature. A cosmological constant implies that the gravitational constant $G$ is
not enough to describe the behaviour of gravity correctly, so that $\Lambda$
must be taken into account as a second constant of nature. The vacuum energy,
on the other hand, is related to the energy content of the universe,
associated with zero-point fluctuations, which, even in the absence of matter,
are capable of curving space-time. As these two cases cannot be distinguished,
we will not include explicit terms for both in our equations, but rather we
will treat this case simply as a dark-energy component characterised by the
parameter $w_{\mathrm{DE}}=-1$.

Setting $w_{\mathrm{DE}}=-1$ in Eq. (13) shows that the corresponding energy
density, $\rho_{\mathrm{DE}}$, remains constant as the universe expands.
Introducing this behaviour into Friedmann’s equations, we obtain that, in this
case, the time evolution of the scale factor is given by

$a(t)\propto\mathrm{exp}(H_{0}t),$ (35)

i.e. the universe expands exponentially.

Before ending this section, we mention two alternative models for dark energy.
One is quintessence, in which dark energy is given by a scalar field. The
other is known as phantom energy, which not only violates the strong energy
condition, but also the so-called weak energy condition: $\rho+p\geq 0$. In
this model, $\rho_{\mathrm{DE}}$ increases as the universe expands, and it is
characterised by a strange property, the Big Rip, in which $a(t)$ diverges at
a finite time.

###  5 Distances in Cosmology

In this section, we study different distance measurements than can be used as
means to describe the Universe. These distance definitions only rely on the
assumption that the metric takes the FRW form. So far, we have used natural
units in which $c=1$, convenient for theoretical purposes. However, they are
not the most practical choice when dealing with observations. Therefore, we
will go back to traditional units in which $c$ appears explicitly, remembering
that we must replace the time coordinate in Eq. (2) by the product $ct$.

####  1 The comoving distance

The expansion of the universe is contained in the scale factor. The distance
between two objects fixed in comoving coordinates, known as proper distance,
$D_{p}$, changes according to $a(t)$. The proper distance to an object with
coordinates $(r,\theta,\phi)$, can be computed from the spatial part of the
FRW metric given by Eq. (2), taking our position as the origin ($r=0$) and
pointing the coordinate system so that $d\phi=d\theta=0$:

$D_{p}(t)=a(t)\int_{0}^{r}\frac{dr^{\prime}}{\sqrt{1-Kr^{\prime
2}}}=a(t)\chi.$ (36)

We can clearly see the physical meaning of the radial coordinate $\chi$,
defined from Eq. (3), which represents the comoving distance to the object. We
will be mostly interested in the present-day proper distance between two
points in space, which is given then by the corresponding comoving distance,
since we have set $a_{0}=1$.

####  2 The cosmological redshift

The cosmological redshift, $z$, represents the change in the wavelength of
radiation due to the expansion of the universe. It is defined as

$z:=\frac{\lambda_{0}-\lambda_{e}}{\lambda_{e}},$ (37)

where $\lambda_{0}$ is the wavelength observed at a time $t_{0}$, and
$\lambda_{e}$ is the wavelength emitted at a time $t_{e}<t_{0}$. The redshift
is a very important physical quantity as it is directly accessible through
spectroscopic observations.

We can find a relation between the redshift and the scale factor at the time a
source emits a light beam by studying the propagation of the light ray in an
expanding universe described by an FRW metric. For this, let us consider a ray
propagating radially from $r=0$ to $r=r_{0}$, and let us orient the coordinate
system so that $d\phi=d\theta=0$. The origin is now at the emitting source. As
light travels along null geodesics ($ds=0$), we find that

$\frac{c\leavevmode\nobreak\ dt}{a(t)}=d\chi=\frac{dr}{\sqrt{1-Kr^{2}}}.$ (38)

Integrating this relation from $r=0$ to $r=r_{0}$, and considering the
emission and reception times between two contiguous peaks of the light wave,
it can be seen that

$\frac{\lambda_{e}}{\lambda_{0}}=\frac{a(t_{e})}{a(t_{0})}.$ (39)

In an expanding universe, $a(t_{0})>a(t_{e})$, which means that the wavelength
increases. This demonstrates that the cosmic expansion leads to a redshift of
spectral lines. If $t_{0}$ represents the present time, so that $a(t_{0})=1$,
we get the following relation between the cosmological redshift and the scale
factor:

$a=\frac{1}{1+z}.$ (40)

This means that the light we receive from a distant galaxy at a redshift $z$
left this galaxy when the scale factor was a fraction $1/(1+z)$ of its present
value.

As the redshift $z$ is a direct observable, it is often a more convenient
variable than the time $t$. From Eq. (38), and the following relation:

$H(z)=-\frac{1}{(1+z)}\frac{dz}{dt},$ (41)

obtained in turn from Eqs. (16) and (40), we can rewrite the comoving distance
in terms of $z$:

$\chi(z)=c\int_{0}^{z}\frac{dz^{\prime}}{H(z^{\prime})},$ (42)

with

$H(z)=H_{0}\left[\sum_{i=1}^{N}\Omega_{i}(1+z)^{3(w_{i}+1)}+\Omega_{K}(1+z)^{2}\right]^{1/2}.$
(43)

Note that for small redshifts, $z\ll 1$,

$\chi(z)\approx\frac{c\leavevmode\nobreak\ z}{H_{0}}.$ (44)

We recovered Hubble’s law. Here, $cz$ is the apparent receding velocity of a
source.

The redshift is a direct observable quantity, but the distance is not.
Nevertheless, the redshift is also a distance indicator. In fact, the comoving
distance to a source can be calculated from its redshift by means of Eq. (42).
However, this calculation depends on the Hubble parameter, given by Eq. (43),
which in turn depends on the parameters $H_{0}$ (or $h$), $\Omega_{i}$ and
$\Omega_{K}$, the so-called cosmological parameters. Therefore, these
parameters must be known in order to estimate distances. We will now see that
there are different observational definitions of distance that are closely
related to $\chi$, and that can be used to design tests in order to estimate
the value of these parameters. These are not truly distances, but recipes to
compute different physical aspects of a source, like its light or its size.

####  3 The angular-diameter distance

Let us first consider an extended object at a redshift $z$ with an angular
diameter $\Delta\phi$. Using Eq. (2), and setting $d\chi=d\theta=0$, we can
compute the physical (proper) size of the object as

$\Delta l=a(t)S_{K}(\chi)\Delta\phi.$ (45)

In an Euclidean universe, the corresponding relation is

$\Delta l=D\Delta\phi,$ (46)

where $D$ denotes the distance to the object. A comparison between these two
expressions leads to the definition of the angular-diameter distance:

$D_{A}(z):=\frac{\Delta l}{\Delta\phi}=\frac{S_{K}(\chi(z))}{1+z}.$ (47)

For a flat universe, $S_{K}(\chi(z))=\chi(z)$, which corresponds to the
comoving distance (as we have set $a_{0}=1$, it also corresponds to the true
physical distance).

Figure 1 shows the redshift evolution of all types of distances defined in
this work for a universe similar to ours with parameters $h=0.72$,
$\Omega_{K}=0$, $\Omega_{m}=0.31$ and $\Omega_{\Lambda}=0.69$. Note, in
particular, that the angular-diameter distance of an object is systematically
smaller than its actual distance. In fact, although for nearby objects both
are similar, $D_{A}$ goes to zero as $z$ tends to infinity. This means that,
beyond a turning point, objects that are further away have larger angular
sizes. This apparently counter-intuitive behaviour can be understood if we
notice that we are using as a reference an object of fixed proper size, which
corresponds to a larger comoving size as $z$ increases.

Figure 1:  Redshift evolution of the comoving distance (green solid curve, Eq.
42), the angular-diameter distance (blue dashed curve, Eq. 47), and the
luminosity distance (red dot-dashed curve, Eq. 52) for a universe similar to
ours with parameters $h=0.72$, $\Omega_{K}=0$, $\Omega_{m}=0.31$ and
$\Omega_{\Lambda}=0.69$.

####  4 The luminosity distance

Let us now consider a source of light at a redshift $z$. For this object, we
define the luminosity distance, denoted by $D_{L}$, by means of the relation
between the observed bolometric flux, $S$, and its intrinsic bolometric
luminosity, $L$, in an Euclidean space-time:

$D_{L}:=\sqrt{\frac{L}{4\pi S}}.$ (48)

Two effects must be taken into account to understand the redshift evolution of
the luminosity distance. First, individual photons lose energy due to the
cosmological redshift, and second, photons arrive less frequently as the
universe expands. In more detail, the bolometric luminosity of a source at a
time $t_{1}$ is given by

$L=\int N_{1}(\nu_{1})h_{P}\nu_{1}d\nu_{1},$ (49)

where $N_{1}(\nu_{1})$ is the total number of photons of frequency $\nu_{1}$
emitted from the source per unit time, and $h_{P}$ is the Planck constant. If
these photons are emitted isotropically, the flux observed at a time $t_{0}$
at a comoving distance $\chi$ from the source will be given by

$S=\int n_{0}(\nu_{0})h_{P}\nu_{0}d\nu_{0},$ (50)

where $n_{0}(\nu_{0})$ is the received number of photons of frequency
$\nu_{0}$ per unit area per unit time, related to $N_{1}(\nu_{1})$ by

$n_{0}(\nu_{0})d\nu_{0}=\frac{N_{1}(\nu_{1})}{4\pi
a_{0}^{2}\chi^{2}}\frac{d\nu_{0}}{d\nu_{1}}d\nu_{1},$ (51)

with $\nu_{1}=\nu_{0}a(t_{0})/a(t_{1})$. Putting these ingredients together
into Eq. (48), we finally get

$D_{L}(z)=(1+z)\chi(z).$ (52)

This means that, as the redshift reduces the apparent luminosity of distant
objects, they appear to be further away than they really are (see Figure 1).

###  6 The redshift-distance relation

Let us assume that we could simultaneously measure the distances and redshifts
of some objects of the same type distributed throughout the Universe. Then, we
could build a redshift-distance relation, also known as a Hubble diagram.
Measuring redshifts from the shift of spectral lines is straightforward, the
hard part is determining distances from an independent method. However, based
on the definitions of the luminosity and angular-diameter distances given
before, together with their dependence on the comoving distance, the redshift-
distance relation can then be used to derive constraints on the cosmological
parameters involved, and in this way, to quantify the expansion rate and
energy budget of the Universe.

####  1 Standard candles

If we could identify a type of objects of the same intrinsic luminosity $L$,
and measure their fluxes $S$ at different redshifts $z$, then it would be
possible to build a Hubble diagram using the luminosity distance: $D_{L}(z)$.
Such an object is known as a standard candle.

In practice, what is relevant is that any difference between the apparent
brightness (measured in terms of astronomical magnitudes) of two such objects
is a direct result of their different distances from us. In this way, the
Hubble diagram is represented in terms of a redshift-magnitude relation. In
some cases, the method is generalised to find a correlation between an
observable and intrinsic brightness. For example, Cepheid variables are stars
for which their intrinsic brightness is tightly related to their period. The
Hubble Space Telescope was able to measure the periods of thousands of Cepheid
variables in galaxies as far away as $20\leavevmode\nobreak\ \mathrm{Mpc}$.
With distances to these galaxies fixed, several secondary distance indicators
could be calibrated to go much further, as far away as
$400\leavevmode\nobreak\ \mathrm{Mpc}$. In this way, it was possible to
calibrate the Hubble constant (??).

The standard candle that can be seen at largest distances is SN Ia. Since they
are so bright, SNe Ia can be used to extend the Hubble diagram to very large
redshifts ($z>1$), a regime where the simple linear Hubble’s law (Eq. 44)
ceases to work, and therefore they can be used to test the redshift evolution
of the luminosity distance. SNe Ia have provided key measurements for the
discovery of the current phase of accelerated expansion of the Universe
(??????).

####  2 Standard rulers: the Alcock-Paczyński test

We turn now to the case in which we could identify a type of objects of the
same physical size $\Delta l$, and measure their angular sizes $\Delta\phi$ at
different redshifts $z$. Then, it would be possible to build a Hubble diagram
using the angular-diameter distance: $D_{A}(z)$. Such an object is known as a
standard ruler. One example is the signal of the baryon acoustic oscillations
(BAO) imprinted on the clustering of galaxies, a relic of the early Universe,
whose pattern is due to the acoustic waves experimented by the primordial
photo-baryonic fluid. See for instance the work of ??).

We can do even more with a standard ruler. Let us assume that, instead of
being extended across the sky, it is located along the line of sight, and it
is large enough for us to measure the redshift difference between its two
endpoints, $\Delta z$. Then, according to Eqs. (36), (40) and (42), and
assuming that the scale factor and the Hubble parameter remain almost constant
between the endpoints, we find that the physical size of the object is given
by

$\Delta l=\frac{c\Delta z}{(1+z)H(z)}.$ (53)

Therefore, measurements of this kind can be used to probe the redshift
evolution of the Hubble factor as $H(z)=c\Delta z/(1+z)\Delta l$, providing a
direct measurement of the expansion rate of the Universe.

These tests require the knowledge of the true physical size $\Delta l$ of our
standard ruler. Even if this is not the case, we can still infer cosmological
information from these objects. What we need is to identify them in both
directions, i.e. some of them across the sky and some of them along the line
of sight. In other words, we can exploit the measurements of the angular and
redshift extent of the ruler to obtain cosmological information. By equating
Eqs. (45) and (53), we obtain the following relation:

$F_{\mathrm{AP}}(z):=\frac{\Delta z}{\Delta\phi}=\frac{(1+z)}{c}D_{A}(z)H(z).$
(54)

This is the Alcock-Paczyński test (hereinafter AP test), first proposed by
??). The combination $F_{\mathrm{AP}}$ is referred to as the Alcock-Paczyński
parameter. This test will be fundamental for our study of voids.

###  7 Our Universe: flat-$\Lambda$CDM

In this section, we summarise the main characteristics of our Universe, like
its energy content and geometry, provided by cosmological experiments.

  1. 1.

Expansion rate. Current observations show that $h=0.74\pm 0.01$ (??). This
means that the critical density is approximately 6 protons per cubic metre
(Eq. 19). Nevertheless, a tension has been detected between this value,
obtained from observations of the local Universe, and that obtained from
studies of the early Universe, which suggest a lower value of $h=0.674\pm
0.005$ (??). In view of this uncertainty in the value of $h$, it is a common
practice to express distances and masses in units of $h^{-1}\mathrm{Mpc}$ and
$h^{-1}\mathrm{M}_{\odot}$, respectively.

  2. 2.

Radiation. We can estimate the current energy density of radiation from the
temperature of CMB photons, Eq. (1):

$\rho_{\mathrm{\gamma,0}}=\frac{\pi^{2}k_{B}^{4}}{15\hbar_{P}^{3}c^{3}}T_{\mathrm{CMB}}^{4}=4.5\times
10^{-34}\frac{\mathrm{g}}{\mathrm{cm}^{3}},$ (55)

where $k_{B}$ is the Boltzmann constant. This implies that

$\Omega_{\gamma}=2.4\times 10^{-5}\leavevmode\nobreak\ h^{-2}.$ (56)

In principle, three massless neutrino species would contribute a similar
amount. Beyond this aspect, it is then possible to neglect the contribution of
radiation to the present-day energy budget of the Universe.

In view of this, we will hereinafter focus on models containing only matter
and dark energy, and also assume that $w_{\mathrm{DE}}=-1$. In this case, Eq.
(23) can be rewritten as

$\dot{a}^{2}=H_{0}^{2}[\Omega_{m}a^{-1}+\Omega_{\mathrm{DE}}a^{2}+(1-\Omega_{m}-\Omega_{\mathrm{DE}})].$
(57)

This relation also provides an expression for the age of the Universe:

$t_{0}=\frac{1}{H_{0}}\int_{0}^{1}\frac{da}{\sqrt{\Omega_{m}a^{-1}+\Omega_{\mathrm{DE}}a^{2}+(1-\Omega_{m}-\Omega_{\mathrm{DE}})}}.$
(58)

It is useful to study the behaviour of various models in the
$\Omega_{m}-\Omega_{\mathrm{DE}}$ plane, as the one shown in Figure 2, which
can be divided into regions with different properties. This figure contains
the parameter constraints from three fundamental cosmological experiments: the
study of CMB anisotropies, the extended Hubble diagram using SNe Ia, and the
AP test using the BAO signal on the galaxy clustering. The importance of these
results resides in the fact that the three methods are completely independent
from each other, and yet they are consistent. But more important is that the
confidence regions have substantially different orientations, hence their
combination provides a much tighter constraint than each method by itself.

Figure 2:  Allowed regions of the parameter space spanned by $\Omega_{m}$ and
$\Omega_{\Lambda}$, as derived from three fundamental cosmological
experiments: the study of CMB anisotropies, the extended Hubble diagram using
SNe Ia, and the AP test using the BAO signal on the galaxy clustering. Since
the confidence regions overlap and have substantially different orientations
in this plane, their combination provides a much better constraint than each
method by itself. Void-based experiments are promising cosmological probes
that will contribute to these constraints even further. Figure credit: ??).

  1. 3.

Curvature. Current observations suggest that the Universe is flat to a high
precision: $|\Omega_{K}\leq 10^{-5}|$ (??).

  2. 4.

Baryonic matter. This is ordinary matter, the one that interacts with light.
There are four established ways of measuring the baryon density: (i) observing
the baryon content directly (the greatest contribution to the density comes,
not from stars in galaxies, but rather from gas in groups and clusters of
galaxies), (ii) looking at spectra of distant quasars (the amount of light
absorbed from these beacons is a measure of the intervening hydrogen), (iii)
the analysis of anisotropies in the CMB, and (iv) the analysis of the light-
elements abundance according to the predictions of BBN. Remarkably, these
estimates with very different techniques all agree. They all place the baryon
density at $5\%$ of the critical density.

  3. 5.

Dark matter. All of the methods mentioned above involve the interaction of
matter and radiation. The methods that do not rely on this interaction usually
involve measuring the gravitational field in a given system, thereby inferring
information about the total mass responsible for that field. We mention four
methods: (i) studying the spatial distribution of galaxies by means of the
power spectrum, (ii) measuring the cosmic velocity field and its relation with
the observed galaxy distribution, (iii) picking out observations sensitive to
$\Omega_{b}/\Omega_{m}$, such as the ratio of the mass of gas in clusters to
the total mass by looking for X-ray emission, and (iv) the analysis of
anisotropies in the CMB. All these experiments favour a value
$\Omega_{m}=0.31$, which means that the total matter density is equal to
approximately $31\%$ of the critical density. We therefore have an enormous
amount of evidence telling us that the total matter density is approximately
five times larger than the baryon density. Most of the matter in the Universe
must be in an unknown form referred to as dark matter. Moreover, it is said
that dark matter must be cold (CDM) in order to explain the observed large-
scale structure, in contrast to hot or warm dark matter. The differences
between them lie in the thermal velocities of their constituents when
radiation and matter had equal density. The particles of CDM were non-
relativistic at that time. Currently, the most plausible candidate for dark
matter is a weakly interacting massive particle.

  4. 6.

Dark energy. In recent years, three methods have provided a solid evidence
about the existence of a dark-energy component responsible for the current
phase of accelerated expansion: (i) the extended Hubble diagram using distant
SNe Ia, (ii) the analysis of anisotropies in the CMB, and (iii) the AP test
using the BAO signal imprinted on the galaxy clustering. They all suggest a
value $\Omega_{\mathrm{DE}}=0.69$ for the density, and $w_{\mathrm{DE}}=-1$
for the equation of state. Hereinafter, we will consider that dark energy is
described by a cosmological constant, and hence we will use the notation
$\Omega_{\Lambda}$ to refer to its density.

  5. 7.

Age of the Universe. According to Eq. (58), the present age of the Universe is
estimated to be $t_{0}=13.8\leavevmode\nobreak\ \mathrm{Gyr}$, a time by which
the Universe is entering into the dark-energy dominated era, and hence,
approaching the exponential expansion.

The cosmological model presented in this chapter, with all the characteristics
describing our Universe, is called the flat-$\Lambda$CDM model. The left-hand
panel of Figure 3 shows the evolution of the densities of radiation, total
matter and dark energy as a function of the scale factor, which follow the
relations described in Sections 3 and 4. It is also shown the evolution of the
total density. The right-hand panel shows the corresponding evolution of the
scale factor as a function of time. In both panels, three different epochs in
the history of the Universe can be seen, in which its energy budget is
dominated by radiation during the early phase, matter at later times, and dark
energy today.

Figure 3:  Left-hand panel. Evolution of the energy density as a function of
the scale factor for the different constituents of our Universe: radiation
(red dashed line), matter (black long-dashed line) and dark energy in the form
of a cosmological constant (green dot-dashed line). The total contribution is
represented by the blue solid curve. Right-hand panel. Evolution of the scale
factor (or equivalently, the redshift) as a function of time. In both panels,
it can be appreciated the different cosmological eras. Radiation dominated at
early times, whereas matter dominated at later times. Apparently, dark energy
is the dominant component nowadays.

##  Chapter 2 Structures in the Universe

In this chapter, we develop the relevant statistical tools for the analysis of
the large-scale structure of the Universe, and in this way, be able to tackle
the study of cosmic voids in the next chapter. The main aspect to consider is
to avoid scales dominated by non-linearities. While perturbations to matter on
small scales, less than $10\leavevmode\nobreak\ h^{-1}\mathrm{Mpc}$, have
grown non-linear, larger-scale perturbations are still small, hence linear
theory can be assumed.

###  1 The density field and the power spectrum

The statistical analysis of the large-scale structure is based on the density
contrast, $\delta(\mathbf{x})$, which describes the density fluctuation field
of the Universe, and is defined as

$\delta(\mathbf{x}):=\frac{\rho(\mathbf{x})-\bar{\rho}}{\bar{\rho}},$ (1)

where $\bar{\rho}$ represents the mean density of the Universe, and
$\rho(\mathbf{x})$, the local density at position $\mathbf{x}$. In order to
distinguish the density contrast of the total matter (dark plus baryonic) from
that of galaxies, we will use the notations $\delta_{m}$ and $\delta_{g}$,
respectively. The same will apply to any other quantity to be defined. If no
index is specified, an arbitrary distribution of objects will be implicitly
assumed.

The analysis of small density fluctuations in linear theory is simpler in
Fourier space, since different Fourier modes evolve independently. Our
convention will be

$\delta(\mathbf{x})=\frac{1}{(2\pi)^{3}}\int\delta(\mathbf{k})e^{i\mathbf{k}.\mathbf{x}}d^{3}k.$
(2)

Here, $\mathbf{k}$ is a wave vector whose modulus, $k=|\mathbf{k}|$,
characterises a given scale of space.

A useful tool to characterise fluctuations is the power spectrum, $P(k)$,
defined via the following relation:

$\langle\delta(\mathbf{k})\delta^{*}(\mathbf{k}^{\prime})\rangle=(2\pi)^{3}\delta_{D}(\mathbf{k}-\mathbf{k}^{\prime})P(k),$
(3)

where the angular brackets denote an average over the whole distribution, and
$\delta_{D}$, the Dirac delta function. Qualitatively speaking, the power
spectrum is the spread or variance of the amplitude distribution of
fluctuations on a given scale. If there are a lot of prominent underdense and
overdense regions, the power spectrum will be large, whereas it will be small
if the distribution is smooth.

###  2 The two-point correlation function

The power spectrum is a useful statistical tool for theoretical grounds.
However, in practice, it is often simpler to use other methods and relate them
to the power spectrum. Of particular importance is the two-point correlation
function, defined as

$\xi(\mathbf{r}):=\langle\delta(\mathbf{x})\delta(\mathbf{x}-\mathbf{r})\rangle.$
(4)

As clustering is statistically isotropic, the correlation function must solely
depend on the separation $r=|\mathbf{r}|$ between two points, i.e.
$\xi(\mathbf{r})=\xi(r)$. Expressing the density fluctuations in terms of
their Fourier transform, we find that

$\xi(r)=\frac{1}{(2\pi)^{3}}\int P(k)e^{i\mathbf{k}.\mathbf{r}}d^{3}k,$ (5)

which shows that the correlation function is simply the Fourier transform of
the power spectrum. As the power spectrum is also isotropic, we can perform
the angular part of the integral independently. Using the fact that $\xi(r)$
must be a real function, the correlation function can be finally written as

$\xi(r)=\frac{1}{2\pi^{2}}\int P(k)\frac{\mathrm{sin}(kr)}{kr}k^{2}dk.$ (6)

Although the general shape of the correlation function is complex, it can be
empirically described by a power law over a limited range of scales. Indeed,
measurements of the correlation function from early galaxy surveys were
correctly described by fitting functions of the form

$\xi_{g}(r)=\left(\frac{r}{r_{0}}\right)^{\alpha}.$ (7)

Typical values of these parameters are $\alpha\approx-1.8$ and $r_{0}\approx
5h^{-1}\mathrm{Mpc}$. As larger galaxy surveys provided more accurate
measurements of $\xi_{g}(r)$, deviations from this simple behaviour were
detected, mostly caused by non-linear effects on small scales.

####  1 Estimators

An alternative definition of the correlation function for a discrete set of
point particles is related to the probability of finding pairs of such
particles as a function of their separation. For a completely homogeneous
distribution, the probability of finding a pair of particles in two volume
elements $dV_{1}$ and $dV_{2}$ separated by a distance $r_{12}$ is given by

$dP_{12}=\bar{\rho}^{2}dV_{1}dV_{2}.$ (8)

For a general distribution, this probability can be computed as

$dP_{12}=\langle\rho_{1}\rho_{2}\rangle dV_{1}dV_{2}.$ (9)

Expressing the densities according to Eq. (1) as
$\rho_{i}=\bar{\rho}(1+\delta_{i})$, this expression can be written as

$dP_{12}=\bar{\rho}^{2}(1+\langle\delta_{1}\delta_{2}\rangle)dV_{1}dV_{2}=\bar{\rho}^{2}(1+\xi(r_{12}))dV_{1}dV_{2}.$
(10)

This means that the correlation function, $\xi(r)$, gives the excess
probability of finding pairs of particles separated by a distance $r$ with
respect to a homogeneous distribution.

The estimation of the correlation function for the galaxy distribution depends
then on the probability of finding galaxy pairs separated by distances between
$r$ and $r+dr$, $DD(r)$. That is,

$DD(r)=\frac{N_{\mathrm{pairs}}(r)}{N_{\mathrm{tot}}},$ (11)

where $N_{\mathrm{pairs}}(r)$ is the number of galaxy pairs inside the volume
element defined by $r$ and $r+dr$, a spherical shell, and $N_{\mathrm{tot}}$
is the total number of pairs. In turn, $N_{\mathrm{tot}}=N_{g}(N_{g}-1)/2$,
where $N_{g}$ is the total number of galaxies. The probability of finding
pairs of galaxies in a homogeneous distribution, $RR(r)$, can be computed in
an analogous way by means of a random catalogue. From Eqs. (8) and (10), we
can see that both probabilities are related by $DD(r)=RR(r)(1+\xi_{g}(r))$. We
can then estimate the two-point correlation function for the galaxy
distribution as

$\hat{\xi}_{g}(r)=\frac{DD(r)}{RR(r)}-1,$ (12)

which is known as the natural estimator.

A number of alternative estimators based on pair counts have been developed. A
few examples are:

$\hat{\xi}_{g}(r)=\frac{DD(r)}{DR(r)}-1,$ (13)

from ??), and

$\hat{\xi}_{g}(r)=\frac{DD(r)-2DR(r)+RR(r)}{RR(r)},$ (14)

from ??). These estimators depend on the quantity $DR(r)$, which measures the
probability of finding pairs with one element in the galaxy sample and the
other element in the random catalogue (in which case
$N_{\mathrm{tot}}=N_{g}N_{\mathrm{ran}}$, with $N_{\mathrm{ran}}$ the total
number of random particles). The first two estimators are more sensitive to
edge effects in the sample. The performance of the last one is the most common
choice in current clustering analyses.

###  3 The bias parameter

Measurements of the correlation function of galaxy samples of varying colour
or luminosity exhibit different amplitudes. This is just an indication that
the galaxy clustering does not necessarily match that of the underlying matter
distribution. In general, we can assume that the galaxy density fluctuation
field is a function of the local matter density fluctuation field, which can
be expanded as

$\delta_{g}(\mathbf{x})=\sum_{k=1}^{\infty}\frac{b_{k}}{k!}\delta_{m}^{k}(\mathbf{x}),$
(15)

where the coefficients $b_{k}$ are known as the bias parameters.

In the linear regime, where $\delta_{m}\ll 1$, we can then expect a simple
linear relation of the form

$\delta_{g}(\mathbf{x})=b_{1}\delta_{m}(\mathbf{x}),$ (16)

in which case the linear bias parameter is simply denoted as $b$. In this
case, the galaxy and matter correlation functions are related by

$\xi_{g}(r)=b^{2}\xi_{m}(r).$ (17)

Galaxies form in dark-matter haloes. It is possible to derive analytical and
semianalytical recipes for the dependence of the bias parameter with halo
mass: $b(M)$. Then, assuming a given value for the amplitude of the scalar
power spectrum (see Section 3), the amplitude of the correlation function of
different galaxy samples can provide information about the mass of the dark-
matter haloes in which they live, or more generally, the way in which they
populate haloes of different masses. This information is valuable for galaxy
formation models, but is of little cosmological interest. Our current
understanding of the physical processes that control the formation and
evolution of galaxies is not detailed enough to predict the bias of a given
galaxy population with high accuracy. This means that no cosmological
information can be extracted from the amplitude of the correlation function or
the power spectrum. Only the shape of these statistics is used when
constraining the cosmological parameters.

###  4 The evolution of density fluctuations

The structure that we observe today evolved from small density fluctuations in
the early Universe. The process that describes this evolution is gravitational
instability. As time evolves, matter accumulates in initially overdense
regions. Complementarily, initially underdense regions become emptier. These
are the seeds of cosmic voids. Gravitational instability can be schematically
depicted in the following way:

$\ddot{\delta}+[\mathrm{pressure}-\mathrm{gravity}]\delta=0.$ (18)

The two basic forces, gravity and pressure, act in opposite directions. There
are different manifestations of this basic idea, where different ambient
cosmological conditions alter the growth rate. For example, in the matter-
dominated era, $\delta$ grows as a power of time, whereas in the radiation-
dominated era, the growth is logarithmic.

####  1 The evolution equations

In order to find the equations that govern the evolution of small departures
from the smooth FRW cosmological models, we will assume that the metric tensor
can be written as

$g_{\mu\nu}=\bar{g}_{\mu\nu}+\delta g_{\mu\nu},$ (19)

where $\bar{g}_{\mu\nu}$ represents the FRW metric, and $\delta g_{\mu\nu}$
characterises the small deviations in this metric. In the same way, we must
include small deviations in the energy-momentum tensor from that of an ideal
fluid, $\delta T_{\mu\nu}$. Perturbations in the metric and the energy-
momentum tensors are related by Einstein’s field equations, which, after
subtracting the terms corresponding to the homogeneous background model, and
keeping only linear terms in the metric perturbations, they result in the
linearised field equations:

$\delta G_{\mu\nu}=8\pi G\delta T_{\mu\nu}.$ (20)

Working in the longitudinal gauge, also known as the conformal Newtonian
gauge, the metric perturbations are characterised by the gauge invariant
variables $\Phi$ and $\Psi$, so that the metric takes the form

$ds^{2}=a^{2}(\tau)[-(1+2\Phi)d\tau^{2}+(1-2\Psi)\gamma_{ij}dx^{i}dx^{j}].$
(21)

In the absence of $\Phi$ and $\Psi$, this expression is simply the FRW metric.
Similarly, in the absence of expansion $(a=1)$, this metric describes a weak
gravitational field. The perturbations have the following interpretation:
$\Phi$ corresponds to the Newtonian potential, whereas $\Psi$, the
perturbation to the spatial curvature. A comment worth mentioning. Eq. (21)
only contains scalar perturbations. In principle, it is possible that the
metric also has vector or tensor perturbations. If so, $g_{\mu\nu}$ would
require other functions besides $\Phi$ and $\Psi$ to fully describe all
perturbations. However, scalar, vector and tensor perturbations are decoupled:
each one evolves independently of the others. Therefore, we will solely focus
on scalar perturbations, these are the only ones that couple to matter
perturbations.

In this context, the linearised field equations are given by

$\nabla^{2}\Phi-3\mathcal{H}(\Phi^{\prime}+\mathcal{H}\Phi)=4\pi
Ga^{2}\delta\rho,$ (22) $\nabla_{i}(\Phi^{\prime}+\mathcal{H}\Phi)=4\pi
Ga^{2}(\bar{\rho}+\bar{p})v_{i}$ (23)

and

$\Phi^{\prime\prime}+3\mathcal{H}\Phi^{\prime}+(2\mathcal{H}^{\prime}+\mathcal{H}^{2})\Phi=4\pi
Ga^{2}\delta p.$ (24)

Here, the fluctuation in density, $\delta\rho$, the fluctuation in pressure,
$\delta p$, and the peculiar velocity of the fluid,

$v^{i}:=\frac{dx^{i}}{d\tau},$ (25)

are the perturbative variables relevant in $\delta T_{\mu\nu}$. Moreover,
$\mathcal{H}:=a^{\prime}/a$ is the Hubble parameter defined in terms of
conformal time $\tau$. Note that $\Psi$ does not appear in these equations.
This is because we have assumed that $\Phi\approx\Psi$, an approximation valid
in the limit when there are no quadrupole moments in the photon distribution.
On subhorizon scales, the second term of Eq. (22) can be neglected, and the
equation takes then the form of Poisson’s equation.

Rather than the full linearised field equations, it is more convenient to use
the energy conservation equations: $\nabla^{\mu}T_{0,\mu}=0$ and
$\nabla^{\mu}T_{i,\mu}=0$, which correspond to the general relativistic
versions of the continuity and Euler’s equations, respectively. More
specifically, we need their linearised versions. Using the relation between
pressure and density in the homogeneous case, $\bar{p}=w\bar{\rho}$, and
relating their perturbations by means of the sound speed, $c_{s}^{2}=\delta
p/\delta\rho$, we find that

$\delta^{\prime}+3\mathcal{H}(c_{s}^{2}-w)\delta+(1+w)(\nabla_{i}v^{i}-3\Phi^{\prime})=0$
(26)

and

$v^{\prime}_{i}+\mathcal{H}(1-3c_{s}^{2})v_{i}+\frac{c_{s}^{2}}{1+w}\nabla_{i}\delta+\nabla_{i}\Phi=0.$
(27)

Let us consider the case of a universe containing only dark matter ($w_{m}=0$)
and radiation ($w_{\gamma}=1/3$) for this analysis, and let us focus on the
scalar part of the velocity field: $v^{i}=\nabla^{i}v$. The system is then
completely specified by the density fluctuations, $\delta_{m}$ and
$\delta_{\gamma}$, their respective velocity potentials, $v_{m}$ and
$v_{\gamma}$, and the potential $\Phi$. After Fourier-transforming, the
continuity and Euler’s equations for the matter and radiation components take
the form

$\delta^{\prime}_{m}-k^{2}v_{m}=3\Phi^{\prime},$ (28)
$\frac{3}{4}\delta^{\prime}_{\gamma}-k^{2}v_{\gamma}=3\Phi^{\prime},$ (29)
$v^{\prime}_{m}+\mathcal{H}v_{m}=-\Phi$ (30)

and

$v^{\prime}_{\gamma}+\frac{1}{4}\delta_{\gamma}=-\Phi.$ (31)

In order to find solutions for the five perturbative variables $\delta_{m}$,
$v_{m}$, $\delta_{\gamma}$, $v_{\gamma}$ and $\Phi$, we need to complement
these equations with one of the linearised field equations.

####  2 The growth factor

Using the continuity and Euler’s equations for the matter component, and
Poisson’s equation in the limit $k\gg\mathcal{H}$ (neglecting the contribution
$\delta_{\gamma}$ from radiation), we get the following second order equation:

$\delta^{\prime\prime}_{m}+2\mathcal{H}\delta^{\prime}_{m}-4\pi
Ga^{2}\rho\delta_{m}=0.$ (32)

It is remarkable that neither does this equation contain derivatives with
respect to the spatial coordinates, nor do the coefficients depend on them.
Therefore, this equation has solutions of the form

$\delta_{m}(k,a)=D(a)\tilde{\delta}_{m}(k),$ (33)

i.e. the spatial and temporal dependences factorise. This differential
equation has two linearly independent solutions. One of them increases with
time, whereas the other decreases. If, at some early time, both functional
dependences were present, the increasing solution dominates at later times,
whereas the decreasing solution becomes irrelevant. Therefore, we will only
consider the increasing solution, characterised by the so-called growth factor
denoted by $D_{+}(a)$.

This mathematical aspect allows us to draw two important conclusions. First,
the type of solution indicates that, in linear perturbation theory, the
spatial shape of the density fluctuations is frozen in comoving coordinates,
only their amplitude increases. Second, the growth factor of the amplitudes,
$D_{+}(a)$, follows a simple ordinary differential equation that is easily
solvable for any cosmological model. This is important because it allows us to
incorporate cosmological models with dark energy. In fact, it can be shown
that for arbitrary values of the density parameter of matter and dark energy,
the growth factor has the form

$D_{+}(a)=\frac{5\Omega_{m}H(a)}{2H_{0}}\int_{0}^{a}\frac{d\tilde{a}}{(\tilde{a}H(\tilde{a})/H_{0})^{3}},$
(34)

which is normalised to give $D_{+}(a)=a$ during matter domination.

####  3 The matter power spectrum

According to the law of linear structure growth of Eq. (33), the corresponding
linear evolution of the matter power spectrum can also be factorised into a
spatial and temporal dependence:

$P_{m}(k,a)=D^{2}_{+}(a)P_{p}(k)T^{2}(k).$ (35)

The spatial part, in turn, is made up of two components: the primordial power
spectrum, $P_{p}(k)$, and the transfer function, $T(k)$.

Today, the most accepted theory that explains the origin of density
fluctuations in the Universe is Inflation. This theory predicts that quantum-
mechanical perturbations in the very early Universe were first produced when
the relevant scales were casually connected. Then, these scales were whisked
outside the horizon by inflation, a brief period of exponential expansion,
only to reenter much later to serve as initial conditions for the growth of
structure. Multiple inflationary models differing on the details of how
inflation starts and ends have been proposed, but almost all models where
inflation is caused by a single scalar field lead to some generic predictions:
(i) the amplitude of the density fluctuations follow a Gaussian probability
distribution, (ii) the density fluctuations are adiabatic, (iii) inflation
generates a background of gravitational waves, and (iv) the density
fluctuations are characterised by a nearly scale-invariant power spectrum.
This last aspect implies that

$P_{p}(k)\propto k^{n_{s}},$ (36)

with small deviations from the value $n_{s}=1$. This parameter is known as the
spectral index, and a power spectrum of the form $P_{p}(k)\propto k$ is also
known as a Harrison-Zeldovich-Peebles spectrum, crediting the people who first
proposed it, a proposal that predates inflation by many years.

As we mentioned at the beginning of this section, different ambient
cosmological conditions alter the growth rate, and hence, the shape of the
power spectrum. Initially, the wavelength of a fluctuation is larger than the
horizon and its amplitude remains constant. Once a given mode enters the
horizon, it begins to grow. This growth is logarithmic during radiation
domination, and linear with the scale factor during matter domination. The
transfer function describes the evolution during these epochs, particularly,
during horizon-crossing and the radiation-matter transition. It is not easy to
describe the behaviour of $T(k)$ analytically, except for two limit cases:
when perturbations enter the horizon well inside the radiation-dominated era,
for which $k\gg k_{\mathrm{eq}}$, and for those that enter well inside the
matter-dominated era, for which $k\ll k_{\mathrm{eq}}$. Here,
$k_{\mathrm{eq}}$ represents the characteristic scale that crosses the horizon
exactly in the matter-radiation equality ($\rho_{m}=\rho_{\gamma}$), usually
referred to as the turn-over scale. Specifically,

$T(k)\propto\left\\{\begin{array}[]{ll}1,&\mathrm{if}\leavevmode\nobreak\ k\ll
k_{\mathrm{eq}}\\\ k^{-2}\mathrm{ln}(k),&\mathrm{if}\leavevmode\nobreak\ k\gg
k_{\mathrm{eq}}.\end{array}\right.$ (37)

In turn, the turn-over scale is given by

$k_{\mathrm{eq}}=0.073\Omega_{m}h^{2}\mathrm{Mpc}^{-1}.$ (38)

Several more detailed approximations to the transfer function have been
developed by ??) and ??), but these recipes are not accurate enough for the
analysis of data from current galaxy surveys. More accurate transfer functions
can be numerically computed using codes such as the software Code for
Anisotropies in the Microwave Background (??, CAMB).

We can see then that the evolution of the Universe through the radiation-
matter transition imprints a characteristic scale $k_{\mathrm{eq}}$ on the
power spectrum, even though the primordial power spectrum is scale invariant.
As the power spectrum is usually expressed in units of $h\leavevmode\nobreak\
\mathrm{Mpc}^{-1}$, the parameter that sets the position of the turn over
scale is the shape parameter: $\Gamma:=\Omega_{m}h$. This means that the value
of $\Omega_{m}$ is imprinted on the large-scale distribution of matter, hence
the importance of dark matter in the formation of structures in the Universe.

In terms of the growth factor, the transfer function and the spectral index,
the matter power spectrum can be written as

$P_{m}(k,a)=A\left[\frac{D_{+}(a)}{D_{+}(1)}\right]^{2}\left(\frac{k}{k_{0}}\right)^{n_{s}}T^{2}(k),$
(39)

where $k_{0}$ is known as the pivot scale, and $A$ is a parameter setting the
global amplitude of the fluctuations at present epoch.

####  4 The peculiar velocity field

The evolution of the peculiar velocity field is closely related to the growth
of density fluctuations. The continuity equation for matter on subhorizon
scales can be written as

$\delta^{\prime}_{m}-k^{2}v_{m}=0,$ (40)

where we used the fact that the gravitational potential is constant during
matter domination. In this regime, density fluctuations grow as
$\delta_{m}\propto D_{+}(a)$, which then gives

$v_{m}=\frac{\delta_{m}(k,\tau)}{k^{2}D_{+}(\tau)}D^{\prime}_{+}(\tau).$ (41)

We define now the logarithmic growth rate factor:

$f:=\frac{a}{D_{+}}\frac{dD_{+}}{da}=\frac{d\mathrm{ln}D_{+}}{d\mathrm{ln}a}.$
(42)

Using this definition, and the relation $d/d\tau=a^{2}Hd/da$, we can express
the velocity potential in terms of the scale factor and the parameter $f$:

$v_{m}(k,a)=\frac{1}{k^{2}}aHf(a)\delta_{m}(k,a).$ (43)

The velocity field is given by the gradient of the potential, so that

$v_{m_{j}}(k,a)=i\frac{k_{j}}{k^{2}}aHf(a)\delta_{m}(k,a).$ (44)

This relation is the starting point of the study of redshift-space
distortions. The growing mode of the velocity field represents matter flowing
to overdense regions and from underdense regions, increasing in this way the
amplitude of perturbations.

####  5 Non-linear evolution of density fluctuations

The linear regime in the cosmological perturbation theory has a limited range
of applicability. For example, the evolution of structures like clusters of
galaxies cannot be treated within this framework. For instance, the fluid
approximation is no longer valid if gravitationally bound systems form because
multiple streams of matter occur. One possibility is to evolve the system of
equations to higher orders in the small variables $\delta$ and $v^{i}$, and so
consider a non-linear perturbation theory. However, the achievements of this
theory do not in general justify the large mathematical effort.

There is, though, one case of non-linear evolution that can be treated
analytically: the spherical evolution model. Spherical collapse allows us to
understand how dark-matter haloes form, and hence to characterise some of
their main statistical properties, such as their associated mass function.
Analogously, spherical expansion helps us to understand some aspects of the
evolution of voids. We will review its main characteristics in the next
chapter.

In general, studying the non-linear structure evolution requires the use of
numerical methods to simulate structure formation. The results of these
simulations, when compared to observations, have substantially contributed to
establishing the standard cosmological model, allowing us to make accurate
quantitative comparisons between different models. The enormous development in
computing power rendered corresponding progress in simulations possible. In
addition, the continuous improvement of numerical algorithms and computer
hardware has made it possible to constantly improve the spatial resolution of
simulations.

Since the Universe is mainly dominated by dark matter and dark energy, it is
often sufficient to compute only the behaviour of these two components, and
thus to consider gravitational interactions solely. Recently, however,
computing power increased to a level where hydrodynamic processes can also be
taken into account approximately, so that the baryonic component of the
Universe can be traced as well. In addition, radiative transfer can be
included in such simulations, hence the influence of radiation on the heating
and cooling of the baryonic component can also be examined. One of the biggest
simulations ever built to date is the Millennium simulation (??). In this
work, we use an extension of it. We will review its main properties in Chapter
4.

###  5 Redshift-space distortions

In spectroscopic surveys, distances are inferred from the measured redshifts
(Section 5). In a homogeneous universe, the redshift of a galaxy would be
purely given by the cosmological expansion, providing an accurate estimate of
its distance. In reality, the presence of inhomogeneities induce peculiar
velocities to the galaxies, which introduce an extra component in the measured
redshifts, and give rise to a difference between the real and apparent
positions of them. This leads to a change in the shape and amplitude of the
measured two-point statistics. This effect, known as redshift-space
distortions (RSD), complicates the cosmological interpretation of galaxy
clustering measurements. However, as peculiar velocities are gravitationally
induced by the inhomogeneities in the density field, the pattern of RSD
contains information on the underlying matter distribution that can be used to
probe the rate at which cosmic structures grow, offering a test to detect
potential deviations from the predictions of General Relativity.

####  1 The observed redshift

The total observed redshift of a galaxy, $z_{\mathrm{obs}}$, is given by the
combination of two contributions:

$1+z_{\mathrm{obs}}=(1+z_{\mathrm{cos}})(1+z_{\mathrm{pec}}),$ (45)

where $z_{\mathrm{cos}}$ represents the cosmological component due to the
expansion of the Universe, and $z_{\mathrm{pec}}=v_{g\parallel}/c$, the
contribution due to the Doppler effect caused by $v_{g\parallel}$, the
component of the peculiar velocity of the galaxy along the line-of-sight
direction (hereinafter LOS). This additional component changes the position of
the galaxies along the LOS, introducing an anisotropic pattern in their
spatial distribution.

Figure 1 shows a schematic representation of the effect of RSD on large and
linear scales (left-hand and central panels). In the upper panels, we see that
the velocity field (represented by arrows) is mostly given by the bulk flow of
matter towards overdense regions. In the linear regime, the divergence of the
velocity field is proportional to the density fluctuations, but with an
opposite sign. This implies that galaxies move away from underdense regions
(left-hand panel) and are directed towards overdense regions (central panel).
In the lower panels, we see how these regions are actually observed. They
appear elongated and flattened along the LOS, respectively, introducing an
anisotropic pattern in the clustering of galaxies. By way of comparison, the
right-hand panel shows the effect of RSD in a smaller virialised region like a
cluster. Here, the galaxies are spread out in observations due to the large
velocity dispersion, yielding large radial patterns also known as the fingers-
of-God effect.

Figure 1:  Left-hand and central panels. Schematic representation of the
effect of redshift-space distortions on large and linear scales. As galaxies
move away from underdense regions, and are directed towards overdense regions,
these regions appear elongated and flattened along the line of sight when seen
in redshift space, respectively. Right-hand panel. Representation of the RSD
effect in a smaller and virialised region like a cluster. Here, the galaxies
are spread out in redshift space due to the large velocity dispersion,
yielding large radial patterns in the wedge diagrams, the so-called fingers-
of-God effect.

In observations, the line of sight is a preferred direction, hence the
correlation function and the power spectrum are no longer functions of $r$ and
$k$ alone, respectively. The full characterisation of the anisotropies caused
by RSD requires us to express the correlation function in terms of the
components of the observational separation vector, $\mathbf{s}$, which can be
decomposed in the directions perpendicular and parallel to the LOS111We will
use the notation $\sigma$ and $\pi$, instead of the more intuitive $s_{\perp}$
and $s_{\parallel}$, as they have been traditionally used in the literature,
and also to be in agreement with the works of ??) and ??) on which part of
this thesis is based., $\sigma$ and $\pi$. That is,
$\xi^{s}_{g}(\mathbf{s})=\xi^{s}_{g}(\sigma,\pi)$. The spatial configuration
spanned by these coordinates is called redshift space, and we will refer to
quantities in this space with the superscript $s$. The same applies to the
power spectrum, which needs to be expressed as a function of the components of
the wave vector $\mathbf{k}$ in the transverse and LOS directions:
$P^{s}_{g}(k_{\perp},k_{\parallel})$. Alternatively, the correlation function
can be expressed in polar coordinates as $\xi^{s}_{g}(s,\mu)$, where
$s=|\mathbf{s}|$ is the radial coordinate, and $\mu$, the cosine of the angle
between $\mathbf{s}$ and the LOS direction.

By contrast, the spatial configuration spanned by the true positions of the
galaxies is called real space, free of RSD. Sometimes, we will need to compare
$\xi^{s}_{g}(\sigma,\pi)$ or $\xi^{s}_{g}(s,\mu)$ with their two-dimensional
real-space counterpart, for which we will use the notation
$\xi_{g}(\mathbf{r})=\xi_{g}(r_{\perp},r_{\parallel})=\xi_{g}(r)$, where
$\mathbf{r}$, $r_{\perp}$ and $r_{\parallel}$ are the real-space counterparts
of the redshift-space quantities $\mathbf{s}$, $\sigma$ and $\pi$,
respectively. The last equality holds since $\xi_{g}$ must exhibit spherical
symmetry in real space. With this notation,
$|\mathbf{r}|=r=\sqrt{r_{\perp}^{2}+r_{\parallel}^{2}}$,
$|\mathbf{s}|=s=\sqrt{\sigma^{2}+\pi^{2}}$ and $\mu=\pi/s$.

####  2 The impact of peculiar velocities

The first step to model the effect of RSD on two-point clustering measurements
is to study the mapping between the real-space position $\mathbf{r}$ and its
redshift-space counterpart $\mathbf{s}$. The following analysis will be
applied to the total matter distribution. From Eq. (45), we see that the
observed redshift is given by

$z_{\mathrm{obs}}=z+(1+z)\frac{v_{m\parallel}}{c},$ (46)

where we have dropped the “cos” subscript of the cosmological redshift for
simplicity. When using the observed redshift to infer a distance, we must
replace the true comoving distance, $\chi(z)$, by $\chi(z_{\mathrm{obs}})$.
Assuming that the peculiar redshifts are small, we can write

$\chi(z_{\mathrm{obs}})=\chi(z)+\frac{d\chi}{dz}(z)\Delta
z=\chi(z)+\frac{1+z}{H(z)}v_{m\parallel}.$ (47)

The observed position in redshift space corresponds to the true position with
an additional term proportional to the peculiar velocity.

We are now in a position to study the impact of RSD on the correlation
function. We will work in the linear regime and assume the distant observer
approximation, the assumption that the objects we are considering cover a
small region of the sky and are far away. In consequence, we can treat the LOS
direction as one of the Cartesian directions. In this context, Eq. (47)
implies that

$\mathbf{s}=\mathbf{r}+\frac{1+z}{H(z)}v_{m\parallel}\leavevmode\nobreak\
\mathbf{\hat{x}_{\parallel}},$ (48)

where $\mathbf{\hat{x}_{\parallel}}$ is a unit vector oriented along the LOS
direction. Equivalently, in terms of the components,

$\sigma=r_{\perp}$ (49)

and

$\pi=r_{\parallel}+\frac{1+z}{H(z)}v_{m\parallel}.$ (50)

Note that only $\pi$ is affected by RSD. It is worth pointing out that,
expressed in this way, $\sigma$ and $\pi$ correspond to comoving distances.

The next step is to consider that the total amount of matter in a given volume
element is conserved. In terms of the density contrast, this relation implies
that

$(1+\delta^{s}_{m}(\mathbf{s}))d^{3}s=(1+\delta_{m}(\mathbf{r}))d^{3}r,$ (51)

where we have assumed that RSD do not change the mean density of the sample:
$\bar{\rho}^{s}_{m}(\mathbf{s})=\bar{\rho}_{m}(\mathbf{r})$. The volume
elements in real and redshift space are related by the Jacobian of the
transformation, given by Eq. (48): $J=|\partial r_{i}/\partial s_{j}|$, in
such a way that

$d^{3}r=Jd^{3}s.$ (52)

Explicitly, this Jacobian is

$J=\left(1+\frac{1}{\mathcal{H}(z)}\frac{\partial v_{m\parallel}}{\partial
r_{\parallel}}\right)^{-1}\approx\left(1-\frac{1}{\mathcal{H}(z)}\frac{\partial
v_{m\parallel}}{\partial r_{\parallel}}\right).$ (53)

Inserting this result into Eq. (51) leads to

$1+\delta^{s}_{m}(\mathbf{s})=(1+\delta_{m}(\mathbf{r}))\left(1-\frac{1}{\mathcal{H}(z)}\frac{\partial
v_{m\parallel}}{\partial r_{\parallel}}\right).$ (54)

Distributing and keeping only linear terms in the perturbations, we find that

$\delta^{s}_{m}(\mathbf{s})=\delta_{m}(\mathbf{r})-\frac{1}{\mathcal{H}(z)}\frac{\partial
v_{m\parallel}}{\partial r_{\parallel}}.$ (55)

Fourier-transforming this relation, and using Eq. (44), we finally find that

$\delta^{s}_{m}(\mathbf{k})=\delta_{m}(\mathbf{k})(1+f(z)\mu_{k}^{2}),$ (56)

where $\mu_{k}:=k_{\parallel}/k$ represents the cosine of the angle between
$\mathbf{k}$ and the LOS direction. With this relation, the anisotropic power
spectrum is given by

$P_{m}^{s}(k,\mu_{k})=(1+f(z)\mu_{k}^{2})^{2}P_{m}(k).$ (57)

This result was first derived by ??).

Eq. (56) shows that, in the linear regime, the density fluctuations observed
in redshift space correspond to those in real space with a correction factor.
Since $f(z)\mu_{k}^{2}\geq 0$, this additional factor leads to an enhancement
of the density fluctuations. This enhancement depends on $\mu_{k}$: it is
maximum when $\mu_{k}=1$, which corresponds to the LOS direction, and it is
zero when $\mu_{k}=0$, which corresponds to the transverse directions, which
define what is called the plane of the sky (hereinafter POS).

####  3 The RSD parameter

Eqs. (56) and (57) imply that, by studying the anisotropic patterns caused by
RSD, it is possible to measure the logarithmic growth rate, $f(z)$. Within the
context of General Relativity, the growth rate can be accurately described by
the following empirical relation:

$f(z)=\Omega_{m}^{\gamma}(z),$ (58)

where $\Omega_{m}(z):=\rho_{m}(z)/\rho_{c}(z)$ is the generalisation of the
parameter $\Omega_{m}$ for any redshift $z$, and $\gamma=0.55$ (with a weak
dependence on other parameters such as the dark-energy equation of state).
This means that a measurement of $f(z)$ provides additional information on the
matter density parameter. Perhaps more interesting is the fact that such a
measurement also provides us with a test of General Relativity. The exponent
$\gamma$ can be treated as a free parameter to be constrained. A measurement
of $\gamma\neq 0.55$ would be a signature of a deviation from the predictions
of General Relativity.

Nevertheless, there is a drawback here. Eqs. (56) and (57) are only valid for
the total matter distribution. However, what we observe are galaxies.
Therefore, we need to take into account the bias relation studied in Section
3. For simplicity, we will assume the linear bias relation of Eq. (16).
Moreover, we will consider that the velocity field of the galaxies corresponds
exactly to that of the total matter distribution:
$\mathbf{v}_{g}=\mathbf{v}_{m}$, i.e. there is no velocity bias. For this
reason, we will hereinafter drop the subindices in the velocity vector. In
this case, Eq. (56) implies that

$\delta_{g}^{s}(\mathbf{k})=\delta_{g}(\mathbf{k})(1+\beta(z)\mu_{k}^{2}),$
(59)

where we have defined the quantity

$\beta(z):=\frac{f(z)}{b},$ (60)

known as the RSD parameter. The redshift-space galaxy power spectrum will be
then given by

$P_{g}^{s}(k,\mu_{k})=(1+\beta(z)\mu_{k}^{2})^{2}b^{2}P_{m}(k).$ (61)

This means that, by measuring the anisotropic patterns in the two-point
statistics caused by RSD, it is possible to constrain the parameter $\beta$, a
combination of $f$ and $b$. This will be a fundamental cosmological parameter
in our study of voids.

###  6 The fiducial cosmology

An important fact is that a measurement of the two-point statistics requires
the assumption of a fiducial cosmology to transform the observed angles and
redshifts of galaxies provided by a spectroscopic survey into distances
expressed in physical units, like the Megaparsec. This is evident in Eqs.
(42), (47) and (52), for instance. These expressions need to assume certain
values for the cosmological parameters $H_{0}$, $\Omega_{m}$ and
$\Omega_{\Lambda}$. In this selection, any deviation between the chosen
parameters and the true underlying ones will skew the distance estimations,
inducing in this way, additional distortion patterns in the spatial
distribution of the galaxies. This has an impact on the clustering
measurements. This phenomenon is a manifestation of the AP effect studied in
Section 2. AP distortions due to the fiducial cosmology must be taken into
account in order to constrain unbiased cosmological parameters.

Given two galaxies, their spatial separation can be estimated by two direct
measurable quantities: an angle $\Delta\phi$ measured across the POS, and a
redshift difference $\Delta z$, an indicator of their separation along the
LOS. Then, according to Eqs. (47) and (53), it is possible to estimate their
separation vector $\mathbf{s}$ by means of its components:

$\sigma=D_{\mathrm{M}}^{\mathrm{fid}}(z)\Delta\phi$ (62)

and

$\pi=\frac{c}{H_{\mathrm{fid}}(z)}\Delta z,$ (63)

where $D_{M}(z):=(1+z)D_{A}(z)$ is the comoving angular-diameter
distance222Note that $D_{M}(z)=\chi(z)$ for a flat universe.. Here, the legend
“fid” refers to quantities calculated with the fiducial cosmology. Analogue
equations follow using the (unknown) true cosmology for the true separations
$r_{\perp}$ and $r_{\parallel}$ (using the legend “true”). Therefore, a simple
comparison leads to

$\sigma=q_{\mathrm{AP}}^{\perp}r_{\perp}$ (64)

and

$\pi=q_{\mathrm{AP}}^{\parallel}r_{\parallel},$ (65)

where the AP factors $q_{\mathrm{AP}}^{\perp}$ and
$q_{\mathrm{AP}}^{\parallel}$ are defined by the following relations:

$q_{\mathrm{AP}}^{\perp}:=\frac{D_{\mathrm{M}}^{\mathrm{fid}}(z)}{D_{\mathrm{M}}^{\mathrm{true}}(z)}$
(66)

and

$q_{\mathrm{AP}}^{\parallel}:=\frac{H_{\mathrm{true}}(z)}{H_{\mathrm{fid}}(z)}.$
(67)

Figure 2 shows a visual impression of the two types of spatial distortions
studied in this chapter. The left-hand panel shows the real-space distribution
of the matter in a simulation, where a hypothetical observer is located at the
centre. The central panel shows how RSD induce radial patterns in the
distribution due to the peculiar velocity of galaxies. It is particularly
evident the fingers-of-God effect. The right-hand panel, on the other hand,
shows the AP distortions induced by the selection of a fiducial cosmology. In
reality, both effects are coupled in observations.

Figure 2:  Visual impression of the types of distortions in the spatial
distribution of galaxies. Left-hand panel. Real-space distribution of the
matter in a simulation. A hypothetical observer is located at the centre.
Central panel. Redshift-space distortions induced by the peculiar velocity of
the galaxies, a dynamical effect. Right-hand panel. Alcock-Paczyński
distortions induced by the selection of a fiducial cosmology, a geometrical
effect. Both effects are coupled in observations. Figure credit: B. Thomas
<https://www.washburn.edu/faculty/bthomas/work.html>.

##  Chapter 3 Cosmic voids

Cosmic voids are vast underdense regions of the Universe. Since their
discovery (??????), they have been recognised as powerful cosmological
laboratories (????). Most of the cosmic volume is occupied by voids (??),
making them the largest observable structures. Hence, they encode key
information about the expansion history and geometry of the Universe.

Large-scale underdensities arise naturally as the result of structure growth.
As the Universe evolves, galaxies progress towards matter concentrations by
the action of gravity, forming diverse structures like groups, clusters and
filaments. In this process, galaxies also dissipate from underdense regions,
which become even emptier, forming in this way cosmic voids. This interplay in
the formation of voids and structures allows us to think of them as
complementary. Moreover, cosmic voids can be related to the small density
fluctuations seen in the early Universe, manifested in the small anisotropies
in the CMB (????).

The statistical properties of voids depend on two conditioning factors: (i)
the matter tracers used to map the large-scale structure, namely, galaxies,
dark-matter haloes, dark-matter particles, etc., and (ii) the method used to
identify them from the spatial distribution of these tracers. There are
different classes of void finders. For a comparison of different techniques
see ??). In this work, we use the so-called spherical void finder, originally
developed by ??), which is based on the integrated density contrast of
underdense regions assuming spherical symmetry. We leave for Chapter 4 the
details of the identification process using this method. As examples of
different techniques, we can mention the ZOBOV111Acronym for ZOnes Bordering
On Voidness. void finder (??), based on the watershed algorithm, and the
method of ??), based on the dynamical properties of tracers.

Despite the intrinsic differences between all the methods, there is a
consensus on the basic statistical properties of voids. For instance, and
roughly speaking, voids are underdense regions with densities as low as
$10-20\%$ of the average of the Universe, with several tens of Megaparsecs in
diameter. It is also important to stress the fact that galaxies and haloes
trace the void distribution in a similar fashion. ??) found that they have
comparable statistical and dynamical properties, such as their abundance,
correlation function and velocity field.

###  1 Void evolution

Initial overdense regions expand slightly less rapidly than the homogeneous
background, reach a maximum size, and then turn around to finally collapse
(this is strictly true only in an EdS or closed universe). In contrast,
initial underdense regions do not turn around, they undergo simple expansion
(more rapidly than the background). This expansion is simple and linear until
matter from the interior overtakes the initially outer shells. The generic
characteristics of these evolutionary paths are best appreciated in terms of
the evolution of isolated spherically symmetric density perturbations, either
overdense or underdense, in an otherwise homogeneous and expanding background
universe. This spherical model provides a key reference for understanding and
interpreting more complex situations. As a result of the spherical symmetry,
the problem is essentially one-dimensional, allowing a full analytical
treatment.

??) explained that the most basic and universal properties of evolving
spherical voids are the following.

  1. 1.

Expansion. Voids expand, in contrast to overdense regions, which collapse.

  2. 2.

Evacuation. As they expand, the density within them decreases continuously. To
first order, the density decrease is a consequence of the redistribution of
mass over the expanding volume.

  3. 3.

Spherical shape. Outward expansion makes voids evolve towards a spherical
geometry.

  4. 4.

Reverse top-hat density profile. The effective repulsion of the matter
interior to the void decreases with the distance from the centre, so the
matter distribution evolves into a reverse top-hat.

  5. 5.

Super-Hubble velocity field. Consistent with its homogeneous interior density
distribution, the peculiar velocity field in voids has a constant Hubble-like
interior velocity divergence. Thus, voids evolve into genuine super-Hubble
bubbles.

  6. 6.

Suppressed structure growth. Density inhomogeneities in the interior are
suppressed and, as the void begins to resemble an underdense universe,
structure formation within it gets frozen-in.

  7. 7.

Boundary ridge. As matter from the interior accumulates near the boundary, a
ridge develops around the void.

  8. 8.

Shell crossing. The transition from a quasi-linear towards a mature non-linear
stage occurs as inner shells pass across outer shells. When this process
occurs, a void, as a distinctive entity, is considered to have formed.

A detailed study about the spherical evolution of voids in different
cosmological contexts can be found in ??).

In reality, voids are not isolated structures, but rather part of a complex
network. ??) showed that the formation and evolution of voids is strongly
affected by their surrounding large-scale environment, resulting in a more
complicated structure and dynamics than the simplistic isolated expansion.
Essentially, the hierarchy of voids arises by the assembly of matter in the
growing nearby structures. There are two scenarios for void evolution. In the
first case, voids embedded in an environment similar to the global background
density expand and remain as underdense regions following the so-called void-
in-void mode. This is the case when small voids merge at an early epoch with
another void to form a larger void at a later epoch. It is analogous to the
cloud-in-cloud problem associated with the formation of dark-matter haloes and
clusters of galaxies. In the second case, voids surrounded by an overdense
shell shrink at later times due to the gravitational collapse of the dense
structure surrounding them. This is the void-in-cloud mode, and this is
generally the case of small voids. Note however that virialised haloes within
voids are not likely to be torn apart as the void expands around them. Thus,
the cloud-in-void phenomenon is irrelevant for halo formation. Therefore, the
asymmetry between the void-in-cloud and cloud-in-void processes induce a
symmetry breaking between the emerging halo and void populations: although
they evolve out of the same symmetric initial conditions, the two populations
are expected to have different statistical properties.

###  2 The abundance of voids

The excursion set theory provides a useful framework for describing the
formation history of dark-matter haloes in scenarios of hierarchical structure
formation (????). It provides analytical approximations for the distribution
of halo masses (known as mass function), merger rates and formation times that
are quite accurate. The key ingredient is the assumption that the final
virialised state of haloes form from a smooth spherical collapse. The
analogous spherical expansion can likewise be used to make excursion set
predictions for voids (??). Particularly, we are interested in the void size
function (hereinafter VSF), a description of the abundance of voids by means
of the distribution of their sizes. This is analogous to the mass function in
the halo case. In this work, voids will be defined as underdense spheres with
a well defined radius $R_{\mathrm{v}}$ (see Chapter 4 for more details), so
that the VSF is essentially a radius distribution.

One of the main features of the spherical model is that the evolution does not
depend on the initial size $R$ or enclosed mass $M$ of the region, but only on
the amplitude of the initial top-hat density fluctuation. Therefore, $R$ and
$M$ can be treated indistinctly by means of their relation with the volume of
the region. In the case of collapse, it occurs when the linear density
fluctuation reaches a critical barrier $\Delta_{c}$. The excursion set
formalism predicts then the fraction of trajectories $df$ that cross this
barrier for the first time within some scale interval $d\mathrm{ln}\sigma$,
accounting in this way for the cloud-in-cloud process:

$f_{\mathrm{ln}\sigma}(\sigma):=\frac{df}{d\mathrm{ln}\sigma}=\sqrt{\frac{2}{\pi}}\frac{\Delta_{c}}{\sigma}\mathrm{exp}\left[-\frac{\Delta_{c}^{2}}{2\sigma^{2}}\right].$
(1)

A trajectory is a sequence of overdensities given by subsequent increases in a
smoothing scale $\mathcal{R}$. It is common to relate this smoothing scale to
the corresponding variance of the linear density field:

$\sigma^{2}(\mathcal{R})=\int\frac{k^{2}}{2\pi^{2}}P_{m}(k)|W(k,\mathcal{R})|^{2}\leavevmode\nobreak\
dk,$ (2)

where $W(k,\mathcal{R})$ is a filter function. Note the dependence of
$\sigma^{2}(\mathcal{R})$ on the matter power spectrum, $P_{m}(k)$. The strong
cosmological dependence of calculating abundances is thus evident. Since both
mass and number of particles are conserved during collapse, the linear mapping
$\sigma(M)$ carries over to the non-linear regime, so that the mass function
expressed by means of the comoving differential number density of haloes is

$\frac{dn}{d\mathrm{ln}M}=\frac{\rho_{m}}{M}f_{\mathrm{ln}\sigma}(\sigma)\frac{d\mathrm{ln}\sigma^{-1}}{d\mathrm{ln}M}.$
(3)

We can extend this model to underdense regions in the initial density field.
These are naturally associated with voids in the evolved density field today.
In this case, the critical density threshold is defined to be when the
expanding shells cross. For an EdS universe, this occurs when the non-linear
average density contrast within the void reaches the value $-0.8$, which
corresponds to a linearly extrapolated value of $\Delta_{\mathrm{v}}=-2.71$.
??) showed that this value is still valid for alternative cosmologies, such as
the flat-$\Lambda$CDM model. Therefore, similarly to the cloud-in-cloud
process, we can then follow the excursion set formalism for determining the
fraction of trajectories which pierce the underdense barrier
$\Delta_{\mathrm{v}}$, thus taking into account the void-in-void mode.

Nevertheless, this approach will not give an accurate prediction, since we
cannot assume spherical evolution in isolation. This makes sense for
collapsing objects, since the comoving volume occupied shrinks. In contrast,
voids evolve in a more complex way according to both the void-in-void and
void-in-cloud evolution modes. The underdense barrier $\Delta_{\mathrm{v}}$
only accounts for the former. In order to consider the void-in-cloud mode as
well, ??) propose that there must be a second overdense barrier $\Delta_{c}$
as in the halo case. In calculating the first crossing distribution, we need
to determine the largest scale at which a trajectory crosses the barrier
$\Delta_{\mathrm{v}}$ given that it has not crossed $\Delta_{c}$ on any larger
scale. They posit that this threshold should lie somewhere in between
$\Delta_{c}=1.06$, the value at turnaround in the spherical collapse model,
and $\Delta_{c}=1.686$, the value at the point of collapse. By the same
reasoning as applied to haloes, the abundance of voids in the linear regime is
given by

$\frac{dn_{\mathrm{v}}}{d\mathrm{ln}R_{\mathrm{v}}}=\frac{f_{\mathrm{ln}\sigma}(\sigma)}{V(R_{\mathrm{v}})}\frac{d\mathrm{ln}\sigma^{-1}}{d\mathrm{ln}R_{\mathrm{v}}},$
(4)

where $V(R_{\mathrm{v}})=4/3\pi R_{\mathrm{v}}^{3}$ is the volume of a
spherical void of radius $R_{\mathrm{v}}$, and

$f_{\mathrm{ln}\sigma}(\sigma)=2\sum_{j=1}^{\infty}\mathrm{exp}\left[-\frac{(j\pi
x)^{2}}{2}\right]j\pi x^{2}\mathrm{sin}(j\pi\mathcal{D}),$ (5)

with $\mathcal{D}:=|\Delta_{v}|/(\Delta_{c}+|\Delta_{v}|)$ and
$x:=\mathcal{D}\sigma/|\Delta_{v}|$. Eq. (4) is known as the linear model.
Actually, the spherical evolution model predicts that a void expands from its
linear radius at the epoch of shell crossing by a factor approximately equal
to $1.7$. Hence, the radius in Eq. (4) must be replaced with
$1.7R_{\mathrm{v}}$. With this replacement, we get the Sheth $\&$ van de
Weygaert (SvdW) model.

The key assumption of the SvdW model is that the comoving number density of
voids is conserved during the evolution, so that only the size changes with
respect to the linear prediction. Unfortunately, this assumption is not valid
for large voids. For instance, the cumulative comoving volume fraction in
voids exceeds unity. ??) propose a simple fix to this problem. They require
that the comoving volume fraction is fixed during the evolution, instead. In
this picture, when a void expands, it combines with its neighbours to conserve
volume and not number. In this way, the abundance of voids becomes

$\frac{dn_{\mathrm{v}}}{d\mathrm{ln}R_{\mathrm{v}}}=\frac{f_{\mathrm{ln}\sigma}(\sigma)}{V(R_{\mathrm{v}})}\frac{d\mathrm{ln}\sigma^{-1}}{d\mathrm{ln}R_{\mathrm{v}}^{L}}\frac{d\mathrm{ln}R_{\mathrm{v}}^{L}}{d\mathrm{ln}R_{\mathrm{v}}},$
(6)

where the superscript $L$ refers to the linear radius. Eq. (6) is known as the
volume-conserving (Vdn) model. Figure 1 shows the general characteristics of
the three models described here: linear, SvdW and Vdn.

Figure 1:  Void abundance model predictions: linear (blue curve), SvdW
(orange) and Vdn (grey). In the SvdW model, the comoving number density
remains unchanged and only the size changes (arrow to orange curve). In the
Vdn model, the number density also changes so as to conserve the comoving
volume fraction in voids, lowering the amplitude at fixed shape (arrow to grey
curve). The shaded regions correspond to a variation in the overdense barrier
according to $1.06\leq\Delta_{c}\leq 1.686$, which is significant only for
small voids. Figure credit: ??).

###  3 The density field in voids

Section 1 provides a useful framework to characterise the density and velocity
fields around voids. For instance, property (3) implies that the density
fluctuation field around voids, Eq. (1), can be characterised with a radial
profile $\delta_{\mathrm{v}g}(r)$, where $r$ denotes distance to the void
centre. Actually, this is not a good description for voids taken individually,
but it works in a statistical sense when a stacked sample of similar voids is
considered, since then individual irregularities are erased. We will get rid
of the subscript “vg” for simplicity.

??) showed that the integrated density contrast is a good descriptor of the
void environment, since it is tightly connected to the evolution modes. This
profile is obtained by integrating $\delta(r)$ over a volume $V(r)$ given by
successive void-centric spheres of radius $r$:

$\Delta(r):=\frac{1}{V(r)}\int_{V}\delta(r^{\prime})dV=\frac{3}{r^{3}}\int_{0}^{r}\delta(r^{\prime})r^{\prime
2}dr^{\prime}.$ (7)

These authors examined the distribution of galaxies around voids in the SDSS
and obtained two characteristic types of voids according to their integrated
density profiles: (i) R-type voids, characterised by an increasing profile
that tends to zero (the mean value of the Universe) at large distances, and
(ii) S-type voids, characterised by a profile with a noticeable and positive
peak near the typical radius of the sample that decreases later tending to
zero at large distances. Moreover, small voids are more frequently of S-type,
whereas large voids are more likely classified as R-type. The key aspect is
that R-type voids match with the void-in-void mode, whereas S-type voids, with
the void-in-cloud mode. The lower panels of Figure 2 show the measured
integrated density profiles for different samples of voids identified in the
SDSS survey with different radii ranges taken from ??). The main
characteristics described here for the two types of voids are clearly
distinguished.

Figure 2:  Integrated density contrast profiles (lower panels) and velocity
profiles (upper panels) of galaxies in the SDSS for different S-type (left-
hand panels) and R-type (right-hand panels) samples of voids. Different void
radius ranges are indicated: $6-8\leavevmode\nobreak\ h^{-1}\mathrm{Mpc}$
(downward triangles), $8-9\leavevmode\nobreak\ h^{-1}\mathrm{Mpc}$ (circles)
and $10-14\leavevmode\nobreak\ h^{-1}\mathrm{Mpc}$ (upward triangles). Error
bars indicate the region enclosing all curves within $68.3\%$ uncertainty.
Figure credit: ??).

Property (4) concerning the reverse top-hat density profile gives a hint about
how to model $\delta(r)$. The top-hat profile is a good idealisation to
understand the process of void evolution qualitatively, but it is inaccurate
to use in practice. At the moment, there is not a successful model derived
from first principles, but it is a common practice to use empirical and
parametric approaches. For instance, ??) provides a parametric model for the
two types of voids. Actually, they model $\Delta(r)$. For R-type voids, the
provide a functional form based on the error function:

$\Delta_{\mathrm{R}}(r)=\frac{1}{2}[\mathrm{erf}(S\mathrm{log}_{10}(r/R))-1].$
(8)

This model has two parameters: a scale $R$ characterising the void radius, and
a steepness coefficient $S$. S-type voids, on the other hand, are a bit more
complex and require an extra term in order to account for the overdense shell
surrounding them:

$\Delta_{\mathrm{S}}(r)=\frac{1}{2}[\mathrm{erf}(S\mathrm{log}_{10}(r/R))-1]+P\mathrm{exp}\left[-\frac{\mathrm{log}_{10}^{2}(r/R)}{2\Theta^{2}(r)}\right],$
(9)

where in turn,

$\Theta(r):=\left\\{\begin{array}[]{ll}1/\sqrt{2S}\leavevmode\nobreak\
\mathrm{if}\leavevmode\nobreak\ r<R\\\ 1/\sqrt{2W}\leavevmode\nobreak\
\mathrm{if}\leavevmode\nobreak\ r>R.\end{array}\right.$ (10)

The peak on density due to this shell is modelled with two semi-Gaussians.
This model has two extra parameters: an amplitude $P$, and a semi-width $W$
(the other semi-width is given by $S$).

It is worth keeping in mind that void properties depend on the void
identification method. For instance, the works of ??) and ??) cited before
used a spherical void finder, from which the models given by Eqs. (8) and (9)
were tested. To mention another example, ??) provide a universal parametric
model for $\delta(r)$ (without distinguishing between types of voids) based on
a modified version of the ZOBOV void finder that works under the watershed
algorithm:

$\delta(r)=\delta_{ct}\frac{1-(r/r_{s})^{\alpha}}{1+(r/r_{v})^{\beta}}.$ (11)

This model has five parameters: the central density contrast $\delta_{ct}$,
the effective void radius $r_{v}$, a scale radius $r_{s}$ where the void
density equals the mean density, and two exponents, $\alpha$ and $\beta$, that
determine the inner and outer slope of the compensation wall, respectively.

###  4 The velocity field in voids

Properties (1) to (3) of Section 1 imply that the peculiar velocity field
around voids can also be characterised with a one-dimensional radial profile
$v(r)$. As before, this actually applies to a stacked sample of similar voids.
The upper panels of Figure 2 show the corresponding velocity profiles for the
SDSS’ voids of ??). Note that R-type voids exhibit expansion velocities only,
since $v(r)$ is positive at all scales $r$. In the inner parts, the velocity
increases and reaches a peak near the typical radius of the sample, whereas in
the outer parts, it decreases, tending towards zero at large distances, the
mean global value of the Universe. S-type voids, on the other hand, exhibit
expansion velocities only in the inner parts and near the void walls. The
velocity increases at first, reaches a peak near the typical radius of the