sample, and then decreases in the outer parts becoming negative, a signal of
the gradual shrinking of the surrounding overdense shell that characterises
this type of voids. Finally, the velocity increases again, tending to zero at
large distances.

In contrast to the density field, it is possible to derive an analytical
expression for $v(r)$ following linear theory in the evolution of density
perturbations. The starting point is Eq. (44), from which we obtain

$v(r)=-\frac{1}{3}\frac{H(z)}{(1+z)}\beta(z)r\Delta(r).$ (12)

Note that the RSD parameter, $\beta=f/b$, appears here because we assumed the
linear bias relation of Eq. (16) between the matter and galaxy void-centric
density fields: $\delta_{\mathrm{v}m}(r)=\delta_{\mathrm{v}g}(r)/b$. Eq. (12)
is the basis for studying redshift-space distortions around voids, and hence
constitutes one of the most important results on which most of our work is
based.

###  5 The void-galaxy cross-correlation function

In Section 1, we defined the correlation function for a discrete set of
points, which is related to the probability of finding pairs of points as a
function of their separation. We can extend that discussion to consider a
cross-correlation between two discrete sets of points. Particularly, we are
interested in cross-correlating void centres and galaxies. In this case, Eq.
(10) becomes

$dP_{\mathrm{v}g}=\bar{\rho}_{\mathrm{v}}\bar{\rho}_{g}(1+\xi_{\mathrm{v}g}(r))dV_{\mathrm{v}}dV_{g},$
(13)

where $\xi_{\mathrm{v}g}(r)$, known as the void-galaxy cross-correlation
function, gives the excess probability of finding void-galaxy pairs separated
by a distance $r$ with respect to a homogeneous distribution of such pairs. As
before, we will get rid of the subscript “vg” for simplicity.

The estimation of the void-galaxy cross-correlation function can still be
accounted for with the natural estimator of Eq. (12), but their components now
have a different meaning. On the one hand, $DD(r)$ is the probability of
finding void-galaxy pairs separated by distances between $r$ and $r+dr$, also
given by Eq. (11), but this time $N_{\mathrm{pairs}}$ is the number of void-
galaxy pairs inside the volume element defined by $r$ and $r+dr$, and
$N_{\mathrm{tot}}$ is the total number of pairs. In this case,
$N_{\mathrm{tot}}=N_{\mathrm{v}}N_{g}$, where $N_{\mathrm{v}}$ is the total
number of voids. On the other hand, the probability of finding pairs in a
homogeneous distribution of galaxies and voids, $RR(r)$, can be computed in an
analogous way by means of a random catalogue. The alternative estimators given
by Eqs. (13) and (14) can also be applied for the case of a cross correlation
(see for instance ??)).

An important relation must be noted. In the case of a stacked sample of voids,
the void-galaxy cross-correlation function is equivalent to the density
contrast profile:

$\xi(r)=\delta(r),$ (14)

hence we will refer to them indistinctly.

###  6 Redshift-space distortions around voids

The fundamental assumption of spherical symmetry breaks in observations due to
the presence of redshift-space distortions, which modify the spatial
distribution of the galaxies around voids. In Section 5, we showed that the
line of sight is a preferred direction in such a way that the spherical
symmetry reduces to a cylindrical symmetry along this direction. Therefore, it
is instructive to visualise the stacked density and velocity fields of a void
sample in a void-centric reference system whose coordinates are determined by
the comoving distances $\sigma$ and $\pi$, which are perpendicular and
parallel to the LOS direction, respectively. In this system, it is obtained
the redshift-space void-galaxy cross-correlation function,
$\xi^{s}(\sigma,\pi)$, a function that can be represented as a two-dimensional
map.

Figure 3 shows the redshift-space void-galaxy cross-correlation function for
the SDSS’ voids of ??), distinguishing both types of voids: S-type on the
left, R-type on the right. The upper panels show the measurements, whereas the
lower panels are the best-fitting predictions of a model of distortions that
these authors developed, which will be explained in Section 2. The general
characteristics of RSD discussed in Section 1, and represented in Figure 1
schematically, can be observed for both types of voids: elongation of the
inner contour levels (low density regime) and flattening of the outer contour
levels (high density regime). In particular, there is a red cloud near the
$\pi$-axis, a manifestation of the enhanced density in this direction
explained in Section 2.

Figure 3:  Upper panels. Redshift-space distortions exhibited by the void-
galaxy cross-correlation function measured for small S-type voids
($6<R_{\mathrm{v}}/h^{-1}\mathrm{Mpc}<8$) and large R-type voids
($10<R_{\mathrm{v}}/h^{-1}\mathrm{Mpc}<20$), both selected from the SDSS.
Lower panels. Best-fitting model predictions (Section 2). Notice that the
spatial and colour scales change for both types of voids. Figure credit: ??).

The anisotropic patterns on the correlation function encode valuable
information about the dynamics of the galaxies around voids. Below, we will
explain two main approaches to model the RSD effect. Eq. (12) for the radial
velocity profile will be a fundamental component of these models.

####  1 Linear model

The first model is equivalent to the Kaiser model for RSD in the galaxy
autocorrelation discussed in Section 2, and was first derived for the case of
voids by ??). We will refer to it as the linear model. We review here its main
assumptions and characteristics.

The starting point is the assumption that the number of void-galaxy pairs is
conserved under the redshift-space mapping:

$(1+\xi^{s}(\mathbf{s}))d^{3}s=(1+\xi(\mathbf{r}))d^{3}r,$ (15)

in analogy with Eq. (51), but here $\mathbf{r}$ and $\mathbf{s}$ denote the
vectors from the void centre to the galaxy position in real and redshift
space, respectively. The difference between the galaxy-galaxy and void-galaxy
cases, and also a fundamental assumption, is that we are considering the
relative peculiar velocities with respect to one central point, the void
centre. The bulk motion of voids will therefore not affect the correlation
function within the scales where the bulk velocity field can be considered to
be coherent. In this context, and within the plane-parallel approximation, it
is only the void-centric velocity $\mathbf{v}$ of a galaxy that is relevant in
this mapping, recovering Eq. (48). A further key assumption is that the
velocity field is isotropic and radially directed, such that

$\mathbf{v}=v(r)\mathbf{\hat{r}},$ (16)

where $v(r)$ is given by Eq. (12), and $\mathbf{\hat{r}}$ is a unit vector
oriented radially. With these assumptions, we can rewrite Eq. (15) as

$1+\xi^{s}(\mathbf{s})=(1+\xi(\mathbf{r}))\left[1+\frac{v(r)(1+z)}{rH(z)}+\frac{(v^{\prime}(r)-v(r)/r)(1+z)}{H(z)}\mu^{2}\right]^{-1},$
(17)

where the prime denotes a derivative with respect to $r$. Here, the term in
square brackets represents the Jacobian of the coordinate transformation under
the mapping from $\mathbf{r}$ to $\mathbf{s}$. This expression can be expanded
to linear order in the densities to obtain the basic linear model for the
redshift-space correlation function:

$1+\xi^{s}(s,\mu)=(1+\xi(r))\left[1+\frac{1}{3}\beta(z)\Delta(r)+\beta(z)\mu^{2}[\xi(r)-\Delta(r)]\right],$
(18)

where the radial separations in real and redshift space are related by

$r=s\left(1+\frac{1}{3}\beta(z)\Delta(s)\mu^{2}\right).$ (19)

A further approximation can be made by dropping terms of order $\xi\Delta$ and
$\xi^{2}$:

$\xi(s,\mu)=\xi(r)+\frac{1}{3}\beta(z)\Delta(r)+\beta(z)\mu^{2}[\xi(r)-\Delta(r)].$
(20)

####  2 Gaussian streaming model

The linear model is valid on the assumption that $|\delta|\ll 1$ and that any
random dispersion in velocity is small. These assumptions can be relaxed,
leading to a quasi-linear model.

Following ??), $\xi^{s}(\sigma,\pi)$ can be computed as the convolution of the
real-space correlation, $\xi(r)$, and a pairwise velocity distribution of
void-galaxy pairs, $g(\mathbf{r},\mathbf{v})$:

$1+\xi^{s}(\sigma,\pi)=\int[1+\xi(r)]\leavevmode\nobreak\
g(\mathbf{r},\mathbf{v})\leavevmode\nobreak\ d^{3}v.$ (21)

The pairwise velocity distribution can be chosen as a Maxwell-Boltzmann
distribution. Given that only $\pi$ is affected by peculiar velocities, Eq.
(21) reduces to a one-dimensional integral via the replacements
$g(\mathbf{r},\mathbf{v})\rightarrow g(r,r_{\parallel},v_{\parallel})$ and
$d^{3}v\rightarrow dv_{\parallel}$, where $g$ reduces to a Gaussian
distribution centred on the radial velocity profile $v(r)$, with a constant
velocity dispersion $\sigma_{\mathrm{v}}$. In this way222$\hat{\pi}$ refers to
the irrational number pi: 3.14159…, to avoid confusion with the notation of
the $\pi$-coordinate.,

$1+\xi^{s}(\sigma,\pi)=\int_{-\infty}^{\infty}[1+\xi(r)]\frac{1}{\sqrt{2\hat{\pi}}\sigma_{\mathrm{v}}}\mathrm{exp}\left[-\frac{(v_{\parallel}-v(r)\frac{r_{\parallel}}{r})^{2}}{2\sigma_{\mathrm{v}}^{2}}\right]\mathrm{d}v_{\parallel}.$
(22)

The relations between $(r_{\perp},r_{\parallel})$ and $(\sigma,\pi)$, needed
in this equation, are given by Eqs. (49) and (50). When $|\delta|\ll 1$ and
$\sigma_{\mathrm{v}}$ is small, the linear model is well recovered. This model
was first derived for the case of voids by ??), and we will refer to it as the
Gaussian streaming model (hereinafter GS model).

###  7 Cosmological relevance of voids

The physical description developed throughout this chapter of the evolution of
voids, their abundance and the RSD around them has a strong cosmological
dependence. Therefore, cosmic voids emerge as natural cosmological probes to
test the standard model. Moreover, voids 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, and 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
voids (????????????????????????????????). For instance, some of these
alternative models postulate that a fifth force is responsible for the
accelerated expansion, which is unscreened in low density environments. In
view of this, cosmic voids are ideal for testing different dark-energy models.
The potential of voids has also increased recently with the development of the
new generation of spectroscopic surveys, such as BOSS, eBOSS, HETDEX, DESI and
Euclid, 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 expansion history and
geometry of the Universe with high precision.

As we have seen, there are two primary statistics in void studies: the void
size function and the void-galaxy cross-correlation function. The basic
process consists of measuring these statistics from galaxy spectroscopic
surveys, and then comparing them with model predictions, such as the ones
developed in Sections 2 and 6. In this way, we are able to constrain the
intervening cosmological parameters that best fit the data, together with the
corresponding confidence regions. The ultimate goal is to add void-based tests
to the cosmological toolkit for evaluating dark-energy models, together with
the experiments that involve the CMB anisotropies, SNe Ia and the BAO signal.
Even more important is to make joint analyses combining the results of
different experiments, such as the one shown in Figure 2, and thus see if even
tighter constraints can be achieved.

With respect to the void size function, there are many works in the literature
that attempt to lay the foundations for the design of cosmological tests using
this statistic (??????????????????). In general, the community has
concentrated their efforts on modelling the VSF with the excursion set
formalism and the spherical evolution. Nevertheless, the RSD and AP
distortions have an impact on the VSF, and they lead to biased cosmological
constraints if they are not taken into account properly. In the present work,
we will tackle this problematic in Chapter 7. This is the first time that
redshift-space systematicities on the VSF are treated.

Cosmic voids have been explored as standard rulers as well, since a stacked
sample of voids must show a spherical shape in real space, and hence an AP
test can be performed by analysing the deviations from this spherical symmetry
(????????). However, it is not easy to take into account the anisotropic
patterns induced by the dynamical RSD effect with this approach.

It is better to use the correlation function. We know that in real space, the
correlation function must manifest spherical symmetry, hence the observed
anisotropic patterns are a clear evidence of the presence of geometrical and
dynamical distortions. In this way, the correlation function can be used as an
AP test by quantifying these anisotropic patterns. Moreover, the dynamical
models of Section 5 allow us to take into account the RSD contribution to the
anisotropic patterns from physical grounds. We highlight that the fundamental
ingredient is Eq. (12) for the void-centric radial velocity profile, which is
not only important from a dynamical point of view, but it is also important
for cosmology too, since it depends on the parameters $H(z)$ and $\beta(z)$.
Recall that $H(z)$ depends in turn on the background cosmological parameters
like $h$, $\Omega_{m}$ and $\Omega_{\Lambda}$ (Eq. 43). This approach has been
thoroughly investigated in the literature, both in simulations and with
observational data (????????????????????????????????). The study of the void-
galaxy cross-correlation function as a cosmological probe is one of the main
topics of the present work.

##  Chapter 4 Data set

In this chapter, we introduce the data set to be used throughout all this
work. Specifically, we describe the numerical simulations and the resulting
simulated catalogues used to calibrate the different tests and models. We also
explain the void finding method, and finally, the galaxy spectroscopic
catalogue used to compare observational results with the corresponding
theoretical predictions.

###  1 Millennium XXL simulation

For most of the work, we used the Millennium XXL N-body simulation (??,
hereinafter MXXL), which extends the previous Millennium (??) and Millennium-
II (??) simulations, and follows the non-linear evolution of $6720^{3}$ dark-
matter particles inside a periodic cubic box of length
$3000\leavevmode\nobreak\ h^{-1}\mathrm{Mpc}$. The particle mass is
$8.456\times 10^{9}\leavevmode\nobreak\ h^{-1}\mathrm{M}_{\odot}$. It was
designed under a flat-$\Lambda$CDM cosmology with the same cosmological
parameters as the previous runs: $\Omega_{m}=0.25$, $\Omega_{\Lambda}=0.75$,
$\Omega_{b}=0.045$, $\Omega_{\nu}=0.0$, $h=0.73$, $n_{s}=1.0$ and
$\sigma_{8}=0.9$. We used the snapshots belonging to the redshifts $z=0.51$,
$0.99$, and $1.50$, which will be assumed as the mean redshifts of the samples
selected for the analyses.

The MXXL is one of the largest cosmological simulations ever performed. The
simulated volume is equivalent to that of the whole observable Universe up to
redshift $0.72$, is more than $200$ times larger than that of the previous
Millennium simulation, and about $7$ times larger than the volume of the
Baryon Oscillation Spectroscopic Survey, making it ideal for conducting
feasibility studies for cosmological tests. This is particularly suitable for
voids studies. The mass resolution is sufficient to identify dark-matter
haloes hosting galaxies with stellar masses of $1.5\times
10^{10}\leavevmode\nobreak\ h^{-1}\mathrm{M}_{\odot}$, considerably smaller
than the Milky Way, and also to robustly predict the internal properties of
haloes corresponding to very massive clusters, which are represented by more
than $10^{5}$ dark-matter particles. Figure 1 shows the mass density field in
the MXXL at $z=0$, with zooms into different scales.

Figure 1:  The mass density field in the Millennium XXL simulation focusing on
the most massive halo present at $z=0$. Each inset zooms by a factor of $8$
from the previous one; the side-length varies from $4.1\leavevmode\nobreak\
\mathrm{Gpc}$ down to $8.1\leavevmode\nobreak\ \mathrm{Mpc}$. All these images
are projections of a thin slice through the simulation of thickness
$8\leavevmode\nobreak\ \mathrm{Mpc}$. Figure credit: R. Angulo
<https://wwwmpa.mpa-
garching.mpg.de/mpa/research/current_research/hl2011-9/hl2011-9-en.html>.

Dark-matter haloes were chosen as matter tracers, which were identified as
groups of more than $60$ particles using a friends-of-friends algorithm with a
linking length parameter of $0.2$ times the mean interparticle separation. In
all cases, we selected a lower mass cut of $5\times
10^{11}\leavevmode\nobreak\ h^{-1}\mathrm{M}_{\odot}$.

The simulation provides the position
$\mathbf{x}=(x_{1},x_{2},x_{3})\leavevmode\nobreak\ [h^{-1}\mathrm{Mpc}]$ and
peculiar velocity $\mathbf{v}=(v_{1},v_{2},v_{3})\leavevmode\nobreak\
[\mathrm{km}\leavevmode\nobreak\ \mathrm{s}^{-1}]$ of each halo in a Cartesian
comoving coordinate system. In order to simulate the RSD effect, we treated
the $x_{3}$-axis of the simulation box as the line-of-sight direction,
assuming the distant observer approximation. For the purposes of this work,
this is a fair assumption for two main reasons. On the one hand, the redshifts
of the snapshots can be considered far enough with respect to the void-centric
distances that we treat. On the other hand, although the box length is large,
the simulation box is taken as a simple mock survey, which is periodic,
complete in volume and without any complicated selection functions. The volume
of the simulation simply allows us to have a large sample of voids to detect
the different redshift-space systematicities with a good signal. According to
Eq. (48), we applied the following equation to shift the LOS coordinates of
haloes from real to redshift space:

$\tilde{x}_{3}=x_{3}+\frac{1+z}{H(z)}v_{3},$ (1)

where $\tilde{x}_{3}$ denotes the shifted $x_{3}$-coordinate. According to the
MXXL cosmology, the Hubble parameter can be expressed in terms of the
cosmological parameters as follows:

$H(z)=100\leavevmode\nobreak\ h\leavevmode\nobreak\
\sqrt{\Omega_{m}(1+z)^{3}+\Omega_{\Lambda}}.$ (2)

For our flat case,

$\Omega_{\Lambda}=1-\Omega_{m}.$ (3)

###  2 Spherical void finder

In order to identify voids from the spatial distribution of the MXXL haloes,
we applied the spherical void finder developed by ??), based on the original
method developed by ??). This is a method based on the integrated density
contrast of underdense regions assuming spherical symmetry. The void
identification is a fundamental step for our work, hence we detail below the
main steps of this procedure.

  1. 1.

Voronoi tessellation. A Voronoi tessellation is performed to obtain an
estimation of the density field: each halo has an associated cell with volume
$V_{\mathrm{cell}}$, and a density given by the inverse of that volume:
$\rho_{\mathrm{cell}}=1/V_{\mathrm{cell}}$. We used a parallel version of the
public library voro++ (??).

  2. 2.

Selection of candidates. A first selection of underdense regions is done by
selecting all Voronoi cells that satisfy the following criterion in local
density contrast: $\delta_{\mathrm{cell}}<-0.7$. Each underdense cell is
considered the centre of a potential void.

  3. 3.

Growth of spheres. Centred on each candidate, the integrated density contrast,
$\Delta(r)$, is computed in spheres of increasing radius $r$ until the overall
density contrast satisfies a redshift-dependent threshold of
$\Delta_{\mathrm{id}}$, obtained from the spherical evolution model (????) by
fixing a final spherical perturbation of $\Delta_{\mathrm{id,0}}=-0.9$ for
$z=0$.

  4. 4.

Optimisation. Once these first void candidates are identified, step $3$ is
repeated, but now starting in a randomly displaced centre proportional to
$0.25$ times the radius of the candidate. Then, the void centre is updated to
a new position if the new radius is larger than the previous one. This process
is repeated iteratively until convergence to a sphere of maximum radius is
achieved. We adopted the criterion that the optimal sphere is obtained if the
algorithm can not find a bigger one during a lapse of $50$ iterations. This
normally takes between $200$ and $300$ iterations in total. In this way, this
procedure mimics a random walk around the original centre in order to obtain
the largest possible sphere in that local minimum of the density field.

  5. 5.

Overlap filtering. Finally, the list of void candidates is cleaned so that
each resulting sphere does not overlap with any other. This cleaning is done
by ordering the list of candidates by size and starting from the largest one.
The final result is a catalogue of non-overlapping spherical voids with a
well-defined centre, a radius $R_{\mathrm{v}}$, and overall density contrast
$\Delta(R_{\mathrm{v}})=\Delta_{\mathrm{id}}$.

An important comment about step $3$. The choice of $\Delta_{\mathrm{id,0}}$ is
motivated by previous studies that use voids identified from biased tracers,
namely, haloes or galaxies, which keep a bias relation with respect to the
total matter. For dark-matter voids, i.e. those identified from the spatial
distribution of dark-matter particles, there is a theoretical threshold:
$\Delta_{\mathrm{id,0}}=-0.8$, which corresponds to the moment of shell-
crossing in the expansion process, taken as the moment of void formation
(????, see Section 1). However, it is not trivial to extrapolate this value
for the case of voids identified from biased tracers. Some studies provide a
firm evidence that there is still a linear bias relation between dark matter
and tracers in voids (??????????????????). As pointed out by ??), assuming
that voids from dark-matter particles and haloes have the same radii when the
phenomenon of shell-crossing occurs, implies that the latter have a lower
embedded density contrast. Therefore, if the bias is greater than $1$, then
the density contrast can reach values so low that the phenomenon of shell-
crossing might not even happen. Hence, it is a common practice to define voids
as empty as possible. Previous works using the spherical void finder have
demonstrated that choosing $\Delta_{\mathrm{id,0}}=-0.9$ (and extrapolating
this value to the corresponding redshift according to the spherical model)
give a sample of voids with a well characterised dynamics and suitable for
cosmological studies (see for instance ??) and ??)). In the next chapter, we
will reinforce this aspect by presenting a new cosmological test based on
voids identified in this manner.

###  3 Void catalogues I

For the analysis of Chapter 5, where we design a new cosmological test, we
identified voids using the three snapshots of the MXXL simulation, namely,
$z=0.51$, $0.99$ and $1.50$. To do this, we adopted the same cosmology of the
simulation in order to compute distances and densities, needed in void
definition. Moreover, the identification was performed in real space. The
reasons for this choice will be clear in that chapter and the following ones.
Table 1 shows the main characteristics of the halo and void catalogues used
for this analysis, and Figure 2, the corresponding void radius distribution.

Table 1:  Main characteristics of the halo and void catalogues used in Chapter 5. From left to right: MXXL snapshot, number of dark-matter haloes, density threshold criterion for void identification and number of identified voids.  Snapshot | Haloes | $\Delta_{\mathrm{id}}$ | Voids
---|---|---|---
0.51 | 136 993 439 | -0.8764 | 333 741
0.99 | 133 688 808 | -0.8533 | 305 082
1.50 | 118 244 901 | -0.8302 | 254 993
Figure 2:  Radius distribution corresponding to the void catalogues of Table
1. The vertical dashed lines delimit the void samples used to perform the
cosmological test presented in Chapter 5.

In order to perform the cosmological test, we selected a sample of voids for
each MXXL snapshot with sizes between $20\leq
R_{\mathrm{v}}/h^{-1}\mathrm{Mpc}\leq 25$. This is shown with vertical dashed
lines in the figure. Our results are not sensitive to the size of the voids
used, as we have verified. However, large void sizes ensure that they are
mostly of R-type, which simplifies the modelling of the density profiles (see
Section 1 for more details). The number of voids found in each sample are the
following: $10157$ for $z=0.51$, $13703$ for $z=0.99$ and $21034$ for
$z=1.50$.

###  4 Void catalogues II

In Chapters 6, 7 and 8, we study the effects of RSD and AP distortions on the
void identification process itself, and the subsequent impact on the void size
function and the void-galaxy cross-correlation function as cosmological tests.
For this analysis, we focused only on the $z=0.99$ snapshot as representative,
since the remaining snapshots revealed similar results. We applied the void
finder in two ways, resulting in two types of catalogues.

In the first case, we adopted the same cosmology of the MXXL simulation as
before. In turn, in order to study the impact of RSD, we performed the
identification in both spatial configurations: real and redshift space. We
will refer to these catalogues as the true-cosmology (TC) void catalogues: TC-
rs for the real-space version, and TC-zs for the redshift-space one.

In the second case, in order to study the impact of the combined RSD and AP
effects, we modified the coordinate system of the simulation according to two
different fiducial cosmologies. Specifically, we fixed all the MXXL global
parameters with the exception of $\Omega_{m}$ and $\Omega_{\Lambda}$, in such
a way that the cosmology is still flat, i.e. Eq. (3) is still valid.
Concretely, we chose two fiducial values for $\Omega_{m}$: a lower value with
respect to the true one, $\Omega_{m}^{l}=0.20$, and a higher value,
$\Omega_{m}^{u}=0.30$. In this case, the identification was performed only in
redshift space for reasons that will be clear in Chapter 7. We will refer to
these catalogues as the fiducial-cosmology (FC) void catalogues: FC-l for the
$\Omega_{m}^{l}=0.20$ version, and FC-u for the $\Omega_{m}^{u}=0.30$ one.

Table 2 shows the main characteristics of the void catalogues used for this
analysis. According to Table 1, these catalogues were identified with a
density threshold of $\Delta_{\mathrm{id}}=-0.8533$ for $z=0.99$. In Chapter
6, we will need to define two subcatalogues from the TC ones. The original
catalogues, as they were presented up to here, will be referred to as the full
catalogues: TC-rs-f for the real-space version, and TC-zs-f for the redshift-
space one. The subcatalogues will be precisely defined in Section 2, and will
be referred to as the bijective catalogues: TC-rs-b for the real-space
version, and TC-zs-b for the redshift-space one. Essentially, bijective voids
are a subset of the full voids where each void has a unique counterpart in the
other spatial configuration spanning the same region of space.

Note that the TC-rs-f catalogue must be then equivalent to that of the second
entry of Table 1. Nevertheless, this catalogue has many more voids. This is
because, in order to build it, we relaxed the criterion of step $2$ in the
void identification process to $\delta_{\mathrm{cell}}<-0.4$. This allowed us
to obtain a more complete sample, an important aspect when analysing void
abundances. However, this modification is not that important, since it only
results in an increase of smaller voids, which are generally irrelevant from a
statistical point of view (in Section 2 we provide a more detailed
discussion).

Table 2:  Main characteristics of the void catalogues used in Chapters 6, 7 and 8. These catalogues correspond to the MXXL snapshot $z=0.99$, and were identified with a density threshold of $\Delta_{\mathrm{id}}=-0.8533$. From left to right: catalogue name, cosmology used in void definition, $\Omega_{m}$ choice, spatial configuration where the identification was performed, subcatalogue regarding the bijective condition, number of voids and type of systematic effects considered.  Catalogue | Cosmology | $\Omega_{m}$ | Space | Subcatalogue | Voids | Systematic effects
---|---|---|---|---|---|---
TC-rs-f | MXXL | 0.25 | real | full | 463 690 | none
TC-rs-b | MXXL | 0.25 | real | bijective | 318 784 | none
TC-zs-f | MXXL | 0.25 | redshift | full | 455 482 | RSD
TC-zs-b | MXXL | 0.25 | redshift | bijective | 318 784 | RSD
FC-l | Fiducial | 0.20 | redshift | full | 375 560 | AP + RSD
FC-u | Fiducial | 0.30 | redshift | full | 526 552 | AP + RSD

In order to visualise how the void finder works, Figure 3 shows two slices of
the MXXL simulation box, where the spatial distribution of its haloes and
voids can be appreciated, the latter obtained from the TC catalogues of Table
2. For this representation, we used Eq. (1) to simulate the RSD effect. The
left-hand panel shows a slice in the range $500\leq
x_{1}/h^{-1}\mathrm{Mpc}\leq 1000$, $500\leq x_{2}/h^{-1}\mathrm{Mpc}\leq
1000$ and $95\leq x_{3}/h^{-1}\mathrm{Mpc}\leq 105$. Hence, it is a
representation of the POS distribution of haloes and voids. The right-hand
panel, on the other hand, shows a slice in the range $500\leq
x_{1}/h^{-1}\mathrm{Mpc}\leq 1000$, $95\leq x_{2}/h^{-1}\mathrm{Mpc}\leq 105$
and $500\leq x_{3}/h^{-1}\mathrm{Mpc}\leq 1000$, i.e. it shows the LOS
distribution of haloes and voids. In both cases, the real-space void centres
were represented with blue dots, whereas the redshift-space centres, with red
squares. The circles surrounding them represent the intersections of the
spherical voids with the midplane of the slice, but they were plotted only for
the special class of bijective voids.

Figure 3:  Slices of the MXXL simulation box showing the spatial distribution
of its haloes and voids, the latter selected from the TC catalogues of Table
2. Both slices have a thickness of $10\leavevmode\nobreak\
h^{-1}\mathrm{Mpc}$. Eq. (1) was used to simulate the RSD effect. Real-space
voids are represented in blue, whereas the redshift-space ones, in red.
Bijective voids, in turn, are represented with circles, the intersections of
the spherical voids with the midplane of the slice. Each pair of bijective
voids span the same region of space. Left-hand panel. Representation of the
plane-of-sky distribution. Right-hand panel. Representation of the line-of-
sight distribution.

###  5 The Baryon Oscillation Spectroscopic Survey

The Baryon Oscillation Spectroscopic Survey (??, BOSS) is part of a six-year
program of the SDSS-III (mid-2008 to mid-2014) that used the wide-field
$2.5\leavevmode\nobreak\ \mathrm{m}$-telescope at Apache Point Observatory in
New Mexico, United States of America, to carry out four surveys on three
scientific themes: (i) dark energy and cosmological parameters, (ii) the
structure, dynamics, and chemical evolution of the Milky Way, and (iii) the
architecture of planetary systems. BOSS was designed for the first purpose.
Specifically, BOSS measured the redshift of $1.5$ million luminous red
galaxies and Lyman-$\alpha$ absorption towards $160000$ high redshift quasars.
The main scientific project is to detect the characteristic scale imprinted by
the baryon acoustic oscillations in the early Universe (about
$150\leavevmode\nobreak\ \mathrm{Mpc}$) and use it as a physically calibrated
ruler to determine the cosmic distance scale with a precision of $1.0\%$ at
$z=0.35$, $1.1\%$ at $z=0.6$, and $1.5\%$ at $z=2.5$, achieving tight
constraints on the equation of state of dark energy. The high-precision
clustering measurements over a wide range of redshifts and length scales also
provides rich insights into the origin of cosmic structure and the matter
content of the Universe. Therefore, besides BAO analyses, the characteristics
of this survey make it ideal for similar tests with cosmic voids. The right-
hand panel of Figure 1 shows the coverage of the Universe planned by BOSS and
its extended survey, eBOSS.

In Chapter 9, we will present some preliminary results obtained from this
survey. Specifically, we will analyse the void-galaxy cross-correlation
function measured from these data. For this part of the work, we used the
CMASS and LOWZ samples from the Data Release 12 (BOSS DR12). They contain
$864464$ and $333082$ galaxies in the northern and southern hemispheres,
respectively.

###  6 The MultiDark Patchy mock surveys

In order to interpret the measurements from BOSS data, we used the MultiDark
Patchy mocks (????), a series of mock light cones that reproduce the DR12
galaxy clustering catalogue with high fidelity on all relevant scales in order
to allow a robust analysis of BAO and redshift-space distortions. These mocks
test the cosmic evolution in the redshift range from $0.15$ to $0.75$. They
have been calibrated using a reference galaxy catalogue based on the halo
abundance matching modelling of the BOSS DR12 galaxy clustering data and on
the data themselves. Their production follows five steps. First, an accurate
reference catalogue is generated, for which a large N-body simulation capable
of resolving distinct haloes and the corresponding substructures is used. This
technique is applied at different redshift bins to obtain a detailed galaxy
bias evolution spanning the redshift range covered by BOSS DR12 galaxies. In
this way, several mock galaxy catalogues in full cubical volumes of
$2500\leavevmode\nobreak\ h^{-1}\mathrm{Mpc}$ side at different redshifts are
obtained. Second, the method is trained to match the two- and three-point
clustering of the full mock galaxy catalogues for each redshift bin. In the
third step, stellar masses are assigned to the individual objects. In the
fourth step, a light cone is generated, which includes selection effects, the
masking, and combines different boxes at different redshifts. Finally, the
resulting mocks are compared to the observations. The process is iterated
until the desired accuracy for different statistical measures is reached. For
our analysis, we made use of $2048$ mocks for each hemisphere.

The reference catalogues were extracted from one of the Big MultiDark
simulations (??), which was performed using the GADGET-2 code with $3840^{3}$
particles on a volume of a cubic box of side length $2500\leavevmode\nobreak\
h^{-1}\mathrm{Mpc}$ with periodic boundary conditions, assuming a
flat-$\Lambda$CDM cosmology from Planck with the following parameters:
$\Omega_{m}=0.307115$, $\Omega_{b}=0.048206$, $h=0.6777$, $n_{s}=0.9611$ and
$\sigma_{8}=0.8288$. Dark-matter haloes were defined based on the BoundDensity
Maximum halo finder, and the halo abundance matching technique was used to
connect haloes to galaxies.

###  7 Void catalogues III

In order to identify voids from the Patchy mocks and BOSS DR12 galaxies, we
applied our spherical void finder described in Section 2 using, in both cases,
the fiducial cosmology of the Patchy mocks and a density threshold of
$\Delta_{\mathrm{id}}=-0.9$. From the BOSS survey, we found $4784$ and $1873$
voids in the northern and southern hemispheres, respectively ($6657$ voids in
total). They are distributed in a redshift range between $0.2$ and $0.8$, with
a mean redshift of $z_{\mathrm{BOSS}}=0.48$. Figure 4 shows the radius
distribution of the BOSS DR12 voids. The vertical grey dashed line indicates
the median of the distribution. In order to perform the analysis of the void-
galaxy cross-correlation function, we selected two samples (delimited by the
vertical red solid lines in the figure): one with sizes between $30\leq
R_{\mathrm{v}}/\mathrm{Mpc}\leq 35$, and the other with sizes
$R_{\mathrm{v}}\geq 30\leavevmode\nobreak\ \mathrm{Mpc}$.

Figure 4:  Radius distribution corresponding to the BOSS DR12 voids. The grey
dashed line indicates the median of the distribution. The red solid lines
delimit the samples used for the analysis of the void-galaxy cross-correlation
function in Chapter 9.

##  Chapter 5 Designing a new cosmological test

### Abstract

In this chapter, we present a new cosmological test based on the distribution
of galaxies around cosmic voids without assuming a fiducial cosmology. The
test is based on a physical model for the void-galaxy cross-correlation
function projected along and perpendicular to the line of sight. Correlations
are measured in terms of angular distances and redshift differences between
void-galaxy pairs, hence it is not necessary to assume a fiducial cosmology.
This model reproduces the coupled RSD and AP distortions that affect
correlation measurements. It also takes into account the scale mixing due to
the projection range in both directions. The model is general, it can be
applied to an arbitrary binning scheme, not only to the case of the projected
correlations. It primarily depends on two cosmological parameters:
$\Omega_{m}$ (more sensitive to AP distortions) and $\beta$ (more sensitive to
RSD). The test was calibrated using the MXXL simulation for different
redshifts. The method successfully recovers the cosmological parameters. We
studied the effect of measuring with different projection ranges, finding
robust results up to wide ranges. The data covariance matrices associated with
the method are relatively small, which reduces the noise in the likelihood
analysis, and therefore allows us to use a smaller number of mock catalogues
to estimate them. The performance evaluated in this work indicates that the
developed method is a promising test to be applied on observational data. This
part of the work has been published in the Monthly Notices of the Royal
Astronomical Society Journal (??).

###  1 Foundations of the test

####  1 Geometrical and dynamical distortions

Measuring the void-galaxy cross-correlation function involves measuring
distances between pairs made up of a void centre and a galaxy. Three
observable quantities characterise the separation between such a pair: an
angle $\theta$ subtended by the void centre and the galaxy on the plane of the
sky, the redshift $z$ of the galaxy, and the redshift $z^{\prime}$ of the void
centre provided by the void finder. According to Eqs. (62) and (63), the
respective POS and LOS comoving separations, $\sigma$ and $\pi$, are related
to these observable quantities by

$\sigma=D_{\mathrm{M}}(z^{\prime})\leavevmode\nobreak\ \theta$ (1)

and

$\pi=\left|\chi(z)-\chi(z^{\prime})\right|.$ (2)

Actually, Eq. (2) is a generalisation of Eq. (63), since the void-centric
separations can be large. As we studied in Section 6, it is necessary to
assume fiducial values for the cosmological parameters in order to estimate
$\sigma$ and $\pi$ from the observable quantities, inducing possible
distortions in the spatial distribution of the galaxies around voids due to
the AP effect.

Dynamical distortions, on the other hand, are quantified by Eqs. (49) and
(50), which we rewrite here to have the set of equations relevant for the test
handy:

$\sigma=r_{\perp}$ (3)

and

$\pi=r_{\parallel}+\frac{1+z}{H(z)}v_{\parallel}.$ (4)

Recall that $r_{\perp}$ and $r_{\parallel}$ are the real-space analogues of
$\sigma$ and $\pi$, respectively, whereas $v_{\parallel}$ is the LOS component
of the void-centric peculiar velocity of the galaxy under consideration.

It will be convenient to define a new quantity to express the LOS separation
of a void-galaxy pair in terms of the difference between their redshifts:

$\zeta\leavevmode\nobreak\ :=\leavevmode\nobreak\ |z\leavevmode\nobreak\
-\leavevmode\nobreak\ z^{\prime}|.$ (5)

Throughout this chapter, we will distinguish between three spatial
configurations: (i) the observable space111In this chapter, we will use the
notation $\theta$ and $\zeta$, instead of $\Delta\phi$ and $\Delta z$ that we
have been using, to be in agreement with the notation used in ??), on which
this part of the thesis is based. defined by the coordinates $(\theta,\zeta)$,
where measurements are made; (ii) the real space defined by the coordinates
$(r_{\perp},r_{\parallel})$, free of distortions; and (iii) the redshift space
defined by the coordinates $(\sigma,\pi)$, where the RSD and AP distortions
are jointly observed.

A comment worth mentioning about the redshifts. Strictly speaking, it is not
possible to obtain the redshift $z^{\prime}$ of a void centre directly from
the shift of spectral lines. This is only possible for astrophysical bodies
like galaxies. However, $z^{\prime}$ can be indirectly obtained by means of
the void finder when it localises the corresponding centre. This is achieved
by performing the reversal transformation from comoving coordinates of the
centre to observable-space coordinates. For now, we are only interested in
$z^{\prime}$ for the theoretical modelling. The question of void
identification in a non-fiducial way, namely in observable space, has not been
investigated yet. Whatever the method, this process needs to assume a Mpc-
scale. To make matters worse, the RSD and AP effects have an impact on the
identification process itself, generating additional distortion patterns in
the measurements of the correlation function. We will tackle this problematic
thoroughly in Chapter 6. For this reason, and in order to design and calibrate
the test in this chapter appropriately, we have applied the spherical void
finder to the real-space distribution of the MXXL haloes using the cosmology
of the simulation, leading to the data set detailed in Section 3. In this way,
we make sure to have genuine voids with a well-characterised structure and
dynamics that obey the evolution and physical properties described in Chapter
3. The additional systematicities observed in the correlation function due to
the identification in redshift space will be thoroughly investigated in
Chapter 8.

####  2 Fiducial-free test and mixture of scales

Our method provides two novel aspects. In the first place, we treat
correlations directly in terms of the observable-space quantities $\theta$ and
$\zeta$, thus obtaining the observable-space correlation function,
$\xi^{s}(\theta,\zeta)$. Hence, it is not necessary to assume a fiducial
cosmology, and in this way, the AP effect is taken into account naturally. For
simplicity, we will hereinafter drop the superscript $s$ that refers to a
redshift-space quantity.

The second aspect is related to a third type of systematicity, besides the RSD
and AP effects, that also leads to deviations between observations and
theoretical predictions when modelling the correlation function. Models
evaluate the correlation function on a given point of space. However, in the
measuring process, a binning scheme must be assumed, and hence, several scales
are mixed. This is not a problem if we work with almost differential bins,
nevertheless, this implies a poor signal. On the contrary, increasing the bin
sizes improves the signal, but so does the mixture of scales. Therefore, the
correlation function must then be modelled taking into account this effect,
which finds a description from the volume and geometry of the bins. Our method
provides a treatment for this scale-mixing effect. This allows us to work with
bins of arbitrary sizes, so we can go further and work with fully projected
correlation functions. This variant of measuring correlations constitutes the
basis of the test.

###  2 The projected correlation functions

As we mentioned previously, the correlation function must be isotropic in real
space. However, this is no longer valid in redshift space due to the presence
of the coupled RSD and AP distortions. In this case, the spherical symmetry
breaks into a cylindrical symmetry about an axis oriented along the LOS
direction. Therefore, it is instructive to visualise a void sample as a two-
dimensional stack with cylindrical axes. In view of this symmetry, the
correlation function can be represented as a two-dimensional map
$\xi(\sigma,\pi)$ in the coordinates $\sigma$ and $\pi$.

Working in observable space, the correlation function is then represented as a
two-dimensional map $\xi(\theta,\zeta)$ in the coordinates $\theta$ and
$\zeta$. This statistic is estimated by counting void-galaxy pairs inside
cylindrical bins defined in observable space. In this geometry, a bin is a
cylindrical shell oriented along the LOS with dimensions characterised by an
internal radius $\theta_{\mathrm{int}}$, an external radius
$\theta_{\mathrm{ext}}$, a lower height
$\zeta_{\mathrm{low}}=|z_{\mathrm{low}}-z^{\prime}|$ and an upper height
$\zeta_{\mathrm{up}}=|z_{\mathrm{up}}-z^{\prime}|$. This is schematically
represented in the left-hand panel of Figure 1.

Figure 1:  Binning scheme used to measure the void-galaxy cross-correlation
function in observable space. In each panel, the coordinate system of the MXXL
in the plane-parallel approximation was represented, but with the origin
located at a void centre. The $x_{3}$-axis was chosen as the line-of-sight
direction. The correlation function is estimated by counting galaxies inside
these bins. Left-hand panel. A bin is a cylindrical shell oriented along the
LOS, with internal radius $\theta_{\mathrm{int}}$, external radius
$\theta_{\mathrm{ext}}$, a lower height
$\zeta_{\mathrm{low}}=|z_{\mathrm{low}}-z^{\prime}|$, and upper height
$\zeta_{\mathrm{up}}=|z_{\mathrm{up}}-z^{\prime}|$. Central panel. Binning
scheme for the POS correlation function: a set of nested cylindrical shells
distributed across the POS with dimensions $\theta_{\mathrm{int}}$,
$\theta_{\mathrm{ext}}$, $\zeta_{\mathrm{low}}=0$ and
$\zeta_{\mathrm{up}}=\mathrm{PR_{\zeta}}$. $\mathrm{PR_{\zeta}}$ is the
redshift projection range, whereas
$\delta\theta=\theta_{\mathrm{ext}}-\theta_{\mathrm{int}}$, the corresponding
binning step. Right-hand panel. Binning scheme for the LOS correlation
function: a string of filled cylinders oriented along the LOS with dimensions
$\theta_{\mathrm{int}}=0$, $\theta_{\mathrm{ext}}=\mathrm{PR_{\theta}}$,
$\zeta_{\mathrm{low}}$ and $\zeta_{\mathrm{up}}$. $\mathrm{PR_{\theta}}$ is
the angular projection range, whereas
$\delta\zeta=\zeta_{\mathrm{up}}-\zeta_{\mathrm{low}}$, the corresponding
binning step.

If we project $\xi(\theta,\zeta)$ towards the plane of the sky (towards the
$\theta$-axis) in a given range of redshift separations $\mathrm{PR}_{\zeta}$,
we get the POS correlation function, denoted by $\xi_{\mathrm{pos}}(\theta)$,
which is a function only of the angular distance $\theta$. Conversely, if we
project $\xi(\theta,\zeta)$ towards the line of sight (towards the
$\zeta$-axis) in a given angular range $\mathrm{PR}_{\theta}$, we get the LOS
correlation function, denoted by $\xi_{\mathrm{los}}(\zeta)$, which is a
function only of the redshift difference $\zeta$. These projections of the
correlation function can be considered as special cases in the cylindrical
binning scheme. Specifically, the POS correlation function is measured from a
set of nested cylindrical shells distributed across the POS, whereas the LOS
correlation function is measured from a string of filled cylinders oriented
along the LOS. Concretely, the scheme for the POS projection involves bins
with dimensions $\theta_{\mathrm{int}}$, $\theta_{\mathrm{ext}}$,
$\zeta_{\mathrm{low}}=0$ and $\zeta_{\mathrm{up}}=\mathrm{PR_{\zeta}}$. In
this way, $\delta\theta:=\theta_{\mathrm{ext}}-\theta_{\mathrm{int}}$ is the
POS binning step, assumed equal for all bins. Similarly, the scheme for the
LOS projection involves bins with dimensions $\theta_{\mathrm{int}}=0$,
$\theta_{\mathrm{ext}}=\mathrm{PR_{\theta}}$, $\zeta_{\mathrm{low}}$ and
$\zeta_{\mathrm{up}}$. In a similar way,
$\delta\zeta:=\zeta_{\mathrm{up}}-\zeta_{\mathrm{low}}$ is the LOS binning
step. These two binning schemes are schematically represented in the central
and right-hand panels of Figure 1. The goal of our method is to measure and
model these two complementary functions by taking into account the RSD and AP
distortions, together with the mixture of scales due to the projection range.
In this way, we can perform an AP test to constrain the cosmological
parameters $\Omega_{m}$ and $\beta$, as we will see.

Although the final intention is to apply the method to a galaxy spectroscopic
survey, the aim of this chapter is to evaluate its performance and calibrate
it, for which we have used the dark-matter haloes of the MXXL as matter
tracers. For this simulation study, in which the box is assumed to be a simple
and periodic mock, complete in volume and without any complicated selection
functions, there is no need to employ any estimators involving a random
comparison sample. Therefore, the estimation of the correlation function can
then be thought as analogous to the natural estimator, Eq. (12), in which the
quantity $RR$ can be computed analytically. In this context, given a bin in
observable space labelled by $(i,j)$, the estimator for the correlation
function can be written as

$\hat{\xi}(\theta_{i},\zeta_{j})=\frac{DD_{ij}}{RR_{ij}}-1,$ (6)

where $DD_{ij}$ is computed by counting void-halo pairs inside the bin, and
$RR_{ij}$, the expected number of pairs in a homogeneous distribution, as the
product of the density of tracers, the volume of the bin and the total number
of voids. Here, $(\theta_{i},\zeta_{j})$ denotes the coordinates of the
geometrical centre of the bin, merely in order to label it.

To perform the analysis, we used the void catalogues defined in Section 3 and
summarised in Table 1. Specifically, for each MXXL snapshot, we selected a
void sample with sizes between $20\leq R_{\mathrm{v}}/h^{-1}\mathrm{Mpc}\leq
25$, as indicated in Figure 2. For better clarity in the discussion of the
results, we will express the projection ranges in units of
$h^{-1}\mathrm{Mpc}$ using the MXXL cosmology. In these units, we took equal
projection ranges in both directions, so that we will refer to both with the
common notation PR. We analysed $8$ different projection ranges:
$\mathrm{PR}/h^{-1}\mathrm{Mpc}=1$, $5$, $10$, $20$, $30$, $40$, $50$ and
$60$. As an example, Figure 2 shows the projected correlation functions
$\xi_{\mathrm{pos}}(\theta)$ and $\xi_{\mathrm{los}}(\zeta)$ corresponding to
the void sample taken from the snapshot222In this chapter, $z_{\mathrm{box}}$
will denote the redshift of the MXXL snapshots. $z_{\mathrm{box}}=0.99$, for
the cases $\mathrm{PR}/h^{-1}\mathrm{Mpc}=20$ (blue circles), $40$ (green
squares) and $60$ (red triangles). The POS and LOS binning steps were selected
in such a way that each step has an approximate length of
$1\leavevmode\nobreak\ h^{-1}\mathrm{Mpc}$. The error bars, taken from the
diagonal of the associated covariance matrices (see Section 1), are not shown
because they are smaller than the data points symbols. It can be seen that the
profiles flatten as the PR increases, and that the LOS projection is more
affected by RSD than the POS projection. The remaining snapshots show a
similar behaviour. By way of comparison, we show both the observable (below)
and real-space (above) axes. We also show the theoretical predictions (solid
curves) obtained after the application of the model from Section 3 with the
best fitting parameters obtained from the likelihood analysis of Section 4.

Figure 2:  Plane-of-sky (left-hand panel) and line-of-sight (right-hand panel)
void-halo cross-correlation functions corresponding to the void sample taken
from the MXXL snapshot $z_{\mathrm{box}}=0.99$. Here are shown the cases
$\mathrm{PR}/h^{-1}\mathrm{Mpc}=20$ (blue circles), $40$ (green squares) and
$60$ (red triangles). Both the observable-space (below) and real-space (above)
axes are shown for comparison. The solid curves are the theoretical
predictions obtained after the application of the model from Section 3 with
the best fitted parameters obtained in Section 4.

###  3 Model

In this section, we present a physical model for the observable-space
correlation function for a general cylindrical binning scheme, in which each
bin is characterised by the quantities $\theta_{\mathrm{int}}$,
$\theta_{\mathrm{ext}}$, $\zeta_{\mathrm{low}}$ and $\zeta_{\mathrm{up}}$.
This model takes into account the RSD and AP effects, and also the mixture of
scales due to the geometry and sizes of the bins. In this way, the projected
correlation functions can be treated as special cases with the appropriate bin
limits: $\theta_{\mathrm{int}}$, $\theta_{\mathrm{ext}}$,
$\zeta_{\mathrm{low}}=0$ and $\zeta_{\mathrm{up}}=\mathrm{PR}_{\zeta}$ for the
POS projection; $\theta_{\mathrm{int}}=0$,
$\theta_{\mathrm{ext}}=\mathrm{PR}_{\theta}$, $\zeta_{\mathrm{low}}$ and
$\zeta_{\mathrm{up}}$ for the LOS projection. Firstly, we will give the full
treatment, the one that must be applied when working with a spectroscopic
survey. Then, we will turn to the simplifications used for our simulation
study.

The first step is to model the geometry of an observable-space cylindrical bin
$(i,j)$ and to map the redshift-space scales $\sigma$ and $\pi$ that are
involved. First, note that for a given void centre at redshift $z^{\prime}$,
there are two possible values of the redshift $z$ for each limit in $\zeta$
that match the criterion $\zeta=|z-z^{\prime}|$. Specifically, tracers inside
the bin will have angular coordinates between $\theta_{\mathrm{int}}$ and
$\theta_{\mathrm{ext}}$, and their redshifts $z$ will belong to either one of
these two disjoint intervals:
$(z^{\prime}+\zeta_{\mathrm{low}},z^{\prime}+\zeta_{\mathrm{up}})$ or
$(z^{\prime}-\zeta_{\mathrm{up}},z^{\prime}-\zeta_{\mathrm{low}})$. Therefore,
each cylindrical bin corresponds to two volumes on the data. Given a set of
cosmological parameters, these two regions correspond to two different volumes
in redshift space.

Taking these considerations into account, the expected number of void-tracer
pairs, $DD_{ij}$, can be calculated by expressing Eq. (6) in differential form
and then integrating over the volume of the slice taken from the catalogue and
the volume of the bin, obtaining the following expression:

$\displaystyle DD_{ij}=$ $\displaystyle
2\,\hat{\pi}\int_{z^{\prime}_{\mathrm{min}}}^{z^{\prime}_{\mathrm{max}}}dz^{\prime}\leavevmode\nobreak\
\chi^{2}(z^{\prime})\leavevmode\nobreak\
n_{\mathrm{v}}(z^{\prime})\leavevmode\nobreak\ V_{\mathrm{slice}}$ (7)
$\displaystyle\left[\int_{z^{\prime}+\zeta_{\mathrm{low}}}^{z^{\prime}+\zeta_{\mathrm{up}}}dz\leavevmode\nobreak\
\frac{d\chi}{dz}(z)\leavevmode\nobreak\
n_{\mathrm{t}}(z)\int_{\theta_{\mathrm{int}}}^{\theta_{\mathrm{ext}}}d\theta\leavevmode\nobreak\
\theta\leavevmode\nobreak\ \left[1+\xi(\sigma,\pi)\right]\right.+$
$\displaystyle\left.\int_{z^{\prime}-\zeta_{\mathrm{up}}}^{z^{\prime}-\zeta_{\mathrm{low}}}dz\leavevmode\nobreak\
\frac{d\chi}{dz}(z)\leavevmode\nobreak\
n_{\mathrm{t}}(z)\int_{\theta_{\mathrm{int}}}^{\theta_{\mathrm{ext}}}d\theta\leavevmode\nobreak\
\theta\leavevmode\nobreak\ [1+\xi(\sigma,\pi)]\right],$

where $n_{\mathrm{v}}(z^{\prime})$ is the number density distribution of voids
in the slice $z^{\prime}_{\mathrm{min}}\leq z^{\prime}\leq
z^{\prime}_{\mathrm{max}}$, $V_{\mathrm{slice}}$ the volume of this slice, and
$n_{\mathrm{t}}(z)$ the number density distribution of tracers in the bin.
Here, $\xi(\sigma,\pi)$ is a theoretical correlation function defined in
redshift space that must be modelled considering the RSD effect. Its
arguments, $\sigma=\sigma(\theta,z^{\prime})$ and $\pi=\pi(z,z^{\prime})$,
depend on the observable-space coordinates by means of Eqs. (1) and (2). The
expected number of pairs in a uniform distribution, $RR_{ij}$, is given in a
similar fashion:

$\displaystyle RR_{ij}=$
$\displaystyle\hat{\pi}\left(\theta^{2}_{\mathrm{ext}}-\theta^{2}_{\mathrm{int}}\right)\int_{z^{\prime}_{\mathrm{min}}}^{z^{\prime}_{\mathrm{max}}}dz^{\prime}\leavevmode\nobreak\
\chi^{2}(z^{\prime})\leavevmode\nobreak\
n_{\mathrm{v}}(z^{\prime})\leavevmode\nobreak\ V_{\mathrm{slice}}$ (8)
$\displaystyle\left[\int_{z^{\prime}+\zeta_{\mathrm{low}}}^{z^{\prime}+\zeta_{\mathrm{up}}}dz\leavevmode\nobreak\
\frac{d\chi}{dz}(z)\leavevmode\nobreak\
n_{\mathrm{t}}(z)+\int_{z^{\prime}-\zeta_{\mathrm{up}}}^{z^{\prime}-\zeta_{\mathrm{low}}}dz\leavevmode\nobreak\
\frac{d\chi}{dz}(z)\leavevmode\nobreak\ n_{\mathrm{t}}(z)\right].$

In this way, combining Eqs. (7) and (8) into Eq. (6), we get a theoretical
estimation for $\hat{\xi}(\theta_{i},\zeta_{j})$. This is a general expression
that takes into account the AP effect and all possible mixture of scales due
to the binning scheme.

The scope of this work is to present a novel cosmological test focusing the
analysis on the effects of the different types of distortions that arise in
measurements. In order to test these effects, we used simplified mock
catalogues taken from the MXXL simulation. When applied to observational data,
the full treatment developed above must be considered. For the moment, for our
simulation study, some simplifications can be done. On the one hand,
$n_{\mathrm{t}}$ is a constant function. On the other hand, we have assumed a
unique redshift for void centres, the one corresponding to the MXXL snapshot,
$z_{\mathrm{box}}$. In this way, $n_{\mathrm{v}}(z^{\prime})$ can be thought
of as a Dirac-delta distribution. Therefore, according to Eqs. (7) and (8),
the estimator of the correlation function simplifies to the following
expression:

$\displaystyle\hat{\xi}(\theta_{i},\zeta_{j})=$
$\displaystyle-1+2\leavevmode\nobreak\
\left(\theta^{2}_{\mathrm{ext}}-\theta^{2}_{\mathrm{int}}\right)^{-1}$ (9)
$\displaystyle\left[\int_{z_{\mathrm{box}}+\zeta_{\mathrm{low}}}^{z_{\mathrm{box}}+\zeta_{\mathrm{up}}}\frac{dz}{H(z)}\int_{\theta_{\mathrm{int}}}^{\theta_{\mathrm{ext}}}d\theta\leavevmode\nobreak\
\theta\leavevmode\nobreak\ \left[1+\xi(\sigma,\pi)\right]\right.+$
$\displaystyle\left.\left.\int_{z_{\mathrm{box}}-\zeta_{\mathrm{up}}}^{z_{\mathrm{box}}-\zeta_{\mathrm{low}}}\frac{dz}{H(z)}\int_{\theta_{\mathrm{int}}}^{\theta_{\mathrm{ext}}}d\theta\leavevmode\nobreak\
\theta\leavevmode\nobreak\ [1+\xi(\sigma,\pi)]\right]\right/$
$\displaystyle\left[\int_{z_{\mathrm{box}}+\zeta_{\mathrm{low}}}^{z_{\mathrm{box}}+\zeta_{\mathrm{up}}}\frac{dz}{H(z)}+\int_{z_{\mathrm{box}}-\zeta_{\mathrm{up}}}^{z_{\mathrm{box}}-\zeta_{\mathrm{low}}}\frac{dz}{H(z)}\leavevmode\nobreak\
\right],$

where Eq. (42) was used to express the derivative of the comoving distance.

The next step is to model $\xi(\sigma,\pi)$, the redshift-space correlation
function, needed in Eqs. (7) or (9), that takes into account the RSD effect.
We chose the GS model developed by ??) and discussed in Section 2.
Specifically, what we need is Eq. (22), which we rewrite here as it is a
fundamental result for the development of the test:

$1+\xi(\sigma,\pi)=\int_{-\infty}^{\infty}[1+\xi(r)]\frac{1}{\sqrt{2\hat{\pi}}\sigma_{\mathrm{v}}}\mathrm{exp}\left[-\frac{(v_{\parallel}-v(r)\frac{r_{\parallel}}{r})^{2}}{2\sigma_{\mathrm{v}}^{2}}\right]\mathrm{d}v_{\parallel}.$
(10)

This relation indicates that the computation of $\xi(\sigma,\pi)$ needs a
recipe for the density profile, $\xi(r)$, the velocity profile, $v(r)$, and
the velocity dispersion, $\sigma_{\mathrm{v}}$.

####  1 Density profile

Before proceeding to a model for the real-space density profile, it is
instructive to first study the characteristics of this profile by measuring it
directly from the simulation. This analysis will help us to build such a
model.

The left-hand panel of Figure 3 shows, with black circles, the density profile
corresponding to the void sample taken from the snapshot
$z_{\mathrm{box}}=0.99$. This profile was measured using the prescription of
Eq. (12) by counting void-halo pairs within a spherical binning scheme. In
this geometry, a bin is a void-centric spherical shell with internal radius
$r_{\mathrm{int}}$, external radius $r_{\mathrm{ext}}$, and a binning step
$\delta r:=r_{\mathrm{ext}}-r_{\mathrm{int}}$, equal for all bins. Here, we
used a binning step of $\delta r=1\leavevmode\nobreak\ h^{-1}\mathrm{Mpc}$.
The remaining snapshots show a similar behaviour. Note the three reference
dashed lines in the figure: (i) the horizontal $\xi=-1$ line, which indicates
total emptiness, as is the case near the void centres, (ii) the horizontal
$\xi=0$ line, which is the mean value of the Universe, and (iii) the vertical
$r=r_{\mathrm{cut}}$ line, which indicates the minimum radius of the sample
and can be thought as a representative border between the inner parts of the
voids and their environment. The same panel of the figure also shows, with
light-blue squares, the corresponding integrated density profile, $\Delta(r)$.
As can be seen, the sample is represented by an increasing profile that tends
to the mean density of the Universe at large distances, that is, the sample is
mostly composed of R-type voids. One of the reasons that R-type voids are more
convenient to perform the cosmological test is because they are less sensitive
to non-linear effects compared to the S-type ones.

Figure 3:  Left-hand panel. Density contrast profile (black circles) and its
integration (light-blue squares) corresponding to the void sample taken from
the MXXL snapshot $z_{\mathrm{box}}=0.99$. Right-hand panel. The corresponding
radial velocity profile (black circles). In both panels, the solid curves are
the theoretical predictions obtained after the application of the model from
Section 3 with the best fitted parameters obtained in Section 4 for the case
$\mathrm{PR}=40h^{-1}\mathrm{Mpc}$.

In Section 3, we discussed that there is not a first-principle model for
$\xi(r)$, but there are many empirical and parametric approaches in the
literature. In our case, we developed an own parametric model suitable for our
R-type void samples:

$\xi(r)=\left\\{\begin{array}[]{ll}Ar-1&\mathrm{if}\leavevmode\nobreak\
r<r_{\mathrm{cut}}\\\
-\xi_{0}\left[\left(\frac{r}{r_{0}}\right)^{-3}+\left(\frac{r}{r_{0}}\right)^{-\alpha}\right]&\mathrm{if}\leavevmode\nobreak\
r\geq r_{\mathrm{cut}},\end{array}\right.$ (11)

where $\xi_{0}$, $r_{0}$ and $\alpha$ are the three parameters of the model.
Note that this model is not fixed, in the sense that it depends on the global
cosmology as $r_{\mathrm{cut}}$ and $r$ must be calculated from the relations
given by Eqs. (3) and (4), which in turn are given by Eqs. (1) and (2). Three
ranges can be identified: (i) the void-inner zone $r<r_{\mathrm{cut}}$, (ii)
the void-environment zone $r_{\mathrm{cut}}\leq r\leq 100\leavevmode\nobreak\
h^{-1}\mathrm{Mpc}$, and (iii) the void-outer zone $r>100\leavevmode\nobreak\
h^{-1}\mathrm{Mpc}$.

Let us start with the void-environment zone. This range is quantified by a
double power law with slopes $-3$ and $\alpha$. The first exponent describes
the behaviour near the void walls, whereas the second one, in the more remote
areas. The other two parameters are an amplitude $\xi_{0}$ and a pivot scale
$r_{0}$, where the slope changes. The model fails to describe the void-inner
zone, since this function tends to $-\infty$ as $r$ tends to zero. It also
fails to describe the outer zone beyond $100\leavevmode\nobreak\
h^{-1}\mathrm{Mpc}$ because of the BAO feature.

We now describe the remaining zones. The void-inner zone, on the one hand, is
not relevant in terms of correlation signal and is not trivial to model.
Moreover, the condition $\Delta(R_{\mathrm{v}})\leavevmode\nobreak\
=\leavevmode\nobreak\ \Delta_{\mathrm{id}}$ imposed by the spherical void
finder has a direct impact on the shape of the profile in this zone. This is
evident in Figure 3, where the curves break at $r=r_{\mathrm{cut}}$. For these
reasons, we decided to measure correlations only in the void-environment zone,
as Figure 2 reflects. Nevertheless, the contributions from the inner and outer
zones when reproducing RSD are in fact significant, specially the inner zone.
This is because some scales from the inner zone can be shifted into the
environment zone in redshift space, making a significant contribution to RSD.
In the same way, some scales from the environment zone can be shifted into the
outer zone. Hence, these ranges must be modelled, even though data from there
are not used. For the inner zone, we found that it is sufficient a segment
connecting the points $(r=0,\xi=-1)$ (void centre) and
$(r_{\mathrm{cut}},\xi_{\mathrm{cut}})$ (void wall), where
$\xi_{\mathrm{cut}}:=\xi(r_{\mathrm{cut}})$, in such a way that the segment
and the rest of the curve are connected. The slope of this segment is

$A=\frac{1}{r_{\mathrm{cut}}}\leavevmode\nobreak\ (\xi_{\mathrm{cut}}+1).$
(12)

For the outer zone, on the other hand, it is sufficient to extend the scope of
validity of the double power-law model, finding no significant deviations.
Although the slope $A$ is not a known quantity, since $\xi_{\mathrm{cut}}$ can
only be truly known in real space, in Section 3 we will give an approximate
theoretical value for it.

####  2 Velocity profile

In this subsection, we continue the analysis with the velocity profile of the
void sample, shown in the right-hand panel of Figure 3. As in the case of the
density profile, the remaining snapshots show a similar behaviour. Each data
point represents the average of the void-centric radial component of the
peculiar velocities of the haloes around voids using the same binning scheme
as in the case of the density profile. There are two reference dashed lines:
(i) the horizontal $v=0$ line, which represents the global mean value, and
(ii) the vertical $r=r_{\mathrm{cut}}$ line. As can be seen, the sample is
represented by a profile that shows only expansion velocities, as expected for
R-type voids. The characteristic distance $r_{\mathrm{cut}}$ represents the
change between increasing (inside the voids) and decreasing (outside the
voids) expansion.

In Section 4, we showed that $v(r)$ can be analytically modelled following
linear perturbation theory. What we need is Eq. (12), which we rewrite here as
it is a fundamental result for the development of the test and for the
analysis of the following chapters:

$v(r)=-\frac{1}{3}\frac{H(z)}{(1+z)}\beta(z)r\Delta(r).$ (13)

The RSD parameter, $\beta(z)$, is thus a fundamental cosmological parameter in
the test. The dynamical information of our void sample regarding its type is
contained then in the integrated density contrast profile, $\Delta(r)$, which
we discuss below.

####  3 Integrated density profile

The integrated density contrast profile can be modelled by combining Eqs. (7)
and (11):

$\Delta(r)=\left\\{\begin{array}[]{ll}\frac{3}{4}Ar-1&\mathrm{if}\leavevmode\nobreak\
r<r_{\mathrm{cut}}\\\
\frac{3}{r^{3}}\left[\frac{Ar_{\mathrm{cut}}^{4}}{4}-\frac{r_{\mathrm{cut}}^{3}}{3}+I(r)-I(r_{\mathrm{cut}})\right]&\mathrm{if}\leavevmode\nobreak\
r\geq r_{\mathrm{cut}},\end{array}\right.$ (14)

where $I(r)$ is a primitive function for the integral given in Eq. (7)
(without the prefactor $3/r^{3}$) and with Eq. (11) as integrand:

$I(r)=-\xi_{0}\left[r_{0}^{3}\mathrm{ln}(r)+\frac{r_{0}^{\alpha}}{3-\alpha}r^{3-\alpha}\right]+\mathrm{constant}.$
(15)

To complete the model, we need an approximate value for the slope $A$. For
this, we realised that the true value
$\Delta_{\mathrm{cut}}:=\Delta(r_{\mathrm{cut}})$ can be approximated by the
identification-method value $\Delta_{\mathrm{id}}$ shown in Table 1. Then,
from Eq. (14) for $r<r_{\mathrm{cut}}$,

$A\approx\frac{4}{3r_{\mathrm{cut}}}(\Delta_{\mathrm{id}}+1).$ (16)

From Eqs. (12) and (16), we can also give an approximate value for
$\xi_{\mathrm{cut}}$:

$\xi_{\mathrm{cut}}\approx\frac{4}{3}(\Delta_{\mathrm{id}}+1)-1.$ (17)

The validity of these approximations can be corroborated in Figure 3 visually,
where the approximated values for $\Delta_{\mathrm{cut}}$ and
$\xi_{\mathrm{cut}}$ (where the dashed segments and solid curves match) are
near the corresponding true values (data points at $r=r_{\mathrm{cut}}$).

In the next section, we will evaluate in detail the performance of the models
that we have developed for the correlation function and the real-space density
and velocity profiles. For the moment, we just mention that the solid curves
in Figure 3 are the predictions of Eqs. (11), (13), and (14) for the density
and velocity profiles using the best fitted parameters obtained from the
likelihood analysis of Section 4 applied to the measurements of the projected
correlation functions. The good agreement between the predictions and the data
points shows that our method is capable of recovering these profiles. This is
part of the calibration of the test.

###  4 Testing the method

In this section, we test the performance of our model in reproducing the
observed features on the projected POS and LOS correlation functions, as well
as on the corresponding real-space density and velocity profiles. The aim is
to extract cosmological information from the parameters involved in the model.
For the implementation of the test, we adopted a flat-$\Lambda$CDM cosmology.
Moreover, we considered a fixed $H_{0}$ value, the one corresponding to the
MXXL. When applied to a real data set, $H_{0}$ can be extracted from a
different method. In view of this, the parameters involved can be summarised
in two sets: (i) the cosmological set $\\{\Omega_{m},\beta\\}$ (from Eqs. 1, 2
and 13), and (ii) the nuisance set
$\\{\sigma_{\mathrm{v}},\xi_{0},r_{0},\alpha\\}$ (from Eqs. 10 and 11). We
will focus our analysis mainly on the cosmological set.

A comment worth mentioning about the cosmological set. Within the standard
model, $\Omega_{m}$ and $\beta=f/b$ are not independent, they are connected
via Eqs. (58) and (60). As we mentioned in Section 7, theories of Modified
Gravity predict that deviations from the predictions of General Relativity
should be more pronounced in unscreened low-density environments, making voids
a powerful tool for detecting them. Moreover, a simple linear bias relation
may not be a suitable choice for the case of voids. In such cases, these
relations must be modified. Therefore, in order to detect any tensions with
these standard assumptions, it is instructive to take $\beta$ as a free
parameter of the model, and not to incorporate the explicit dependence on
$\Omega_{m}$ or $b$. In this way, we keep the method more general.

In order to constrain these parameters, we implemented a Bayesian analysis
based on the Markov Chain Monte Carlo (MCMC) technique using a Metropolis-
Hastings sampler (????). Although we adopted a standard flat-$\Lambda$CDM
cosmology, the method can be easily generalised to incorporate other models as
well. In fact, we implemented in our code the analytical and numerical
expressions for the comoving distance in the general non-flat case given in
terms of elliptical functions by ??).

Given that AP distortions are sensitive to the redshift of void identification
(Eqs. 1 and 2), we analysed the performance of our test with the redshift
using the three MXXL snapshots: $z_{\mathrm{box}}=0.51$, $0.99$ and $1.50$.
This is important in view of the new generation of galaxy spectroscopic
surveys, which will in general cover a volume with a median redshift larger
than $0.5$, a significant improvement with respect to available surveys.

####  1 Likelihood analysis and covariance matrices

Let us denote the likelihood function with
$\mathcal{L}(\boldsymbol{\theta}|\boldsymbol{x})$, where
$\boldsymbol{\theta}=\\{\Omega_{m},\beta,\sigma_{\mathrm{v}},\xi_{0},r_{0},\alpha\\}$
represents the parameter space, and $\boldsymbol{x}$, the measured data. The
MCMC chains explore $\mathcal{L}(\boldsymbol{\theta}|\boldsymbol{x})$ until
they reach the equilibrium distribution near its maximum, that is, until they
find the $\boldsymbol{\theta}$-values that make the data most probable with
the corresponding confidence regions. We took the ??) convergence criterion,
which compares the spread of the distribution of the means of the chains with
the variance of the target distribution. Once the chains satisfy this
criterion, the unburned parts are discarded and the remaining ones are used to
sample the likelihood function.

The $\mathcal{L}$-function is obtained by computing the differences between
the measured and theoretical correlations for a given set of parameters:

$\mathrm{ln}(\mathcal{L})=-\boldsymbol{\Delta\xi}^{T}\boldsymbol{C}^{-1}\boldsymbol{\Delta\xi}+\mathrm{constant}.$
(18)

Here, both the measured and theoretical correlations are stored in vectors of
the form $\boldsymbol{\xi}:=(\xi_{\mathrm{pos}},\xi_{\mathrm{los}})$
containing the correlation values of each bin, whereas
$\boldsymbol{\Delta\xi}$ denotes the corresponding difference vector.
$\boldsymbol{C}$ denotes the associated covariance matrix. Each element
$C_{ij}$ is computed on the data by a jackknife resampling using the
multivariate generalisation of ??):

$C_{ij}=\frac{n-1}{n}\sum_{k=1}^{n}\left[\xi_{(k)}-\xi_{(.)}\right]_{i}\left[\xi_{(k)}-\xi_{(.)}\right]_{j},$
(19)

where $n$ is the number of jackknife realisations, $\xi_{(k)}$ the correlation
function for the $k^{\mathrm{th}}$ jackknife realisation, and $\xi_{(.)}$ the
average of $\xi_{(k)}$ over the $n$ realisations.

If $2m$ is the number of bins used to measure correlations ($m$ for the POS
projection and $m$ for the LOS projection), then $2m$ is the length of the
correlation vector containing the measurements: $\boldsymbol{\xi}[m]$, and
therefore $2m\leavevmode\nobreak\ \times\leavevmode\nobreak\ 2m$ is the
dimension of the associated covariance matrix: $\boldsymbol{C}[2m\times 2m]$.
This is by far much smaller than in the traditional case, where the
correlation measurements are contained in a matrix $\boldsymbol{\xi}[m\times
m]$, and the associated covariance is a matrix $\boldsymbol{C}[m^{2}\times
m^{2}]$. This is a key aspect of our method, first because the estimation of
the inverse of a smaller matrix is numerically more stable, and second and
more important, because the propagation of covariance errors into the
likelihood estimates are substantially reduced, allowing us to use a smaller
number of mock catalogues to estimate covariances (????). In this work, we
used $2\cdot 80=160$ bins, which implies correlation vectors of length
$\boldsymbol{\xi}[160]$, with associated covariance matrices of dimension
$\boldsymbol{C}[160\times 160]$.

Figure 4 shows the covariance matrices corresponding to the void sample taken
from the snapshot $z_{\mathrm{box}}=0.99$ for the cases
$\mathrm{PR}/h^{-1}\mathrm{Mpc}\leavevmode\nobreak\ =\leavevmode\nobreak\ 10$,
$30$ and $50$. The remaining snapshots show a similar behaviour. Technically,
the so-called correlation matrices are shown: $C_{ij}/\sqrt{C_{ii}C_{jj}}$,
which acquire absolute values from $0$ to $1$ encoded as a coloured contour
map from red to blue. To avoid confusion of nomenclature with the correlation
function, we will refer to them simply as normalised covariance matrices. The
x- and y-axes are expressed in real-space coordinates for a better comparison.
Note that the matrices are not diagonal since the independence of the
correlation values for bins at different scales can not be guaranteed. In
fact, they show clear patterns. Focusing on an individual matrix, four
distinct quadrants can be seen: (i) the bottom left quadrant is the covariance
submatrix for the LOS projection, (ii) the top right quadrant, the covariance
submatrix for the POS projection, whereas (iii) and (iv) the bottom right and
top left quadrants are the cross and symmetric LOS $\times$ POS covariance
submatrices. The diagonal of the entire matrix is the global variance. The
square root of the values from these diagonals make up the implicit error bars
in Figure 2. Comparing now the three matrices, it can be seen that if the PR
is small, the covariance matrices tend to be diagonal, whereas as the PR
increases, off-diagonal values become more prominent. For instance, note the
increase of the cloud pattern observed on the LOS $\times$ POS quadrants.

Figure 4:  Normalised covariance matrices, shown as coloured contour maps,
corresponding to the void sample taken from the MXXL snapshot
$z_{\mathrm{box}}=0.99$ for the cases $\mathrm{PR}/h^{-1}\mathrm{Mpc}=10$,
$30$ and $50$. Absolute values range from $0$ (red) to $1$ (blue). The axes
are expressed in terms of the real-space coordinates. Four distinct quadrants
can be distinguished on each matrix: the covariance for the LOS projection
(bottom left), the covariance for the POS projection (top right), and the
symmetric and cross $\mathrm{LOS}\times\mathrm{POS}$ covariance (bottom right
and top left). The diagonal indicates the global variance and makes up the
implicit error bars in Figure 2. Clear off-diagonal patterns arise as the PR
increases.

####  2 Cosmological constraints

We now present the results of the likelihood analysis. Since $\Omega_{m}$ and
$\beta$ are the two parameters of interest, we will focus mainly on them. The
goal is to calibrate the method. This is achieved by retrieving at the end of
the process the values of these parameters inherent to the MXXL. On the one
hand, $\Omega_{m}=0.25$ (Section 1). On the other hand, the target
$\beta(z_{\mathrm{box}})$ values were directly inferred by fitting the real-
space velocity profiles measured in the simulation boxes with Eq. (13).

Figure 5 shows the marginalised likelihood distributions over the parameters
$\Omega_{m}$ and $\beta$ for each MXXL snapshot and for each projection range
as $1\sigma$ ($68.3\%$) error bars, since these distributions show a Gaussian
shape. The dashed horizontal lines indicate the MXXL target values. As can be
seen, the target values fall inside the $1\sigma$ error bars in most cases,
and fall inside $3\sigma$ ($99.7\%$) of error in all of them, which is the
consistency check we were looking for to calibrate the method. This is a
consequence of the ability of our model to reproduce the RSD and AP
distortions as well as the mixture of scales. This can also be corroborated by
inspecting Figures 2 and 3, where the theoretical profiles (solid curves)
obtained with the best fitted parameter values from this likelihood analysis
match very well the data points.

Inspecting Figure 5 in more detail, we can make some conclusions from the
behaviour of the constraints. First, note that the true value of $\beta$ is
slightly dependent on the redshift. Note also that the error bars for $\beta$
are almost constant, nearly independent of the PR and $z_{\mathrm{box}}$. In
the case of $\Omega_{m}$, by contrast, the error bars reach a minimum at a PR
between $10$ to $20\leavevmode\nobreak\ h^{-1}\mathrm{Mpc}$, which points out
that there is an optimal range to perform the test. Moreover, they generally
decrease from lower to higher $z_{\mathrm{box}}$, which elucidates that
tighter confidence regions are obtained by performing the test at higher
redshifts. This is due to the fact that the model is more sensitive to the AP
effect at higher redshifts. Note also that, although the MXXL values fall
inside the error bars in most cases, there is an appreciable deviation in
$\beta$ for the case $z_{\mathrm{box}}=0.51$ and
$\mathrm{PR}\leavevmode\nobreak\ \geq\leavevmode\nobreak\
10\leavevmode\nobreak\ h^{-1}\mathrm{Mpc}$ (upper right-hand panel). This is
possibly due to a deficiency in our model for RSD, since this effect prevails
over the AP effect at lower redshifts. Incidentally, ??) and ??) present an
improved RSD model for voids, analysing non-linearities and second order
effects. Alternatively, since $\Omega_{m}$ and $\beta=f/b$ are not
independent, and the constraints are unbiased for the former, the tensions
detected for the latter may come from the assumption of a linear bias
relation. In other words, one should expect the constraints on $f$ to be
unbiased too, as the constraint on $\Omega_{m}$ should translate analytically
into a constraint on $f$, the only additional uncertainty to consider is the
constraint on $b$. We leave this problematic for a future investigation.

Two aspects worth mentioning. First, we are using a high-density halo sample
in a large volume. Therefore, the confidence levels on the estimated
parameters must be understood in a precision limit framework. When applied to
real data, the confidence regions will be larger. Second, for a fixed
snapshot, the error bars do not represent independent estimates, since it is
about measuring the same correlation function for the same void sample but
merely adding more void-halo pairs when increasing the projection range.
Having said this, Figure 5 is a robust confirmation that the test can be
applied with a wide variety of projection ranges.

Figure 5:  Marginalised likelihood distributions over the parameters
$\Omega_{m}$ (left-hand panel) and $\beta$ (right-hand panel) for each MXXL
snapshot and for each projection range, shown as $1\sigma$ ($68.3\%$) error
bars. The dashed horizontal lines indicate the respective MXXL target values.
The test is robust with the redshift and the projection range.

Figure 6 shows, as an example, the two-dimensional likelihood marginalisations
onto the $\Omega_{m}-\beta$ plane for the case
$\mathrm{PR}=40\leavevmode\nobreak\ h^{-1}\mathrm{Mpc}$ for the three MXXL
snapshots. From the inner to the outermost, the coloured contour levels
enclose $1\sigma$ ($68.3\%$), $2\sigma$ ($95.5\%$) and $3\sigma$ ($99.7\%$)
confidence regions. The dashed lines indicate the respective MXXL values,
whereas the white crosses, the best fit values. Note that the target values
fall inside the $1\sigma$ confidence region for medium and high redshifts,
whereas for low redshift the deviation of $\beta$ explained before can be
appreciated. While the best results (tightest constraint and smallest
deviation) were obtained for the case $\mathrm{PR}=5\leavevmode\nobreak\
h^{-1}\mathrm{Mpc}$ case, we decided to show the case
$\mathrm{PR}=40\leavevmode\nobreak\ h^{-1}\mathrm{Mpc}$ to highlight the
robustness of the test with the projection range, which is important because
the wider it is, then more data pairs are counted, and therefore, the measured
signal increases. In this sense, this is a more realistic case applicable to
real data.

For completeness, Figure 7 shows the two-dimensional likelihood
marginalisations of the full parameter space for the case
$z_{\mathrm{box}}=0.99$ and $\mathrm{PR}=40h^{-1}\mathrm{Mpc}$. The
constraints are tight, showing no degeneracies between the parameters.
Moreover, the one-dimensional marginalisations show a Gaussian shape in all
cases. The remaining snapshots and projection ranges show a similar behaviour.

Figure 6:  Two-dimensional marginalised likelihood distributions onto the
plane $\Omega_{m}-\beta$ for each MXXL snapshot and for the case
$\mathrm{PR}=40h^{-1}\mathrm{Mpc}$. From the inner to the outermost, the
coloured contour levels enclose $1\sigma$ ($68.3\%$), $2\sigma$ ($95.5\%$) and
$3\sigma$ ($99.7\%$) confidence regions. The dashed lines indicate the
respective MXXL target values, whereas the white crosses, the best fitted
values.  Figure 7:  Marginalised likelihood distributions of the full
parameter space for the case $z_{\mathrm{box}}=0.99$ and
$\mathrm{PR}=40h^{-1}\mathrm{Mpc}$. The $\Omega_{m}-\beta$ panel is the same
as the central panel of Figure 6. No degeneracies are observed. The individual
distributions follow a Gaussian behaviour.

##  Chapter 6 Redshift-space effects in voids

### Abstract

Voids are promising cosmological probes. Nevertheless, every cosmological test
based on voids must necessarily employ methods to identify them in redshift
space. The dynamical and geometrical distortions have an impact on the void
identification process itself, affecting statistical properties of voids such
as their number, size and spatial distribution. This generates additional
distortion patterns in observations. Using the spherical void finder, we
developed a statistical and theoretical framework to physically describe the
connection between the identification in real and redshift space. We found
that redshift-space voids above the shot-noise level have a unique real-space
counterpart spanning the same region of space. In this context, it is valid to
assume void number conservation under the redshift-space mapping. Moreover,
they are systematically bigger and their centres are preferentially shifted
along the line-of-sight direction. The expansion effect is a by-product of the
RSD induced by tracer dynamics at scales around the void radius (t-RSD),
whereas the off-centring effect constitutes a different class of RSD induced
at larger scales by the global dynamics of the whole region containing the
void (v-RSD). The volume of voids is also altered by the fiducial cosmology
assumed to measure distances, this is the AP change-of-volume effect. We
developed a theoretical framework to model these effects, which was tested
using the MXXL simulation. This description depends strongly on dynamical and
cosmological considerations. This part of the work has been published in the
Monthly Notices of the Royal Astronomical Society Journal (??).

###  1 The standard picture of RSD around voids

In Section 5, we studied the standard picture of RSD on large and linear
scales, represented in Figure 1 schematically. However, recent studies have
demonstrated that this is a description too simplified for the case of voids.
??) and ??) noticed that there are four commonly assumed hypotheses in all RSD
models for voids, which are violated when voids are identified in redshift
space, i.e. from observations.

  1. 1.

Conservation of the number of void-galaxy pairs, under the mapping from real-
space into redshift-space. This condition is expressed in Eq. (15).

  2. 2.

Invariability of void-centre positions. This condition is expressed in Eq.
(48). This means that the centres do not suffer any RSD, and it is only the
velocity of galaxies that are relevant when quantifying the change from
$\mathbf{r}$ to $\mathbf{s}$.

  3. 3.

Isotropy of the velocity field around voids. This fundamental hypothesis is
expressed in Eq. (16). Motivated by the two previous items, this condition
assumes that, from the point of view of the observational centres (identified
in redshift space), the average radial outflow of the galaxies is still
radially directed.

  4. 4.

Isotropy of the density field around voids. As in the previous item, this
condition assumes that, from the point of view of the observational centres,
the surrounding density field in real space has spherical symmetry.

The failure of these hypotheses finds an explanation on the RSD and AP
effects. Traditionally, we have focused only on the spatial distribution of
the galaxies around voids. The truth is that these effects also have a direct
impact on the identification process itself, affecting global properties of
voids such as their number, size and spatial distribution. For this reason, it
is not clear if there is a relation between both void populations: the one
identified from observational data (in redshift space) and the true underlying
one (in real space). The validity of these hypotheses is also discussed in
??).

This problematic is important when designing cosmological tests, since these
void systematicities generate additional deviations and anisotropic patterns
in the measurements of the void size function and the correlation function.
Given the precision achievable with modern surveys nowadays, it is extremely
important to detect and model all these effects in order to obtain unbiased
cosmological constraints. This is the reason why we have used real-space voids
in the previous chapter in order to develop and calibrate our cosmological
test.

??) propose as a solution to use the reconstruction technique, which has been
first applied to the case of BAO analyses (??). Reconstruction is an algorithm
to approximately recover the real-space position of galaxies from redshift
space based on the Zel’dovich approximation. The idea is to apply
reconstruction before performing the void finding step. As this is a
cosmology-dependent procedure, it can be used to measure the growth rate
factor parameter if reconstruction plus the void finding step are applied
iteratively. This method has been recently applied to data (????), showing a
robust power in constraining the cosmological parameters and recovering the
statistical properties of voids. However, it also presents some disadvantages.
One the one hand, it is computationally expensive, since it is an iterative
process. On the other hand, it can be quite redundant, since it requires
combining it with a model for RSD around the recovered voids (after this
effect was previously removed) in order to obtain accurate constraints.
Finally, it does not take advantage of the valuable physical information about
the structure and dynamics of voids contained in these additional
systematicities, which manifest when voids are directly identified from
observations.

In the present work, we propose an alternative approach: to analyse the void
finding process in order to find a physical connection between the resulting
void populations in real and redshift space, and in this way, be able to
detect the physical effects responsible for the observed differences between
their statistical properties. This is the goal of this chapter. Some
conventions before we start. The real space and redshift space will be
referred to as $r$-space and $z$-space, respectively. Moreover, we will
distinguish between real and redshift-space void radii with the notation
$R_{\mathrm{v}}^{\mathrm{rs}}$ and $R_{\mathrm{v}}^{\mathrm{zs}}$,
respectively.

###  2 Bijective mapping

We begin the analysis with a visual inspection of $r$-space and $z$-space
voids, returning back to Figure 3. This figure shows two slices of the MXXL
simulation box using the TC void catalogues. The aim of this section is to
study the impact of RSD alone, postponing the analysis of the combined RSD and
AP effects until the next chapter, where we will use the FC void catalogues.
In Section 4, we discussed that the left-hand panel of this figure is a
representation of the POS distribution of haloes and voids, whereas the right-
hand panel shows the corresponding LOS distribution. There, $r$-space void
centres are represented with blue dots, whereas $z$-space centres, with red
squares. From the figure, it is clear that both types of voids approximately
span the same regions of space.

In order to link $r$-space and $z$-space voids, we looked for a correspondence
between them by cross-correlating catalogues TC-rs-f and TC-zs-f.
Specifically, for each $z$-space centre, we picked the nearest $r$-space
centre with the condition that it must lay inside $1\leavevmode\nobreak\
R_{\mathrm{v}}^{\mathrm{zs}}$. Then, we filtered those voids if no partner
could be found. Note that this mapping from $z$-space into $r$-space is a well
defined function, since the condition of the nearest $r$-space neighbour
assigns only one object to each $z$-space void. Moreover, this mapping is also
injective, since the non-overlapping condition imposed by the void finder
implies that each $r$-space void can only be reached by a single $z$-space
one. Furthermore, the filtering condition guarantees then a one-to-one
relationship between $z$-space and $r$-space voids. For this reason, these
voids constitute what we call the bijective catalogues. In Table 2, these
catalogues were denoted by TC-rs-b and TC-zs-b, the former for the bijective
voids in $r$-space, the latter for the bijective voids in $z$-space. Note
that, by construction, both catalogues have the same number of elements:
$318784$. Moreover, it is ensured in this way that a void and its associated
counterpart span the same region of the simulated universe. In order to
distinguish the bijective catalogues from the original ones, we will refer to
the latter as the full catalogues, denoted by TC-rs-f and TC-zs-f in Table 2.
Going back to Figure 3, bijective voids are represented with circles around
their centres, which correspond to the intersections of their volumes with the
midplane of the slice. The rest are voids of the full catalogues without a
partner in the other space.

In order to enquire deeper into the relation between $r$-space and $z$-space
voids, the left-hand panel of Figure 1 shows the void size functions of the
four TC catalogues (in $r$-space and $z$-space, with their respective full and
bijective versions). The abundances were computed from the radius distribution
of each catalogue, expressing the void counts as comoving differential number
densities, $dn_{\mathrm{v}}$, and normalising them by the logarithmic sizes of
the radius bins, $d\mathrm{ln}R_{\mathrm{v}}$. The solid curves represent the
abundances of the full catalogues, both in $r$-space (blue) and $z$-space
(red), whereas the dashed curves, the abundances of the bijective catalogues.
In all cases, the error bands were calculated from Poisson errors in the void
counting process. The vertical dashed line represents the median of the
$z$-space full catalogue (TC-zs-f), which will serve as a reference line
throughout this chapter. This value is equal to $13.26\leavevmode\nobreak\
h^{-1}\mathrm{Mpc}$ ($2.28$ in units of the mean interparticle separation).
The qualitative behaviour of the VSFs is consistent with previous studies,
where we can distinguish two main behaviours separated by the vertical line:
(i) on the left, small voids dominated by shot noise, and (ii) on the right,
genuine voids decreasing their number as the radius increases with a
functional shape similar to those predicted by the theory (see Figure 1 and
the corresponding discussion in Section 2). Small voids dominated by shot
noise, in this sense, are not reliable for any statistical and cosmological
analyses, hence we will mainly focus throughout this work on larger voids
delimited by the reference line.

Figure 1:  Left-hand panel. Void size functions of the TC void catalogues. The
solid curves correspond to the full catalogues, both in $r$-space (blue) and
$z$-space (red). The dashed curves correspond to the bijective catalogues. The
vertical dashed line is the median of the $z$-space full distribution (TC-
zs-f), which separates the small voids dominated by shot noise (on the left)
from the genuine ones relevant for cosmological analyses (on the right). Upper
right-hand panel. Fractional differences of void counts between the bijective
and full catalogues. It quantifies the void loss under the mapping between
$z$-space and $r$-space. Voids above the shot-noise level are almost
conserved. Lower right-hand panel. Fractional differences of void counts
between the $z$-space and $r$-space catalogues.

Note that the full (solid curves) and bijective (dashed curves) abundances
tend to the same values as the radius increases. This means that the loss of
voids under the mapping between both spatial configurations is only
significant in the region dominated by shot noise, whereas the larger and
relevant voids are almost conserved. This can be better appreciated in the
upper right-hand panel of Figure 1, where we present the corresponding
fractional differences of void counts between the full and bijective
catalogues: $\Delta
N_{\mathrm{v}}/N_{\mathrm{v}}=(N_{\mathrm{v}}^{\mathrm{bij}}-N_{\mathrm{v}}^{\mathrm{full}})/N_{\mathrm{v}}^{\mathrm{full}}$.
For all radii of interest, the loss of voids decreases as the radius
increases, being less than $25\%\leavevmode\nobreak\ (\Delta
N_{\mathrm{v}}/N_{\mathrm{v}}<0.25)$ in the worst case. We arrive here at the
first and one of the most important conclusions of this chapter: voids
identified from an observational catalogue above the shot-noise level are true
voids, i.e. they have a real-space counterpart. Therefore, it is valid to
treat the full and bijective catalogues indistinctly in their statistical
properties.

With this in mind, let us now compare the $r$-space (blue curves) and
$z$-space (red curves) abundances. It is clear that they are very different
from each other. In particular, the corresponding fractional differences
$\Delta
N_{\mathrm{v}}/N_{\mathrm{v}}=(N_{\mathrm{v}}^{\mathrm{zs}}-N_{\mathrm{v}}^{\mathrm{rs}})/N_{\mathrm{v}}^{\mathrm{rs}}$
(lower right-hand panel) increase as the radius increases, and they can be
very high at the largest sizes. However, in the context of the bijective
mapping, as both types of voids are in fact the same entities, this means that
these differences can only be attributed to some physical effect that voids
suffer when they are mapped from $r$-space into $z$-space. Incidentally, note
that for each radius, $z$-space voids are systematically bigger than their
$r$-space counterparts. This hints of an expansion effect.

###  3 Theoretical description

The goal of this section is to theoretically study the possible physical
mechanisms responsible for the transformation of $r$-space voids into their
associated $z$-space counterparts. We will do this in the context of the four
hypotheses commonly assumed to model RSD around voids noticed in Section 1 and
the bijective mapping of the last section. In the next section, we will
provide the corresponding statistical evidence of the framework developed
here.

####  1 Void number conservation

Since the galaxy number is conserved under the redshift-space mapping,
condition (1) concerning the conservation of void-galaxy pairs can be
inspected by analysing the corresponding void number conservation. This is not
trivial, since unlike galaxies, which can be considered as particles that are
totally conserved under this mapping (only their position changes), some voids
can be destroyed while new artificial voids can be created in this process.

Strictly speaking, void number conservation is violated. This is evident when
analysing the full catalogues, since the two versions in $r$-space and
$z$-space have different number of elements (compare the number of voids
between the TC-rs-f and TC-zs-f entries in Table 2) and different abundances
(compare the blue and red curves in Figure 1). Nevertheless, this condition is
not violated in the context of the bijective mapping that we defined. This is
supported by two reasons. First, bijective voids are, by definition, the same
entities spanning the same regions of space. This is why the bijective
catalogues have the same number of voids (compare the number of voids between
the TC-rs-b and TC-zs-b entries in Table 2). Second, voids above the shot-
noise level (characterised by the median of the radius distribution) are
practically conserved under this mapping, as we have demonstrated in the
previous section. For these sizes, the full and bijective catalogues are
equivalent regarding their statistical properties.

Therefore, any difference detected in the statistical properties between
$z$-space and $r$-space voids can only be attributed to some physical effect
that impacts on voids when they are mapped from $r$-space into $z$-space. Such
an effect, then, must be associated with the distortions in the observed
spatial distribution of galaxies. Hence, a physical description must find its
bases on the large-scale dynamics of galaxies.

####  2 Expansion effect

In Section 2, we showed that $z$-space voids are systematically bigger than
their $r$-space counterparts. This suggests that voids expand when they are
mapped from $r$-space into $z$-space. The left-hand panel of Figure 2 depicts
this expansion effect schematically. There, a spherical $r$-space void of
radius $R_{\mathrm{v}}^{\mathrm{rs}}$ (represented with a blue solid circle
with some galaxies) appears elongated along the LOS direction in $z$-space due
to the RSD induced by the LOS component of the peculiar velocities of the
tracers surrounding it. The $r$-space spherical void has been transformed into
a $z$-space ellipsoid (orange dashed ellipse in the figure) with semi-axes
$s_{\perp}$ and $s_{\parallel}$, where $s_{\perp}$ is the POS semi-axis (equal
for both $x_{1}$ and $x_{2}$ directions), and $s_{\parallel}$, the LOS semi-
axis.

Figure 2:  Schematic illustration of the three $z$-space effects in voids. In
all cases, a hypothetical spherical $r$-space void of radius
$R_{\mathrm{v}}^{\mathrm{rs}}$ is represented with a blue solid circle with
some galaxies around. The LOS direction is assumed vertical. Left-hand panel.
Expansion effect. In $z$-space, the void appears elongated along the LOS
(orange dashed ellipse) due to the LOS component of the peculiar velocity of
the galaxies that surround it. The void finder identifies an equivalent and
bigger sphere of radius
$R_{\mathrm{v}}^{\mathrm{zs}}=q_{\mathrm{RSD}}R_{\mathrm{v}}^{\mathrm{rs}}$
(red solid circle). This effect is a by-product of tracer dynamics at scales
around the void radius (t-RSD). Central panel. AP change-of-volume effect. The
dimensions of the void are distorted in both directions due to the AP effect,
manifesting itself again as an ellipsoid (orange dashed ellipse). The void
finder identifies an equivalent sphere of radius
$R_{\mathrm{v}}^{\mathrm{fid}}=q_{\mathrm{AP}}R_{\mathrm{v}}^{\mathrm{rs}}$
(red solid circle). Right-hand panel. Off-centring effect. In $z$-space, the
void centre appears shifted along the LOS due to the LOS component of the void
net velocity. This effect is a by-product of the global void dynamics (v-RSD).
The three effects can be treated independently.

We now derive analytical expressions for the semi-axes. We will assume that
RSD do not affect the void dimensions across the POS. Hence, we can consider
that $s_{\perp}=R_{\mathrm{v}}^{\mathrm{rs}}$. An expression for
$s_{\parallel}$, on the other hand, can be obtained by means of Eq. (4), the
expression for the $z$-space void-centric distance along the LOS. In this
case, $\pi$ and $r_{\parallel}$ must be replaced by $s_{\parallel}$ and
$R_{\mathrm{v}}^{\mathrm{rs}}$, respectively. The next ingredient is an
expression for $v_{\parallel}$, which can be obtained from Eq. (13), the
expression for the void-centric radial velocity profile characterising the
peculiar velocity field around voids. Here,
$v_{\parallel}=v(r_{\parallel})=v(R_{\mathrm{v}}^{\mathrm{rs}})$, for which
$\Delta(r)$ must be evaluated at $r=R_{\mathrm{v}}^{\mathrm{rs}}$, which in
turn is equal to the threshold of void identification:
$\Delta(R_{\mathrm{v}}^{\mathrm{rs}})=\Delta_{\mathrm{id}}$. In this way,
combining Eqs. (4) and (13) with the mentioned replacements, we get an
expression for $s_{\parallel}$:

$s_{\parallel}=R_{\mathrm{v}}^{\mathrm{rs}}\left(1-\frac{1}{3}\beta(z)\Delta_{\mathrm{id}}\right).$
(1)

Note that here, we have assumed the validity of hypotheses (3) and (4)
concerning the isotropy of the density and velocity fields in $r$-space in
order to explain a $z$-space phenomenon, even if this isotropy is no longer
valid in this last spatial configuration.

Our void finder identifies spherical regions instead of ellipsoidal ones.
Hence, as a first ansatz, we will assume that $z$-space spherical voids
enclose the same volume of the corresponding ellipsoidal regions. This is
depicted in Figure 2 with a red solid circle. Denoting the radius of this
equivalent sphere by $R_{\mathrm{v}}^{\mathrm{zs}}$, equating both volumes,
and using Eq. (1), we finally get an expression for
$R_{\mathrm{v}}^{\mathrm{zs}}$:

$R_{\mathrm{v}}^{\mathrm{zs}}=q^{s}_{\mathrm{RSD}}R_{\mathrm{v}}^{\mathrm{rs}},$
(2)

where

$q^{s}_{\mathrm{RSD}}=\sqrt[3]{1-\frac{1}{3}\beta(z)\Delta_{\mathrm{id}}}.$
(3)

Note that $q^{s}_{\mathrm{RSD}}$, which we will call the RSD factor, is simply
a scale-independent proportionality constant. Moreover, it has a strong
cosmological dependence, since it depends on $\beta$. To get the explicit
value of $q^{s}_{\mathrm{RSD}}$ inherent to the MXXL simulation, we need the
corresponding values of $\Delta_{\mathrm{id}}$ and $\beta(z)$. As specified in
Table 1, $\Delta_{\mathrm{id}}=-0.853$ for $z=0.99$. The corresponding value
of $\beta$ can be obtained from the analysis of Section 2: ${\beta=0.65}$ (see
the central panel of Figure 5). With these two quantities, we get
$q^{s}_{\mathrm{RSD}}=1.058$. Note that $q^{s}_{\mathrm{RSD}}>1$, hence
$R_{\mathrm{v}}^{\mathrm{zs}}>R_{\mathrm{v}}^{\mathrm{rs}}$, which is in
agreement with our assumption that voids expand when they are mapped from
$r$-space into $z$-space.

In reality, the actual value of $R_{\mathrm{v}}^{\mathrm{zs}}$ is expected to
vary between $R_{\mathrm{v}}^{\mathrm{rs}}$ and $s_{\parallel}$, i.e.
$R_{\mathrm{v}}^{\mathrm{rs}}\leq R_{\mathrm{v}}^{\mathrm{zs}}\leq
s_{\parallel}$. In order to test for possible deviations in the predictions of
Eq. (2), we also considered a more general approach by introducing a new
variable $\delta R_{\mathrm{v}}$ that quantifies the variation in radius:

$\delta
R_{\mathrm{v}}:=\frac{R_{\mathrm{v}}^{\mathrm{zs}}-R_{\mathrm{v}}^{\mathrm{rs}}}{s_{\parallel}-R_{\mathrm{v}}^{\mathrm{rs}}}.$
(4)

Considering that
$q_{\mathrm{RSD}}:=R_{\mathrm{v}}^{\mathrm{zs}}/R_{\mathrm{v}}^{\mathrm{rs}}$,
and combining Eqs. (1) and (4), we obtain the following linear relation:

$q_{\mathrm{RSD}}=1-\frac{1}{3}\delta
R_{\mathrm{v}}\beta(z)\Delta_{\mathrm{id}}.$ (5)

Note that $q_{\mathrm{RSD}}\geq 1$ always, since $\Delta_{\mathrm{id}}<0$.
Therefore, $\delta R_{\mathrm{v}}$ is expected to vary in the range $[0,1]$.
One limit case is $\delta R_{\mathrm{v}}=0$ ($q_{\mathrm{RSD}}=1$), which
corresponds to the unlikely case of no expansion at all, i.e.
$R_{\mathrm{v}}^{\mathrm{zs}}=R_{\mathrm{v}}^{\mathrm{rs}}$. The other limit
case is $\delta R_{\mathrm{v}}=1$ ($q_{\mathrm{RSD}}=1.185$), which
corresponds to the case where the $z$-space void is characterised by a sphere
of radius $R_{\mathrm{v}}^{\mathrm{zs}}=s_{\parallel}$. This last case is also
unlikely because it would mean that RSD affect the dimensions of voids equally
in all directions. In particular, the theoretical prediction of Eq. (2)
corresponds to the case $\delta R_{\mathrm{v}}^{s}=0.315$.

In Section 1, we will show that the theoretical prediction
$q^{s}_{\mathrm{RSD}}$ fits very well the median of the overall
$R_{\mathrm{v}}^{\mathrm{zs}}/R_{\mathrm{v}}^{\mathrm{rs}}$ ratio distribution
(see Figure 3). However, we found that voids well above the shot-noise level
respond better to the value $\delta R_{\mathrm{v}}^{l}=0.5$, i.e. to a
$z$-space radius that is the mean between $R_{\mathrm{v}}^{\mathrm{rs}}$ and
$s_{\parallel}$. We will discuss these aspects in more detail in Sections 4
and in the next chapter. In this way, we get a prediction slightly different
from that of Eq. (2):

$R_{\mathrm{v}}^{\mathrm{zs}}=q^{l}_{\mathrm{RSD}}R_{\mathrm{v}}^{\mathrm{rs}},$
(6)

where

$q^{l}_{\mathrm{RSD}}=1-\frac{1}{6}\beta(z)\Delta_{\mathrm{id}},$ (7)

with an explicit value of $q^{l}_{\mathrm{RSD}}=1.092$ inherent to the MXXL
simulation at $z=0.99$.

The discrepancies between Eqs. (2) and (6) can be attributed to the way in
which the void finder performs the average spherical integration of the
density field in an ellipsoidal underdense region. Hence, the optimal value of
$q_{\mathrm{RSD}}$ will depend on the shape and slope of the real-space
density profiles in the inner parts of voids. We leave for a future
investigation a deeper analysis about the derivation of $q_{\mathrm{RSD}}$
considering these aspects.

####  3 Alcock-Paczyński change-of-volume effect

Up to here, a true distance scale was implicitly assumed. Note, however, that
the only information available from observational catalogues are angular
positions and redshifts of astrophysical objects like galaxies. These
observable quantities must be transformed into a physical distance scale,
which involves the assumption of a fiducial cosmology. A deviation between the
true and fiducial cosmologies will lead to additional distortions in the
spatial distribution of galaxies. Hence, the AP effect will also affect the
volume of voids. To understand the impact of this effect, we will consider for
the following analysis the distribution of galaxies in $r$-space, free of RSD.

In principle, the size of a spherical void can be quantified by two directly
measurable quantities: an angular radius111We use $\Delta\theta$ instead of
$\Delta\phi$ to be in agreement with the notation used in ??), from which this
part of the work is based. on the plane of the sky, $\Delta\theta$, and a
redshift radius along the line of sight, $\Delta z$. According to Eqs. (62)
and (63), these observable quantities are related to their respective physical
dimensions $R_{\mathrm{v\perp}}$ and $R_{\mathrm{v\parallel}}$ by the
following transformation equations:

$R_{\mathrm{v\perp}}=D_{\mathrm{M}}(z)\Delta\theta$ (8)

and

$R_{\mathrm{v\parallel}}=\frac{c}{H(z)}\Delta z.$ (9)

Note that if one knew the true cosmology, then it would not be necessary to
distinguish between the POS and LOS dimensions. Both would be equal to the
$r$-space void radius:
$R_{\mathrm{v}}^{\mathrm{rs}}=R_{\mathrm{v\parallel}}=R_{\mathrm{v\perp}}$.
However, assuming a fiducial cosmology leads to possible discrepancies between
both quantities, therefore a spherical void will appear again as an ellipsoid
in the underlying coordinate system. Nevertheless, unlike the RSD-ellipsoids
from the expansion effect, the AP-ellipsoids are distorted in both the POS and
LOS directions. Furthermore, the net result is not necessarily an expansion,
it can also be a contraction, it all depends on the chosen cosmology. This
additional AP change-of-volume effect is schematically depicted in the central
panel of Figure 2.

We can describe this AP-volume effect following a similar approach to that
used for the expansion effect. Considering that the ellipsoid has semi-axes
$s_{\perp}^{\mathrm{fid}}$ and $s_{\parallel}^{\mathrm{fid}}$, given by Eqs.
(8) and (9) with fiducial values $H_{\mathrm{fid}}$ and
$D_{M}^{\mathrm{fid}}$, then a direct comparison with its true dimension,
$R_{\mathrm{v}}^{\mathrm{rs}}$, also given by the same expressions but with
the true values $H_{\mathrm{true}}$ and $D_{M}^{\mathrm{true}}$, leads to the
following relations:

$s_{\perp}^{\mathrm{fid}}=q_{\mathrm{AP}}^{\perp}R_{\mathrm{v}}^{\mathrm{rs}}$
(10)

and

$s_{\parallel}^{\mathrm{fid}}=q_{\mathrm{AP}}^{\parallel}R_{\mathrm{v}}^{\mathrm{rs}},$
(11)

where $q_{\mathrm{AP}}^{\perp}$ and $q_{\mathrm{AP}}^{\parallel}$ are given by
Eqs. (66) and (67), respectively. Here, we adopted the index “true” to refer
to quantities based on the true underlying cosmology, although unknown.
Finally, considering the equivalent sphere with the same volume of the
ellipsoid, and calling this new radius by $R_{\mathrm{v}}^{\mathrm{fid}}$, we
get an expression similar to Eq. (2):

$R_{\mathrm{v}}^{\mathrm{fid}}=q_{\mathrm{AP}}R_{\mathrm{v}}^{\mathrm{rs}},$
(12)

where

$q_{\mathrm{AP}}=\sqrt[3]{(q_{\mathrm{AP}}^{\perp})^{2}q_{\mathrm{AP}}^{\parallel}}.$
(13)

Like the RSD factor, the AP factor is also a proportionality constant, scale-
independent and cosmology-dependent. However, there is an interesting
difference between both. On the one hand, $q_{\mathrm{AP}}$ depends only on
the background cosmological parameters, such as $\Omega_{m}$,
$\Omega_{\Lambda}$ and $H_{0}$, hence it encodes information about the
expansion history and geometry of the Universe. On the other hand,
$q_{\mathrm{RSD}}$ depends only on $\beta$, hence it is related to the
dynamics and growth rate of cosmic structures.

For the development of the next chapter, we will need the explicit values of
the AP factor for the two FC void catalogues defined in Section 4. Catalogue
FC-l, which assumes a fiducial value of $\Omega_{m}^{l}=0.20$, has a
corresponding value of $q_{\mathrm{AP}}^{l}=1.046$. Catalogue FC-u, with
$\Omega_{m}^{u}=0.30$, has a corresponding value of
$q_{\mathrm{AP}}^{u}=0.960$. Note that $q_{\mathrm{AP}}^{l}>1$ for the former,
hence according to Eq. (12), it is expected an expansion of these fiducial
voids. Conversely, $q_{\mathrm{AP}}^{u}<1$ for the latter, hence a contraction
is expected.

####  4 AP and RSD combined contributions

The volume of a void will be affected by the combined contributions of the AP
and RSD effects, which are indistinguishable in observations. A priori, it is
not trivial to ensure that both effects can be treated independently as we
did. However, in the next chapter we will provide statistical evidence of
this. From a theoretical point of view, the fact that the factors
$q_{\mathrm{RSD}}$ and $q_{\mathrm{AP}}$ encode different cosmological
information is a good sign of this assumption.

Assuming this independence, we can relate the $z$-space and $r$-space void
radii making a two-step correction: first, we apply Eq. (12) to correct for
the AP effect, and then, we apply Eq. (2) (or Eq. 6) to correct for the
expansion effect. Note that this correction can be performed in reverse order
as well. Therefore, the final ratio between the radii in both spatial
configurations is obtained by combining both expressions:

$R_{\mathrm{v}}^{\mathrm{zs}}=q_{\mathrm{AP}}\leavevmode\nobreak\
q_{\mathrm{RSD}}\leavevmode\nobreak\ R_{\mathrm{v}}^{\mathrm{rs}}.$ (14)

####  5 Off-centring effect

A simple visual inspection of Figure 3 shows that $z$-space void centres are
shifted with respect to their $r$-space counterparts. This off-centring effect
is a direct consequence of the failure of hypothesis (2) concerning the
invariability of centre positions when voids are mapped from $r$-space into
$z$-space. ??) remark that this hypothesis is equivalent to assuming that void
positions do not suffer any RSD themselves. On the other hand, ????) and ??)
demonstrated that voids move through space as whole entities with a net
velocity $\textbf{V}_{\mathrm{v}}$. Inspired by these results, then the off-
centring effect can be simply understood as a new kind of RSD induced by the
global dynamics of voids, and therefore it is expected that their centres
appear preferentially shifted along the LOS when they are identified in
$z$-space, in the same way as tracers do. The right-hand panel of Figure 2
depicts this effect schematically. We will provide a solid statistical
evidence of this effect in Section 2.

We can make an analytical prediction of this effect by considering it as a
dynamical phenomenon. The void finder used in this work provides the position
$\textbf{X}_{\mathrm{v}}=(X_{\mathrm{v1}},X_{\mathrm{v2}},X_{\mathrm{v3}})\leavevmode\nobreak\
[h^{-1}\mathrm{Mpc}]$ and peculiar velocity
$\textbf{V}_{\mathrm{v}}=(V_{\mathrm{v1}},V_{\mathrm{v2}},V_{\mathrm{v3}})\leavevmode\nobreak\
[\mathrm{km}\leavevmode\nobreak\ \mathrm{s}^{-1}]$ of void centres (see the
next section for more details about how these velocities are calculated).
Therefore, in order to account for the LOS shifting of the centres, it is only
necessary to write an expression equivalent to Eq. (1) but applied to voids:

$\tilde{X}_{\mathrm{v3}}=X_{\mathrm{v3}}+\frac{1+z}{H(z)}V_{\mathrm{v3}},$
(15)

where $\tilde{X}_{\mathrm{v3}}$ denotes the shifted
$X_{\mathrm{v3}}$-coordinate. As before, it is not trivial to know if the
volume and off-centring effects are independent from each other. In Section 3
we will provide evidence of this. In this case, Eq. (4) is still valid
provided that $r_{\parallel}=|X_{\mathrm{v3}}-x_{3}|$ and
$v_{\parallel}=|V_{\mathrm{v3}}-v_{3}|$.

Before ending this section, a brief reflection about the $z$-space effects in
voids described here. One the one hand, the expansion effect is a by-product
of the RSD induced by tracer dynamics at scales around the void radius. At
these scales, the velocity field of tracers responds to a divergence
originated in the local density minimum that defines the void. On the other
hand, the off-centering effect is also a result of the RSD effect, but induced
at larger scales. Its source is the bulk motion of galaxy tracers in the whole
region containing the void following the large-scale dynamics of the
gravitational field (??). This last aspect resembles a void dynamics, that
presents itself as a different kind of RSD. Therefore, it is expected that
both effects leave a footprint on the cosmological statistics, such as the
void size function and the void-galaxy cross-correlation function.

Hereinafter, we will refer to the expansion effect with the acronym t-RSD,
whereas we will refer to the off-centring effect with the acronym v-RSD. This
is motivated by the fact that, although both effects are a consequence of the
dynamical distortions present in the spatial distribution of galaxies, the net
effect manifests at different scales: the prefix t- refers to tracer dynamics
itself (classic RSD effect), whereas the prefix v- refers to void dynamics.
This will be particularly useful to distinguish the different contributions to
the deviations and anisotropic patterns observed on the void size function and
the correlation function.

###  4 Statistical analysis

The statistical analysis of this section aims to provide evidence about the
expansion and off-centring effects postulated in the last section. In the next
chapter, we will complete the analysis by incorporating the additional AP
change-of-volume effect. For this reason, we will continue using the TC
catalogues, specifically the bijective versions (TC-rs-b and TC-zs-b of Table
2).

This analysis is based on looking for correlations between three statistics
that characterise the volume alteration and movement of a void: (i) the
$z$-space to $r$-space radius ratio
$R_{\mathrm{v}}^{\mathrm{zs}}/R_{\mathrm{v}}^{\mathrm{rs}}$, (ii) the
$z$-space displacement of the centre
$\textbf{d}_{\mathrm{v}}=(d_{\mathrm{v1}},d_{\mathrm{v2}},d_{\mathrm{v3}})$,
and (iii) its $r$-space net velocity
$\textbf{V}_{\mathrm{v}}=(V_{\mathrm{v1}},V_{\mathrm{v2}},V_{\mathrm{v3}})$.
Specifically, $\textbf{d}_{\mathrm{v}}$ is calculated as the displacement of a
void centre in going from $r$-space into $z$-space normalised to the $r$-space
radius:

$\textbf{d}_{\mathrm{v}}:=\frac{\tilde{\mathbf{X}}_{\mathrm{v}}-\textbf{X}_{\mathrm{v}}}{R_{\mathrm{v}}^{\mathrm{rs}}}.$
(16)

The net velocity, on the other hand, is computed as the sum of all the
individual velocities of the haloes that fall inside a spherical shell with
dimensions $0.8\leq r/R_{\mathrm{v}}^{\mathrm{rs}}\leq 1.2$. This is an
unbiased and fair estimation of the bulk flow velocity of the void, as was
demonstrated in ??) (see their Fig. 1).

####  1 Correlations between $r$-space and $z$-space void radii

The left-hand panel of Figure 3 shows the two-dimensional distribution
$(R_{\mathrm{v}}^{\mathrm{rs}},\delta R_{\mathrm{v}})$ as a heat map. From
blue to red, the colours span from low to high void counts $N_{\mathrm{v}}$.
These counts are presented in a logarithmic scale in order to highlight the
patterns of the distribution at different scales. The right-hand axis shows
the equivalent scale based on the ratio
$R_{\mathrm{v}}^{\mathrm{zs}}/R_{\mathrm{v}}^{\mathrm{rs}}$, related to
$\delta R_{\mathrm{v}}$ via Eq. (5). In order to study the evolution of this
distribution with void radius, we computed the median and interquartile range
(IQR) taking bins of width $2\leavevmode\nobreak\ h^{-1}\mathrm{Mpc}$ in the
range $10\leq R_{\mathrm{v}}^{\mathrm{rs}}/h^{-1}\mathrm{Mpc}\leq 32$,
represented in the figure with black dots and error bars. The horizontal lines
indicate the predictions of Eqs. (2) ($\delta R_{\mathrm{v}}^{s}=0.315$,
$q^{s}_{\mathrm{RSD}}=1.058$, dashed line) and (6) ($\delta
R_{\mathrm{v}}^{l}=0.5$, $q^{l}_{\mathrm{RSD}}=1.092$, solid line). Note that
$q^{s}_{\mathrm{RSD}}$ is a better predictor of the median for smaller voids,
whereas $q^{l}_{\mathrm{RSD}}$ is more suitable for larger voids, the ones
more relevant for cosmological studies. The right-hand panel of the figure
shows the two-dimensional distribution
$(R_{\mathrm{v}}^{\mathrm{rs}},R_{\mathrm{v}}^{\mathrm{zs}})$. There is a
clear linear trend between both radii, whose slope can be correctly described
by the RSD factors $q^{s}_{\mathrm{RSD}}$ (dashed line) and
$q^{l}_{\mathrm{RSD}}$ (solid line). As before, $q^{l}_{\mathrm{RSD}}$ is more
suitable for larger voids.

From this analysis, we arrive at the second important conclusion of this
chapter: voids expand when they are mapped from $r$-space into $z$-space, and
this expansion can be statistically quantified as an increase of void radius
by a factor $q_{\mathrm{RSD}}$ ($q^{s}_{\mathrm{RSD}}$ for smaller voids or
$q^{l}_{\mathrm{RSD}}$ for larger voids). These results give support to the
t-RSD expansion effect postulated in Section 2.

Figure 3:  Left-hand panel. Two-dimensional distribution
$(R_{\mathrm{v}}^{\mathrm{rs}},\delta R_{\mathrm{v}})$ shown as a heat map.
From blue to red, the colours span from low to high number of void counts in a
logarithmic scale. The right-hand axis shows the equivalent scale based on the
ratio $R_{\mathrm{v}}^{\mathrm{zs}}/R_{\mathrm{v}}^{\mathrm{rs}}$, related to
$\delta R_{\mathrm{v}}$ via Eq. (5). The black dots with error bars show the
evolution of the median and interquartile range of this distribution with void
radius. The horizontal lines indicate the predictions of Eqs. (2) ($\delta
R_{\mathrm{v}}^{s}=0.315$, $q^{s}_{\mathrm{RSD}}=1.058$, dashed line) and (6)
($\delta R_{\mathrm{v}}^{l}=0.5$, $q^{l}_{\mathrm{RSD}}=1.092$, solid line).
Right-hand panel. Two-dimensional distribution
$(R_{\mathrm{v}}^{\mathrm{rs}},R_{\mathrm{v}}^{\mathrm{zs}})$. There is a
linear trend, whose slope is correctly described by the factors
$q^{s}_{\mathrm{RSD}}$ (dashed line) and $q^{l}_{\mathrm{RSD}}$ (solid line).
In both panels, it is clear that $q^{s}_{\mathrm{RSD}}$ is a better predictor
of the median for smaller voids, whereas $q^{l}_{\mathrm{RSD}}$ is more
suitable for the larger ones. This analysis constitutes a statistical
demonstration of the t-RSD expansion effect.

####  2 Correlations between displacement of centres and net velocity

Figure 4 shows the two-dimensional distribution
$(|\textbf{V}_{\mathrm{v}}|,|\textbf{d}_{\mathrm{v}}|)$, where the module of
these vectors were taken. This distribution contains information about the
dynamics of voids as whole entities. The cross in the figure indicates the
mode of the distribution, which shows that voids tend to move with a speed of
$290\leavevmode\nobreak\ \mathrm{km}\leavevmode\nobreak\ \mathrm{s}^{-1}$, and
their centres tend to displace an amount of $0.17\leavevmode\nobreak\
R_{\mathrm{v}}^{\mathrm{rs}}$. It is clear then, that voids cannot be
considered at rest.

Figure 4:  Two-dimensional distribution
$(|\textbf{V}_{\mathrm{v}}|,|\textbf{d}_{\mathrm{v}}|)$. The cross indicates
the two-dimensional mode, which shows that voids tend to move with a speed of
$290\leavevmode\nobreak\ \mathrm{km}\leavevmode\nobreak\ \mathrm{s}^{-1}$, and
their centres tend to shift an amount of $0.17\leavevmode\nobreak\
R_{\mathrm{v}}^{\mathrm{rs}}$.

Concerning the velocities, Figure 5 shows the distribution of the components
of $\textbf{V}_{\mathrm{v}}$ along the three directions of the simulation box.
They all show Gaussian shapes, centred at $0\leavevmode\nobreak\
\mathrm{km}\leavevmode\nobreak\ \mathrm{s}^{-1}$ with a dispersion of
$231\leavevmode\nobreak\ \mathrm{km}\leavevmode\nobreak\ \mathrm{s}^{-1}$.
This was expected, since there must not be any privileged direction of motion
for voids according to the cosmological principle.

Figure 5:  Distribution of the components of $\textbf{V}_{\mathrm{v}}$ along
the three directions of the simulation box. They show Gaussian shapes, centred
at $0\leavevmode\nobreak\ \mathrm{km}\leavevmode\nobreak\ \mathrm{s}^{-1}$,
with a dispersion of $231\leavevmode\nobreak\ \mathrm{km}\leavevmode\nobreak\
\mathrm{s}^{-1}$. This is a manifestation of the isotropic movement of voids.

Concerning the displacements, the left-hand panel of Figure 6 shows the
distribution of the components of $\textbf{d}_{\mathrm{v}}$ along the three
directions of the simulation box. They all show Gaussian shapes centred at
$0$. However, unlike the velocities, displacements manifest a difference
depending on the direction. On the one hand, the POS distributions (green
dotted curve and blue dashed curve) are almost identical, as expected, with a
dispersion of $0.25$. On the other hand, the LOS distribution (red solid
curve) has a dispersion of $0.3$, different to the other two. However, after
correcting the LOS displacements with Eq. (15), a distribution that
practically coincides with the previous ones is obtained, as shown in the
right-hand panel. Nevertheless, the three still show a residual and isotropic
displacement. This can be attributed to Poisson noise when the void finder
tries to localise the optimum centre. This is step (4) in the void
identification process (Section 2).

Figure 6:  Left-hand panel. Distribution of the components of
$\textbf{d}_{\mathrm{v}}$ along the three directions of the simulation box.
They show Gaussian shapes centred at $0$. The POS distributions (green dotted
and blue dashed curves) are almost identical with a dispersion of $0.25$,
whereas the LOS distribution (red solid curve) is very different, with a
dispersion of $0.3$. Right-hand panel. The three distributions are almost
identical after correcting the LOS displacements with Eq. (15). This is a
statistical demonstration of the v-RSD off-centring effect.

The phenomenon described in the last paragraph is more evident in the left-
hand panel of Figure 7, where the two-dimensional distribution
$(V_{\mathrm{v3}},d_{\mathrm{v3}})$ is shown, i.e. for the LOS components of
the velocity and displacement. There is a linear trend between both
quantities, which is correctly described by Eq. (15), represented by the
dashed line. Specifically, the slope of this line is given by the term
$(1+z)/H(z)$. The right-hand panel of the figure shows that, after correcting
the LOS displacements with this equation, the correlation disappears, leading
to a two-dimensional distribution that is almost identical to the
corresponding ones for the POS components: $(V_{\mathrm{v1}},d_{\mathrm{v1}})$
and $(V_{\mathrm{v2}},d_{\mathrm{v2}})$ (not shown here).

Figure 7:  Left-hand panel. Two-dimensional distribution
$(V_{\mathrm{v3}},d_{\mathrm{v3}})$. There is a linear trend that is correctly
described by the linear relation given by Eq. (15) (dashed line). Right-hand
panel. The correlation disappears after correcting the LOS displacements with
this equation. This is a statistical demonstration of the v-RSD off-centring
effect.

From this analysis, we arrive at the third important conclusion of this
chapter: void centres shift systematically along the line of sight when they
are mapped from $r$-space into $z$-space. This is an RSD-type displacement
that can be statistically quantified by means of Eq. (15). These results give
support to the v-RSD off-centring effect postulated in Section 5.

####  3 Cross correlations

Since the ratio $R_{\mathrm{v}}^{\mathrm{zs}}/R_{\mathrm{v}}^{\mathrm{rs}}$
(or equivalently $\delta R_{\mathrm{v}}$) characterises the change of volume
in voids, the statistical analysis of Section 1 gives support to the t-RSD
expansion effect postulated in Section 2. On the other hand, as the
displacement $\textbf{d}_{\mathrm{v}}$ and net velocity
$\textbf{V}_{\mathrm{v}}$ characterise the movement of voids, the statistical
analysis of Section 2 gives support to the v-RSD off-centring effect
postulated in Section 5. It only remains to test if both effects are
statistically independent by looking for cross correlations between these
quantities. Figure 8 shows the two-dimensional distributions
$(d_{\mathrm{v3}},R_{\mathrm{v}}^{\mathrm{zs}}/R_{\mathrm{v}}^{\mathrm{rs}})$
(left-hand panel) and
$(V_{\mathrm{v3}},R_{\mathrm{v}}^{\mathrm{zs}}/R_{\mathrm{v}}^{\mathrm{rs}})$
(right-hand panel). The horizontal lines are the theoretical predictions given
by $q^{s}_{\mathrm{RSD}}$ (dashed line) and $q^{l}_{\mathrm{RSD}}$ (solid
line). No correlations are observed, giving support to the postulated