Fractal Analysis of the UltraVISTA Galaxy Survey–Fractal Analysis of the
UltraVISTA Galaxy Survey

# Fractal Analysis of the UltraVISTA Galaxy Survey

Sharon Teles  1 Amanda R. Lopes E-mail<EMAIL_ADDRESS>2${}^{\mbox{\thanks{E-mail<EMAIL_ADDRESS>and Marcelo B.
Ribeiro1,3${}^{\mbox{\thanks{E-mail<EMAIL_ADDRESS>author}}}$
1Valongo Observatory E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>author Universidade Federal do Rio de Janeiro
Brazil
2Department of Astronomy  Observatório Nacional  Rio de Janeiro  Brazil
3Physics Institute  Universidade Federal do Rio de Janeiro  Brazil

(2020)

###### Abstract

This paper seeks to test if the large-scale galaxy distribution can be
characterized as a fractal system. Tools appropriate for describing galaxy
fractal structures with a single fractal dimension $D$ in relativistic
settings are developed and applied to the UltraVISTA galaxy survey. A graph of
volume-limited samples corresponding to the redshift limits in each redshift
bins for absolute magnitude is presented. Fractal analysis using the standard
$\Lambda$CDM cosmological model is applied to a reduced subsample in the range
$0.1\leq z\leq 4$, and the entire sample within $0.1\leq z\leq 6$. Three
relativistic distances are used, the luminosity distance
$d_{\scriptscriptstyle L}$, redshift distance $d_{z}$ and galaxy area distance
$d_{\scriptscriptstyle G}$, because for data at $z\gtrsim 0.3$ relativistic
effects are such that for the same $z$ these distance definitions yield
different values. The results show two consecutive and distinct redshift
ranges in both the reduced and complete samples where the data behave as a
single fractal galaxy structure. For the reduced subsample we found that the
fractal dimension is $D=\left(1.58\pm 0.20\right)$ for $z<1$, and
$D=\left(0.59\pm 0.28\right)$ for $1\leq z\leq 4$. The complete sample yielded
$D=\left(1.63\pm 0.20\right)$ for $z<1$ and $D=\left(0.52\pm 0.29\right)$ for
$1\leq z\leq 6$. These results are consistent with those found by Conde-
Saavedra et al. (2015), where a similar analysis was applied to a much more
limited survey at equivalent redshift depths, and suggest that either there
are yet unclear observational biases causing such decrease in the fractal
dimension, or the galaxy clustering was possibly more sparse and the universe
void dominated in a not too distant past.

###### keywords:

cosmology – fractals – galaxy distribution – UltraVISTA survey

##  1 Introduction

The fractal galaxy distribution hypothesis is an approach for the description
of the large-scale structure of the Universe which assumes that this
distribution is formed by a fractal system. This approach characterizes the
system by means of its key feature, the fractal dimension $D$, which is
basically a way of quantifying the irregularity of the distribution
(Mandelbrot, 1983). In the context of the large-scale structure of the
Universe, $D$ essentially measures galactic clustering sparsity or,
complementary, the dominance of voids. Values of $D$ smaller than 3, the
topological dimension where the fractal structure is embedded, means irregular
patterns in the structure. The smaller the values of $D$, the more sparse, or
void dominated, is the galactic clustering.

The determination of the possible fractal properties of a given galaxy
distribution with data gathered from a galactic survey is the subject of
fractal analysis, where standard techniques of fractal geometry are applied to
a given galaxy distribution dataset in order to calculate the fractal
dimension (Pietronero, 1987). There are two ways in which this analysis can be
performed: the single fractal approach or the multifractal one (Coleman &
Pietronero, 1992; Ribeiro & Miguelote, 1998).

Describing a fractal system by means of a single fractal dimension is the
simplest approach, since it basically reduces the quantification of the
irregular patterns within the system by means of a unique value for $D$. The
single fractal approach is also capable of describing more complex
distributions, since a system can exhibit different values of $D$ at different
distance ranges, that is, different scaling ranges may possess different
single values for $D$. This situation means a succession of single fractal
systems at different data ranges (Sylos Labini et al., 1998).

Differently from the single fractal approach, the multifractal one
characterizes the fractal system by several fractal dimensions in the same
scaling range, that is, a whole spectrum of dimensions whose maximum value
corresponds to the single fractal dimension the structure would have if it
were treated as a single fractal. The multifractal approach is applied when
quantities like galactic luminosity or mass have a distribution, that is, they
range between very different values. Hence, this is a generalization of $D$
that includes such distributions and the maximum value of the multifractal
spectrum corresponds to the single fractal dimension the system would have if
the studied quantity did not range, that is, as if it did not exhibit a
multifractal pattern (Gabrielli et al., 2005).

The fractal analysis of galaxy surveys whose redshift measurements are bigger
than $z\approx 0.1-0.3$ cannot be done without considering relativistic
effects. This is because $D$ is determined from plots of volume density vs.
distance, which means volume-limited samples, and when one determines galaxy
distances at those ranges relativistic effects become strong enough so that
one is faced not with one distance value, but several different ones whose
difference increases as $z$ increases. That happens because relativistic
cosmological models do not possess a single distance definition (Ellis, 1971,
2007; Holanda et al., 2010) and, therefore, different distance values can be
assigned for a galaxy with an empirically determined value of $z$, and the
range where those differences become significant also depends on the chosen
cosmological model. As a result, relativistic effects cannot be neglected when
performing fractal analysis of galaxy distributions whose redshift data are
gathered beyond those redshift ranges (Rangel Lemos & Ribeiro, 2008).

In addition, as a consequence of relativistic effects even spatially
homogeneous cosmological models like the standard Friedmann-Lemaître-
Robertson-Walker (FLRW) will present observational inhomogeneities because
observations are performed along the past light cone and at high redshift
ranges distance measures required in the determination of volume densities
will necessarily depart the local spatially homogeneous hypersurfaces of these
models (Ribeiro, 1992b, 1995, 2001b).

Fractal cosmology, that is, modelling the large-scale structure of the
Universe by assuming that the galaxy distribution forms a fractal system, is
an old subject previously known in the literature as hierarchical cosmology.
Discussions regarding the possible hierarchical structuring of the Universe go
as far back as the beginning of the 20th century (Charlier, 1908, 1922;
Einstein, 1922; Selety, 1922; Amoroso Costa, 1929), and attempts to
theoretically describe and empirically characterize this hierarchical galaxy
structure followed suit in later decades (Carpenter, 1938; De Vaucouleurs,
1960, 1970; Wertz, 1970, 1971; Haggerty & Wertz, 1972). The appearance of
fractal geometry in the 1980s showed that the hierarchical galaxy structure
discussed in these earlier studies has essentially the same features of galaxy
distribution models that considered this distribution as a fractal system
(Mandelbrot, 1983; Pietronero, 1987). In fact, the basic theoretical framework
of these early hierarchical models turns out to lead to the same expressions
as the ones originated from a fractal galaxy distribution model (Ribeiro,
1994; Ribeiro & Miguelote, 1998).

Newtonian cosmology was used in earlier hierarchical cosmology models (Wertz,
1970, 1971; Haggerty & Wertz, 1972), as well as in more recent fractal
cosmology ones (Abdalla et al., 1999; Abdalla & Chirenti, 2004). Relativistic
cosmological models, both spatially homogeneous and inhomogeneous, which
considered a fractal system embedded in a 4-dimensional spacetime along the
observer’s past light cone, producing then relativistic fractal cosmologies,
were developed later (Ribeiro, 1992a, 1993; Abdalla et al., 2001; Ribeiro,
2001a, b, 2005). More recently, several authors discussed relativistic
cosmological models with fractal features either theoretically (Mureika &
Dyer, 2004; Mureika, 2007; Sylos Labini, 2011; Nogueira, 2013; Hossienkhani et
al., 2018; Sadri et al., 2018; Cosmai et al., 2019; Jawad et al., 2019) or by
attempting to apply fractal features to different observational scenarios
using Newtonian or relativistic models (Jones et al., 1988; Martínez, 1990;
Pan & Coles, 2000; Gaite, 2007; Stahl, 2016; Raj et al., 2019; Souza, 2020).

Of particular interest to this paper is the study presented by Conde-Saavedra
et al. (2015, from now on CS2015), which stands on the middle ground between
the two previously mentioned types of works in the sense that it developed
both theoretical tools capable of characterizing a fractal system at high
redshift by fully considering relativistic effects, and also performed data
analysis of high redshift galaxy data in order to actually measure the single
fractal dimension of the structure at different scale ranges having deep
redshift values, that is, far from our present spatial foliation of spacetime
as described in a 3+1 formalism of general relativity. This study concluded
that for $z\lesssim 1.3-1.9$ the average single fractal dimension that
considered all distance definitions used in the paper resulted in
$D=1.4^{\scriptscriptstyle+0.7}_{\scriptscriptstyle-0.6}$, whereas for
redshift values higher than this approximate threshold they obtained
$D=0.5^{\scriptscriptstyle+1.2}_{\scriptscriptstyle-0.4}$.

This paper aims at applying the fractal analysis methodology used by CS2015 to
a different galaxy sample, but with two major differences. First, the fractal
analysis is applied to the UltraVISTA galaxy survey, which has measured
redshift values of about 220k galaxies, a considerably larger galaxy sample
than the FORS Deep Field dataset of 5.5k galaxies used in CS2015. Second,
samples are obtained directly from measured redshift data instead of the
indirect luminosity function methodology employed by CS2015. A graph of
absolute magnitudes in terms of redshifts shows that the UltraVISTA galaxies
scale with redshift bins, providing then a volume-limited subsample
appropriate for a fractal analysis. Nevertheless the whole survey data was
also subject to fractal analysis for comparison purposes.

As a consequence of these different approaches, the results obtained here are
clearly improved once compared to the ones achieved by CS2015, namely, a
better defined threshold for low and high scaling ranges, smaller
uncertainties and results more in line with each other considering all
cosmological distance definitions used in both studies. Our calculations
showed that summing up all results and uncertainties obtained with the
employed distance definitions we concluded that both the subsample and entire
survey data of the UltraVISTA catalog can be well characterized as a fractal
galaxy distribution system possessing two consecutive scaling ranges with the
following single fractal dimensions. The subsample resulted in
$D=\left(1.58\pm 0.20\right)$ for $z<1$, and $D=\left(0.59\pm 0.28\right)$ for
$1\leq z\leq 4$, whereas the complete sample yielded $D=\left(1.63\pm
0.20\right)$ for $0.1<z<1$, and $D=\left(0.52\pm 0.29\right)$ for $1\leq z\leq
6$.

The plan of the paper is as follows. Section 2 develops standard tools of
fractal geometry necessary for modelling the large-scale galaxy distribution
as a fractal system in both Newtonian and relativistic frameworks, comprising
review material extensively discussed and developed elsewhere plus a few
additional remarks. This is included here for a self-contained presentation.
Section 3 describes the observational details of the UltraVISTA galaxy survey
relevant to this work, and discusses the data handling required for the
application of fractal tools to this specific dataset. Section 4 presents the
results of the fractal analysis of the UltraVISTA galaxy distribution.
Discussions and conclusions are the subject of Section 5.

##  2 Fractal cosmology

It has been known since Mandelbrot’s (1983) original studies that fractal
systems are characterized by power-laws. In fact, the early hierarchic
cosmological models were connected to fractals exactly because galaxy density
distribution data showed power-law features (e.g., De Vaucouleurs, 1970).
Hence, the early definitions and concepts used in the hierarchical cosmologies
are the appropriate ones to start with. As it turns out these quantities form
a set of very simple concepts and definitions able to characterize single-
fractal-dimension galaxy distributions. Moreover, they are easily and
straightforwardly adapted to a relativistic setting, albeit with some
limitations.

###  2.1 Newtonian hierarchical (fractal) cosmology

Let $V_{{\textstyle{\scriptscriptstyle\rm obs}}}$ be the observational volume
defined by the expression below,

$V_{{\textstyle{\scriptscriptstyle\rm
obs}}}=\frac{4}{3}\pi{(d_{{\textstyle{\scriptscriptstyle\rm obs}}})}^{3},$ (1)

where $d_{{\textstyle{\scriptscriptstyle\rm obs}}}$ is the observational
distance. The observed volume density
$\gamma_{{\textstyle{\scriptscriptstyle\rm obs}}}^{\ast}$ is defined as
follows,

$\gamma_{{\textstyle{\scriptscriptstyle\rm
obs}}}^{\ast}=\frac{N_{{\textstyle{\scriptscriptstyle\rm
obs}}}}{V_{{\textstyle{\scriptscriptstyle\rm obs}}}},$ (2)

where $N_{{\textstyle{\scriptscriptstyle\rm obs}}}$ is the observed cumulative
number counts of cosmological sources, that is, galaxies. Clearly
$\gamma_{{\textstyle{\scriptscriptstyle\rm obs}}}^{\ast}$ gives the number of
sources per unit of observational volume out to a distance
$d_{{\textstyle{\scriptscriptstyle\rm obs}}}$.

The key hypothesis underlining the hierarchical (fractal) galaxy distribution
relates the cumulative number counts of observed cosmological sources and the
observational distance by a phenomenological equation called the number-
distance relation (Wertz, 1970; Pietronero, 1987), whose expression yields,

$N_{{\textstyle{\scriptscriptstyle\rm
obs}}}=B\,{(d_{{\textstyle{\scriptscriptstyle\rm obs}}})}^{D},$ (3)

where $B$ is a positive constant and $D$ is the fractal dimension. This
expression forms the basic hypothesis of the Pietronero-Wertz hierarchical
(fractal) cosmology (Ribeiro, 1994; Ribeiro & Miguelote, 1998, and references
therein).

Note that since $N_{{\textstyle{\scriptscriptstyle\rm obs}}}$ is a cumulative
quantity, if beyond a certain distance there are no longer galaxies then
$N_{{\textstyle{\scriptscriptstyle\rm obs}}}$ no longer increases with
$d_{{\textstyle{\scriptscriptstyle\rm obs}}}$. If instead objects are still
detected and counted then it continues to increase. Observational biases may
possibly affect its rate of growth, leading to an intermittent behavior,
however, $N_{{\textstyle{\scriptscriptstyle\rm obs}}}$ must grow or remain
constant, and thus the exponent in Eq. (3) must be positive or zero.

One may also define a second density in this context, the  observed
differential density $\gamma_{{\textstyle{\scriptscriptstyle\rm obs}}}$. Its
expression may be written as follows (Wertz, 1970, 1971),

$\gamma_{{\textstyle{\scriptscriptstyle\rm
obs}}}=\frac{1}{4\pi{(d_{{\textstyle{\scriptscriptstyle\rm
obs}}})}^{2}}\frac{\mathrm{d}N}{\mathrm{d}(d_{{\textstyle{\scriptscriptstyle\rm
obs}}})}.$ (4)

From this definition it is clear that
$\gamma_{{\textstyle{\scriptscriptstyle\rm obs}}}$ gives the rate of growth in
number counts, or more exactly in their density, as one moves along the
observational distance $d_{{\textstyle{\scriptscriptstyle\rm obs}}}$.

Substituting Eqs. (1) and (3) into Eqs. (2) and (4) we respectively reach at
two forms of the De Vaucouleurs density power-law (Pietronero, 1987; Ribeiro,
1994),

$\gamma_{{\textstyle{\scriptscriptstyle\rm
obs}}}^{\ast}=\frac{3B}{4\pi}{(d_{{\textstyle{\scriptscriptstyle\rm
obs}}})}^{D-3},$ (5) $\gamma_{{\textstyle{\scriptscriptstyle\rm
obs}}}=\frac{DB}{4\pi}{(d_{{\textstyle{\scriptscriptstyle\rm obs}}})}^{D-3}.$
(6)

Thus, if the observed galaxy distribution behaves as a fractal system with
$D<3$, that is, if it follows the number-distance relation (3), both
observational densities above decay as power-laws. If $D=3$ the distribution
is said to be observationally homogeneous, as both densities become constant
and distance independent. Note that the two power-laws above allow the
empirical determination of different single fractal dimensions in two or more
scaling ranges dependent on the observational distance.

The ratio between the two forms of the De Vaucouleurs density power-law
yields,

$\frac{\gamma_{{\textstyle{\scriptscriptstyle\rm
obs}}}}{\gamma_{{\textstyle{\scriptscriptstyle\rm obs}}}^{\ast}}=\frac{D}{3}.$
(7)

For an observationally homogeneous galaxy distribution this ratio must be
equal to one, whereas an irregular distribution forming a single fractal
system will have $0\leq\left(\gamma_{{\textstyle{\scriptscriptstyle\rm
obs}}}\big{.}\big{/}\gamma_{{\textstyle{\scriptscriptstyle\rm
obs}}}^{\ast}\right)<1$.

There are two important remarks to be made regarding the densities defined by
Eqs. (2) and (4). First, both $\gamma_{{\textstyle{\scriptscriptstyle\rm
obs}}}^{\ast}$ and $\gamma_{{\textstyle{\scriptscriptstyle\rm obs}}}$ are
radial quantities and, therefore, should not be understood in statistical
sense because they do not average all points against all points. Second,
although both quantities are in principle equally applicable to cosmological
objects, as pointed out by Conde-Saavedra et al. (2015, Secs. 4.2.1, 4.2.2)
$\gamma_{{\textstyle{\scriptscriptstyle\rm obs}}}$ is unsuitable for high
redshift measures because the term
$\mathrm{d}N_{{\textstyle{\scriptscriptstyle\rm
obs}}}/\mathrm{d}(d_{{\textstyle{\scriptscriptstyle\rm obs}}})$ in Eq. (4)
increases, reaches a maximum and then decreases, affecting the exponent at
high values of $z$ in such a way as producing spurious negative values for $D$
at such ranges. As noted above, negative values for $D$ are not possible due
to the very definition of the number-distance relation (3) and, hence,
$\gamma_{{\textstyle{\scriptscriptstyle\rm obs}}}$ can only be safely used at
relatively low redshift values, that is, $z\lesssim 1$ or at ranges smaller
than the observed galaxy distribution histogram of galaxy numbers per redshift
bins reaches its maximum.

So, for the reasons exposed above, from now on we shall only consider the
volume density $\gamma_{{\textstyle{\scriptscriptstyle\rm obs}}}^{\ast}$ in
our calculations, since this density is not contaminated by spurious effects
at $z>1$.

###  2.2 Relativistic fractal cosmology

The expressions above can be applied as such in Newtonian cosmologies, but as
far as relativistic cosmological models are concerned two important conceptual
issues must be considered which alter the expressions above in specific ways.

First, in relativistic cosmology observations are located along the observer’s
past light cone. This means that even spatially homogeneous cosmologies like
the FLRW will not produce observationally constant volume densities at high
redshift values because it is theoretically impossible to expect that the
observed volume density will become constant even at moderate redshift values
in FLRW cosmologies (Conde-Saavedra et al., 2015, Sec. 2.1). The basic point
here is that observational and spatial homogeneities are different concepts in
relativistic cosmology, so it is theoretically possible to have a
cosmological-principle-obeying spatially homogeneous cosmological model
exhibiting observational inhomogeneity, as extensively discussed elsewhere
(Ribeiro, 1992b, 1994, 1995, 2001b, 2005; Rangel Lemos & Ribeiro, 2008).

Second, both $\gamma_{{\textstyle{\scriptscriptstyle\rm obs}}}^{\ast}$ and
$\gamma_{{\textstyle{\scriptscriptstyle\rm obs}}}$ are average densities
defined in the fractal cosmology context, and, therefore, should not be
confused with the local density appearing on the right hand side of Einstein
equations. Moreover, densities in fractal cosmology are defined in terms of
observational distances, which means that at high redshift
$d_{{\textstyle{\scriptscriptstyle\rm obs}}}$ will have different values for
each distance definition at the same redshift value $z$. In other words, as
distance in relativistic cosmology is a concept not uniquely defined (Ellis,
1971, 2007; Holanda et al., 2010) we need to replace
$d_{{\textstyle{\scriptscriptstyle\rm obs}}}$ for $d_{i}$ in the equations
above, where the index will indicate the observed distance definition chosen
to be calculated with specific redshift values. In this case the applicable
distance definitions are the redshift distance $d_{z}$, luminosity distance
$d_{\scriptscriptstyle L}$ and galaxy area distance $d_{\scriptscriptstyle
G}$, also known as transverse comoving distance. The last two are connected by
the Etherington reciprocity law (Etherington, 1933; Ellis, 2007) which reads
as follows,

$d_{\scriptscriptstyle L}=(1+z)\,d_{\scriptscriptstyle G}.$ (8)

The redshift distance yields,

$d_{z}=\frac{c\,z}{H_{0}},$ (9)

where $c$ is the light speed and $H_{0}$ is the Hubble constant. This
definition of $d_{z}$ is, of course, only valid in the FLRW metric. Albani et
al. (2007) and Iribarrem et al. (2012) showed that within the FLRW cosmology
the densities defined by both $d_{\scriptscriptstyle L}$ and $d_{z}$ have
empirical power-law properties, the same applying for $d_{\scriptscriptstyle
G}$.

Another distance measure that can be defined in this context is the angular
diameter distance $d_{\scriptscriptstyle A}$, also known simply as area
distance, which is also connected to the quantities above by the reciprocity
law, also known in the literature as the cosmic distance duality relation
(Holanda et al., 2010, 2011, 2012; Zheng et al., 2020), which reads as
follows,

$d_{\scriptscriptstyle L}={(1+z)}^{2}d_{\scriptscriptstyle A}.$ (10)

However, densities defined with $d_{\scriptscriptstyle A}$ have the odd
behavior of increasing as $z$ increases, making this distance unsuitable to
use in the context of a fractal analysis of the galaxy distribution (Ribeiro,
2001b, 2005; Rangel Lemos & Ribeiro, 2008).

Bearing these points in mind, the expressions above become applicable to
relativistic cosmology models once they are rewritten as below,

$d_{{\textstyle{\scriptscriptstyle\rm obs}}}=d_{i},$ (11)
$V_{{\textstyle{\scriptscriptstyle\rm
obs}}}=V_{i}=\frac{4}{3}\pi{(d_{i})}^{3},$ (12)
$N_{{\textstyle{\scriptscriptstyle\rm obs}}}=N_{i}=B_{i}\,{(d_{i})}^{D_{i}},$
(13) $\gamma_{{\textstyle{\scriptscriptstyle\rm
obs}}}^{\ast}=\gamma^{\ast}_{i}=\frac{N_{i}}{V_{i}}=\frac{3B_{i}}{4\pi}{(d_{i})}^{D_{i}-3},$
(14)

where $i=({\scriptscriptstyle L}$, ${\scriptstyle z}$, ${\scriptscriptstyle
G}$) according to the distance definition used to calculate the volume
density. The proportionally constant $B_{i}$ of the number-distance relation
will, therefore, be attached to each specific distance definition, this being
also true for the fractal dimension $D_{i}$, because $N_{i}$ is counted
considering the limits of each distance definition. Hence, for a given $z$
each $d_{i}$ yields its respective $V_{i}$, $N_{i}$, $B_{i}$ and $D_{i}$,
which means that all quantities become attached to a certain distance
definition.

As final points, first it is important to mention that although the fractal
analysis discussed above can be performed using $d_{\scriptscriptstyle L}$ in
any cosmological model, the same is not true for $d_{z}$ because its
definition in Eq. (9) is restricted to FLRW cosmologies. Regarding
$d_{\scriptscriptstyle G}$, it has been previously shown that calculating the
volume density in the Einstein-de Sitter cosmology using
$d_{\scriptscriptstyle G}$ results in $\gamma_{\scriptscriptstyle
G}^{\ast}=\mbox{constant}$ (Ribeiro, 2001b, pp. 1718, 1723, 1724), which
seemed to indicate $d_{\scriptscriptstyle G}$ as an unsuitable distance
definition to be used in fractal analysis. Nevertheless, later works showed
that such behavior is cosmology dependent, not being valid in other FLRW
models Albani et al., 2007, fig. 7; Iribarrem et al., 2012, figs. 2-5; Conde-
Saavedra et al., 2015, figs. 3-4. This finding justifies the inclusion of
$\gamma_{\scriptscriptstyle G}^{\ast}$ in the present study.

Second, the concepts above indicate that reasoning that regards the fractal
approach to the galaxy distribution only at low redshift ranges are not
applicable to the analysis performed in this paper. As the light cone is a
relativistic concept, confusion arises if one does not acknowledge the
difference between spatial and observational volume densities. It is in the
latter spacetime locus that astronomy is made and where fractality in the
sense of this work may be detected. Hence, observed fractal features appear
only by correctly manipulating the FLRW observational quantities along the
observer’s past light cone at ranges where the null cone effects become
relevant as far as distance definitions are concerned (Ribeiro, 1995, 2001b,
2005; Rangel Lemos & Ribeiro, 2008). These effects form the very core of the
present analysis.

##  3 Fractal analysis

As this study seeks to empirically ascertain whether or not the fractal galaxy
distribution hypothesis holds at very large-scales of the observed Universe,
we chose to perform a fractal analysis with the data provided by the
UltraVISTA galaxy survey, since it contains hundreds of thousands of galaxies
with measured redshifts. Let us present next some details about this survey
and how the fractal analysis was performed in their data.

###  3.1 The UltraVISTA galaxy survey

Our data are based on the first data release of the UltraVISTA galaxy survey,
which is centered on the COSMOS field (Scoville et al., 2007) with an
effective area of $1.5\,\mbox{deg}^{\,2}$. Observations were made in 4 near
infrared filters, $Y$, $J$, $H$ and $K_{\mathrm{S}}$, described in McCracken
et al. (2012). Photometric redshifts were calculated by Ilbert et al. (2013)
applying the SED fitting technique to 29 bands, the ones from UltraVISTA and a
complementary set of broad and narrow bands from other surveys encompassing
the ultraviolet, optical, near infrared and mid infrared regimes. The initial
dataset consisted in a $K_{\mathrm{S}}$-band selected ($K_{\mathrm{S}}<24$)
sample of about 220k galaxies in the redshift range of $0.1\leq z\leq 6$.
Although the sample was originally divided into quiescent and star forming
galaxies, such grouping is unimportant for the purposes of this study and,
hence, all galaxies of both types were included in the subsample selection and
analysis performed here. Fig. 1 shows the UltraVISTA survey’s galaxy numbers
distribution in terms of redshift. Figs. 2 and 3 respectively show the galaxy
numbers distribution in terms of right ascension and declination.

Figure 1: Histogram showing the galaxy distribution numbers in terms of
redshift for the UltraVISTA survey dataset studied here. This graph has
$\Delta z=0.03$ as the redshift bins’ size. Figure 2: Histogram showing the
galaxy distribution numbers in terms of right ascension (deg) for the
UltraVISTA survey dataset studied here. Figure 3: Histogram showing the galaxy
distribution numbers in terms of declination (deg) for the UltraVISTA survey
dataset studied here.

###  3.2 Data selection

The fractal analytical tools discussed above require the establishment of
volume limited samples. However, galaxy surveys are limited by apparent
magnitude, so some methodology is needed to ascertain that reduced subsamples
follow increasing redshift bins in order to render the reduced galaxy data
distributed over limited volume bins. One way of doing this is by plotting the
galaxies’ absolute magnitudes in terms of their respective measured redshifts
and then selecting galaxies below a certain absolute magnitude threshold
defined by the limiting apparent magnitude of the survey. This can be done by
using the usual expression below,

$M=m-5\log d_{\scriptscriptstyle L}(z)-25,$ (15)

where $M$ is the absolute magnitude, $m$ is the apparent magnitude and
$d_{\scriptscriptstyle L}$ is given in Mpc. Next one needs to choose the
apparent magnitude threshold $m$ and its passband, as well as verifying if the
resulting data is indeed, at least generally, distributed along the measured
redshift bins, and then possibly establishing a redshift window for the final
subsample distribution.

The UltraVISTA survey furnished absolute magnitudes calculated in the $NUV$,
$B$, $R$ and $J$ passbands. The $J$-band is the best choice among those for
our purposes here, since this near infrared filter is less affected by dust
extinction. Fig. 4 shows an histogram of the UltraVISTA galaxy numbers in
terms of apparent magnitudes in the $K_{\mathrm{S}}$ and $J$ passbands.
Clearly the number distribution peaks at the apparent magnitude value 24 for
both wavebands, a fact that led us to choose $J=24$ as our apparent magnitude
threshold. Therefore, Eq. (15) can be rewritten according to the expression
below, which provides the dividing line between the selected and unselected
galaxies.

Figure 4: UltraVISTA galaxy numbers plotted in terms of apparent magnitudes in
both the $J$ and $K_{\mathrm{S}}$ passbands. The number distribution for both
bandwidths peaks at apparent magnitudes equal to 24 indicated by the vertical
line.

$M_{J}=24-5\log d_{\scriptscriptstyle L}(z)-25.$ (16)

Fig. 5 shows a plot of the absolute magnitudes of 219300 UltraVISTA galaxies
in the $J$-band against their respective measured redshifts. Green filled
circles are those whose absolute magnitudes provided by the survey are smaller
than $M_{J}$ in Eq. (16), and are then inside the reduced subsample, whereas
the open gray circles are outside this threshold. We noticed that the absolute
magnitude cut based on the $J$-band corresponds to a volume-limited subsample
because the data generally follow the redshift bins of increasing values. In
addition, we also noticed that there are few galaxies at the tail of the
distribution, so our subsample suffered a further cut at $z=4$.

Hence, we end up with a reduced UltraVISTA subsample of 166566 galaxies cut by
$J$-band absolute magnitudes $M_{J}$ and limited up to $z=4$. The remaining
52734 galaxies outside this subsample, that is, below the dividing line in
Fig. 5 and having $z>4$, were disregarded.

Figure 5: Absolute magnitudes in the $J$-band vs. redshift for all galaxies of
the UltraVISTA survey. Green filled circles are galaxies having $M_{J}$
smaller than the blue line cut given by Eq. (16), whereas open gray circles
have bigger values for $M_{J}$.

###  3.3 Data analysis

To obtain the observational distances $d_{i}\,(i={\scriptscriptstyle G}$,
${\scriptscriptstyle L}$, ${\scriptstyle z})$ from the calculated photometric
redshift values one needs to choose a cosmological model. We assumed the FLRW
cosmology with $\Omega_{m_{\scriptscriptstyle 0}}=0.3$,
$\Omega_{\Lambda_{\scriptscriptstyle 0}}=0.7$ and
$H_{0}=70\;\mbox{km}\;{\mbox{s}}^{-1}\;{\mbox{Mpc}}^{-1}$.

The next steps were the establishment of the minimum redshift value
$z_{\scriptscriptstyle 0}$ from where to start the analysis, the respective
minimum distances $d_{i_{0}}=d_{i_{0}}(z_{\scriptscriptstyle 0})$, and the
incremental distance interval $\Delta d_{i}$. The data sorting process was
initiated by counting the number of observed galaxies
$N_{i_{\scriptscriptstyle\rm 1}}$ in the first interval
$d_{i_{1}}=d_{i_{0}}+\Delta d_{i}$ and calculating the respective volume
density $\gamma_{i_{1}}^{\ast}$. This first interval defined the first bin.
Then, the size of the bin was increased by $\Delta d_{i}$ and values for
$N_{i_{\scriptscriptstyle\rm 2}}$ and $\gamma_{i_{2}}^{\ast}$ were obtained at
the distance $d_{i_{2}}=d_{i_{0}}+2\Delta d_{i}$. This algorithm was repeated
$n$ times until the last, and farthest, group of galaxies were included and
all relevant quantities were also counted and calculated.

Different bin size increments $\Delta d_{i}$ were tested for each distance
definition in order to find out whether or not that would affect the results.
This test turned out negative, which means that the obtained results are
independent of bin size increment. We then chose $\Delta d_{i}=200$ Mpc, value
which was applied to all calculations and provided in the end a very
reasonable amount of data points for each quantity from where simple linear
regression analyses were able to be performed.

The final step was the determination of the fractal dimension itself. If the
galaxy distribution really formed a fractal system, according to Eq. (14) the
graphs of $\gamma_{i}^{\ast}$ vs. $d_{i}$ would behave as decaying power-law
curves, and whose linear fit slopes in log-log plots allow for the fractal
dimensions $D_{i}$ of the distribution to be straightforwardly determined.

##  4 Results

###  4.1 Reduced subsample

Graphs for log-log values of $\gamma_{i}^{\ast}$ vs. $d_{i}$ showed that to a
good approximation the reduced UltraVISTA galaxy survey subsample sorted
according to the criteria set at Sec. 3.2 conforms with what is predicted as
if the galaxy distribution does form a fractal system. Moreover, two power-law
decaying regions were observed in the data, for $z<1$ and $z>1$, producing
different single fractal dimensions.

Figs. 6 to 8 present the results for all distance definitions adopted here,
from where it can be concluded that for $z<1$ the fractal dimension is in the
range $1.38-1.78$, whereas for $1\leq z\leq 4$ the resulting range is
significantly smaller, $0.31-0.87$. Table 1 collects all results.

Figure 6: Graph showing the log-log results for $\gamma_{\scriptscriptstyle L}^{\ast}$ vs. $d_{\scriptscriptstyle L}$ obtained with the reduced UltraVISTA galaxy redshift survey dataset (see Sec. 3.2). The dotted line is the straight line fit for galaxies having $z<1$, whereas the dashed line is for those with $z>1$. The error area is in gray. According to Eq. (14) the fractal dimensions obtained from these data are $D_{\scriptscriptstyle L}=(1.40\pm 0.02)$ for $z<1$ and $D_{\scriptscriptstyle L}=(0.32\pm 0.01)$ for $1\leq z\leq 4$. Figure 7: Graph showing the log-log results for $\gamma_{\scriptscriptstyle z}^{\ast}$ vs. $d_{z}$ obtained with the reduced UltraVISTA galaxy redshift survey dataset (see Sec. 3.2). The dotted line is the straight line fit for galaxies having $z<1$, whereas the dashed line is for those with $z>1$. The error area is in gray. According to Eq. (14) the fractal dimensions obtained from these data are $D_{\scriptscriptstyle z}=(1.61\pm 0.02)$ for $z<1$ and $D_{\scriptscriptstyle z}=(0.38\pm 0.02)$ for $1\leq z\leq 4$. Figure 8: Graph showing the log-log results for $\gamma_{\scriptscriptstyle G}^{\ast}$ vs. $d_{\scriptscriptstyle G}$ obtained with the reduced UltraVISTA galaxy redshift survey dataset (see Sec. 3.2). The dotted line is the straight line fit for galaxies having $z<1$, whereas the dashed line is for those with $z>1$. The error area is in gray. According to Eq. (14) the fractal dimensions obtained from these data are $D_{\scriptscriptstyle G}=(1.75\pm 0.03)$ for $z<1$ and $D_{\scriptscriptstyle G}=(0.81\pm 0.06)$ for $1\leq z\leq 4$. Table 1: Results in two redshift scales of the UltraVISTA galaxy survey fractal analysis in the reduced subsample (see Sec. 3.2). The single fractal dimensions $D_{\scriptscriptstyle L}$, $D_{\scriptscriptstyle z}$ and $D_{\scriptscriptstyle G}$ were obtained from this galaxy distribution respectively using the luminosity distance $d_{\scriptscriptstyle L}$, redshift distance $d_{z}$ and galaxy area distance (transverse comoving distance) $d_{\scriptscriptstyle G}$. | $D_{\scriptscriptstyle L}$ | $D_{\scriptscriptstyle z}$ | $D_{\scriptscriptstyle G}$
---|---|---|---
$z<1$ | $1.40\pm 0.02$ | $1.61\pm 0.02$ | $1.75\pm 0.03$
$1\leq z\leq 4$ | $0.32\pm 0.01$ | $0.38\pm 0.02$ | $0.81\pm 0.06$

In summary, the results above indicate that the UltraVISTA galaxy survey
provided a galaxy distribution subsample dataset that can be described as a
fractal system with the following two consecutive and approximate single
fractal dimensions values: $D\,(z<1)=(1.58\pm 0.20)$ and $D\,(1\leq z\leq
4)=(0.59\pm 0.28)$. The possible reasons as why the fractal dimension is so
much reduced at the deepest range $z>1$ will be discussed below.

###  4.2 Complete (unselected) survey data

The fractal analysis of the UltraVISTA galaxy subsample, as defined in Sec.
3.2, were based in the plot of absolute magnitudes in the $J$-band in terms of
the measured redshifts of the galaxies shown in Fig. 5. However, this plot
also shows that even the galaxies outside the absolute magnitude cut are also
distributed along increasing redshift bins, a fact that suggests that the
whole sample may also be volume-limited. Hence, it is interesting to apply the
same fractal methodology developed above not only to the subsample, but also
to the whole survey data in order to compare the results.

Figs. 9-11 show the results of the fractal analysis of the complete UltraVISTA
survey data. It is clear that the galaxy distribution also presents fractal
features in two regions below and above the redshift value $z=1$. The
corresponding single fractal dimensions for each distance definitions adopted
here were found to lie in the range $1.42-1.83$ for $z<1$, whereas for $1\leq
z\leq 6$ the dimension is significantly smaller, in the range $0.23-0.81$.
Table 2 collects these results and a comparison with the ones for the reduced
subsample presented in Table 1 indicated results very much alike, although
their respective uncertainties do not overlap.

Figure 9: Graph showing the log-log results for $\gamma_{\scriptscriptstyle
L}^{\ast}$ vs. $d_{\scriptscriptstyle L}$ obtained with the complete
UltraVISTA galaxy survey dataset. The dotted line is the straight line fit for
galaxies having $z<1$, whereas the dashed line is for those with $z>1$. The
error area is in gray. According to Eq. (14) the fractal dimensions obtained
from these data are $D_{\scriptscriptstyle L}=(1.44\pm 0.02)$ for $z<1$ and
$D_{\scriptscriptstyle L}=(0.24\pm 0.01)$ for $1\leq z\leq 6$. Figure 10:
Graph showing the log-log results for $\gamma_{\scriptscriptstyle z}^{\ast}$
vs. $d_{z}$ obtained with the complete UltraVISTA galaxy survey dataset. The
dotted line is the straight line fit for galaxies having $z<1$, whereas the
dashed line is for those with $z>1$. The error area is in gray. According to
Eq. (14) the fractal dimensions obtained from these data are
$D_{\scriptscriptstyle z}=(1.65\pm 0.02)$ for $z<1$ and $D_{\scriptscriptstyle
z}=(0.30\pm 0.02)$ for $1\leq z\leq 6$. Figure 11: Graph showing the log-log
results for $\gamma_{\scriptscriptstyle G}^{\ast}$ vs. $d_{\scriptscriptstyle
G}$ obtained with the complete UltraVISTA galaxy survey dataset. The dotted
line is the straight line fit for galaxies having $z<1$, whereas the dashed
line is for those with $z>1$. The error area is in gray. According to Eq. (14)
the fractal dimensions obtained from these data are $D_{\scriptscriptstyle
G}=(1.80\pm 0.03)$ for $z<1$ and $D_{\scriptscriptstyle G}=(0.75\pm 0.06)$ for
$1\leq z\leq 6$.

So, it seems that the whole UltraVISTA galaxy survey can also be described as
a fractal system having two consecutive single fractal dimensions:
$D\,(z<1)=(1.63\pm 0.20)$ and $D\,(1\leq z\leq 6)=(0.52\pm 0.29)$.

Table 2: Results in two redshift scales of the UltraVISTA galaxy survey fractal analysis in the complete sample. The single fractal dimensions $D_{\scriptscriptstyle L}$, $D_{\scriptscriptstyle z}$ and $D_{\scriptscriptstyle G}$ were obtained from this galaxy distribution respectively using the luminosity distance $d_{\scriptscriptstyle L}$, redshift distance $d_{z}$ and galaxy area distance (transverse comoving distance) $d_{\scriptscriptstyle G}$. | $D_{\scriptscriptstyle L}$ | $D_{\scriptscriptstyle z}$ | $D_{\scriptscriptstyle G}$
---|---|---|---
$z<1$ | $1.44\pm 0.02$ | $1.65\pm 0.02$ | $1.80\pm 0.03$
$1\leq z\leq 6$ | $0.24\pm 0.01$ | $0.30\pm 0.02$ | $0.75\pm 0.06$

##  5 Conclusions

This paper sought to empirically test if the large-scale galaxy distribution
can be described as a fractal system. Tools originally developed for Newtonian
hierarchical cosmology were extended and applied to relativistic cosmological
models in order to possibly describe galaxy fractal structures by means of
single fractal dimensions at deep redshift values. These tools were applied to
the UltraVISTA galaxy survey dataset comprising 220k objects spanning the
redshift interval of $0.1\leq z\leq 6$.

A reduced subsample of the survey was established by plotting the galaxies’
absolute magnitudes in the $J$-band against their respective redshifts and
setting an absolute magnitude cut with the apparent magnitude of $J=24$, as
well as a redshift cut at $z=4$. Since this subsample showed that its galaxies
followed increasing redshift bins, they were considered as effectively being a
volume-limited distribution.

Fractal analysis of the reduced subsample was carried out using the standard
$\Lambda$CDM relativistic cosmological model. As relativistic cosmologies have
several definitions of observed distance, only three distinct ones were used
here, namely the luminosity distance $d_{\scriptscriptstyle L}$, redshift
distance $d_{z}$ and galaxy area distance $d_{\scriptscriptstyle G}$, also
known as transverse comoving distance. The use of several cosmological
distance measures is due to the fact that relativistic effects become strong
enough for redshift ranges larger than $z\gtrsim 0.3$, so these distance
definitions produce different results for the same redshift value at those
ranges. An algorithm for sorting the data according to the analytical
developments required for testing an observed fractal structure as discussed
here was also detailed.

The results indicate that the UltraVISTA subsample has two consecutive
redshift ranges behaving as single fractal structures. For $z<1$ the derived
fractal dimension is well approximated by $D=(1.58\pm 0.20)$, whereas for
$1\leq z\leq 4$ this dimension decreased to $D=(0.59\pm 0.28)$. For
comparison, the same fractal analysis was also carried out in the complete
survey data, yielding as well two consecutive redshift ranges characterized as
single fractal systems: $D=(1.63\pm 0.20)$ for $z<1$ and $D=(0.52\pm 0.29)$ in
the range $1\leq z\leq 6$. Both results, from the reduced and complete data,
are consistent with those found by Conde-Saavedra et al. (2015), although here
the conclusions were reached by a different, and simpler, methodology, and by
applying this methodology to a numerically much larger galaxy sample in both
cases.

The most obvious question regarding these results is why there is such a
significant decrease in the fractal dimension for redshift values larger than
the unity. Conceivably this might be due to an observational bias caused by
the simple fact that many galaxies located beyond $z=1$ are not being
detected, decreasing then the observed galaxy clustering and, therefore, the
associated fractal dimension at those scales. One might also consider the
possibility of a bias in the galaxy number counts due to the small angular
area of this survey. This means that its observational area might not provide
a representative measurement of the entire sky distribution. Moreover it is
important to consider the choice of observational field. If the size of the
observed area is small and contains galaxy clusters, some galaxy concentration
peaks can lead to higher fractal dimensions at low $z$.

It is also conceivable that the data used to obtain our results suffer from a
detection bias related to the cumulative measure
$N_{{\textstyle{\scriptscriptstyle\rm obs}}}$. Unbiassing this quantity might
depend on comparing observed $N_{{\textstyle{\scriptscriptstyle\rm obs}}}$
data with simulations derived from cosmological models’ subsets and halo
occupation distribution models (Berlind et al., 2003). One must also remember
that the UltraVISTA photometric redshifts were obtained using SED fitting
analysis, which means that the errors in the values are entirely dependent on
the input assumptions used to derive these photo-$z$, and, therefore, it is
not clear how the use of different types of photometric modeling might affect
our results. This possible error source may be mitigated by the use of
restricted galaxy samples, say, containing only luminous red galaxies such
that photometric redshift uncertainties are minimized (Rozo et al., 2016;
Cawthon et al., 2018).

Another possible source of data uncertainty that might affect our results is
the tension among the several measures of the Hubble constant. Here we have
assumed the standard FLRW cosmological model with
$H_{0}=70\;\mbox{km}\;{\mbox{s}}^{-1}{\mbox{Mpc}}^{-1}$, nevertheless
different Hubble constant predictions range this quantity by about $\pm
4\;\mbox{km}\;{\mbox{s}}^{-1}{\mbox{Mpc}}^{-1}$. Relativistic cosmological
models are known to be highly nonlinear, so how much this uncertainty in the
Hubble constant might affect our results, if at all, cannot be quantified
beforehand. So, to ascertain the possibly negligible, or otherwise, impact of
this uncertainty in our results can only be concluded by actually carrying out
the calculations with this parameter range and performing comparisons.

Aside from possible observational biases and yet unknown impact due to error
sources, one might also attribute the decrease in the fractal dimension to a
real physical effect. Perhaps, galaxy evolution dynamics is at play in causing
such a decrease in the sense that there might be indeed much less galaxies at
high $z$, meaning that the Universe was void dominated at those epochs since
galaxies were much more sparsely distributed and in smaller numbers. Only
further work with different high-$z$ galaxy samples and in different regions
of the sky may clarify this issue.

## Acknowledgments

We are grateful to the referee for useful comments. S.T. thanks the
Universidade Federal do Rio de Janeiro for a PIBIC scholarship. A.R.L.
acknowledges Brazil’s Federal Funding Agency CNPq for the financial support
with a PCI fellowship.

## References

  * Abdalla & Chirenti (2004) Abdalla E., Chirenti C. B. M. H., 2004, Physica A, 337, 117
  * Abdalla et al. (1999) Abdalla E., Afshordi N., Khodjasteh K., Mohayaee R., 1999, A&A, 345, 22
  * Abdalla et al. (2001) Abdalla E., Mohayaee R., Ribeiro M. B., 2001, Fractals, 9, 451
  * Albani et al. (2007) Albani V. V. L., Iribarrem A. S., Ribeiro M. B., Stoeger W. R., 2007, ApJ, 657, 760
  * Amoroso Costa (1929) Amoroso Costa M., 1929, Annals of Brazilian Acad. Sci. (in Portuguese), 51, 1
  * Berlind et al. (2003) Berlind A. A., et al., 2003, ApJ, 593, 1
  * Carpenter (1938) Carpenter E. F., 1938, ApJ, 88, 344
  * Cawthon et al. (2018) Cawthon R., et al., 2018, MNRAS, 481, 2427
  * Charlier (1908) Charlier C. V. L., 1908, Ark. Mat. Astr. Fys., 4, 1
  * Charlier (1922) Charlier C. V. L., 1922, Ark. Mat. Astr. Fys., 16, 1
  * Coleman & Pietronero (1992) Coleman P. H., Pietronero L., 1992, Phys. Rep., 213, 311
  * Conde-Saavedra et al. (2015) Conde-Saavedra G., Iribarrem A., Ribeiro M. B., 2015, Physica A, 417, 332
  * Cosmai et al. (2019) Cosmai L., Fanizza G., Sylos Labini F., Pietronero L., Tedesco L., 2019, Class. Quantum Grav., 36, 045007
  * De Vaucouleurs (1960) De Vaucouleurs G., 1960, ApJ, 131, 585
  * De Vaucouleurs (1970) De Vaucouleurs G., 1970, Science, 167, 1203
  * Einstein (1922) Einstein A., 1922, Ann. Phys., 69, 436
  * Ellis (1971) Ellis G. F. R., 1971, in Sachs R. K., ed., General Relativity and Cosmology: Proceedings of the International School of Physics “Enrico Fermi”, Course 47\. Academic Press, New York, pp 104–182. Reprinted in Gen. Rel. Grav., 41 (2009) 581–660, doi:10.1007/s10714-009-0760-7
  * Ellis (2007) Ellis G. F. R., 2007, Gen. Rel. Grav., 39, 1047
  * Etherington (1933) Etherington I. M. H., 1933, Phil. Mag., 15, 761. Reprinted in Gen. Rel. Grav., 39 (2007) 1055
  * Gabrielli et al. (2005) Gabrielli A., Sylos Labini F., Joyce M., Pietronero L., 2005, Statistical Physics for Cosmic Structures. Springer, Berlin
  * Gaite (2007) Gaite J., 2007, ApJ, 658, 11
  * Haggerty & Wertz (1972) Haggerty M. J., Wertz J. R., 1972, MNRAS, 155, 495
  * Holanda et al. (2010) Holanda R. F. L., Lima J. A. S., Ribeiro M. B., 2010, ApJL, 722, L233
  * Holanda et al. (2011) Holanda R. F. L., Lima J. A. S., Ribeiro M. B., 2011, A&A, 528, L14
  * Holanda et al. (2012) Holanda R. F. L., Lima J. A. S., Ribeiro M. B., 2012, A&A, 538, A131
  * Hossienkhani et al. (2018) Hossienkhani H., Yousefi H., Azimi N., 2018, Int. J. Geometric Methods Modern Phys., 15, 1850200
  * Ilbert et al. (2013) Ilbert O., et al., 2013, A&A, 556, A55
  * Iribarrem et al. (2012) Iribarrem A. S., Lopes A. R., Ribeiro M. B., Stoeger W. R., 2012, A&A, 539, A112
  * Jawad et al. (2019) Jawad A., Butt S., Rani S., Asif K., 2019, Eur. Phys. J. C, 79, 926
  * Jones et al. (1988) Jones B. J. T., Martínez V. J., Saar E., Einasto J., 1988, ApJL, 332, L1
  * Mandelbrot (1983) Mandelbrot B. B., 1983, The Fractal Geometry of Nature. Freeman, New York
  * Martínez (1990) Martínez V. J., 1990, Vistas in Astronomy, 33, 337
  * McCracken et al. (2012) McCracken H. J., et al., 2012, A&A, 544, A156
  * Mureika (2007) Mureika J. R., 2007, J. Cosmology Astroparticle Phys., 2007, 021
  * Mureika & Dyer (2004) Mureika J. R., Dyer C. C., 2004, Gen. Rel. Grav., 36, 151
  * Nogueira (2013) Nogueira F. A. M. G., 2013, Master’s thesis, Valongo Observatory, Universidade Federal do Rio de Janeiro (arXiv:1312.5005)
  * Pan & Coles (2000) Pan J., Coles P., 2000, MNRAS, 318, L51
  * Pietronero (1987) Pietronero L., 1987, Physica A, 144, 257
  * Raj et al. (2019) Raj V., Swapna M. S., Soumya S., Sankararaman S., 2019, Indian J. Phys.
  * Rangel Lemos & Ribeiro (2008) Rangel Lemos L. J., Ribeiro M. B., 2008, A&A, 488, 55
  * Ribeiro (1992a) Ribeiro M. B., 1992a, ApJ, 388, 1
  * Ribeiro (1992b) Ribeiro M. B., 1992b, ApJ, 395, 29
  * Ribeiro (1993) Ribeiro M. B., 1993, ApJ, 415, 469
  * Ribeiro (1994) Ribeiro M. B., 1994, in Hobill D. W., Burd A., Coley A., eds, B Vol. 332, Deterministic Chaos in General Relativity. Plenum, New York, p. 269 (arXiv:0910.4877), doi:10.1007/978-1-4757-9993-4_15
  * Ribeiro (1995) Ribeiro M. B., 1995, ApJ, 441, 477
  * Ribeiro (2001a) Ribeiro M. B., 2001a, Fractals, 9, 237
  * Ribeiro (2001b) Ribeiro M. B., 2001b, Gen. Rel. Grav., 33, 1699
  * Ribeiro (2005) Ribeiro M. B., 2005, A&A, 429, 65
  * Ribeiro & Miguelote (1998) Ribeiro M. B., Miguelote A. Y., 1998, Brazilian J. Phys., 28, 132
  * Rozo et al. (2016) Rozo E., et al., 2016, MNRAS, 461, 1431
  * Sadri et al. (2018) Sadri E., Khurshudyan M., Chattopadhyay S., 2018, Astrophys. Space Sci., 363, 230
  * Scoville et al. (2007) Scoville N., et al., 2007, Astrophys. J. Supp. Series, 172, 1
  * Selety (1922) Selety F., 1922, Ann. Phys., 68, 281
  * Souza (2020) Souza B. J., 2020, Master’s thesis, Physics Institute, Universidade Federal do Rio de Janeiro (in Portuguese)
  * Stahl (2016) Stahl C., 2016, Int. J. Modern Phys. D, 25, 1650066
  * Sylos Labini (2011) Sylos Labini F., 2011, Class. Quantum Grav., 28, 164003
  * Sylos Labini et al. (1998) Sylos Labini F., Montuori M., Pietronero L., 1998, Phys. Rep., 293, 61
  * Wertz (1970) Wertz J. R., 1970, PhD thesis, University of Texas at Austin
  * Wertz (1971) Wertz J. R., 1971, ApJ, 164, 227
  * Zheng et al. (2020) Zheng X., Liao K., Biesiada M., Cao S., Liu T.-H., Zhu Z.-H., 2020, ApJ, 892, 103