Mon. Not. R. Astron. Soc. 000, 1–7 (0000)

Printed 26 August 2013

(MN LATEX style file v2.2)

On de-Sitter Geometry in Cosmic Void Statistics
G.
W. Gibbons1, M. C. Werner2,4⋆ , N. Yoshida2,3 and S. Chon3
1

Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK

2 Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, 277-8583, Japan,
3 Department of Physics, University of Tokyo, 7-3-1 Hongo, Tokyo, 113-0033, Japan,
4 Department of Mathematics, Duke University, Durham, NC 27708, USA

This draft 26 August 2013

ABSTRACT

Starting from the geometrical concept of a 4-dimensional de-Sitter configuration of spheresin Euclidean 3-space and modelling voids in the Universe as spheres, we show that a uniform
distribution over this configuration space implies a power-law for the void number density
which is consistent with results from the excursion set formalism and from data, for an intermediate range of void volumes. We also discuss the effect of restricting the survey geometryon the void statistics. This work is a new application of de-Sitter geometry to cosmology andalso provides a new geometrical perspective on self-similarity in cosmology.
Key words: cosmology:theory–large-scale structure of Universe

1 INTRODUCTION
While the existence and characteristic distributions of empty regions are already implicit in early self-similar models of structure
hierarchy in the Universe (cf. Mandelbrot (1977)), it was the observational discovery of large, approximately spherical regions almost
devoid of visible galaxies known as voids and supervoids (e.g., Kirshner et al. (1981)) that sparked further theoretical work on underdense regions in the Universe. This included analytical studiesof the dynamical evolution of individual voids and shell-crossing(Peebles (1982); Sato (1982)) as well as statistical properties of thevoid distribution. Applying a simple sign-reversal argument, Icke
(1984) pointed out that the sphericity of underdense regions increases due to gravitational dynamics, so that it is in fact natural toexpect approximately spherical voids. Early theoretical approaches
to void statistics used Poisson statistics of empty regions, as developed by Politzer & Preskill (1986), Voronoi tesselations (Icke &van de Weygaert (1987)), and structure formation theory in analogy to the statistics of overdensity peaks (Bardeen et al. (1986))
in Gaussian primordial density fluctuations (e.g., Betancourt-Rijo(1990)). Self-similar features in the void size distribution were also
noted (e.g., Einasto, Einasto & Gramann (1989)). Of course, muchof this earlier work concentrated on Einstein-de-Sitter cosmology
with ΩM = 1, as was favoured then. Especially after the discoveryof the near-isotropic CMB, it was realized that the largest observable voids were difficult to explain theoretically and may therefore
provide important cosmological constraints (e.g., Blumenthal et al.
(1992)). Attention has also been drawn to galaxy properties within
voids (e.g., the void phenomenon of Peebles (2001)). Now, withthe advent of the ΛCDM paradigm, detailed numerical studies ofvoid statistics (e.g., Colberg et al. (2005) for a comparison) and⋆E-mail:marcus.werner@ipmu.jp

c 0000 RAS

analytical studies of void shape evolution in redshift space (e.g.,
Maeda, Sakai & Triay (2011)) have been conducted. Moreover,the void formation theory based on the excursion set formalismof primordial density fluctuations has been refined to include thenotion of hierarchy (Sheth & van de Weygaert (2004)). This formalism was extended more recently to investigate effects of nonGaussianity (e.g., D’Amico et al. (2011)) and modified theories ofgravity (e.g., Clampitt, Cai & Li (2013)) on voids. Following earlier work on self-similarity, Gaite (2009) applied advanced fractal
structures to void statistics. On the observational side, a void catalogue extracted from the SDSS Data Release 7 has recently been
published (Pan et al. (2012)), which illustrates the strong dependence of void statistics on the underlying void definition. The possibility of using voids for precision cosmology has also been explored (e.g., Lavaux & Wandelt (2010)). A fitting formula whichmatches the distributions of voids extracted from 2dF Galaxy Redshift Survey observations and from the Millennium Simulation has
been proposed by von Benda-Beckmann & Müller (2008), who
also discussed self-similarity in the void size distribution. Higuchi,
Oguri & Hamana (2013) have shown how weak lensing observations may be used to measure the mass distribution of voids.
In this paper, we propose a new geometrical approach to void statistics based on the idea that voids can be modelled well by spheres inthe Euclidean 3-space. Each such sphere is represented by a pointin the 4-dimensional configuration space of spheres, with radiusand centre position as coordinates. Since the spatial contact between these spheres is physically important (e.g., in the evolutionof voids), it is interesting to consider transformations of the configuration space that preserve such contact relations, namely conformal transformations, in order to define a distance measure on theconfiguration space. Given these assumptions, it turns out that theconfiguration space has a de-Sitter geometry, with timelike radiiand spacelike centre positions of the spheres, which emerges from

2

G. W. Gibbons, M. C. Werner, N. Yoshida and S. Chonclassical sphere geometry and is different from cosmological deSitter spacetime. One can now study the size distribution of spherical voids in the Universe in terms of their distribution over this deSitter configuration space. The approach presented here has been
inspired by other notions of configuration space measures in cosmology (e.g., Gibbons & Turok (2008)), and another applicationof this de-Sitter configuration space of spheres in a different astronomical context has been proposed recently (Gibbons & Werner(2013)).
The outline of this work is as follows. In Section 2, we give a brief
review of the de-Sitter configuration space for the general case ofn-spheres in n + 1-dimensional Euclidean space and indicate its
conformal structure. This is followed by a discussion of a uniform
distribution over the 4-dimensional de-Sitter configuration spaceused to model voids, that is n = 2, in Section 3. We shall derive, firstly, the corresponding void size distribution for voids in thewhole (unrestricted) 3-space and find a self-similar power-law. Secondly, the effect of restricting void positions in 3-space due to survey geometry is studied, which naturally breaks the self-similarityof the size distribution. This is carried out analytically for a survey
volume shaped as a spherical sector. Then in Section 4, we investigate how the void size distributions predicted by the unrestricted
and restricted uniform de-Sitter distributions compare with actual
void size distributions. We consider, firstly, theoretical models of
self-similarity and the excursion set formalism, to find that thepower-law from the unrestricted uniform distribution is, to an extent, consistent with the expectation from structure formation theory. To illustrate this and the effect of survey geometry restriction
in a data set, we apply an N-body simulation using the GADGET2(Springel (2005)) code and extract spherical voids. A comparison
with the fitting formula of von Benda-Beckmann & Müller (2008)is also given. Section 5 contains uur conclusions and comments on
possible further applications of this idea.

2 DE-SITTER CONFIGURATION SPACE

Sn1
R1

R2

Sn2

D
x1

x2

Figure 1. Angle between spheres. The intersection of two n-spheresnn+1 , here illustrated for the case n = 1
Sn
1 (R1 , x1 ), S1 (R1 , x1 ) in E
and, in projection, n = 2, can be characterized by the Euclidean anglen∆(Sn
1 , S2 ) given in eq. (3), which is an invariant of the isometries of thede-Sitter configuration space.

Therefore, the space of unoriented spheres is in fact given by half ofthe full de-Sitter quadric, sometimes referred to as de-Sitter spacemodulo the antipodal map X µ 7→ −X µ . Geometrically, the coordinates X µ of a sphere can be regarded as a form of Lie cycle coordinates for the Laguerre cycle representing the sphere (see, e.g.,
the appendix of Gibbons & Werner (2013) for more mathematicaldetails). Taking y i = (R, x), 0 6 i 6 n + 1, as coordinates of thede-Sitter configuration space, its metric g induced by the ambient
Minkowski metric in the usual way can be read off from the line
elementds2 =−(dX 0 )2 + (dX)2 + (dX n+2 )2 |dS n+2
1−dR2 + dx2=
R2= gij dy i dy j
(1)
so that, with respect to the coordinate-induced basis,


1 1
1
gij = diag − 2 , 2 , . . . , 2 .
R R
R

(2)

We begin by briefly recapitulating the notion of a configuration
space of spheres in Euclidean space and its de-Sitter geometry, as
discussed in some more detail in Gibbons & Werner (2013); seealso Zee (2013). Consider an n-dimensional unoriented sphere
Sn (R, x) in n + 1-dimensional Euclidean space En+1 given interms of its radius R > 0 and its centre at x = (x1 , . . . xn+1 ) ∈En+1 . It turns out that, to each such sphere, one can uniquely assign a point X µ = (X 0 , X, X n+2 ), 0 6 µ 6 n + 2, in n + 3dimensional Minkowski space E1,n+2 endowed with the usual metric ηµν = diag(−1, 1, . . . , 1), with

1 x2 − 1
X0 = −−R ,2R
x
X = − ,R

1 x2 + 1
n+2
X
= −
−R ,
2
R

Hence, this metric measures the distance between spheres in termsof their configuration, that is, their radii and positions in space.
Isometries of this de-Sitter space can be identified with conformaltransformations of spheres, for the following reason. Consider twoµ
nintersecting spheres Sn
1 (R1 , x1 ) and S2 (R2 , x2 ) given by X1 andX2µ , respectively. Thenon the n + 2-dimensional hypersurfaceds2

ηµν X µ X ν = 1,
which is a de-Sitter quadric. In other words, each sphere witha given centre and radius corresponds to a point on an n + 2dimensional de-Sitter space dS n+2 which can therefore be interpreted as a configuration space of spheres. Since for unoriented
1
spheres one may set R > 0, we have X 0 − X n+2 = R> 0.

(ηµν X1µ X2ν ) |dS n+2==

R12 + R22 − (x1 − x2 )2
2R1 R2ncos ∆(Sn1 , S2 ),(3)nso that isometries leave the Euclidean angle ∆(Sn
1 , S2 ) between

the two spheres invariant which, as illustrated in Fig. 1, measures
their contact. Indeed, it provides a geometrical interpretation for theline element (1), since for two spheres that are infinitesimally closein the configuration space so that they are nearly identical in theirradii and positions, one finds,==

1(−dR2 + dx2 )R2d∆2 (Sn (R, x), Sn (R + dR, x + dx)).

Finally, a measure of the number of spheres having radii within[R, R + dR] and centres in the volume element dx1 . . . dxn+1 at
x ∈ En+1 is provided by the volume element,p− det gij dy 0 . . . dy n+1
dV n+2 =c 0000 RAS, MNRAS 000, 1–7

On de-Sitter Geometry in Cosmic Void Statistics=1
dRdx1 . . . dxn+1 ,
Rn+2

(4)

3

ΘS

of the n + 2-dimensional de-Sitter configuration space.
Θ2

3 UNIFORM DISTRIBUTION OVER dS 4

O

r1

3.1 Without survey geometry restriction
In order to study the distribution of cosmic voids from the geometrical perspective of the de-Sitter configuration space, we shall model
them as unoriented 2-spheres in E3 . Hence specializing the previous discussion to the case n = 2, the corresponding 4-dimensional
configuration space has the de-Sitter geometry dS 4 with metricgiven by (2). Then the configuration space volume element forvoids with radii in [R, R + dR] and with centres in the volume
element dx1 dx2 dx3 at x ∈ E3 becomesdV 4 =

1
dRdx1 dx2 dx3 ,R4(5)

Θ=0, rS

10 000

100
nH>vL

(6)

In this section, we will consider the simplest distribution in thisframework, corresponding to a uniform distribution of voids overthe de-Sitter configuration space such thatr2

Figure 2. Survey geometry. In a survey region shaped as spherical sector
with r 6 rS , θ 6 θS centred at the observer O, the limits r1 (R), r2 (R)and θ2 (r, R) for voids of radius R (solid circles) are illustrated, see eq.s(10). The largest void for the given survey geometry, at r0 with radius R0
given by eq. (11), is also shown (dashed circle).

from eq. (4). Also, the infinitesimal number of voids dN with radiiin [R, R + dR] centred within the volume element at x can be
described in terms of a non-negative distribution function over theconfiguration space, f : dS 4 → R+ , so thatdN = f (R, x)dRdx1 dx2 dx3 .

r1

dN ∝ dV 4 .
Clearly, this uniform distribution has the property that all voids arecounted regardless of their position in E3 : there is no restriction
due to survey geometry and any mutual overlap of voids that may
occur is ignored. The corresponding distribution function f is independent of the position in space,1f (R, x) ∝ 4 ,R(7)from eq. (5) and (6). Defining the void number density as the number of void centres per volume element in E3 , one can compute thecumulative number density n(> R) of voids with radius greaterthan R that corresponds to the uniform distribution,Z ∞1n(> R) =f (R′ )dR′ ∝ 3 ,RR
by integrating (6) using (7). Expressing this in terms of void volume
V = 4πR3 /3,n(> V ) ∝

1
.
V

(8)

A more realistic modification of this cumulative number density
should take into account that any actual void survey can, of course,
only encompass a finite subregion of E3 . We shall investigate thisin the following.

0.01
0.01

0.1
v

0.5

1

Figure 3. Cumulative number density of voids. Double logarithmic plots ofn(> v), the integral (15) with the same constant of proportionality and rS
set to unity, as functions of the dimensionless volume parameter 0 6 v 6 1
defined in (12), for three choices of the opening angle θS of the survey
geometry: a solid curve for θS = π/2, the half sphere; a dashed curve forθS = π/6; and a dotted curve for θS = π/18.

of dn void centres per volume in E3 , per void radius dR or void
volume dV , respectively, again by eq. (6) and (7). Now supposethat a survey counts only those voids which are wholly within
the survey region VS . For simplicity, consider a survey geometry
shaped as a spherical sector with opening angle θS and maximal(comoving) distance rS from an observer who is situated at its
apex and coordinate origin. Then, in terms of spherical polar coordinates, the survey region can be described byVS :0 6 r 6 rS ,0 6 θ 6 θS ,0 6 φ 6 2π,

and we let θS 6 π/2. The total volume of the survey is thenVS (rS , θS ) =3.2 With survey geometry restriction

0.05

2π 3
rS (1 − cos θS ).
3

The assumption of a uniform distribution of voids over their deSitter configuration space results in a differential number density

A given void of radius R will be counted in the distribution if it lies
wholly within VS , that is, if the coordinates of its centre (r, θ, φ)satisfy the conditionsdn
1= f (R) ∝ 4dR
RVS (R) : r1 (R) 6 r 6 r2 (R), 0 6 θ 6 θ2 (r, R), 0 6 φ 6 2π,

⇒

dn
dR dn
1
=
∝ 2
dV
dV dR
V

c 0000 RAS, MNRAS 000, 1–7

(9)

4

G. W. Gibbons, M. C. Werner, N. Yoshida and S. Chonwhere the lower and upper coordinate limits are determined byvoids touching the boundary of VS . The conditions for these canbe read off from Fig. 2,
r1 (R) sin θS = R, r2 (R) + R = rS , r sin(θS − θ2 (r, R)) = R,=

r2 (R)=

θ2 (r, R)=R,
sin θSrS − R,(10)R
θS − arcsin .
r

The largest void which fits into VS is situated at r0 on the polar axis
and has radius R0 , whose values can be obtained from the condition
r1 (R0 ) = r2 (R0 ). Hence, using eq. (10),
r0 =

rS
,
1 + sin θS

R0 =rS sin θS.
1 + sin θS(11)

In order to compare different survey geometries, it will be convenient to express void volume V in terms of the volume V0 of
the largest void admitted by a given survey geometry (rS , θS ), andhence define the dimensionless quantity 0 6 v 6 1,v=

VV0 (rS , θS )4.1 From theoretical models
4.1.1 Self-similarity
As shown in Section 3.1, the uniform de-Sitter distribution without survey geometry restriction yields a power-law size distribution
given bywhence we haver1 (R)

4 COMPARISON WITH VOID DISTRIBUTIONS

such thatR=rS sin θS 13v
1 + sin θS(12)from eq. (11). The differential number density of voids within therange [v, v + dv] of the volume parameter can then be expressed as
Z
1
dN,
dn =
VS (rS , θS ) VS (v)so that by eq. (6) in spherical polar coordinates,
Z
dn
dR
1
f (v, r, θ, φ)r 2 sin θdrdθdφ.
=dv
VS (rS , θS ) dv VS (v)(13)

Using the limits of VS (v) for voids with volume parameter v givenby (10) and the uniform distribution (7), whose constant of proportionality is kept, we can recast (13) to obtain
Z r2 Z θ21(1 + sin θS )3dn∝
r 2 sin θdθdr
dv
sin3 θS (1 − cos θS )rS6 v 2 r1 0

(1 + sin θS )31
1=
3 sin3 θS (1 − cos θS )rS3 v 2
3 sin θS (2 + sin θS ) 31
sin2 θS  32
v
−
v +
3v −
2
1 + sin θS
1 + sin θS23 #2 sin θS 31 2
− cos θS 1 −v.
(14)
1 + sin θS
This expression can in turn be integrated to yield the corresponding
cumulative number density,
Z 1
dn ′
dv ,
(15)
n(> v) =′dvv
and Fig. 3 shows it for three choices of the opening angle θS of
the survey geometry. These results for the number density (14) andcumulative number density of voids illustrate how the intrinsic distribution and the observed distribution can differ substantially due
to the restrictions imposed by the survey geometry. Small voids, ofcourse, tend to be affected less by the survey geometry and therefore approach asymptotically the power-law (8) of the intrinsic distribution.

ν(V ) :=

dn
1
∝ 2
dV
V

(16)

for the differential number density of voids, from eq. (9). This is
obviously self-similar in the sense that a change in volume scale,Ṽ = sV, s = const., also implies
ν(Ṽ ) ∝1.
Ṽ 2

Moreover, since, for a given volume in E3 , the integer rank R ofa void in an ordered sequence decreasing in size from largest one
is proportional to n(> V ), one can see from (8) that Zipf’s law
is satisfied. The applicability of fractal structures and Zipf’s law tovoid distributions was investigated in detail by Gaite & Manrubia(2002), showing that, in our notation for a void of volume V in E3 ,3

V (R) ∝ R− D ,where D < 3 is the fractal dimension, so that in terms of the cumulative number density with R ∝ n(> V ),Dn(> V ) ∝ V − 3 .
Hence, the inverse power-law of eq. (8) represents a limiting casethat cannot be described by a fractal set with one dimension D < 3.
In fact, Gaite (2007) shows that the extended notion of a multifractal, a set with multiple fractal dimensions, is useful in this context(see also Gaite (2009); a recent review on scaling laws in the large
scale structure more generally is given by Jones et al. (2004)). Of
course, the property of self-similarity does not fix the power in (16)by itself, which will depend on the underlying physics. It is therefore interesting to note that the self-similar power-law (16) fromthe uniform de-Sitter distribution has also been found to arise in
the excursion set formalism of structure formation, as we shall outline below.

4.1.2 Excursion set formalism
In their pioneering paper, Press & Schechter (1974) point out that,starting from Gaussian density perturbations in a Friedmann cosmology and considering the linear quasi-Newtonian perturbation
analysis of the growing mode, the mass distribution at late times
does not depend on the initial mass distribution. They identify a
simple physical reason for this self-similarity in the existence of
two dimensionless quantities governing the gravitational collapse,
which remain approximately constant during the matter-dominatedphase on subhorizon scales. The Press-Schechter argument yielding a power-law size distribution can be summarized as follows.
Consider massive particles distributed in some comoving volume.
Then the mass density function ρ on this volume is obtained byapplying some smoothing window function with scale length l ateach point. At sufficiently early times, this density function will beapproximately constant,ρ = ρ̄(1 + δ),|δ| ≪ 1.
c 0000 RAS, MNRAS 000, 1–7

On de-Sitter Geometry in Cosmic Void Statistics

These windows will contain slightly different masses, but basicallyM ∝ ρ̄l3 . Also, the variance may depend on the smoothing scaleand hence also on the mass, σ 2 ∝ l−n−3 ∝ M −n/3−1 , for some
spectral index n > −3. Thus, F is a function M as well, and so
the differential number density ν(M ) of ultimately bound objectsobeysM ν(M ) ∝ −ρ̄

dF
dσ dF
= −ρ̄
,
dM
dM dσ

(17)

and hencen3

ν(M ) ∝ M − 6 − 2forM ≪ M0 ,where M0 is the mass scale of the exponential term, which gives
rise to a cutoff at large masses (or volumes). For a scale-invariant
σ, that is n = −3, this yields the inverse square power-law
ν(M ) ∝ M −2 .

(18)

It turns out that this power-law emerges also in the modified PressSchechter approach proposed by Appel & Jones (1990), whichuses an adaptive window scale l. Now in the context of voids, one
can consider underdensity troughs rather than overdensity peaks(as in Bardeen et al. (1986)) and, since Gaussian fluctuations are
symmetric about the average density, the same power-law applies,
as pointed out by Sheth & van de Weygaert (2004). Then sinceM ∝ l3 ∝ V , we recover (16) from (18). Of course, in addition to
the large volume cutoff, this simple excursion set argument ignores
effects of void hierarchy which affect small volumes in particular(i.e., by the void-in-void and void-in-cloud processes, cf. Sheth &van de Weygaert (2004)), so that the power-law will only apply
within a range of volumes.
Hence, at least within a range of volumes, the power-law (16) has
a physical basis in the excursion set formalism: the Gaussian fluctuations producing voids give rise to a self-similar power-law size
distribution which can be described geometrically as a uniform distribution over the de-Sitter configuration space of spheres.

10-3

n(>V)[(h-1Mpc)-3]

However, if the density within a window is greater than some (fixed
dimensionless, say) critical density so that δ > δcrit around overdensity peaks due to fluctuations in the density function, then themass within the window will gravitationally collapse and ultimately
form a bound object. Again, for sufficiently early times, one may
take these fluctuations to be Gaussian, such that the volume fractionof points with collapsing windows is given by


Z
1 ∞
δ
F∝
exp − 2 dδ.
σ δcrit
2σ

5

10-5

10-7

10-9

103

104
V[(h Mpc)3]

105

-1

Figure 4. Simulated void size distributions. Double logarithmic plots ofthe cumulative number density of voids as a function of volume, using thesimulation discussed in Section 4.2: uniform de-Sitter distribution from eq.
(8) without survey geometry restriction (upper solid curve and data set)
and fitting formula of von Benda-Beckmann & Müller (2008) (dotted); uniform de-Sitter distribution from eq. (15) with survey geometry restriction in
spherical sector (lower solid curve and data set) and corresponding powerlaw asymptote (dashed). Poisson errors are indicated.

mass is scaled to match the mass density of the Universe. At redshift z = 0, voids are extracted according to the following algorithm.
Firstly, the discrete simulation data are smoothed to construct thematter density function, using a Gaussian window with adaptive
length scale. This smoothing length is chosen to be the distance tothe 20th nearest particle at each cell. Secondly, a set of sphericalvoids is extracted by defining density minima as void centres anddetermining the radius for each void as the largest radius R centered at the density minimum for which the average density of theenclosed sphere ρ(R) is less than a critical value
δcrit =

ρ(R) − ρ̄= −0.8,ρ̄

relative to the average density ρ̄ of the box. Hence, for our present
purposes, we ignore the overlap of the spherical voids and do not
apply an additional merging algorithm to these voids (cf. Colberg etal. (2005), also called ’protovoids’), which would give rise to nonspherical voids. This should allow for are meaningful comparison
with the uniform distribution over the de-Sitter configuration spaceof spheres, which allows void overlap as noted in Section 3.1.

4.2 From data
4.2.1 An N-body simulation

4.2.2 Discussion

Now in order to compare the void size distributions predicted by
the unrestricted and restricted uniform de-Sitter distributions from
Section 3 with data, we perform an N-body simulation with the parallel Tree-Particle Mesh code, GADGET2 (Springel (2005)), using
osmological parameters consistent with the WMAP seven-year results (Komatsu et al. (2011)), namely spatial curvature k = 0,
dark energy density ΩΛ = 0.7274, Hubble parameter h = 0.704
and spectral index ns = 0.963. The initial condition for the simulation is generated at redshift z0 = 50 using the code developedby Nishimichi et al. (2009), who employ second order Lagrangian
perturbation theory. The simulation is run in a cubic box with side
length 240h−1 Mpc in comoving coordinates with periodic boundary conditions, and 2563 dark matter particles whose individual

The resulting cumulative number density of voids in the simulation
box as a function of void volume is shown in Fig. 4. The largest
void has a volume of Vmax ≃ 4.2 · 104 (h−1 Mpc)3 , which can beinterpreted in terms of the exponential cutoff in the excursion set
formalism mentioned in Section 4.1.2, and whose value is comparable with the simulation results reported in Fig. 7 of Colberg etal. (2005). Until the curve flattens at very small volumes – here,of course, rather due to simulation resolution and smoothing scalethan effects of void hierarchy –, the cumulative number density for
V ≪ Vmax can indeed be approximated by the power-law of eq.
(8) derived from the unrestricted uniform de-Sitter distribution. It
is also instructive to compare this result with the fitting formulaof von Benda-Beckmann & Müller (2008), which was shown toc 0000 RAS, MNRAS 000, 1–7

6

G. W. Gibbons, M. C. Werner, N. Yoshida and S. Chonmodel well the cumulative volume filling factor F (> R) of voids
derived from magnitude-limited samples of galaxies in the 2dF
Galaxy Redshift Survey,
 p1 
p2 
 
R
R
−
,
(19)
F (> R) = exp −
s1 λ
s2 λ
where R = (3V /4π)1/3 is the effective spherical radius of a voidof volume V and any shape, λ denotes the mean separation between
galaxies, and p1 , p2 , s1 , s2 are parameters. In order to convert thisto a cumulative number density as a function of volume, one canrewrite eq. (19) in terms of V . Again ignoring void overlap,
Z ∞dn
F (> V ) =V′
dV ′ ,dV ′V
and we have1 dF (> V )
dn
=−
,
dV
V
dV
so that
Z ∞Z ∞dn
1 dF (> V ′ )′n(> V ) =
dV
=
−
dV ′
′
′
′
dV
V
dV
V
V
can be computed from (19). While a detailed comparison is beyond the scope of this paper, we note that the parameters p1 = 1,
p2 = 4.5, s1 = 1, s2 = 1.5 and λ = 11 h−1 Mpc, which seemappropriate choices given the data in Tables 1 and 2 of von BendaBeckmann & Müller (2008), yield a curve in reasonable agreementwith our simulation data, as seen in Fig. (4), using a factor of 2.5
to correct for the fact that our voids are not merged and therefore
have a higher cumulative number density. This factor is also compatible with the void number densities shown in Fig. 6 and Fig. 7of Colberg et al. (2005).
Given, then, that our simulation appears to produce a realistic size
distribution of spherical voids without merging, which does indeed
agree with the uniform de-Sitter distribution for voids in the range
103 . . . 104 (h−1 Mpc)3 , we shall now turn to the effect of restricting the survey geometry. To this end, we select a survey region in
the simulation box shaped like a spherical sector as in Fig. 2, with
its apex at a corner of the box and its axis oriented along the diagonal. Consider such a spherical sector with an opening angle ofθS = 10◦ and a radius of half the maximum radius within the box,rS ≃ 169h−1 Mpc. Although such a survey volume has only 1.1%of the total box volume, the volume of the largest void within thissurvey region is V0 ≃ 6.5 · 104 (h−1 Mpc)3 , from eq. (11), which isgreater than Vmax . The corresponding cumulative number density
is also shown in Fig. 4, together with the theoretical prediction ofeq. (15) from the restricted uniform de-Sitter distribution. Since, forvoids in the whole box, the unrestricted uniform de-Sitter distribution is a good approximation in the range 103 . . . 104 (h−1 Mpc)3
where v = V /V0 is of the order 0.1, we expect from Fig. 3 thatthe survey geometry restriction causes a significant deviation inthis range from the power-law asymptote, which represents theeffect of the survey volume restriction alone. Furthermore, since
V0 /Vmax ≃ 1.5, we expect no significant effect of the large volume cutoff. Both expectations are indeed borne out by the data, as
can be seen in the figure.

5 CONCLUDING REMARKS“In parallel with efforts to explain, I think it indispensable to describe
clustering, and to mimic reality by purely geometric means.”
Mandelbrot (1977), p. 84

In this article, we have considered an application of the 4dimensional de-Sitter configuration space of 2-spheres in Euclidean
3-space to cosmology. Modelling cosmic voids as spheres and ignoring their overlap, it was shown that a uniform distribution overthis configuration space gives rise to a self-similar power-law size
distribution of voids. It appears to agree well with data in an intermediate range of void volumes, and this can be understood physically from the excursion set formalism as long as the large volume
cutoff and void hierarchy effects at small volumes may be ignored.
We have also seen how restrictions of void positions in 3-space due
to survey geometry significantly affect the size distribution.
Now in order to refine this approach, one might implement void
merging conditions using this configuration space language. Furthermore, since the uniform distribution appears to provide a reasonably good match with actual void distributions of intermediate volume, it may be worthwhile to treat the uniform distribution
as a null hypothesis and study the physical interpretation of nonuniform distributions. Then deviations from uniformity will, as wehave seen, encode physically interesting effects such as void hierarchy processes or the large volume cutoff, which may be particularly
sensitive to the underlying cosmology.
Finally, while de-Sitter geometry plays an important and wellknown rôle in cosmological spacetimes, we would like to emphasize the interesting, and perhaps rather surprising, fact that thepresent application of de-Sitter geometry to cosmology arises ina physically entirely different context. At this stage, our approachis primarily a descriptive device which also offers a novel geometrical interpretation of an aspect of self-similarity in cosmology, inthe spirit of the quotation above.

ACKNOWLEDGMENTS
MCW gratefully acknowledges support from the World PremierInternational Research Center Initiative (WPI Initiative), MEXT,
Japan. NY acknowledges financial support from the Japan Society for the Promotion of Science (JSPS) Grant-in-Aid for Scientific
Research (25287050).

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