A new geometrical approach to void statistics
M. C. Werner
Kavli Institute for the Physics and Mathematics of Universe (WPI)
University of Tokyo
5-1-5 Kashiwanoha, Kashiwa 277-8583, Japan
Department of Mathematics
Duke UniversityScience Drive, Durham NC 27708, USA

Summary. — Modelling cosmic voids as spheres in Euclidean space, the notion ofa de-Sitter configuration space is introduced. It is shown that a uniform distribution
over this configuration space yields a power-law approximating the void size distribution in an intermediate range of volumes, as well as an estimate for the fractal
dimension of the large scale structure.

1. – Introduction
Cosmic voids play an increasingly important rôle in cosmology (see, e.g., [1]). A new
geometrical framework to study the distribution of voids proposed in ref. [2] was presented
at the Enrico Fermi School New Horizons for Observational Cosmology in Varenna. Thisapproach models voids as spheres in 3-dimensional Euclidean space, which are represented
by points in a 4-dimensional configuration space with de-Sitter geometry. It turns outthat a uniform distribution over this configuration space gives a good approximation tothe void size distribution in an intermediate volume range, and it can also be used toc Società Italiana di Fisica

12M. C. Werner

estimate the fractal dimension of the complementary set of the voids, that is, the large
scale structure.
Hence, this is a new application of de-Sitter geometry to cosmology which derives, however, from classical sphere geometry rather than general relativity. Another applicationof this de-Sitter configuration space of spheres to astronomy in a different context is
discussed in ref. [3].
2. – De-Sitter configuration space.
2 1. From sphere geometry to de-Sitter . – In the 3-dimensional Euclidean space E3
endowed with the usual Euclidean metric δij , an unoriented sphere can be uniquely
defind by its centre at x = (xi ) ∈ E3 , 1 ≤ i ≤ 3, and its radius r > 0, so that the tuple
y µ = (r, xi ), 0 ≤ µ ≤ 3, defines its configuration and can be used as a coordinate in a
4-dimensional configuration space. We would like to introduce a notion of distance on
this configuration space such that spheres which overlap in the same way have the same
distance on the configuration space, and this overlap can be characterized by the angleof sphere intersection ∆. Given two spheres (r1 , xi1 ) and (r2 , xi2 ), this is defined as(1)cos ∆ =

r12 + r22 − δij (xi1 − xi2 )(xj1 − xj2 ),2r1 r2and is illustrated in fig. 1. In other words, we seek a metric on the configuration space
whose isometries preserve ∆, and find that this is provided by de-Sitter geometry. To see
this, a short excursion to the sphere geometry of Lie and Laguerre, which studies contact
relations between spheres, is in order. Consider the set of Lie cycles which consists of alloriented spheres, called Laguerre cycles, all oriented hyperplanes, and the object infinity.
It turns out (e.g., [4], proposition 3.56) that one can assign bijectively to every Lie cycle
a homogeneous Lie cycle coordinate
 0 1 2 3 4 5  0
λ , λ , λ , λ , λ , λ = kλ , kλ1 , kλ2 , kλ3 , kλ4 , kλ5 ,any k 6= 0,

which obeys the Lie quadric,(2)−(λ0 )2 + (λ1 )2 + (λ2 )2 + (λ3 )2 + (λ4 )2 − (λ5 )2 = 0.

For a Laguerre cycle given by (r, xi ), r 6= 0, the corresponding Lie cycle coordinate may
be written (cf. [4], p. 154) 2
−r + δij xi xj + 1 1 2 3 −r2 + δij xi xj − 1(3),x ,x ,x ,, −r := X 0 , X 1 , X 2 , X 3 , X 4 , 1 .
22Now suppose that the X A , 0 ≤ A ≤ 4, are components of a coordinate in 5-dimensional
Minkowski space E1,4 , with the standard metric ηAB = diag(−1, 1, 1, 1, 1), then the Lie3A new geometrical approach to void statistics

quadric (2) implies that this denotes a point on a de-Sitter quadric hypersurface in thisMinkowski space,dS 4 :

ηAB X A X B = 1,X = (X A ) ∈ E1,4 ,which defines a 4-dimensional de-Sitter space in the usual way. Returning to our original
problem of unoriented spheres in 3-dimensional Euclidean space, then, we see that eachcan be represented by precisely one point in (half of the full) de-Sitter space. Moreover,given two spheres defined by X1A , X2B , a little algebra shows that, in fact,
ηAB X1A X2B |dS 4 = cos ∆,so that this space has the desired metric property as well. Hence, we can identify 4dimensional de-Sitter space as a suitable configuration space for spheres, as promised.
The metric g on the de-Sitter quadric induced by the ambient Minkowski metric can befound from the line element
ds2 = ηAB dX A dX B |dS 4 =
1−dr2 + δij dxi dxj := gµν dy µ dy ν ,2r

using a chart with configuration space coordinates y µ = (r, xi ), so that the configuration
space metric in the coordinate-induced basis is given by(4)


1 1 1 1gµν = diag − 2 , 2 , 2 , 2 .
r r r rHence, from this point of view, the sphere radius is a timelike coordinate and the spherecentre position a spacelike coordinate.
.
2 2. From the uniform distribution to fractals. – Let us now turn to the uniform
distribution of spheres over this de-Sitter configuration space, such that the number
dN of spheres with radius in the interval [r, r + dr] and centre in the volume element
dv = dx1 dx2 dx3 of Euclidean 3-space is proportional to the volume element of theconfiguration space dvg , which can be computed from eq. (4). Hence,dN ∝ dvg =p1− det gdy 0 dy 1 dy 2 dy 3 = 4 drdv.
rOne can recast this in terms of the sphere volume V = 4πr3 /3 to find the differential
number density ν(V ) in Euclidean 3-space of spheres with volumes in [V, V + dV ], as
well as the corresponding cumulative number density n(> V ),(5a)ν(V ) =(5b)n(> V ) =dN
1∝ 2,VZdV dv∞

ν(V ′ )dV ′ :=VC,V

4

M. C. Wernerrespectively, where C > 0 is a dimensionless constant. Our uniform distribution allows
any sphere position and, therefore, any overlap. Hence, it is instructive to compare it toMandelbrot’s random spherical cutout model, which provides a geometrical interpretationof the constant C in eq. (5b). Consider a homogeneously filled cube of volume V0 in E3 , inwhich a sequence of spherical regions of volume Vi are emptied, whose centres are placed
randomly (distributed uniformly and independently) in the cube with faces identified.
Now if, in an infinite sequence, the sum of volumes (regardless of overlap) diverges only
slowly, namely as a harmonic series,(6)

Vi =C̃V0,
iso thatkX

Vi → C̃V0 ln kask → ∞,i=1

then the cube will not be emptied entirely and there remains, in general, a complementary
set of fractal dimension D (cf. [5], proposition 8.8, for the technical argument in E1 ).
We can estimate D by applying a box-counting algorithm, assuming for simplicity that,
1at the kth stage, V0 is divided into k equal boxes of edge length proportional to k − 3 ,and that overlap is still negligible. Then the remaining volume of the cube is V (k) ≃V0 (1 − C̃ ln k), C̃ ln k ≪ 1, occupying N (k) ≃ k(1 − C̃ ln k) boxes, and so the fractal
dimension is(7)D=log N (k)log k13≃3+3ln(1 − C̃ ln i)≃ 3(1 − C̃).
ln iHence, for a volume V ≪ V0 whose corresponding k ≫ 1, the cumulative number densityof the spherical cutouts (again regardless of overlap) can now be written(8)n(> V ) =C̃
k−1
≃
≃V0V



D 1
1−,
3 V

using eq. (6) and eq. (7), which turns out to hold more generally (cf. [6], p. 302).
Comparing the cumulative number densities of eq. (5b) and eq. (8), we see that theuniform distribution over the de-Sitter configuration space results in the same scaling
behaviour as the random spherical cutout model, both of which allow any sphere overlap.
Thus identifying C̃ = C, we obtain a geometrical interpretation of the proportionality
constant(9)

C =1−D
3in eq. (5b) in terms of the fractal dimension D of the complementary set not containedin the spheres. We shall now discuss the applicability of this geometrical approach tocosmic voids.

5A new geometrical approach to void statistics-310-3r2n(>V)/(h Mpc)

r1-1

Dx1x2-510-710341010-1V/(h Mpc)

Fig. 1. – Intersection angle of spheres.

5103Fig. 2. – Cumulative number density of voids.

3. – Application to cosmology.
3 1. Comparison with structure formation theory. – The excursion set formalism pioneered by Press and Schechter [7] considers small Gaussian fluctuations of an almost
homogeneous density field at a sufficiently early time, in which a window of length scale
L around an overdensity will gravitationally collapse to form an object of mass M ∝ L3
if the density contrast exceeds some critical value, δ(L) > δcrit . The fluctuation variancemay depend on L as well, σ 2 ∝ L−k , k ≥ 0, resulting in the differential number density(10)ν(M ) ∝MM0

 k6 −2" k #M 3,
exp −M0

where M0 is the mass scale of the exponential cutoff. As pointed out by Sheth and van deWeygaert [8], due to the symmetry of Gaussian fluctuations about the average density,
the same reasoning applies also to underdensities and hence to voids of volume V ∝ L3 .
Therefore, given a scale-free variance (k = 0) and void volumes sufficiently smaller thanthe exponential cutoff, we recover the same differential number density of voids fromstructure formation theory,
ν(V ) ∝ V −2 ,as from the uniform de-Sitter distribution, eq. (5a). Of course, this ignores effects ofvoid hierarchy, that is merging of voids in the cosmological evolution (cf. again ref. [8]),which cuts the power-law off at small volumes. Hence, the uniform de-Sitter distribution
should hold approximately for cosmic voids of spherical shape, which is also supported
theoretically (e.g., [9]), in an intermediate range of volumes.
.
3 2. Comparison with data. – To illustrate this with data, an N-body simulation of
3256 particles in a box of edge length 240h−1 Mpc with periodic boundary conditions was6

M. C. Wernerperformed, using the GADGET-2 code [10] and a cosmology consistent with WMAP7. After applying adaptive smoothing, spherical voids were identified and no merging
algorithm was applied (cf. [11]) in order to allow any overlap as discussed above. Theresulting cumulative number density of voids is shown in fig. 2, together with an analyticalfit (dotted curve) adapted from ref. [12] and the power-law (solid line) from the uniform
de-Sitter distribution, eq. (5b) with C = 0.36. Given the approximate agreement withthe data in an intermediate range of volumes, as expected, we can now also compute thefractal dimension of the large scale structure regarded as the complementary set of thevoids, to obtain D ≃ 1.9 from eq. (9) as a reasonable estimate (c.f. [13]). Notice thatthe present approach differs from other fractal models using disjoint cutouts (e.g., [14]).
4. – Outlook
The next step in extending this geometrical idea is to investigate non-uniform distributions over the de-Sitter configuration space, which will result in deviations from thepower-law (5a). As mentioned above, it is those outside of the intermediate power-law
range which will be most interesting for cosmology: the exponential cutoff at large volumes, since the largest voids may be particularly sensitive to cosmological parameters;
and effects of void hierarchy and merging at small volumes.
∗ ∗ ∗This work was supported by the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan.
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