The void probability function and related statistics
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The Void Probability function and related statistics. Sophie Maurogordato CNRS, Observatoire de la Cote d’Azur, France. The Void probability function. Count probability P N (V): probability of finding N galaxies in a randomly chosen volume of size V N= 0: Void Probability Function P 0 (V)

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The Void Probability function and related statistics

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The Void Probability functionand related statistics

Sophie Maurogordato

CNRS, Observatoire de la Cote d’Azur, France

The Void probability function

  • Count probability PN(V): probability of finding N galaxies in a randomly chosen volume of size V

  • N= 0: Void Probability Function P0(V)

  • Related to the hierarchy of n-point reduced correlation functions (White 1979)

Why the VPF ?

  • Statistical way to quantify the frequency of voids of a given size.

  • Complementary information on high-order correlations that low-order correlations do not contain: strongly motivated by the existence of large-scale clustering patterns (walls, voids filaments).

  • Straightforward calculated.

  • But density dependent, denser samples have smaller voids: be careful when comparing samples with different densities.

Scaling properties for correlation functions

Observational evidence for low orders:

  • n=3

    (Groth & Peebles, 1977, Fry & Peebles 1978, Sharp et al 1984)

  • n=4

    (Fry & Peebles 1978)

Hierarchical models

Generalisation for the reduced N-point correlation xN :

a :tree shape L(a ) labellings of a given tree

(Fry 1984, Schaeffer 1984, Balian and Schaeffer 1989)

Scaling invariance expected for the correlation functions of matter

  • In the linear- and mildly non linear regime:

    Evolution under gravitational instability of initial gaussian fluctuation; can be followed by perturbation theory >> predictions for SN’s

    (Peebles 1980, Jusckiewicz, Bouchet & Colombi 1993, Bernardeau 1994, Bernardeau 2002)

    SN independant on W, L and z !

  • In the strongly non-linear regime: solution of the BBGKY equations

Scaling of the VPF under the hierarchical « ansatz »

The reduced VPF writes:

The reduced VPF as a function of Nc is a function of the whole set of SN’s

VPF from galaxy surveys

Zwicky catalog: Sharp 1981

CfA: Maurogordato & Lachièze-Rey 1987

Pisces-Perseus: Fry et al. 1989

CfA2: Vogeley et al. 1991, Vogeley et al. 1994

SSRS: Maurogordato et al.1992, Lachièze-Rey et al. 1992

Huchra’s compilation: Einasto et al. 1991

QDOT: Watson & Rowan-Robinson, 1993

SSRS2: Benoist et al. 1999

2dFGRS: Croton et al. 2004, Hoyle & Vogeley 2004

DEEP2 and SDSS: Conroy et al. 2005

Not exhaustive!

How to compute it ?

  • Select sub-samples of constant density: volume and magnitude limited samples.

  • Randomly throw N spheres of volume V and calculate the whole CPDF: PN(V), P0(V).

  • Nc from the variance of counts.

  • Volume-averaged correlation functions from the cumulants

  • Test for scale-invariance for the VPF and for the reduced volume-averaged correlation functions.

Scaling or not scaling for the VPF ?

  • First generation of catalogs: CfA, SSRS, CfA2, SSRS2

    First evidences of scaling, but not on all samples.

    Large scale structures of size comparable to that of the survey

    Problem of « fair sample »

  • New generation of catalogs: 2dFGRS, SDSS:

    Excellent convergence to a common function corresponding to the negative binomial model.

Statistical analysis of the SSRS

Reduced VPF’s rescales to the same function even for samples with very different amplitudes of the correlation functions.

M>-18, D< 40h-1 Mpc

M>-19, D< 60 h-1 Mpc

M>-20, D < 80h-1 Mpc

From Maurogordato et al. 1992

Void statistics of the CfA redshift Survey

From Vogeley, Geller and Huchra, 1991, ApJ, 382, 44

Scaling of the reduced VPF in the 2DdFGRS

From Croton et al., 2004, MNRAS, 352, 828

Enormous range of Nc tested: up to ~40 !

Excellent agreement with the negative binomial distribution

Converges towards a universal function at z <0.2

Scaling at high redshift



Negative binomial



0.12 < z < 0.5







VPF from DEEP2 (Conroy et al. 2005)

VPF from VVDS (Cappi et al. in prep.)

Seems to work also at high z !

Real/redshift space distorsions

  • Small scales: random pairwise velocities

  • Large scales: coherent infall (Kaiser 1997)

Distorsion on 2-pt correlation from peculiar velocities in the 2dFGRS

From Hawkins et al.,2003

Void statistics in real and redshift space

Vogeley et al. 1994, Little & Weinberg 1994

  • Voids appear larger in redshift space :

    Amplification of large-scale fluctuations

    Model dependant

  • Small scales: VPF is reduced in redshift space due to fingers of God (small effect)

    Howevever difference is smaller than uncertainties on data (Little & Weinberg 1994, Tinker et al. 2006)

Scaling for p-point averaged correlation functions

Well verified in many samples, for instance:


  • APM (Gaztanaga 1994, Szapudi et al.1995, Szapudi et Gaztanaga 1998), EDSGC (Szapudi, Meiksin and Nichol 1996)

  • Deep-range (Postman et al. 1998, Szapudi et al. 2000)

  • SDSS (Szapudi et al. 2002, Gaztanaga 2002)


  • IRAS 1.2 Jy (Bouchet et al. 1993)

  • CFA+SSRS (Gaztanaga et al. 1994)

  • SSRS2 (Benoist et al. 1999)

  • Durham/UKST and Stromlo-APM (Hoyle et al. 2000)

  • 2dFGRS (Croton et al. 2004, Baugh et al. 2004) to p=5!

Skewness and kurtosis (2D) for the Deeprange and SDSS

No clear evolution of S3 and S4 with z

Open: Deeprange

Filled: SDSS

From Szapudi et al. 2002

SN’s for 3D catalogs

Good agreement for S3 and S4 in redshift catalogues

Hierarchical correlations for the VVDS

0.5< z < 1.2

S3 ~ 2

On courtesy of Alberto Cappi and the VVDS consortium

Hierarchical Scaling

  • for VPF in redshift space

  • Valid for samples with different luminosity ranges, redshift ranges, and bias factors

  • for the reduced volume-averaged N-point correlation function

    SN’s roughly constant with scale

    Good agreement for S3 and S4 in different redshift catalogs

    But different amplitudes from 2D and 3D measurement

    (damping of clustering in z space, Lahav et al. 1993)

    Good agreement with evolution of clustering under gravitational instability from initial gaussian fluctuations

The VPF as a tool to discriminate between models of structure formation

  • Can gravity alone create such large voids as observed in redshift surveys ?

  • What is the dependence of VPF on cosmological parameters ?

  • What VPF can tell us about the gaussianity/ non gaussianity of initial conditions ?

  • Can we infer some clue on the biasing scheme necessary to explain them ?

Dependence on model parameters

Einasto et al. 1991, Weinberg and Cole 1992, Little and Weinberg 1994, Vogeley et al. 1994,…

  • For unbiased models:

    weak dependance on n (VPF when n )

    Insensitive to W and L

    Good discriminant on the gaussianity of initial conditions

  • For biased models: sensitive to biasing prescription

    VPF is higher for higher bias factor

What can we learn from VPF (and SN’s) about « biasing » ?

In the « biased galaxy formation » frame, galaxies are expected

to form at the high density peaks of the matter density field

(Kaiser 1984, Bond et al. 1986, Mo and White 1996,..)

Observations show multiple evidences of bias: luminosity, color, morphological bias

Variation of the amplitude of the auto-correlation function

(Benoist et al. 1996, Guzzo et al. 2000, Norberg et al 2001, Zehavi et al. 2004, Croton et al. 2004)

Luminosity bias from galaxy redshift surveys

From Norberg et al 2001

Testing the bias model with SN’s

  • Linear bias hypothesis:

Inconsistency between the the measured values of SN’s towards the expected values from the correlation functions under the linear bias hypothesis (Benoist et al. 1999, Croton et al. 2004)

High order statistics in the SSRS2

S3 should be lower for more luminous (more biased) samples, which is not the case !

From Benoist et al. 1999

Non-linear local bias and high-order moments

This local biasing transformation preserves the hierarchical structure in the regime of small

Presence of secondary order terms in SN’s:

Fry and Gatzanaga 1993

Gatzanaga et al 1994, 1995

Benoist et al. 1999

Hoyle et al. 2000

Croton et al. 2004

Constraining the biasing scheme

Galaxy distribution results from gravitational evolution of dark matter coupled to astrophysical processes: gas cooling, star formation, feedback from supernovae…

  • Large-scales: bias is expected to be linear

  • Small scales: bias reflects the physics of galaxy formation, so can be scale-dependant

    Recent progress in modelling the non-linear clustering:

    HOD >> bias at the level of dark matter halos

    (Benson et al. 2001, Berlind & Weinberg 2002, Kravtsov et al. 2004, Conroy et al. 2005, Tinker, Weinberg & Warren 2006)

Constraining the HOD parameters

Berlind and Weinberg 2002, Tinker, Weinberg & Warren 2006

Void statistics expected to be sensitive to HOD at low halo masses

BW02: <N>M =(M/M1)a with a lower cutoff Mmin

Strong correlation between the minimum mass scale Mmin / size of voids

TWW06: <N>M = <Nsat>M + <Ncen>M

Once fixed the constraints on parameters from galaxy number density + projected correlation functions, VPF does not add much more

But: very sensitive to minimum halo mass scale between low and high density region







d < dc , Mmin = fmin x Mmin

fmin= ∞

From Tinker, Weinberg, Warren 2006


  • Convergence of observational results from existing redshift surveys:

  • scale-invariance of the reduced VPF

  • Hierarchical behaviour of N-point averaged correlation functions

  • More: the shape for the reduced VPF, and the amplitudes of S3 and S4 are consistent for the different samples.

    Good agreement with the gravitational instability model.

  • VPF in recent surveys + state of the art HOD

    very promising to constrain the non linear bias

Testing a prescription for bias ?

  • LCDM + semi-analytic model (Benson et al 2002)

  • Galaxy distribution show more large voids than dark matter.

  • Matching the VPF >> constrain the feedback mechanisms

Benson et al. 2003

The effect of introducing biasing on VPF

  • Strongly discriminant Gaussian/non Gaussian if non biasing

  • Biasing creates large voids in all models

  • Non gaussinaity is not required to explain current observations

Weinberg and Cole 1992

Little & Weinberg 1994

Models for the VPF

  • BBGKY (Fry 1984)

  • Thermodynamical model (Saslaw & Hamilton 1984)

  • Binomial model (Carruthers & Shih 1983)

  • Log-normal model (Coles & Jones 1991)

  • What « empty » regions can tell us about « filled » ones ?

  • How both are connected ?

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