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The Void Probability function and related statistics

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Title: The Void Probability function and related statistics


1
The Void Probability functionand related
statistics
  • Sophie Maurogordato
  • CNRS, Observatoire de la Cote dAzur, France

2
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)

3
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.

4
Scaling properties for correlation functions
  • Observational evidence for low orders
  • n3
  • (Groth Peebles, 1977, Fry Peebles 1978,
    Sharp et al 1984)
  • n4
  • (Fry Peebles 1978)

5
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)

6
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 gtgt predictions for SNs
  • (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

7
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 SNs
8
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
  • Huchras 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!

9
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.

10
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.

11
Statistical analysis of the SSRS
Reduced VPFs rescales to the same function even
for samples with very different amplitudes of the
correlation functions.
Mgt-18, Dlt 40h-1 Mpc Mgt-19, Dlt 60 h-1 Mpc Mgt-20, D
lt 80h-1 Mpc
From Maurogordato et al. 1992
12
Void statistics of the CfA redshift Survey
From Vogeley, Geller and Huchra, 1991, ApJ, 382,
44
13
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 lt0.2
14
Scaling at high redshift
Gaussian Thermodynamic Negative binomial
Different colors
0.12 lt z lt 0.5 Mgt-19.5 Mgt-20 Mgt-20.5 Mgt-21
Different Luminosities
VPF from DEEP2 (Conroy et al. 2005)
VPF from VVDS (Cappi et al. in prep.)
Seems to work also at high z !
15
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
16
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)

17
Scaling for p-point averaged correlation functions
  • Well verified in many samples, for instance
  • 2D
  • 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)
  • 3D
  • IRAS 1.2 Jy (Bouchet et al. 1993)
  • CFASSRS (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
    p5!

18
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
19
SNs for 3D catalogs
SN Gatzanaga et al. 1994 CFA SSRS Benoist et al. 1999 SSRS2 Hoyle et al 2000 Stomlo-APM Durham/UKST Baugh et al. 2004 2dFGRS
N3 1.86 0.07 1.80 0.2 1.8-2.2 0.4 1.95 0.18
N4 4.15 0.6 5.50 3.0 5.0 3.8 5.50 1.43
N5 17.8 10.5
N6 16.3 50
Good agreement for S3 and S4 in redshift
catalogues
20
Hierarchical correlations for the VVDS
0.5lt z lt 1.2 S3 2
On courtesy of Alberto Cappi and the VVDS
consortium
21
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
  • SNs 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

22
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 ?

23
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

24
What can we learn from VPF (and SNs) 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)

25
Luminosity bias from galaxy redshift surveys
From Norberg et al 2001
26
Testing the bias model with SNs
  • Linear bias hypothesis

Inconsistency between the the measured values of
SNs towards the expected values from the
correlation functions under the linear bias
hypothesis (Benoist et al. 1999, Croton et al.
2004)
27
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
28
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 SNs

Fry and Gatzanaga 1993
Gatzanaga et al 1994, 1995 Benoist et al.
1999 Hoyle et al. 2000 Croton et al. 2004
29
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 gtgt 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)

30
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 ltNgtM (M/M1)a with a lower cutoff Mmin
  • Strong correlation between the minimum mass
    scale Mmin / size of voids
  • TWW06 ltNgtM ltNsatgtM ltNcengtM
  • 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

31
fmin2
fmin4
dc-0.2 dc-0.4 dc-0.6 dc-0.8
d lt dc , Mmin fmin x Mmin
fmin 8
From Tinker, Weinberg, Warren 2006
32
Conclusions
  • 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

33
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35
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 gtgt constrain the feedback
    mechanisms

Benson et al. 2003
36
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
37
Little Weinberg 1994
38
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39
Models for the VPF
  • BBGKY (Fry 1984)
  • Thermodynamical model (Saslaw Hamilton 1984)
  • Binomial model (Carruthers Shih 1983)
  • Log-normal model (Coles Jones 1991)

40
  • What  empty  regions can tell us about
     filled  ones ?
  • How both are connected ?
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