Download The Void Probability Function and Related Statistics

The Void Probability
function
and 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)
  (-n) i

P0 (V)  exp 
...  ( x1 , x2 ,..., xi )dV1...dVi 

 i 1 i!

Why the VPF ?

Statistical way to quantify the frequency of voids of a given size.

Complementary information on high-order correlations that loworder 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
 (r1 , r2 , r3 )  Q3[13   23  13 23 ]
(Groth & Peebles, 1977, Fry & Peebles 1978, Sharp et al
1984)
 n=4
 4 (r1 , r2 , r3 , r4 )  Q4 (12 2334  perm.)  Q4 (121314  perm.)
(Fry & Peebles 1978)
Hierarchical models
Generalisation for the reduced N-point correlation N :
N 
 N (r1 , r2 ,..., rN )   Qa  ij
a
N
L (a 
a :tree shape
L(a ) labellings of a given tree
(Fry 1984, Schaeffer 1984, Balian and Schaeffer 1989)
1
 N  N  d 3r1d 3r2 ...d 3rN  N (r1 ,..., rN )
V V
N 
( N 1)
SN
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 »

(nV ) ( N 1) 

P0  exp   S N

N!
 N 1


N
The reduced VPF writes:
Nc  nV 
Log ( P0 )
  N c 
nV

SN
N 1
  N c )    N c 
N 1 N !
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.
nV 
Randomly throw N spheres of volume VN 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.
c
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
Scaling at high redshift
Gaussian
Different
colors
Thermodynamic
Negative binomial
0.12 < z < 0.5
M>-19.5
M>-20
Different
M>-20.5
Luminosities
M>-21
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:
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)
 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
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
N=3
1.86 ± 0.07
1.80 ± 0.2
1.8-2.2 ± 0.4
1.95 ±0.18
N=4
4.15 ± 0.6
5.50 ± 3.0
5.0± 3.8
5.50 ±1.43
N=5
17.8 ±10.5
N=6
16.3 ±50
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

  b 
Linear bias hypothesis:
 b 
2
N  b 

SN 
N
1
b
N 2
S


N
N
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
bk k
 g  k  0 
k!

Fry and Gatzanaga 1993
This local biasing transformation preserves the hierarchical

structure in the regime of small
Presence of secondary order terms in SN’s:
1
S  ( S 3  3c2 )
b1
Gatzanaga et al 1994, 1995
1
S  2 ( S 4  12c2 S 3  4c3  12c22 )
b1
Hoyle et al. 2000
g
3
g
4
Benoist et al. 1999
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
fmin=2
fmin=4
c=-0.2
c0.4
fmin= ∞
c0.6
c0.8
 < c , Mmin = fmin x Mmin
From Tinker, Weinberg,
Warren 2006
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
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
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 ?