The Void Probability The Void Probability functionfunction

and related statisticsand related statistics

Sophie MaurogordatoSophie Maurogordato

CNRS, Observatoire de la Cote d’Azur, CNRS, Observatoire de la Cote d’Azur, FranceFrance

The Void probability The Void probability functionfunction

Count probability PCount probability PNN(V): probability of finding N (V): probability of finding N galaxies in a randomly chosen volume of size Vgalaxies in a randomly chosen volume of size V

N= 0: Void Probability Function PN= 0: Void Probability Function P00(V)(V)

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

1i121

i

0 ...),...,,(...i!

(-n) exp V)(P ii dVdVxxx

Why the VPF ?Why the VPF ? Statistical way to quantify the frequency of voids of a Statistical way to quantify the frequency of voids of a

given size.given size.

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

Straightforward calculated.Straightforward calculated.

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

Scaling properties for Scaling properties for correlation functionscorrelation functions

Observational evidence for low orders:Observational evidence for low orders: n=3n=3

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

n=4n=4

(Fry & Peebles 1978)(Fry & Peebles 1978)

][),,( 231323133321 Qrrr

.)(.)(),,,( 1413124342312443214 permQpermQrrrr

Hierarchical models Hierarchical models

Generalisation for the reduced N-point correlation Generalisation for the reduced N-point correlation : :

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

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

(

21 N),...,,(

LijN Qrrr

),...,(...1

13

23

13

NN

VNN rrrdrdrd

V

N

N

N S)1(

Scaling invariance expected Scaling invariance expected for the correlation for the correlation functions of matterfunctions of matter

In the In the linear- and mildly non linear regimelinear- and mildly non linear regime: : Evolution under Evolution under gravitational instability of gravitational instability of

initial gaussianinitial gaussian fluctuation; can be followed by fluctuation; can be followed by perturbation theory >> predictions for Sperturbation theory >> predictions for SNN’s’s

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

SSN N independant on independant on and z ! and z !

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

Scaling of the VPF under Scaling of the VPF under the hierarchical « ansatz »the hierarchical « ansatz »

The reduced VPF writes: The reduced VPF writes:

nVNc

)1(

10 !

)(exp

N

N

N

N N

nVSP

1

1 !)

Nc

N

Nc N

N

S

cNnV

PLog )( 0

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

VPF from galaxy surveysVPF from galaxy surveys

Zwicky catalog: Sharp 1981Zwicky catalog: Sharp 1981CfA: Maurogordato & Lachièze-Rey 1987CfA: Maurogordato & Lachièze-Rey 1987Pisces-Perseus: Fry et al. 1989Pisces-Perseus: Fry et al. 1989CfA2: Vogeley et al. 1991, Vogeley et al. 1994CfA2: Vogeley et al. 1991, Vogeley et al. 1994SSRS: Maurogordato et al.1992, Lachièze-Rey et al. SSRS: Maurogordato et al.1992, Lachièze-Rey et al.

19921992Huchra’s compilation: Einasto et al. 1991Huchra’s compilation: Einasto et al. 1991QDOT: Watson & Rowan-Robinson, 1993QDOT: Watson & Rowan-Robinson, 1993SSRS2: Benoist et al. 1999SSRS2: Benoist et al. 19992dFGRS: Croton et al. 2004, Hoyle & Vogeley 20042dFGRS: Croton et al. 2004, Hoyle & Vogeley 2004DEEP2 and SDSS: Conroy et al. 2005DEEP2 and SDSS: Conroy et al. 2005

Not exhaustive!Not exhaustive!

How to compute it ?How to compute it ?

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

Randomly throw Randomly throw N N spheres of volume V and spheres of volume V and calculate the whole CPDF: Pcalculate the whole CPDF: PNN(V), P(V), P00(V). (V).

NNcc from the variance of counts. from the variance of counts. Volume-averaged correlation functions from Volume-averaged correlation functions from

the cumulantsthe cumulants Test for scale-invariance for the VPF and for Test for scale-invariance for the VPF and for

the reduced volume-averaged correlation the reduced volume-averaged correlation functions.functions.

nVNc

Scaling or not scaling for Scaling or not scaling for the VPF ?the VPF ?

First generation of catalogs: CfA, SSRS, CfA2, First generation of catalogs: CfA, SSRS, CfA2, SSRS2SSRS2

First evidences of scalingFirst evidences of scaling, but not on all samples. , but not on all samples. Large scale structures of size comparable to that of Large scale structures of size comparable to that of

the surveythe surveyProblem of Problem of « fair sample »« fair sample »

New generation of catalogs: 2dFGRS, SDSS:New generation of catalogs: 2dFGRS, SDSS:

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

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

From Maurogordato et al. 1992

Statistical analysis of the SSRS

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

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

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

Void statistics of the CfA redshift Void statistics of the CfA redshift SurveySurvey

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

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 of the reduced VPF in the 2DdFGRS

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

Scaling at high redshift Scaling at high redshift

VPF from DEEP2 (Conroy et al. 2005)

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

Gaussian

Thermodynamic

Negative binomial

0.12 < z < 0.5

M>-19.5

M>-20

M>-20.5

M>-21

Differentcolors

Different

Luminosities

Seems to work also at high z !

Real/redshift space Real/redshift space distorsions distorsions

Small scales: random pairwise Small scales: random pairwise velocitiesvelocities

Large scales: coherent infall (Kaiser Large scales: coherent infall (Kaiser 1997)1997)

From Hawkins et al.,2003

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

Void statistics in real and Void statistics in real and redshift spaceredshift space

Vogeley et al. 1994, Little & Weinberg 1994Vogeley et al. 1994, Little & Weinberg 1994

Voids appear larger in redshift space :Voids appear larger in redshift space :Amplification of large-scale fluctuationsAmplification of large-scale fluctuationsModel dependant Model dependant Small scales: VPF is reduced in redshift Small scales: VPF is reduced in redshift

space due to fingers of God (small effect)space due to fingers of God (small effect) Howevever difference is smaller than Howevever difference is smaller than

uncertainties on data (Little & Weinberg uncertainties on data (Little & Weinberg 1994, Tinker et al. 2006)1994, Tinker et al. 2006)

Scaling for p-point Scaling for p-point averaged correlation averaged correlation

functionsfunctionsWell verified in many samples, for instance:Well verified in many samples, for instance:

2D:2D: APM (Gaztanaga 1994, Szapudi et al.1995, Szapudi et APM (Gaztanaga 1994, Szapudi et al.1995, Szapudi et

Gaztanaga 1998), EDSGC (Szapudi, Meiksin and Nichol Gaztanaga 1998), EDSGC (Szapudi, Meiksin and Nichol 1996)1996)

Deep-range (Postman et al. 1998, Szapudi et al. 2000)Deep-range (Postman et al. 1998, Szapudi et al. 2000) SDSS (Szapudi et al. 2002, Gaztanaga 2002)SDSS (Szapudi et al. 2002, Gaztanaga 2002)3D:3D: IRAS 1.2 Jy (Bouchet et al. 1993)IRAS 1.2 Jy (Bouchet et al. 1993) CFA+SSRS (Gaztanaga et al. 1994)CFA+SSRS (Gaztanaga et al. 1994) SSRS2 (Benoist et al. 1999)SSRS2 (Benoist et al. 1999) Durham/UKST and Stromlo-APM (Hoyle et al. 2000)Durham/UKST and Stromlo-APM (Hoyle et al. 2000) 2dFGRS (Croton et al. 2004, Baugh et al. 2004) to p=5!2dFGRS (Croton et al. 2004, Baugh et al. 2004) to p=5!

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

SDSS SDSS

From Szapudi et al. 2002

No clear evolution of S3 and S4 with z

Open: Deeprange

Filled: SDSS

SSNN’s for 3D catalogs’s for 3D catalogs

SSNN Gatzanaga et al. Gatzanaga et al. 19941994

CFA+ SSRSCFA+ SSRS

Benoist et al. Benoist et al. 19991999

SSRS2SSRS2

Hoyle et al Hoyle et al 20002000

Stomlo-APMStomlo-APM

Durham/Durham/UKSTUKST

Baugh et al. Baugh et al. 20042004

2dFGRS2dFGRS

N=N=33

1.86 1.86 ± 0.07± 0.07 1.80 1.80 ± 0.2± 0.2 1.8-2.2 1.8-2.2 ± 0.4± 0.4 1.95 1.95 ±±0.180.18

N=N=44

4.15 4.15 ± 0.6± 0.6 5.50 5.50 ± 3.0± 3.0 5.05.0± 3.8± 3.8 5.50 5.50 ±±1.431.43

N=N=55

17.8 17.8 ±±10.510.5

N=N=66

16.3 16.3 ±±5050

Good agreement for S3 and S4 in redshift catalogues

Hierarchical correlations Hierarchical correlations for the VVDSfor the VVDS

0.5< z < 1.2

S3 ~ 2

On courtesy of Alberto Cappi and the VVDS consortium

Hierarchical Scaling Hierarchical Scaling for VPF for VPF in redshift spacein redshift space- Valid for samples with Valid for samples with different luminosity ranges, different luminosity ranges,

redshift ranges, and bias factorsredshift ranges, and bias factors

for the reduced volume-averaged N-point correlation for the reduced volume-averaged N-point correlation functionfunction

SSNN’s roughly constant with scale’s roughly constant with scaleGood agreement for S3 and S4 in different redshift Good agreement for S3 and S4 in different redshift

catalogscatalogsBut different amplitudes from 2D and 3D measurementBut different amplitudes from 2D and 3D measurement(damping of clustering in z space, Lahav et al. 1993)(damping of clustering in z space, Lahav et al. 1993)

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

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

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

What is the dependence of VPF on What is the dependence of VPF on cosmological parameters ?cosmological parameters ?

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

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

Dependence on model Dependence on model parametersparameters

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

For unbiased models:For unbiased models:weak dependance on n (weak dependance on n (VPF when n )VPF when n )Insensitive to Insensitive to and and

Good discriminant on the gaussianity of initial Good discriminant on the gaussianity of initial conditionsconditions

For biased models: For biased models: sensitive to biasing prescriptionsensitive to biasing prescription

VPF is higher for higher bias factor VPF is higher for higher bias factor

What can we learn from What can we learn from VPF (and SN’s) about VPF (and SN’s) about

« biasing » ?« biasing » ?In the In the « biased galaxy formation »« biased galaxy formation » frame, galaxies are frame, galaxies are

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

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

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

Variation of the amplitude of the auto-correlation Variation of the amplitude of the auto-correlation function function

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

Luminosity bias from galaxy Luminosity bias from galaxy redshift surveysredshift surveys

From Norberg et al 2001

Testing the bias model Testing the bias model with Swith SNN’s’s

Linear bias hypothesis:Linear bias hypothesis: b

2b NNb

NNN Sb

S

2

1

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)

From Benoist et al. 1999

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

High order statistics in the SSRS2

Non-linear local bias and Non-linear local bias and high-order momentshigh-order moments

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

Presence of secondary order terms in SPresence of secondary order terms in SNN’s:’s:

k

kk

g k

b

0 !Fry and Gatzanaga 1993

)12412(1

)3(1

2233242

14

231

3

ccScSb

S

cSb

S

g

g

Gatzanaga et al 1994, 1995

Benoist et al. 1999

Hoyle et al. 2000

Croton et al. 2004

Constraining the biasing Constraining the biasing schemescheme

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

- Large-scales: bias is expected to be linearLarge-scales: bias is expected to be linear- Small scalesSmall scales: bias reflects the physics of galaxy : bias reflects the physics of galaxy

formation, so formation, so can be scale-dependantcan be scale-dependant

Recent progress in modelling the non-linear clustering:Recent progress in modelling the non-linear clustering:

HOD >> bias at the level of dark matter halosHOD >> bias at the level of dark matter halos (Benson et al. 2001, Berlind & Weinberg 2002, (Benson et al. 2001, Berlind & Weinberg 2002,

Kravtsov et al. 2004, Conroy et al. 2005, Tinker, Kravtsov et al. 2004, Conroy et al. 2005, Tinker, Weinberg & Warren 2006)Weinberg & Warren 2006)

Constraining the HOD Constraining the HOD parametersparameters

Berlind and Weinberg 2002, Tinker, Weinberg & Warren 2006Berlind and Weinberg 2002, Tinker, Weinberg & Warren 2006

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

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

Strong correlation between the minimum mass scale MStrong correlation between the minimum mass scale Mminmin / size of voids / size of voids

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

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

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

fmin=2

fmin=4

fmin= ∞

c=-0.2

c

c

c

From Tinker, Weinberg, Warren 2006

cMmin = fmin x Mmin

ConclusionsConclusions Convergence of observational results from existing Convergence of observational results from existing

redshift surveys:redshift surveys:

- scale-invariance of the reduced VPFscale-invariance of the reduced VPF- Hierarchical behaviour of N-point averaged correlation Hierarchical behaviour of N-point averaged correlation

functionsfunctions- More: the shape for the reduced VPF, and the More: the shape for the reduced VPF, and the

amplitudes of Samplitudes of S33 and S and S44 are consistent for the different are consistent for the different samples.samples.

Good agreement with the gravitational instability model.Good agreement with the gravitational instability model.

VPF in recent surveys + state of the art HODVPF in recent surveys + state of the art HODvery promising to constrain the non linear very promising to constrain the non linear

bias bias