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The Void Probability function and related statistics Sophie Maurogordato CNRS, Observatoire de la Cote d’Azur, France

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The Void probability function Count probability P N (V): probability of finding N galaxies in a randomly chosen volume of size V 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) N= 0: Void Probability Function P 0 (V) Related to the hierarchy of n-point reduced correlation functions (White 1979) Related to the hierarchy of n-point reduced correlation functions (White 1979)

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Why the VPF ? Statistical way to quantify the frequency of voids of a given size. 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). 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. Straightforward calculated. But density dependent, denser samples have smaller voids: be careful when comparing samples with different densities. But density dependent, denser samples have smaller voids: be careful when comparing samples with different densities.

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Scaling properties for correlation functions Observational evidence for low orders: n=3 n=3 (Groth & Peebles, 1977, Fry & Peebles 1978, Sharp et al 1984) (Groth & Peebles, 1977, Fry & Peebles 1978, Sharp et al 1984) n=4 n=4 (Fry & Peebles 1978) (Fry & Peebles 1978)

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Hierarchical models Generalisation for the reduced N-point correlation : tree shape L( ) labellings of a given tree (Fry 1984, Schaeffer 1984, Balian and Schaeffer 1989) (Fry 1984, Schaeffer 1984, Balian and Schaeffer 1989)

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Scaling invariance expected for the correlation functions of matter In the linear- and mildly non linear regime: In the linear- and mildly non linear regime: Evolution under gravitational instability of initial gaussian fluctuation; can be followed by perturbation theory >> predictions for S N ’s Evolution under gravitational instability of initial gaussian fluctuation; can be followed by perturbation theory >> predictions for S N ’s (Peebles 1980, Jusckiewicz, Bouchet & Colombi 1993, Bernardeau 1994, Bernardeau 2002) (Peebles 1980, Jusckiewicz, Bouchet & Colombi 1993, Bernardeau 1994, Bernardeau 2002) S N independant on and z ! In the strongly non-linear regime: solution of the BBGKY equations In the strongly non-linear regime: solution of the BBGKY equations

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Scaling of the VPF under the hierarchical « ansatz » The reduced VPF writes: The reduced VPF as a function of N c is a function of the whole set of SN’s

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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!

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How to compute it ? Select sub-samples of constant density: volume and magnitude limited samples. Select sub-samples of constant density: volume and magnitude limited samples. Randomly throw N spheres of volume V and calculate the whole CPDF: P N (V), P 0 (V). Randomly throw N spheres of volume V and calculate the whole CPDF: P N (V), P 0 (V). N c from the variance of counts. N c from the variance of counts. Volume-averaged correlation functions from the cumulants Volume-averaged correlation functions from the cumulants Test for scale-invariance for the VPF and for the reduced volume-averaged correlation functions. Test for scale-invariance for the VPF and for the reduced volume-averaged correlation functions.

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Scaling or not scaling for the VPF ? First generation of catalogs: CfA, SSRS, CfA2, SSRS2 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: New generation of catalogs: 2dFGRS, SDSS: Excellent convergence to a common function corresponding to the negative binomial model.

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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

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Void statistics of the CfA redshift Survey From Vogeley, Geller and Huchra, 1991, ApJ, 382, 44

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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

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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 Different colors Different Luminosities Seems to work also at high z !

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Real/redshift space distorsions Small scales: random pairwise velocities Small scales: random pairwise velocities Large scales: coherent infall (Kaiser 1997) Large scales: coherent infall (Kaiser 1997) From Hawkins et al.,2003 Distorsion on 2-pt correlation from peculiar velocities in the 2dFGRS

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Void statistics in real and redshift space Vogeley et al. 1994, Little & Weinberg 1994 Voids appear larger in redshift space : 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) 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) Howevever difference is smaller than uncertainties on data (Little & Weinberg 1994, Tinker et al. 2006)

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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) 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) 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: 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!

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Skewness and kurtosis (2D) for the Deeprange and SDSS From Szapudi et al. 2002 No clear evolution of S3 and S4 with z Open: Deeprange Filled: SDSS

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S N ’s for 3D catalogs SNSNSNSN Gatzanaga et al. 1994 CFA+ SSRS Benoist et al. 1999 SSRS2 Hoyle et al 2000 Stomlo-APMDurham/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

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Hierarchical correlations for the VVDS 0.5< z < 1.2 S3 ~ 2 On courtesy of Alberto Cappi and the VVDS consortium

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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 S N ’s roughly constant with scale S N ’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 Good agreement with evolution of clustering under gravitational instability from initial gaussian fluctuations

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The VPF as a tool to discriminate between models of structure formation Can gravity alone create such large voids as observed in redshift surveys ? Can gravity alone create such large voids as observed in redshift surveys ? What is the dependence of VPF on cosmological parameters ? What is the dependence of VPF on cosmological parameters ? What VPF can tell us about the gaussianity/ non gaussianity of initial conditions ? 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 ? Can we infer some clue on the biasing scheme necessary to explain them ?

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Dependence on model parameters Einasto et al. 1991, Weinberg and Cole 1992, Little and Weinberg 1994, Vogeley et al. 1994,… Einasto et al. 1991, Weinberg and Cole 1992, Little and Weinberg 1994, Vogeley et al. 1994,… For unbiased models: For unbiased models: weak dependance on n (VPF when n ) Insensitive to and Good discriminant on the gaussianity of initial conditions Good discriminant on the gaussianity of initial conditions For biased models: sensitive to biasing prescription For biased models: sensitive to biasing prescription VPF is higher for higher bias factor

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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 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)

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Luminosity bias from galaxy redshift surveys From Norberg et al 2001

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Testing the bias model with S N ’s Linear bias hypothesis: 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)

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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

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Non-linear local bias and high-order moments This local biasing transformation preserves the hierarchical structure in the regime of small This local biasing transformation preserves the hierarchical structure in the regime of small Presence of secondary order terms in S N ’s: Presence of secondary order terms in S N ’s: Fry and Gatzanaga 1993 Gatzanaga et al 1994, 1995 Benoist et al. 1999 Hoyle et al. 2000 Croton et al. 2004

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Constraining the biasing scheme Galaxy distribution results from gravitational evolution of dark matter coupled to astrophysical processes: gas cooling, star formation, feedback from supernovae… 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: Recent progress in modelling the non-linear clustering: HOD >> bias at the level of dark matter halos 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) (Benson et al. 2001, Berlind & Weinberg 2002, Kravtsov et al. 2004, Conroy et al. 2005, Tinker, Weinberg & Warren 2006)

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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: M =(M/M1) with a lower cutoff M min Strong correlation between the minimum mass scale M min / size of voids TWW06: M = M + 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

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fmin=2 fmin=4 fmin= ∞ c=-0.2 c c c From Tinker, Weinberg, Warren 2006 c M min = f min x M min

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Conclusions Convergence of observational results from existing redshift surveys: 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 S 3 and S 4 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

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