PONT 2011 Avignon - 20 Apr 2011 Non-Gaussianities in Halo Clustering properties. Andrea De Simone

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1 PONT 2011 Avignon - 20 Apr 2011 Non-Gaussianities in Halo Clustering properties Andrea De Simone Based on: ADS, M. Maggiore, A. Riotto ADS, M. Maggiore, A. Riotto - MNRAS ( )

2 OUTLINE Excursion Set Theory Path Integral formulation Computation of NG corrections to: Halo mass function Halo bias Halo formation time distribution 1/19

3 Introduction The formation and evolution of structures is a very complex phenomenon. Clustering properties of DM haloes (e.g. mass function) can be sensitive probes of NG and will be tested in the near future. At present, quantitative knowledge comes mainly from N-body sims. A full theoretical understanding is still lacking. A successful theory of structure formation must be able to make predictions. Need for an analytical control. Focus on: analytical description of the formation of DM haloes and impact of NG. Excursion Set Theory 2/19

4 Excursion Set Theory It allows to map the statistics of initial conditions with the subsequent formation of structures. Smooth out the density pert. on a sphere of radius R δ(r, x) = d 3 x W ( x x,r) δ(x ) Study the evolution of δ as a function of R At R=, δ(r)=0. Non-Gaussianities in halo clustering properties δ = δρ ρ [Bond, Cole, Efstathiou, Kaiser 1991] [Peacock, Heavens 1990] R 3/19

5 Excursion Set Theory It allows to map the statistics of initial conditions with the subsequent formation of structures. δ = δρ ρ Smooth out the density pert. on a sphere of radius R δ(r, x) = d 3 x W ( x x,r) δ(x ) Study the evolution of δ as a function of R At R=, δ(r)=0. Lowering R, δ(r) evolves stochastically and performs a random walk. [Bond, Cole, Efstathiou, Kaiser 1991] [Peacock, Heavens 1990] 3/19

6 Excursion Set Theory It allows to map the statistics of initial conditions with the subsequent formation of structures. δ = δρ ρ Smooth out the density pert. on a sphere of radius R δ(r, x) = d 3 x W ( x x,r) δ(x ) Study the evolution of δ as a function of R At R=, δ(r)=0. Lowering R, δ(r) evolves stochastically and performs a random walk. [Bond, Cole, Efstathiou, Kaiser 1991] [Peacock, Heavens 1990] Use S= σ 2 (R) as time. S increases as R decreases. 3/19

7 δ(s) S = η(s) Non-Gaussianities in halo clustering properties Excursion Set Theory d 3 k δ(r, x = 0) = (2π) δ 3 k W (k,r) stochastic variable (Langevin eq.) noise Ex: for sharp k-space filter η(s 1 )η(s 2 ) = δ D (S 1 S 2 ) white noise Window function ( filter ) 4/19

8 Excursion Set Theory d 3 k δ(r, x = 0) = (2π) δ 3 k W (k,r) δ(s) S = η(s) stochastic variable (Langevin eq.) noise A region collapses if it is dense enough: when δ crosses δc the first time Window function ( filter ) Halo formation probability is mapped into a first - passage time problem 4/19

9 Excursion Set Theory Problem: find the probability that a particle subject to a random walk passes for the first time through a given point Π(δ, S): prob. that the density contrast arrives for the first time at δ in a time S Knowledge of Π solves the problem and allows computation of the mass function First-crossing rate: F(S) = S Halo mass function: δc dδ Π(δ, S) dn dm dm = ρ M F(S) number of trajectories that did not cross the barrier before S ds dm dm 5/19

10 Excursion Set Theory Assumptions: 1. Top-hat filter in k-space: W (k,r)=θ(r 1 k ) 2. Spherical collapse (δc) 3. Gaussian initial conditions 6/19

11 Excursion Set Theory Assumptions: 1. Top-hat filter in k-space: W (k,r)=θ(r 1 k ) 2. Spherical collapse (δc) 3. Gaussian initial conditions solution Π(δ, S) = 1 S 2 Π(δ c,s)=0 2 Π(δ, S) δ 2 (Fokker-Planck eq.) Π(δ, S) = 1 2πS e δ2 /(2S) e (2δ c δ) 2 /(2S) F(S) = δ c 2πS 3 e δ2 c /(2S) (Press-Schechter) 6/19

12 νf(ν) =2δ c F(δ c /ν) Sheth-Tormen, 1999 Jenkins et al., 2001 PS Tinker et al., 2008 ν = δ c /S = δ c /σ 2 At large masses, the PS theory underestimates the halo masses by a factor ~ 10; at small halo masses it overestimates them by a factor ~ 2 7/19

13 Excursion Set Theory 1. Top-hat filter in k-space? Non-Gaussianities in halo clustering properties Unphysical, one may not identify well-defined mass. N-body sim. use top-hat filter in real space. Smoothing with the top-hat filter in real space and/or dealing with non-gaussianities makes the various random steps correlated: the dynamics is non-markovian and memory effects are introduced. 8/19

14 2. Spherical collapse? Non-Gaussianities in halo clustering properties Excursion Set Theory Formation of DM haloes proceeds through an ellipsoidal collapse along each of the principal ellipsoidal axes under the action of external tides (DM haloes carry angular momentum) One can describe it by changing the collapse barrier into a moving collapse barrier δ c B(S) 0.6 S B(S) =δ c [Sheth, Tormen 2001] It tends to sph. coll. in large mass limit δ 2 c 9/19

15 Excursion Set Theory 3. Gaussian initial conditions? Non-Gaussianities in halo clustering properties Want to include effects of primordial NG on structure formation. NG introduces non-markovian dynamics and memory effects 10/19

16 Excursion Set Theory 3. Gaussian initial conditions? Non-Gaussianities in halo clustering properties Want to include effects of primordial NG on structure formation. If any of the assumptions is not met, FP eq is non-local. Need to go to a more fundamental level. Excursion Set Theory has stuck for years because of this technical difficulty. NG introduces non-markovian dynamics and memory effects 10/19

17 Path Integral Formulation [Maggiore, Riotto 2009] Take ensemble of trajectories of the smoothed density contrast ξ(s) and follow them for a time S. S 0 11/19

18 Path Integral Formulation [Maggiore, Riotto 2009] Take ensemble of trajectories of the smoothed density contrast ξ(s) and follow them for a time S. S 0 Discretize S: Sk=k ε. ξ(sk)=δk. A given trajectory is defined by the set {δ1,..., δn}. The prob. density in the space of trajectories is W (δ 0 ; δ 1,,δ n ; S n ) δ D (ξ(s 1 ) δ 1 ) δ D (ξ(s n ) δ n ) 11/19

19 Path Integral Formulation [Maggiore, Riotto 2009] Take ensemble of trajectories of the smoothed density contrast ξ(s) and follow them for a time S. S 0 Discretize S: Sk=k ε. ξ(sk)=δk. A given trajectory is defined by the set {δ1,..., δn}. The prob. density in the space of trajectories is W (δ 0 ; δ 1,,δ n ; S n ) δ D (ξ(s 1 ) δ 1 ) δ D (ξ(s n ) δ n ) Π is constructed by summing over all paths that never exceed the threshold Π (δ 0 ; δ n,s n )= δc dδ 1 δc dδ n 1 W (δ 0 ; δ 1,,δ n 1,δ n ; S n ) The problem is reduced to the evaluation of a path integral with boundaries. 11/19

20 Π (δ 0 ; δ n,s n ) = Non-Gaussianities in halo clustering properties Path Integral Formulation δc dδ 1 dδ n 1 e i n i=1 λ iδ i + p=2 ( i) p p! dλ 1 dλ n 2π 2π [Maggiore, Riotto 2009] n n i 1 =1 ip=1 λ i 1 λ i p ξ i 1 ξ i p c e Z Z = i i λ i δ i 1 2 i.j λ i λ j ξ(s i )ξ(s j ) c + ( i)3 3! λ i λ j λ k ξ(s i )ξ(s j )ξ(s k ) c + i,j,k Generating functional of connected correlators! 12/19

21 Π (δ 0 ; δ n,s n ) = Non-Gaussianities in halo clustering properties Path Integral Formulation δc dδ 1 dδ n 1 e i n i=1 λ iδ i + p=2 ( i) p p! dλ 1 dλ n 2π 2π [Maggiore, Riotto 2009] n n i 1 =1 ip=1 λ i 1 λ i p ξ i 1 ξ i p c e Z Z = i i λ i δ i 1 2 i.j λ i λ j ξ(s i )ξ(s j ) c + ( i)3 3! λ i λ j λ k ξ(s i )ξ(s j )ξ(s k ) c + i,j,k Generating functional of connected correlators! NG: ξ(s i )ξ(s j )ξ(s k ) c =0 Non-spherical collapse: δ c B(S) in the integrals 12/19

22 Halo Mass Function [DS, Maggiore, Riotto 2010] Using path integrals, we computed the halo mass function with and without NG, for a generic barrier B(S). Sheth&Tormen (2002) showed that the empirical formula fits well numerical sims: (S)/(2S) 5 ( S) p p B(S) F ST (S) = e B2 2πS 3/2 p! S p p=0 For NG=0, recover (numerically) the ST ansatz (better than 10%) and put it on firmer grounds Our result f NL =0 F(S) = B(S) (S)/(2S) 2πS 3/2 e B2 B (S) e B(S)2 /(2S) 2πS + B (S) 4π 2πSe B(S) 2 B(S) /(2S) πb(s)erfc + 2S Ν fν ST Ν c Σ 13/19

23 Halo Mass Function [DS, Maggiore, Riotto 2010] Using path integrals, we computed the halo mass function with and without NG, for a generic barrier B(S). Sheth&Tormen (2002) showed that the empirical formula fits well numerical sims: (S)/(2S) 5 F ST (S) = e B2 ( S) p p B(S) 2πS 3/2 p! S p p=0 For NG=0, recover (numerically) the ST ansatz (better than 10%) and put it on firmer grounds. F(S) = B(S) (S)/(2S) 2πS 3/2 e B2 B (S) e B(S)2 /(2S) 2πS + B (S) 4π 2πSe B(S) 2 B(S) /(2S) πb(s)erfc + 2S For NG 0, ~10% corrections f NG f G f NL = Ν c Σ 13/19

24 Halo Mass Function with NG The halo mass function with NG is usually computed hoping that F NG (f NL,S) = F NG(f NL,S) F G (S) F G (S) R(S) PS No rigorous justification. [DS, Maggiore, Riotto 2010] We computed directly FNG for ellipsoidal collapse, with saddle-point improvement [D Amico, Musso, Norena, Paranjape 2010] Proved that the above ansatz is incorrect. F NG F ST R NG Lo Verde et al. 2008, z1 Lo Verde et al. 2008, z2 Matarrese et al. 2000, z1 Matarrese et al. 2000, z M M h 1 14/19

25 Halo Mass Function with NG Conditional probabilities are useful for formation history. Given a halo of mass M0 at redshift za, how was its mass partitioned among smaller haloes of mass M at redshift zb >za. In Excursion Set Theoy: two-barrier problem. [Lacey, Cole 1993] Conditional mass function for generic barrier, with/without NG dn dln M M 0 NG dn dln M M 0 G M 0 = h 1 M z =0.3 f NL = MM 0 [DS, Maggiore, Riotto 2010] 15/19

26 Halo Bias The bias is the proportionality of halo overabundance wrt matter overdensity: δρ ρ galaxies = b δρ ρ mass b(s) =1+ ν(s)2 1 δ c =1+ δ c S 1 [Cole, Kaiser 1989] δ c [Mo, White 1996] Our result for generic barrier, without NG: b(s) = 1+ B(S) S 1 B(S)+ p=1 ( S) p p! p B(S) S p b f NL =0 spherical collapse Our result Sheth-Mo-Tormen Fit to N-body (Tinker et al. 2010) Ν = δ c /σ 16/19

27 Halo Bias with NG We computed the (linear and quadratic) bias for generic barrier and with NG. For ellipsoidal collapse: b (1) NG 1 6 S 3 3aν 2 1.6(aν 2 ) 0.4 S3 = δ 3 (S)/S 2 17/19

28 Halo Bias with NG We computed the (linear and quadratic) bias for generic barrier and with NG. For ellipsoidal collapse: b (1) NG 1 6 S 3 3aν 2 1.6(aν 2 ) 0.4 S3 = δ 3 (S)/S 2 Factor 2/3 discrepancy with uses the form factor prescription. Our calculation is from first principles. [Desjacques, Marian, Smith 2009] which Non-Markov corrections due to filter: [Ma, Maggiore, Riotto, Zhang 2010] N-body sims began to study the bias with NG. [Wagner, Verde 2011] 17/19

29 Halo Formation Time with NG Formation time: the earliest time when at least half of its mass was assembled into a single progenitor. p(z b )=2ω(z b )Erfc Non-Gaussianities in halo clustering properties ω(zb ) 2 dω(zb ) dz b ω(z b )= δ c (z b ) δ c (z a ) S(M0 /2) S(M 0 ) [Lacey, Cole 1993] P (> z) 1.5 Spherical collapse with barrier a c M 0 = h 1 M Our computation: prob. of formation redhifts with NG. We also found an analytical generalization of Lacey&Cole for NG. Caveat on S(M) [Giocoli, Moreno, Sheth, Tormen 2006] z dpdz b fnl=0 fnl= z b < 10% shift S(M) numerical S(M)~M -1 18/19

30 Conclusions Excursion Set Theory (with path integrals) is a convenient and powerful framework to compute properties of LSS analytically. It allowed a consistent derivation of the effects of NG. First-principlesʼʼ calculation of halo mass function, bias and formation time for generic barrier, with and without NG. Look forward to comparing with N-body sims and new data. 19/19

31 Conclusions Excursion Set Theory (with path integrals) is a convenient and powerful framework to compute properties of LSS analytically. It allowed a consistent derivation of the effects of NG. First-principlesʼʼ calculation of halo mass function, bias and formation time for generic barrier, with and without NG. Look forward to comparing with N-body sims and new data. Thank you! 19/19