Following DM Haloes with the Time-RG

Transcription

1 Following DM Haloes with the Time-RG Massimo Pietroni - INFN Padova In collaboration with A. Elia, S. Kulkarni, C. Porciani, S. Matarrese Cosmo-CosPa, Tokyo

2 Outline Failures of Perturbation Theory Resummations and the TRG Introducing DM haloes Results

3 The future of precision cosmology: non-linear scales matter density ρ(x, τ) ρ(τ)[1 + δ(x, τ)] power spectrum δ(k, τ) δ(k, τ) = P (k, τ)δ (3) (k + k ) `size of the fluctuations at different scales/epochs: 2 (k, τ) =4πk 3 P (k, τ) Baryonic Acoustic Oscillations (BAO) Neutrino mass bounds f ν =0.01 (m ν, tot =0.12 ev) f ν =0.1 (m ν, tot =1.2 ev)

4 Theoretical tools Linear perturbation theory badly fails for z<2-3 and k> 0.05h/Mpc N-body simulations In principle ok, but need large volumes/resolutions: - practically impossible to scan over cosmological models; - non-standard but interesting scenarios are problematic: (massive neutrinos, non-gaussianity, DE-DM coupling...)

5 Fluid equations for Cold Dark Matter δ τ + [(1 + δ)v] = 0, In Fourier space, ( defining ), δ(k, τ) τ + θ(k, τ) + θ(x, τ) v(x, τ) v τ + Hv + (v )v = φ 2 φ = 3 2 Ω M H 2 δ d 3 k 1 d 3 k 2 δ D (k k 1 k 2 )α(k 1, k 2 )θ(k 1, τ)δ(k 2, τ) = 0 θ(k, τ) τ + H θ(k, τ) Ω M H 2 δ(k, τ) + d 3 k 1 d 3 k 2 δ D (k k 1 k 2 )β(k 1, k 2 )θ(k 1, τ)θ(k 2, τ) = 0 mode-mode coupling controlled by: α(k 1, k 2 ) (k 1 + k 2 ) k 1 k 2 1 β(k 1, k 2 ) k 1 + k 2 2 (k 1 k 2 ) 2k 2 1 k2 2

6 δ(k, τ) τ θ(k, τ) τ linear approximation: + θ(k, τ) = 0 + H θ(k, τ) Ω M H 2 δ(k, τ) = 0 α(k 1, k 2 ) = β(k 1, k 2 ) = 0 no mode-mode coupling Ω M = 1 H a 1/2 δ(k, τ) = δ(k, τ i ) θ(k, τ) H = m δ(k, τ) m a(τ) a(τ i ) m = growing mode decaying mode

7 Compact Perturbation Theory Consider again the continuity and Euler equations Crocce, Scoccimarro 05 δ τ + [(1 + δ)v] = 0, v τ + Hv + (v )v = φ, define ϕ1 (η, k) ϕ 2 (η, k) e η δ(η, k) θ(η, k)/h with η = log a(τ) a(τ i ) Ω = 1 1 3/2 3/2 then we can write (we assume an EdS model): (δ ab η + Ω ab ) ϕ b (η, k) = e η γ abc (k, k 1, k 2 ) ϕ b (η, k 1 ) ϕ c (η, k 2 ) and the only non-zero components of the mode-mode coupling are γ 121 (k 1, k 2, k 3 ) = γ 112 (k 1, k 3, k 2 ) = δ D (k 1 + k 2 + k 3 ) α(k 2, k 3 ) 2 γ 222 (k 1, k 2, k 3 ) = δ D (k 1 + k 2 + k 3 ) β(k 2, k 3 )

8 Perturbation Theory: Feynman Rules a b propagator (linear growth factor): i g ab (η a, η b ) a b power spectrum: P L ab(η a, η b ; k) a interaction vertex: i e η γ abc (k a, k b, k c ) c b Example: 1-loop correction to the density power spectrum: Linear Power spectrum P22 All known results in cosmological perturbation theory are expressible in terms of diagrams in which only a trilinear fundamental interaction appears P13

9 Problems with PT 1-loop large k: k k G ab (k; η a, η b ) = g ab (η a, η b ) σ d 3 q P 0 (q) q 2 1 k 2 σ 2 (eη a e η b ) 2 2 (σ e η a ) h Mpc 1 in the BAO range! + O(k 4 σ 4 ) 2-loop the PT series blows up in the BAO range

10 But it can be resummed!! (Crocce-Scoccimarro 06)... k k k k G(k; η, η in )= δ(k, η)δ(k, η in) δ(k, η in )δ(k, η in ) e k 2 σ 2 2 e 2η physically, it represents the effect of multiple interactions of the k-mode with the surrounding modes: memory loss `coherence momentum k ch = (σ e η ) h Mpc 1 damping in the BAO range! RPT: use G, and not g, as the linear propagator

11 Partial (!) list of contributors to the field traditional P.T.: see Bernardeau et al, Phys. Rep. 367, 1, (2002), and refs. therein; Jeong-Komatsu; Saito et al; Sefusatti;... resummation methods: Valageas; Crocce-Scoccimarro; McDonald; Matarrese-M.P.; Matsubara; Taruya- Hiratamatsu; M.P.; Bernardeau- Valageas; Bernardeau-Crocce- Scoccimarro;...

12 Time-RG (M.P. 08) (also for cosmologies with D ± = D ± (k, z) ) (δ ab η + Ω ab ) ϕ b (η, k) = e η γ abc (k, k 1, k 2 ) ϕ b (η, k 1 ) ϕ c (η, k 2 ) η ϕ = Ω ϕ + e η γϕϕ η ϕϕ = Ω ϕϕ + e η γ ϕϕϕ η ϕϕϕ = Ω ϕϕϕ + e η γ ϕϕϕϕ infinite tower of equations can be obtained from the RG-like physical requirement d Z[J, Λ; η in ]=0 η in

13 Advantages Works also for cosmologies with Ω ab = Ω ab (k, η) not only for Ω ab = 1 1 3/2 3/2 e.g. massive neutrinos, Scalar-tensor theories Power spectrum ( ϕϕ) and bispectrum ( ϕϕϕ ) from a single run! Systematic approximation scheme straightforward

14 Equations to solve: initial conditions given at η =0, corresponding to z = z in Only approximation: T abcd =0

15 Full equation: numerical results M. Pietroni (JCAP) initial conditions: P ab (k, 0) = P Lin (k,z in )u a u b B abc (k, q, q k, 0) = 0

16 Comparison with other methods Carlson, White, Padmanabhan, 09 TRG Fractional difference w.r.t. high resolution N-body below 2% in the BAO range down to z=0!

17 From DM fluid to DM haloes Cole & Kaiser 88 Mo & White 96 Porciani (PhD thesis) 99 Catelan et al, 98 the bias of DM haloes is stocastic, non-linear, and non-local

18 The dynamics of proto-haloes δ h τ + [(1 + δ h)v h ]=0, v h τ + Hv h +(v h )v h = φ 2 φ = 3 2 Ω M H 2 δ proto-haloes are conserved! gravitational potential determined by the DM fluid v h v 1/a 3-fluid system: δ, v, δ h alternative approach: follow the peak distribution (Desjacques et al 10)

19 Compact notation

20 Initial conditions DM haloes identified at z=0 and traced back to zin=50 Two possibilities: 1) assume/fit the functional relation: δ h (k,z in )=F[δ m ] 2) assume/fit all the n-point (cross)correlators: P m (k, z in ),P mh (k, z in ),P h (k, z in ) B mmm (k 1,k 2,k 3 ; z in ),B mmh (k 1,k 2,k 3 ; z in ), D 4 B mmm D 3 B h D 2 initial values for B h are irrelevant!

21 Initial conditions: lagrangian halo bias M> M DM haloes identified at z=0 and traced back to zin=50 initial (lagrangian) bias well reproduced by the non-local relation: P mh (k) =(b 1 + b 2 k 2 )P m (k)e k2 R 2 /2 (Matsubara 99, Desjacques 08)

22 Linear approximation: debiasing

23 Beyond linear order: Large-k resummation for the propagator... k k k k G ab (k; η) =(g ab (η)+δ a3 f b (2) g 31 (k; η)) exp[ k 2 σ 2 (e η 1) 2 2 ] 1-lo op σ d 3 q P 0 (q) q 2 The exact result of Crocce and Scoccimarro generalizes to the protohalo propagator

24 Cross-correlation at different times

25 Power spectrum Unlike the propagator, it cannot be resummed analytically. Use the TRG Independence from the initial mixed bispectra

26 Power spectrum Comparison with simulations (Pillepich et al 10)

27 Power spectrum Comparison with simulations

28 Conclusions Mildly non-linear scales are an unique opportunity to look for deviations from vanilla ΛCDM (w -1, massive ν s, NonGaussianity, DE-DM interactions, exotic DM,...) Semi-analytic methods are needed to go beyond linear PT in a more transparent, flexible, and fast (!) way than NBody s First step towards bias. Open issues: velocity bias, from protohaloes to real haloes

29 Time as the flow parameter η G(k; η, η )= Ω G(k; η, η )+ η η ds Σ(k; η,s) G(k; s, η ) exact evolution equation for the propagator η k k k k η = Ω + ds η η η η η s s η η η η ds large-momentum factorization k k η η s s η η η η large k ds s k η 1-lo op! k 2 σ 2 e 2η reproduce the Crocce-Scoccimarro resummation:g = e k2 σ 2 2 e 2η and allows to go beyond CS... (Anselmi, MP, in preparation)

30 More General Cosmologies δ τ v τ + [(1 + δ)v] = 0, deviation from geodesic (e.g. DM-scalar field interaction) + H(1 + A(x, τ))v +(v )v = φ, deviation from Poisson (e.g. scale-dep. growth factor) (δ ab η + Ω ab (η, k)) ϕ b (η, k) =e η γ abc (k, k 1, k 2 ) ϕ b (η, k 1 ) ϕ c (η, k 2 ) Ω ab = 2 φ =4πG (1 + B(x, τ)) ρ a 2 δ Ω M (1 + B(η, k)) 2 + H H + A(η, k) (η = log a) Ex: Scalar-Tensor: A = α dϕ/d log a B =2α 2 α 2 =1/(2ω + 3) see Saracco, MP, Tetradis, Pettorino, Robbers 09