Modeling the clustering of dark-matter haloes in resummed - PowerPoint PPT Presentation

Modeling the clustering of dark-matter haloes in resummed perturbation theories arxiv:astro-ph/1012.4833 Suchita Kulkarni kulkarni@th.physik.uni-bonn.de Bethe Center for Theoretical Physics Universit¨ at Bonn Bonn, Germany Suchita Kulkarni, BCTP – p. 1

Outline Introduction: - Linearized theory of large scale structure formation - Standard treatment - Validity Need for going to non-linearities: current approaches Renormalized Perturbation theory Going beyond matter power spectrum - issue of bias Results for bias predictions Summary Suchita Kulkarni, BCTP – p. 2

Introduction - from small to large scale einen kurzen Kino Zeit Suchita Kulkarni, BCTP – p. 3

Large Scale Structure (LSS) Inhomogeneous Universe on small scales - Galaxy clusters and groups - Positions are correlated Great wall - scale of 100 h − 1 Mpc - Universe on length scales > 200 h − 1 Mpc smooth Qn: Can we predict/ understand the structures and their evolution? [SDSS + CfA1] Suchita Kulkarni, BCTP – p. 4

LSS - understanding the formation ∂ρ ∂t + ∇ · ( ρ v ) = 0 Continuity equation ∂ v ∂t + ( v · ∇ ) v + ∇ P = −∇ Φ Euler equation ρ ∇ 2 Φ = 4 πGρ Poisson equation System of non-linear coupled differential equations, no analytical solution in general Continuity equation :- matter conservation Euler equation :- momentum conservation Poisson equation:- gravitational potential Our interest: chasing the evolution of inhomogeneities Suchita Kulkarni, BCTP – p. 5

Linear Theory of LSS Velocity field = homogeneous expansion + peculiar velocity v Density field = Average density field + density contrast δ ( x , t ) ∂ v ∂t + ˙ a a v + 1 − 1 a ( v · ∇ ) v = a ∇ Φ ∂δ ∂t + 1 a ∇ · [(1 + δ ) v ] = 0 3 H 2 Ω m ∇ 2 Φ = δ 2 a Linearize and combine: ∂t − 3 H 2 ∂ 2 δ 0 Ω m ∂t 2 + 2˙ a ∂δ δ = 0 2 a 3 a Has two solutions: δ ∝ a ( t ) Growing mode δ ∝ a − 3 / 2 ( t ) Decaying mode Suchita Kulkarni, BCTP – p. 6

LSS - power spectrum Impossible to exactly simulate our universe - exact initial conditions unknown Statistically predict the properties Correlation function - measure of the statistical properties Two point correlation function - the matter power spectrum Three point correlation function - bi- spectrum . . . P ( k ) δ D ( k + k ′ ) ≡ � δ ( k ) δ ( k ′ ) � Suchita Kulkarni, BCTP – p. 7

Non-linearities - where and why? Suchita Kulkarni, BCTP – p. 8

Non-linearities - where and why? BAO at k ∼ 0 . 1 h/Mpc Potential to constrain expansion history Can differentiate between differ- ent DE models Suchita Kulkarni, BCTP – p. 9

Non-linearities - where and why? BAO at k ∼ 0 . 1 h/Mpc Potential to constrain expansion history Can differentiate between different DE models Model w 0 w m a m ∆ m INV1 -0.4 -0.27 0.18 0.5 INV2 -0.79 -0.67 0.29 0.4 SUGRA -0.82 -0.18 0.1 0.7 2EXP -1.0 0.01 0.19 0.043 AS -0.96 -0.01 0.53 0.13 CNR -1.0 0.1 0.15 0.016 Jennings et.al. 1998 arXiv:0908.1394 Suchita Kulkarni, BCTP – p. 10

Non-linearities - where and why? (Shamelessly) stolen from Y. Wong’s talk given at neutrino 2010 Suchita Kulkarni, BCTP – p. 11

Non-linearities - current approaches Zel’dovich approximation :- d 3 r � (2 π ) 3 e ι k · r � e − [ k 2 σ 2 v − I ( k , r )] − 1 � P ( k ) = PS in Zel’dovich approximation � d 3 q ( k · q ) 2 cos( k · q ) P L ( q ) /q 4 I ( k , r ) ≡ I ( k, 0) /k 2 σ v = Effective expansion in the amplitude of PS Delicate cancellations between different orders Improvement over linear theory Suchita Kulkarni, BCTP – p. 12

Non-linearities - current approaches Numerical simulations Power spectrum (normalized to smooth) 0.01 k [ h / Mpc ] 0.10 1.00 0.01 k [ h / Mpc ] 0.10 1.00 0.10 0.10 0.10 0.10 lin ) log ( ∆ 2 ( k ) / ∆ 2 0.05 0.05 0.05 0.05 0.00 lin ) 0.00 2 log ( ∆ 2 ( k ) / ∆ 2 2 lin ) 0.00 log ( ∆ ( k ) / ∆ ) ( k ) / ∆ 2 log ( ∆ 2 0.00 lin -0.05 -0.05 -0.05 -0.05 -0.10 z = 127.00 -0.10 -0.10 z = 14.87 -0.10 0.01 0.01 k [ h / Mpc ] 0.10 k [ h / Mpc ] 0.10 0.01 1.00 1.00 0.01 0.10 0.10 k [ h / Mpc ] k [ h / Mpc ] 1.00 1.00 2.7 2 0.10 0.10 0.10 0.10 lin ) log ( ∆ 2 ( k ) / ∆ 2 0.05 0.05 0.05 0.05 0.00 2 lin ) log ( ∆ 2 ( k ) / ∆ 2 0.00 0.00 log ( ∆ ) ( k ) / ∆ 2 lin ) 0.00 ( k ) / ∆ 2 lin log ( ∆ 2 -0.05 -0.05 -0.05 -0.05 -0.10 z = 7.02 -0.10 -0.10 z = 3.06 -0.10 0.01 0.01 0.10 k [ h / Mpc ] k [ h / Mpc ] 0.10 0.01 1.00 1.00 0.10 0.1 k [ h / Mpc ] k [ h / Mpc ] 1.0 1.00 1.15 2 0.10 0.10 0.10 0.10 lin ) log ( ∆ 2 ( k ) / ∆ 2 0.05 0.05 0.05 0.05 0.92 2 0.00 log ( ∆ 2 ( k ) / ∆ 2 0.00 lin ) lin ) 2 ( k ) / ∆ 2 0.00 log ( ∆ 0.00 ) lin ( k ) / ∆ 2 log ( ∆ 2 -0.05 -0.05 -0.05 -0.05 -0.10 z = 0.98 -0.10 -0.10 z = 0.00 -0.10 0.01 0.10 1.00 0.10 1.00 k [ h / Mpc ] k [ h / Mpc ] Actual realization of initial fluctuations Millennium simulation dark matter, galaxies Scatter in initial realization due to finite number of modes Springel et. al. 2005 arxiv:astro-ph/0504097 Suchita Kulkarni, BCTP – p. 13

Perturbation Theory for cosmology ∂δ ∂ v ∇ 2 φ = 3 ∂τ + ∇ · [(1 + δ ) v ] = 0; ∂τ + H v + ( v · ∇ ) v = −∇ φ ; 2 Ω m H δ In Fourier space with θ ( x, τ ) ≡ ∇ · v ( x , τ ) ∂ δ ( k , τ ) � d 3 q d 3 p δ D ( k − q − p ) α ( q , p ) θ ( q , τ ) δ ( p , τ ) = 0 + θ ( k , τ ) + ∂ τ ∂ θ ( k , τ ) + H θ ( k , τ ) + 3 � 2 H 2 δ ( k , τ ) + d 3 q d 3 p δ D ( k − q − p ) β ( q , p ) θ ( q , τ ) θ ( p , τ ) = 0 ∂ τ Mode - mode coupling controlled by:- β ( p , q ) = ( p + q ) 2 p · q α ( p , q ) = ( p + q ) · p , p 2 2 p 2 q 2 Suchita Kulkarni, BCTP – p. 14

Linear approximation α ( p , q ) = β ( p , q ) = 0 No mode-mode coupling ∂ δ ( k , τ ) + θ ( k , τ ) = 0 ∂ τ ∂ θ ( k , τ ) + H θ ( k , τ ) + 3 2 H 2 δ ( k , τ ) = 0 ∂ τ Ω m = 1 → H ∼ a 1 / 2 ↓ � a ( τ )  � m 1 Growing mode  δ ( k , τ ) = δ ( k , τ i )   a ( τ i ) m =  − θ ( k , τ ) − 3  Decaying mode  = mδ ( k , τ ) 2 H Suchita Kulkarni, BCTP – p. 15

Compactifying PT for cosmology ∂δ ∂ v ∇ 2 φ = 3 ∂τ + ∇ · [(1 + δ ) v ] = 0; ∂τ + H v + ( v · ∇ ) v = −∇ φ ; 2 Ω m H δ Define        ϕ 1 ( k , η ) δ ( k , η ) 1 − 1 a  ≡ e − η η = log Ω =     a in ϕ 2 ( k , η ) − θ ( k , η ) / H − 3 / 2 3 / 2 Then (assuming EdS cosmology) we can write:- ( δ ab ∂ η + Ω ab ) ϕ b ( k , η ) = e η γ abc ( k , − p , − q ) ϕ b ( p , η ) ϕ c ( q , η ) , With mode-mode coupling γ abc ( k , p , q ) ( a, b, c, = 1 , 2 ) γ 121 ( k , p , q ) = γ 112 ( k , q , p ) = 1 2 δ D ( k + p + q ) α ( p , q ) , γ 222 ( k , p , q ) = δ D ( k + p + q ) β ( p , q ) , Suchita Kulkarni, BCTP – p. 16

Perturbation Theory for cosmology The action is given by � S = dη [ χ a ( − k , η ) ( δ ab ∂ η + Ω ab ) ϕ b ( k , η ) − e η γ abc ( − k , − p , − q ) χ a ( k , η ) ϕ b ( p , η ) ϕ c ( q , η )] � � D ′′ ϕ a D χ b × Z [ J a , K b ; ϕ a (0)] ≡ D ϕ a ( η f ) � η f � � dη χ a ( δ ab ∂ η + Ω ab ) ϕ b − e η γ abc χ a ϕ b ϕ c + J a ϕ a + K a χ a exp i 0 Averaging the probabilities over the initial conditions with a statistical weight function for the physical fields ϕ a (0) , � Z [ J a , K b ; C ′ s ] = D ϕ a (0) W [ ϕ a (0) , C ′ s ] Z [ J a , K b ; ϕ a (0)] . Gaussian initial conditions, the weight function reduces to the form − 1 � � W [ ϕ a (0) , C ab ] = exp 2 ϕ a ( k , 0) C ab ( k ) ϕ b ( − k , 0) , Suchita Kulkarni, BCTP – p. 17 where

Feynman diagrams and all that . . . Tree level diagrams Propagator - linear evolution Power spectrum - initial conditions Vertex - nonlinearities Loop corrections to power spec- trum P ab 2 Suchita Kulkarni, BCTP – p. 18

Does it work? Propagator in large-k limit k k k k g ab 1 − k 2 σ 2 ( e ηa − e ηb ) 2 � � + O ( k 4 σ 4 ) G ab ( k ; η a , η b ) = g ab ( η a , η b ) 2 d 3 q P 0 ( q ) � σ ≡ 1 � � q 2 3 ( σe η a ) − 1 ≃ 0 . 15 hMpc − 1 , in the BAO range PT blows up in the BAO range Suchita Kulkarni, BCTP – p. 19

Renormalized perturbation theory Different contributions can be resumed − k 2 σ 2 v (exp( η ) − 1) 2 / 2 � � G ab ( k, η ) = g ab ( η ) exp Exponential damping in the BAO range Represents the effect of multiple interactions Memory loss Suchita Kulkarni, BCTP – p. 20

Results Results from PT using Zel’dovich Results from PT using field approximation theory methods Fine cancellation between different loop Clearly improved results orders Suchita Kulkarni, BCTP – p. 21

Matter power spectra RG equations for propagator and PS can be written down Result of using the RG for propagator and PS RG for propagator ∂ 2 W λ ∂J a ( k ,η a ) ∂K b ( k ,η a ) = − δ ( k + k ′ ) ∂ λ G ab,λ ( k, η a , η b ) ∂ λ Suppression of PS from RG at k ∼ 0 . 25 hMpc − 1 due to failure of analytical approx- imations RPT, one loop, linear, simulations, halo approach Suchita Kulkarni, BCTP – p. 22