Halo Power Spectrum and Bispectrum in the Effective Field Theory of - PowerPoint PPT Presentation

Halo Power Spectrum and Bispectrum in the Effective Field Theory of Large Scale Structures Zvonimir Vlah Stanford University & SLAC with: Raul Angulo (CEFCA), Matteo Fasiello (Stanford), Leonardo Senatore (Stanford)

Contents ◮ Clustering of Dark Matter in EFT ◮ Clustering of DM Halos ◮ Earlier approaches ◮ EFT approach ◮ Halo Power Spectrum and Bispectrum Results ◮ Adding baryonic effects and non-Gaussianities ◮ Summary Biased Tracers in the EFT of LSS Contents 2 / 18

& Gravitational clustering of dark matter Evolution of collisionless particles - Vlasov equation: d τ = ∂ f df ∂τ + 1 m p · ∇ f − am ∇ φ · ∇ p f = 0 , and ∇ 2 φ = 3/2 H Ω m δ. Integral moments of the distribution function: mass density field mean streaming velocity field � � d 3 p p i am f ( x , p ) ρ ( x ) = ma − 3 d 3 p f ( x , p ) , v i ( x ) = � d 3 p f ( x , p ) , Biased Tracers in the EFT of LSS Gravitational clustering of dark matter 3 / 18

Gravitational clustering of dark matter Evolution of collisionless particles - Vlasov equation: d τ = ∂ f df ∂τ + 1 m p · ∇ f − am ∇ φ · ∇ p f = 0 , and ∇ 2 φ = 3/2 H Ω m δ. Eulerian framework - fluid approximation: ∂δ ∂τ + ∇ · [(1 + δ ) v ] = 0 ∂ v i ∂τ + H v i + v · ∇ v i = −∇ i φ − 1 ρ ∇ i ( ρσ ij ) , where σ ij is the velocity dispersion. Biased Tracers in the EFT of LSS Gravitational clustering of dark matter 3 / 18

Gravitational clustering of dark matter Evolution of collisionless particles - Vlasov equation: d τ = ∂ f df ∂τ + 1 m p · ∇ f − am ∇ φ · ∇ p f = 0 , and ∇ 2 φ = 3/2 H Ω m δ. Eulerian framework - pressureless perfect fluid approximation: ∂δ ∂τ + ∇ · [(1 + δ ) v ] = 0 ∂ v i ∂τ + H v i + v · ∇ v i = −∇ i φ. Irrotational fluid: θ = ∇ · v . Biased Tracers in the EFT of LSS Gravitational clustering of dark matter 3 / 18

Gravitational clustering of dark matter Evolution of collisionless particles - Vlasov equation: d τ = ∂ f df ∂τ + 1 m p · ∇ f − am ∇ φ · ∇ p f = 0 , and ∇ 2 φ = 3/2 H Ω m δ. EFT approach introduces a tress tensor for the long-distance fluid: ∂δ ∂τ + ∇ · [(1 + δ ) v ] = 0 ∂ v i ∂τ + H v i + v · ∇ v i = −∇ i φ − 1 ρ ∇ j ( τ ij ) , with given as τ ij = p 0 δ ij + c 2 s δρδ ij + O ( ∂ 2 δ, . . . ) [Carrasco et al. 2012] -derived by smoothing the short scales in the fluid with the smoothing filter W (Λ) . Biased Tracers in the EFT of LSS Gravitational clustering of dark matter 3 / 18

EFTofLSS one-loop results for DM k 2 P EFT-1-loop = P 11 + P 1-loop − 2(2 π ) c 2 P 11 s (1) k 2 NL ���� � = ��� ���� � ������ / � ���� ���� ���� ���� � - ���� ��� ���� � - ���� ���� ��� �� - �������� � - ���� ���� �� - �������� ���� ��� ��� ��� ��� ��� ��� � [ � / ��� ] [first by Carrasco et al, 2012] ◮ Well defined and convergent expansion in k / k NL (one parameter). ◮ IR resummation (Lagrangian approach) - BAO peak! [Senatore et al, 2014] Biased Tracers in the EFT of LSS Gravitational clustering of dark matter 4 / 18

Galaxies and biasing of dark matter halos Galaxies form at high density peaks of 3 initial matter density: 2 rare peaks exhibit higher clustering! overdensity 1 0 � 1 � 2 � 3 0 2 4 6 8 10 Λ� 1 � k ◮ Tracer detriments the amplitude: P g ( k ) = b 2 P m ( k ) + . . . ◮ Understanding bias is crucial for understanding the galaxy clustering [Tegmark et al, 2006] Biased Tracers in the EFT of LSS Galaxies and biasing of dark matter halos 5 / 18

Earlier approaches to halo biasing Local biasing model: halo field is a function of just DM density field � � δ 2 �� δ 2 − + c δ 3 δ 3 + . . . δ h = c δ δ + c δ 2 [Fry & Gaztanaga, 1993] Non-local (in space) relation of the halo density field to the dark matter δ h ( x ) = c δ δ ( x ) + c δ 2 δ 2 ( x ) + c δ 3 δ 3 ( x ) [McDonald & Roy, 2008] + c s 2 s 2 ( x ) + c δ s 2 δ ( x ) s 2 ( x ) + c ψ ψ ( x ) + c st s ( x ) t ( x ) + c s 3 s 3 ( x ) + c ǫ ǫ + . . . , with effective ('Wilson') coefficients c l and variables: s ij ( x ) = ∂ i ∂ j φ ( x ) − 1 t ij ( x ) = ∂ i v j − 1 ij δ ( x ) , ij θ ( x ) − s ij ( x ) , 3 δ K 3 δ K ψ ( x ) = [ θ ( x ) − δ ( x )] − 2 7 s ( x ) 2 + 4 21 δ ( x ) 2 , where φ is the gravitational potential, and white noise (stochasticity) ǫ . Biased Tracers in the EFT of LSS Earlier modelling of halo bias 6 / 18

Effective field theory of biasing Non-local (space and time) relation of the halo density field to the dark matter � t [Senatore 2014] dt ′ H ( t ′ ) [¯ δ h ( x , t ) ≃ c δ ( t , t ′ ) : δ ( x fl , t ′ ) : c δ 2 ( t , t ′ ) : δ ( x fl , t ′ ) 2 : +¯ c s 2 ( t , t ′ ) : s 2 ( x fl , t ′ ) : + ¯ c δ 3 ( t , t ′ ) : δ ( x fl , t ′ ) 3 : +¯ c δ s 2 ( t , t ′ ) : δ ( x fl , t ′ ) s 2 ( x fl , t ′ ) : + . . . + ¯ c ǫ ( t , t ′ ) ǫ ( x fl , t ′ ) + ¯ c ǫδ ( t , t ′ ) : ǫ ( x fl , t ′ ) δ ( x fl , t ′ ) : + . . . + ¯ � c ∂ 2 δ ( t , t ′ ) ∂ 2 x fl δ ( x fl , t ′ ) + . . . +¯ k 2 M Novice consideration of non-local in time formation, which depends on fields evaluated on past history on past path: � τ τ ′ d τ ′′ v ( τ ′′ , x fl ( x , τ, τ ′′ )) x fl ( x , τ, τ ′ ) = x − Biased Tracers in the EFT of LSS Effective field theory of biasing 7 / 18

Effective field theory of biasing � 4 Pi � 1/3 , which can be different then k NL . ρ 0 New physical scale k M ∼ 2 π M 3 We look at the correlations at k ≪ k M . Each order in perturbation theory we get new bias coefficients: � � δ h ( k , t ) = c δ, 1 δ (1) ( k , t ) + flow terms � � δ (2) ( k , t ) + flow terms + c δ, 2 + . . . Emergence of degeneracy: choice of most convenient basis Turns out that at one loop 2-pt and tree level 3-pt function LIT and non-LIT are degenerate- this is no longer the case at higher loops or when 4-pt function is considered. Biased Tracers in the EFT of LSS Effective field theory of biasing 8 / 18

Effective field theory of biasing Independent operators in the`Basis of Descendants': � � C (1) (1)st order: δ, 1 � � C (2) δ, 1 , C (2) δ, 2 , C (2) (2)nd order: δ 2 , 1 � � C (3) δ, 1 , C (3) δ, 2 , C (3) δ, 3 , C (3) δ 2 , 1 , C (3) δ 2 , 2 , C (3) δ 3 , 1 , C (3) δ, 3 cs C (3) (3)rd order: s 2 , 2 � � C ǫ , C (1) Stochastic: δǫ, 1 We compare P 1 − loop , P 1 − loop , B tree hhh , B tree hhm , B tree hmm statistics hh hm Renormalization! (takes care of short distance physics has at long distances of interest) In practice, ˜ c δ, 1 is a bare parameter, the sum of a finite part and a counterterm: c δ, 1 = ˜ c δ, 1 , finite + ˜ c δ, 1 , counter , ˜ After renormalization we end up with using 7 finite bias parameters b i (coefficients in EFT). Biased Tracers in the EFT of LSS Effective field theory of biasing 9 / 18

Observables: P hm , P hh , B hmm , B hhm , B hhh Example: Halo-Matter Power Spectrum (one loop) � � d 3 q c (2) (2 π ) 3 F (2) P hm ( k ) = b δ, 1 ( t ) P 11 ( k ) + 2 ( k − q , q ) � δ, 1 , s ( k − q , q ) P 11 ( q ) P 11 ( | k − q | ) s � � � � d 3 q c (3) F (3) +3 P 11 ( k ) ( k , − q , q ) + � δ, 1 , s ( k , − q , q ) P 11 ( q ) s (2 π ) 3 � � � d 3 q c (2) + b δ, 2 ( t ) 2 (2 π ) 3 F (2) ( k − q , q ) F (2) ( k − q , q ) − � δ, 1 , s ( k − q , q ) s s × P 11 ( q ) P 11 ( | k − q | ) � � � d 3 q c (3) + b δ, 3 ( t )3 P 11 ( k ) δ, 3 , s ( k , − q , q ) P 11 ( q ) � (2 π ) 3 � d 3 q + b δ 2 ( t )2 (2 π ) 3 F (2) ( k − q , q ) P 11 ( q ) P 11 ( | k − q | ) s � k 2 � b c s ( t ) − 2(2 π ) c 2 s (1) ( t ) b δ, 1 ( t ) P 11 ( k ) + k 2 NL Biased Tracers in the EFT of LSS Effective field theory of biasing 10 / 18

Error estimates and bias fits Error bars of the theory are given by the higher loop estimates: � � 3 k e.g. ∆ P hm ∼ (2 π ) b 1 P 11 ( k ) . k NL This determines the theory reach k max . k max [ h / Mpc ] bin0 bin1 Fits to N-body simulations: ��� _ �� � ��� = ���� � / ���� � ��� = ���� � / ��� mm 0 . 22 − 0 . 31 0 . 22 − 0 . 31 �� �� ��� ��� ��� χ � � hm + + - - - 0 . 24 − 0 . 35 0 . 22 − 0 . 35 ������ ����� + + + - - ����� ������ hh 0 . 19 − 0 . 32 0 . 17 − 0 . 30 + + - + - ����� ������ mmm 0 . 14 − 0 . 22 0 . 14 − 0 . 22 + + - - + ����� ������ + + + + - hmm 0 . 13 − 0 . 22 0 . 13 − 0 . 22 ����� ������ + + + - + ����� ������ hhm 0 . 13 − 0 . 22 0 . 13 − 0 . 22 + + - + + ���� ������� hhh + + + + + 0 . 13 − 0 . 21 0 . 13 − 0 . 21 ���� ������ Most of the constraint comes form the 3-pt function. Fits to 3-pt and 4-pt function would enable full predictivity for 2-pt function. Biased Tracers in the EFT of LSS Effective field theory of biasing 11 / 18