Advances in the perturbative description of LSS Zvonimir Vlah CERN - PowerPoint PPT Presentation

Advances in the perturbative description of LSS Zvonimir Vlah CERN with: Elisa Chisari, Fabian Schmidt & Patrick McDonald

Understanding galaxy overdensity and shape clustering 2 / LSS using PT Introduction 24

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] 3 / LSS using PT Galaxies and biasing of dark matter halos 24

Earlier approaches to halo biasing Local biasing model: halo field is a function of just DM density field δ 2 − + c δ 3 δ 3 + . . . [Fry & Gaztanaga, 1993] δ h = c δ δ + c δ 2 ( δ 2 ⟩) ⟨ Quasi-local (in space) relation of the halo density field to the dark matter [McDonald & Roy 2008, Assassi et al, 2014] δ h ( x ) = c δ δ ( x ) + c δ 2 δ 2 ( x ) + c δ 3 δ 3 ( x ) + 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 ϵ ϵ + . . . , 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) ϵ . More complex structure (more physical effects) : [Senatore 2014, Mirbabayi et al 2014, Angulo et al 2015, Desjacques et al 2016] 4 / LSS using PT Earlier modeling of halo bias 24

Effective field theory of biasing Non-local (time) and quasi-local (spece) relation of the halo density field to the dark matter ∫ t [Senatore 14, Angulo et al 15, Desjacques et al, 16] 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 ϵ ( t , t ′ ) ϵ ( x fl , t ′ ) + ¯ c ϵδ ( t , t ′ ) : ϵ ( 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 − Alternative - all effects chaptered in Lagrangian approach. Note: Assembly bias effects captured in the scheme. 5 / LSS using PT Effective field theory of biasing 24

tr tr tr tr tr tr tr Effective field theory of biasing Alternatively we can be similarly expand density of tracers as [Desjacques et al, 16] ∑ δ t ( x ) = c o O t ( x ) , O where we list operators O h : Π [1] ] [ ( 1 ) , Π [1] ]) 2 Π [1] ) 2 ] ( [( [ ( 2 ) , , Π [1] ]) 3 Π [1] ) 3 ] Π [1] ) 2 ] ( Π [1] ] Π [1] Π [2] ] [( [( [ [ [ ( 3 ) , tr , , . ij ( k ) = k i k j where Π [1] k 2 δ m ( k ) , with derivative operators R 2 ∗ ∇ 2 tr [ Π [1] ] , . . . . – series allows one to estimate the higher order (theory) errors – coefficients - physics from the R ∗ scale - degeneracies 6 / LSS using PT Effective field theory of biasing 24

TF TF TF tr TF Effective field theory of biasing Expansion of the field of galaxy shapes: ∑ g ij ( x ) = b o O ij ( x ) . O where the list of operators (up to higher derivatives and stochastic contributions) is Π [1] ] [ ( 1 ) ij , Π [1] ) 2 ] Π [2] ] Π [1] ] Π [1] ] [ [( [ [ ( 2 ) ij , TF ij , TF , ij tr Π [3] ] Π [1] Π [2] ] Π [2] ] Π [1] ] [ [ [ [ ( 3 ) ij , TF ij , TF , ij tr Π [1] ]) 2 Π [1] ) 3 ] Π [1] ) 2 ] ( Π [1] ] Π [1] ] [( [( [ [ [ ij , TF , TF . . . ij tr ij Derivative operators relevant for leading power spectrum corrections R 2 ∗ ∇ 2 TF Π [1] ] [ ij . 7 / LSS using PT Effective field theory of biasing 24

Projections onto the sky Master observable correlators P ab ijlm ( k ) = ⟨ Π t , a ij ( k )Π t , b lm ( k ′ ) ⟩ ′ , B abc ijlmrs ( k 1 , k 2 , k 3 ) = ⟨ Π t , a ij ( k 1 )Π t , b lm ( k 2 )Π t , c rs ( k 3 ) ⟩ ′ Isotropy and homogeneity makes the expansion in spherical tensors useful 2 Π ij ( k ) = 1 Π ( m ) 2 ( k ) Y ( m ) 3Π (0) 0 ( k ) δ K ∑ ij + ij m = − 2 different spectra are obtained by applying/subtracting the trace ⟨ δ t , a ( k ) δ t , b ( k ′ ) ⟩ ′ = P ab (0) ( k ) 00 ij ( k ′ ) ⟩ ′ = Y (0) ij P ab (0) ⟨ δ t , a ( k ) g b ( k ) 02 2 lm ( k ′ ) ⟩ ′ = Y (0) lm P ab (0) Y ( q ) ij Y ( − q ) P ab ( q ) ⟨ g a ij ( k ) g b ij Y (0) ∑ ( k ) + 2 ( k ) lm 22 22 q =1 bispectrum B abc , (0) ⟨ δ t , a ( k 1 ) g b ij ( k 2 ) g c lm ( k 3 ) ⟩ = Y (0) ( ˆ Y (0) ( ˆ k 2 ) k 3 ) ( k 1 , k 2 , k 3 ) + . . . , ij lm 022 8 / LSS using PT Effective field theory of biasing 24

Projections onto the sky 3D shape of galaxies get projected onto the onto the sky: ( n ) − 1 ) γ I , ij ( r , z ) = n ) P jl (ˆ n ) P kl (ˆ n ) g kl ( r , z ) , P ik (ˆ 2 P ij (ˆ Using the helicity basis (ˆ n , m + , m − ) intrinsic shape is: γ I , ij ( r , z ) = γ +2 ( r , z ) M (+2) + γ − 2 ( r , z ) M ( − 2) . ij ij { ξ, P } ab t , t = tt { ξ, P } ab , { ξ, P } ab t , ± 2 = M ( ± 2) n ) tg { ξ, P } ab n ) P ijkl (ˆ (ˆ ij , • ij ∆ r 12 r 1 { ξ, P } ab ± 2 , ± 2 = M ( ± 2) n ) M ( ± 2) κ κ 1 κ n ) κ κ κ 2 (ˆ (ˆ ij kl • n n ) gg { ξ, P } ab ˆ n ) P klrs (ˆ × P ijmn (ˆ mnrs , r 2 • z 1 z 2 2 (1 − µ 2 ) e ± 2 i ϕ ,  | q | = 0 , 1  n ) Y ( q )  i M ( ± 2) ( ˆ √ 1 − µ 2 e ± 2 i ϕ , n ) P ijkl (ˆ k | q | = 1 , 2 ( sgn ( q ) ∓ µ ) (ˆ ) = ± √ ij kl 2  1 2 ( sgn ( q ) ∓ µ 2 ) e ± 2 i ϕ | q | = 2 ,  9 / LSS using PT Effective field theory of biasing 24

Effective field theory of biasing Perturbative form of the shear tensor field is given in the form ∞ k − q 1 n K ( n ) ∑ (2 π ) 3 δ D ij ( k ) = ij , bias ( q 1 . . . , q n ) δ L ( q 1 ) . . . δ L ( q n ) , Π t n =1 where kernels K ( n ) ij , bias up to the third order are needed for one loop spectrum. We now apply the decomposition to the PT results up to one-loop power spectum ijlm ( k ) + P ab , ( 22 ) ( k ) + P ab , ( 13 ) ( k ) + P ab , ( 31 ) P ab , one − loop ( k ) = P ab , lin ( k ) , ijlm ijlm ijlm ijlm Linear, and loop (22), (13) contributions ijlm ( k ) = k i k j k l k m P ab , lin c ( a ) Π [1] c ( b ) Π [1] P lin ( k ) , k 4 P ab , ( 22 ) ( k ) = 2 K (2) ij , a ( q , k − q ) K (2) lm , b ( q , k − q ) P lin ( q ) P lin ( | k − q | ) , ijlm k i k j P ab , ( 13 ) ( k ) = 3 c ( a ) k 2 P lin ( k ) K (3) lm , b ( k , q , − q ) P lin ( q ) . ijlm Π [1] 10 / LSS using PT Effective field theory of biasing 24

Effective field theory of biasing ( 4 π ) 1/3 , which can be different then k NL . ρ 0 New physical scale k M ∼ 2 π M 3 Interesting case k NL ≫ k M ! We look at the correlations at k ≪ k M . Each order in perturbation theory we get new bias coefficients: ∫ ∫ [ ] [ ] D t δ (1) ( k ) + flow terms D 2 t δ (2) ( k ) + flow terms O t ( k , t ) = c δ, 1 c δ, 2 ˜ + ˜ + . . . t t [ ] [ ] = c δ, 1 δ (1) ( k ) + flow terms + c δ, 2 δ (2) ( k ) + flow terms + . . . Emergence of degeneracy: choice of most convenient basis Renormalization! (takes care of short distance effects at long distances) 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 7(12) finite bias parameters b i . Observables: P tg , P gg , B ttt , B ttg , B tgg , B ggg 11 / LSS using PT Effective field theory of biasing 24

Effective field theory of biasing ���� � ��� ���� ������ ���������� ��� � � δ ���� ���� � ���� �� Consistency with N-body simulations achieved up to the k < 0 . 3 Mpc / h for ���� ���� the Power Spectra, similar for the Bispectrum k < 0 . 15 Mpc / h [Angulo et al 2015] ���� �� � ��� ���� ���� ��� ��� ��� ��� ��� ��� ���� ���� - ���� ��������� ��� _ � ( � δ = ���� ) � = ��� ���� ( �� ) / � ���� ( �� ) ���� ���� ���� � ��� ���� ���� ��� ��� ��� ��� ��� ��� � [ � / ��� ] ��� � ��� / � ���� � = ���� ∢ �� = ���� π � � � = ��� � � � ��� = � ��� ��� ��� Most of the constraint comes ��� form the 3-pt function ��� ��� ��� ��� ��� ��� ��� [ � � � � � � � � ] If we had the simulations for the 4-pt function 2-pt function would be fully predicted. 12 / LSS using PT Effective field theory of biasing 24