Perturbative approaches to the LSS in CDM and beyond Massimo - PowerPoint PPT Presentation

Perturbative approaches to the LSS in Λ CDM and beyond Massimo Pietroni - INFN, Padova Firenze 6/4/2016

Outline ✤ IR effects on the nonlinear PS ✤ UV effects on the nonlinear PS ✤ Intermediate scales ✤ Putting all together: an improved TRG ✤ Scalar field (axion-like) DM

Linear and non-linear scales linear Power Spectrum @z=0, Λ CDM 100 ∆ ( k ) ≡ k 3 P ( k ) 10 2 π 2 1 0.1 increasing z 0.01 D ( z ) 2 ∆ ( k ) 0.001 0.0001 0.00001 1 x 10 -6 k ( h/ Mpc) 1 x 10 -7 0.0001 0.001 0.01 0.1 1 10 linear non-linear 3

The nonlinear PS a, · · · , d = 1 density a, · · · , d = 2 velocity div. ⌧ δϕ a ( k , z ) � 0 = h ϕ a ( k , z ) ϕ b ( � k , z in ) i 0 propagator G ab ( k ; z ) = + PNG δϕ b ( k , z in ) P lin ( k, z in ) P NL ab ( k, z ) = G ac ( k, z ) G bd ( k, z ) P lin cd ( k, z ) + P MC ( k, z ) ab P P ab ( k ; z ) IR physics intermediate and UV physics

Large scale flows and BAO’s 110 Mpc/h O(10 Mpc) reconstruction displacements Padmanabhan et al 1202.0090

Effect on the Correlation Function ξ lin ⁄ m n = 0.0 eV; z = 0 All the information on the BAO peak 15 is contained in the propagator part ξ P 10 R 2 x 5 The widening of the peak ξ MC can be reproduced by Zel’dovich 0 60 70 80 90 100 110 120 approximation (and improvements of it) R @ Mpc ê h D ξ lin ⁄ m n = 0.0 eV; z = 1 6 ξ P The widening of the peak 4 R 2 x contains physical information 2 (not a parameter to marginalize) ξ MC 0 60 70 80 90 100 110 120 Peloso, MP, Viel, Villaescusa-Navarro, 1505.07477 R @ Mpc ê h D

(simplified) Zel’dovich approximation G Zeld ( k, z ) = e − k 2 σ 2 v ( z ) 2 k 2 σ 2 v ( z ) P P P lin ( k ; z ) ) is the 1-dimensional velocity di 11 ( k, z ) = e − 2 d 3 q P lin ( q, z ) v ( z ) = 1 Z σ 2 . 3 (2 π ) 3 q 2 linear velocity dispersion: contains information on linear PS, growth factor,… Z ✓ sin( qR ) ◆ 1 Z v − 1 ξ 2 ( R ) dq q 2 δ P lin ( q ) e − q 2 σ 2 δξ ( R ) = = 2 π 2 qR 3 q 2 R 2 Peloso, MP, Viel, Villaescusa-Navarro, 1505.07477

How to include Bulk Motions v long ' D α ( τ ) h δ α ( k , τ ) δ α ( k 0 , τ 0 ) i = h ¯ δ α ( k , τ )¯ δ α ( k 0 , τ 0 ) ih e � i k · ( D α ( τ ) � D α ( τ 0 )) i � k 2 σ 2 v ( D ( τ ) � D ( τ 0 ))2 = h ¯ δ α ( k , τ )¯ δ α ( k 0 , τ 0 ) i e 2 Z Λ Z Λ d 3 q P 0 ( q ) 1 = 1 σ 2 d 3 q h v i long ( q ) v i long ( q ) i 0 v = � 3 H 2 f 2 3 q 2 Resummations (~Zel’dovich) take into account the large scale bulk motions

Redshift ratios 0.7 ⁄ m n = 0.0 eV 0.6 z = 0.5 ê z = 0 Ratio x H z Lê x H z = 0 L 0.5 0.4 z = 1 ê z = 0 0.3 0.2 z = 2 ê z = 0 0.1 0.0 60 70 80 90 100 110 120 R @ Mpc ê h D Peloso, MP, Viel, Villaescusa-Navarro, 1505.07477

Massive neutrinos Ratio x H⁄ m n = 0.15 L ê x H⁄ m n = 0 L 1.6 z = 0 1.4 ⁄ m n = 0.15 eV; z = 0 k 2 σ 2 v ( z ) 1.2 P P P lin ( k ; z ) 15 11 ( k, z ) = e − 2 1.0 0.8 10 R 2 z 0.6 0.4 5 0.2 60 80 100 120 140 increasing neutrino masses, R @ Mpc ê h D Ratio x H⁄ m n = 0.3 L ê x H⁄ m n = 0 L 1.6 z = 0 0 Plin decreases, but also 60 70 80 90 100 110 120 1.4 R @ Mpc ê h D 1.2 ⁄ m n = 0.3 eV; z = 0 damping decreases. 1.0 15 0.8 0.6 10 0.4 R 2 x 0.2 60 80 100 120 140 R @ Mpc ê h D 5 Ratio x H⁄ m n = 0.6 L ê x H⁄ m n = 0 L X ↓ 0.6% z = 0 m ν = 0 . 15 eV 0 1.5 60 70 80 90 100 110 120 X R @ Mpc ê h D 1.0 ↑ 1.2% m ν = 0 . 3 eV 0.5 60 80 100 120 140 R @ Mpc ê h D Peloso, MP, Viel, Villaescusa-Navarro, 1505.07477

Massive neutrinos ⁄ m n = 0.0 eV; z = 0 ⁄ m n = 0.0 eV; z = 0 H real space L 25 140 120 20 100 R 2 x s 15 R 2 x hh 80 60 10 40 5 20 0 0 60 70 80 90 100 110 120 60 70 80 90 100 110 120 R @ Mpc ê h D R @ Mpc ê h D ⁄ m n = 0.15 eV; z = 0 H real space L 140 ⁄ m n = 0.15 eV; z = 0 25 120 20 100 R 2 x hh 80 R 2 x s 15 60 10 40 5 20 0 0 60 70 80 90 100 110 120 60 70 80 90 100 110 120 R @ Mpc ê h D R @ Mpc ê h D 25 ⁄ m n = 0.3 eV; z = 0 H real space L ⁄ m n = 0.3 eV; z = 0 150 20 100 15 R 2 x hh R 2 x s 10 50 5 0 60 70 80 90 100 110 120 0 60 70 80 90 100 110 120 R @ Mpc ê h D R @ Mpc ê h D Halos Redshift space Peloso, MP, Viel, Villaescusa-Navarro, 1505.07477

Improving over Zel’dovich 1.0 PP ê Plin, z = 1, L = 0 0.8 PP1loop ê Plin 0.6 PP_CS ê Plin PP_TRG ê Plin 0.4 PP ê Plin H Nbody L 0.2 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ∂ η P P ab ( k ; η , η ) = � Ω ac P P cb ( k ; η , η ) � Ω bc P P ac ( k ; η , η ) Exact equation Z Σ ac ( k ; η , s ) P P cb ( k ; s, η ) + Σ bc ( k ; η , s ) P P ⇥ ⇤ + ac ( k ; η , s ) ds η in Σ ab ( k ; η , s ) → Σ 1 − loop ( k ; η , s ) for k → 0 ab v ( z ) e η + s g ab ( η ; s ) Σ ab ( k ; η , s ) → − k 2 σ 2 for k → ∞ Anselmi, Matarrese, MP, 1011.4477 Peloso, MP, Viel, Villaescusa-Navarro, in preparation

Mode coupling-Response functions The nonlinear PS is a functional of the initial one (in a given cosmology and assuming no PNG): SPT is an expansion around P 0 ( q ) = 0 ∞ δ n P ab [ P 0 ]( k ; η ) � 1 Z X P ab [ P 0 ]( k ; η ) = d 3 q 1 · · · d 3 q n P 0 ( q 1 ) · · · P 0 ( q n ) � � δ P 0 ( q 1 ) · · · δ P 0 ( q n ) n ! � P 0 =0 n =1 n=1 linear order (= “0-loop”) n=2 “1-loop” …

Mode coupling-Response functions Let’s instead expand around a reference PS: P 0 ( q ) = ¯ P 0 ( q ) P ab [ P 0 ]( k ; η ) = P ab [ ¯ P 0 ]( k ; η ) 1 δ n P ab [ P 0 ]( k ; η ) � 1 Z X d 3 q 1 · · · d 3 q n δ P 0 ( q 1 ) · · · δ P 0 ( q n ) , � + � δ P 0 ( q 1 ) · · · δ P 0 ( q n ) n ! � P 0 = ¯ P 0 n =1 Z dq = P ab [ ¯ P 0 ]( k ; η ) + q K ab ( k, q ; η ) δ P 0 ( q ) + · · · , h δ P 0 ( q ) ⌘ P 0 ( q ) � ¯ P 0 ( q ) t (4) δ P ab [ P 0 ]( k ; η ) � Z K ab ( k, q ; η ) ⌘ q 3 � Linear response function: d Ω q � δ P 0 ( q ) � P 0 = ¯ P 0 Non-perturbative (gets contributions from all SPT orders) Key object for more efficient interpolators ?

UV screening Sensitivity of the nonlinear PS at scale k k = 0 . 161 h Mpc − 1 on a change of the initial PS at scale q: K ( k, q ; z ) = q δ P nl ( k ; z ) δ P lin ( q ; z ) . IR: “Galilean invariance” K ( k, q ; z ) ∼ q 3 Peloso, MP 1302.0223 PT overpredicts the effect of UV scales on intermediate ones Nishimichi et al 1411.2970

UV screening The effect of virialized structures on larger scales is screened (Peebles ’80, Baumann et al 1004.2488, Blas et al 1408.2995). However, the departure from the PT predictions starts at small k’s: is it really a virialization effect? q 2 σ 2 q , η in − q , η in damped propagators! v e − 2 (compare SPT: g=O(1)) memory of initial substructures is largely lost k , η − k , η

UV lessons ✤ SPT fails when loop momenta become too high (q ≿ 0.4 h/Mpc) ✤ The real response to modifications in the UV regime is mild ✤ Most of the cosmology dependence is on intermediate scales

Effective approaches to the UV ✤ General idea: take the UV physics from N-body simulations and use (resummed) PT only for the large and intermediate scales

non-linear non-perfect fluid particles 1 2 π k L L UV “PT” ok coarse-grained sources Physics at k is independent on L, L_uv (“Wilsonian approach”) Expansion in sources: h δδ i J = h δδ i J =0 + h δ J δ i J =0 + 1 2 h δ JJ δ i J =0 + · · · measured from computed in PT simulations with cutoff at 1/L

Vlasov Equation Liouville theorem+ neglect non-gravitational interactions:  ∂ ∂τ + p i � d ∂ ∂ x i � am r i d τ f mic = x φ ( x , τ ) f mic ( x , p , τ ) = 0 am moments: Z d 3 pf mic ( x , p , τ ) n mic ( x , τ ) = density 1 Z d 3 p p velocity v mic ( x , τ ) = amf mic ( x , p , τ ) n mic ( x , τ ) d 3 p p i p j velocity 1 Z σ ij mic ( x , τ ) v j amf mic ( x , p , τ ) − v i mic ( x , τ ) = mic ( x , τ ) n mic ( x , τ ) dispersion am …

From particles to fluids Buchert, Dominguez, ’05, Pueblas Scoccimarro, ’09, Baumann et al. ’10 M.P., G. Mangano, N. Saviano, M. Viel, 1108.5203, Carrasco, Hertzberg, Senatore,1206.2976 .... X n mic ( x , τ ) = δ D ( x � x n ( τ )) , n v i n = ˙ x n ( τ ) , a i n = �r i x φ mic ( x , τ ) n, v i , φ , σ ij , . . . L UV f ( x, p, τ ) ≡ 1 Z X d 3 y W ( y/L UV ) f mic ( x + y, p, τ ) Satisfies the “Vlasov f mic ( x, p, τ ) = δ D ( x − x n ( τ )) δ D ( p − p n ( τ )) V eq.” n

Coarse-grained Vlasov equation large scales  ∂ ∂τ + p i � ∂ x φ ( x , τ ) ∂ ∂ x i � am r i f ( x , p , τ ) = ∂ p i am  � h ∂ ∂ p i f mic r i φ mic i L UV ( x , p , τ ) � ∂ ∂ p i f ( x , p , τ ) r i x φ ( x , τ ) am short scales 1 Z d 3 y W ( y/L UV ) g ( x + y ) h g i L UV ( x ) ⌘ V UV φ = h φ mic i L UV f = h f mic i L UV Vlasov equation in the L_uv ➞ 0 limit! Taking moments…