Motivations Present and future probes of DE: BAO, Weak Lensing, Ly , - PowerPoint PPT Presentation

Dark Matter Clustering from the Renormalization Group and implications for cosmic acceleration Massimo Pietroni - Infn Padova (in collaboration with Sabino Matarrese) • Motivations: BAO and all that • Eulerian Perturbation Theory: Traditional and Compact forms. Results. • RG approach: formulation and preliminary results

Motivations Present and future probes of DE: BAO, Weak Lensing, Ly α , 21cm, ... they all require improved computational techniques Ex.: BAO from WFMOS (2M galaxies at 0.5<z<1.3) Goal: predict the LSS power spectrum to % accuracy.

Present Status: Pert. Theory 1-loop PT Jeong Komatsu, ‘06 z=1 z=0 Scoccimarro, ‘04 Non-linearities becomes more and more relevant in the DE-sensitive range 0<z<1

Present Status: N-body simulations+fitting functions Huff et al, ‘06 ~10% discrepancies between fitting functions and simulations redshift-space distorsions quite hard

Goals • Improve Pert. Theory towards lower z and higher k • Study the effect of non-linearities on BAO • Redshift-space distorsions

Dark Matter Hydrodynamics The DM particle distribution function, , obeys the Vlasov equation: f ( x , p , τ ) ∂ f ∂τ + p am · ∇ f − am ∇ φ · ∇ p f = 0 ∇ 2 φ = 3 2 Ω M H 2 δ p = amd x and where d τ � Taking momentum moments, i.e., d 3 p f ( x , p , τ ) ≡ ρ ( x , τ ) ≡ ρ ( τ )[1 + δ ( x , τ )] � d 3 p p i amf ( x , p , τ ) ≡ ρ ( x , τ ) v i ( x , τ ) d 3 p p i p j � a 2 m 2 f ( x , p , τ ) ≡ ρ ( x , τ ) v i ( x , τ ) v j ( x , τ ) + σ ij ( x , τ ) . . . and neglecting and higher moments (single stream approximation), one gets... σ ij

Equations of motion for single-stream cosmology ∂ δ ∂ v ∂ τ + ∇ · [(1 + δ ) v ] = 0 , ∂ τ + H v + ( v · ∇ ) v = −∇ φ In Fourier space, ( defining ), θ ( x , τ ) ≡ ∇ · v ( x , τ ) ∂ δ ( k , τ ) � 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 ∂ θ ( k , τ ) + H θ ( k , τ ) + 3 � 2 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 ∂ τ α ( k 1 , k 2 ) ≡ ( k 1 + k 2 ) · k 1 mode-mode coupling controlled by: k 2 1 β ( k 1 , k 2 ) ≡ | k 1 + k 2 | 2 ( k 1 · k 2 ) 2 k 2 1 k 2 2

Traditional Perturbation Theory fastest growing mode only ∞ � Assume EdS, , then solutions have the form δ ( k , τ ) = a n ( τ ) δ n ( k ) Ω M = 1 n =1 ∞ � θ ( k , τ ) = − H ( τ ) a n ( τ ) θ n ( k ) n =1 fastest growing mode only with � d 3 q 1 . . . d 3 q n δ D ( k − q 1 ... n ) F n ( q 1 , . . . , q n ) δ 0 ( q 1 ) . . . δ 0 ( q n ) δ n ( k ) = � d 3 q 1 . . . d 3 q n δ D ( k − q 1 ... n ) G n ( q 1 , . . . , q n ) δ 0 ( q 1 ) . . . δ 0 ( q n ) θ n ( k ) = The Kernels and satisfy recursion relations, with , and : F n G n δ 1 = θ 1 = δ 0 F 1 = G 1 = 1 n − 1 G m ( q 1 , . . . , q m ) � F n ( q 1 , . . . , q n ) = (2 n + 3)( n − 1) m =1 × [(2 n + 1) α ( k 1 , k 2 ) F n − m ( q m + 1 , . . . , q n ) + 2 β ( k 1 , k 2 ) G n − m ( q m + 1 , . . . , q n )] G n ( q 1 , . . . , q n ) = · · · where , k 1 = q 1 + . . . + q m k 2 = q m + 1 + . . . + q n

Traditional Diagrammar Fry, ‘84 Goroff et al, ‘86 Wise, ‘88 Scoccimarro, Frieman, ‘96 .... An infinite number of basic vertices! very redundant!! Example: 1-loop correction to the density power spectrum: a.k.a. “P13” a.k.a. “P22” bispectrum:

Compact Perturbation Theory Crocce, Scoccimarro ‘05 The hydrodynamical equations for density and velocity perturbations, ∂ v ∂ δ ∂ τ + ∇ · [(1 + δ ) v ] = 0 , ∂ τ + H v + ( v · ∇ ) v = −∇ φ , can be written in a compact form (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 ) (1) � ϕ 1 ( η , k ) � � � δ ( η , k ) � � η = log a 1 − 1 ≡ e − η Ω = where ϕ 2 ( η , k ) − θ ( η , k ) / H − 3 / 2 3 / 2 a in and the only non-zero components of the vertex 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 ) ( k 2 + k 3 ) · k 2 2 k 2 2 γ 222 ( k 1 , k 2 , k 3 ) = δ D ( k 1 + k 2 + k 3 ) | k 2 + k 3 | 2 k 2 · k 3 2 k 2 2 k 2 3

An action principle Matarrese, M.P ., ‘06 Eq. (1) can be derived by varying the action d η 1 d η 2 χ a g − 1 � � d η e η γ abc χ a ϕ b ϕ c S = ab ϕ b − where the auxiliary field has been introduced and is the retarded propagator: χ a ( η , k ) g ab ( η 1 , η 2 ) ( δ ab ∂ η + Ω ab ) g bc ( η , η ′ ) = δ ac δ D ( η − η ′ ) ϕ 0 a ( η , k ) = g ab ( η , η ′ ) ϕ 0 so that is the solution of the linear equation b ( η ′ , k ) � 3 � B = 1 2 B + A e − 5 / 2( η 1 − η 2 ) � η 1 > η 2 3 2 5 Explicitly, one finds: g ( η 1 , η 2 ) = 0 η 1 < η 2 � � A = 1 2 − 2 growing mode − 3 3 5 decaying mode � 1 � � � 1 ϕ 0 b ( η ′ , k ) ∝ u b = , Initial conditions: 1 − 3 / 2

A generating functional The probability of the configuration , given the initial condition , is ϕ a ( η i ) ϕ a ( η f ) P [ ϕ a ( η f ); ϕ a ( η i )] = δ [ ϕ a ( η f ) − ϕ a [ η f ; ϕ a ( η i )]] fixed extrema solution of the e.o.m. � η f � � � D ′′ ϕ a D χ b exp d η χ a [( δ ab ∂ η + Ω ab ) ϕ b − e η γ abc ϕ b ϕ c ] i ∼ η i only tree-level (saddle point) The generating functional at fixed initial conditions is � η f � � � Z [ J a , Λ b ; ϕ c ( η i )] = D ϕ a ( η f ) exp d η ( J a ϕ a + Λ b χ b ) P [ ϕ a ( η f ); ϕ a ( η i )] i η i

We are interested in statistical correlations, not in single solutions: � Z [ J a , Λ b ; K ′ s ] = D ϕ c ( η i ) W [ ϕ c ( η i ); K ′ s ] Z [ J a , Λ b ; ϕ c ( η i )] where all the initial correlations are contained in � � − ϕ a ( η i ; k ) K a ( k ) − 1 W [ ϕ c ( η i ); K ′ s ] = exp 2 ϕ a ( η i ; k a ) K ab ( k a , k b ) ϕ b ( η i ; k b ) + · · · ( K ( k )) − 1 ab = P 0 ab ( k ) ≡ u a u b P 0 ( k ) In the case of Gaussian initial conditions: Putting all together... �� � − 1 � � � � 2 χ g − 1 P L g T − 1 χ + i χ g − 1 ϕ Z [ J , Λ ] = D ϕ D χ exp d η [ e η γ χϕϕ − J ϕ − Λ χ ] d η 1 d η 2 − i g ( η ) P 0 ( k ) g T ( η ′ ) where the initial conditions are encoded in the linear power spectrum: P L � � ab ( η , η ′ ; k ) ≡ ab Derivatives of Z w.r.t. the sources J and Λ give all the N-point correlation functions (power spectrum, bispectrum, ...) and the full propagator (k-dependent growth factor)

Compact Diagrammar propagator (linear growth factor): a b − i g ab ( η a , η b ) P L ab ( η a , η b ; k ) power spectrum: b a a − i e η γ abc ( k a , k b , k c ) interaction vertex: b c Example: 1-loop correction to the density power spectrum: 1 1 1 1 1 1 + + 2 a.k.a. “P22” a.k.a. “P13” All known results in cosmological perturbation theory are expressible in terms of diagrams in which only a trilinear fundamental interaction appears

1-loop PT: how good is it? Makino et al.,’92 P ( k, τ ) = D 2 ( τ ) P 11 ( k ) + D 4 ( τ ) [ P 13 ( k ) + P 22 ( k )] + ... , � ∞ P 13 ( k ) = k 3 P 11 ( k ) � � � 12 r 2 − 158 + 100 r 2 − 42 r 4 + 3 1 + r � � 3 (7 r 2 + 2) ln r 2 − 1 � � � drP 11 ( kr ) � � 252 (2 π ) 2 r 3 1 − r � � 0 (9 d � 1 3 r + 7 x − 10 rx 2 � 2 � ∞ � 1 / 2 � � k 3 � 1 + r 2 − 2 rx � P 22 ( k ) = drP 11 ( kr ) dxP 11 k (1 + r 2 − 2 rx ) 2 98 (2 π ) 0 − 1 Linear growth factor : encodes different cosmologies at best than % level e D ( τ ) = δ 1 ( τ ) / δ initial Ex: (Jeong Komatsu, ‘06) P 22 ( Λ CDM) /P 22 (EdS) ∼ 1 . 006 ( z = 0) Notice: the 1-loop corrections at any time depend on the initial power spectrum ( )! P 11 ( k ) = P 0 ( k ) This will change in the RG...

1-loop PT performs quite well for z >1 (better than halo approach) Baryonic peaks modeled at few % Things get much worse at z<1... Jeong Komatsu, ‘06