Nonlinear dark energy clustering Guillermo Ballesteros Padua University & INFN and Centro Enrico Fermi 22/9/2011 PTChat workshop, IPhT Saclay
Overview 1. Pressure from dark energy 2. Non linearities 3. Conclusions Setup: Mixture of matter and dark energy (DE) Aim: Compute non linearities on the power spectra Motivation: Future high precision cosmological data sets
Pressure perturbations ( ) Energy density: ρ(τ, x i ) = ρ(τ)+δρ(τ, x i ) = ρ(τ) 1+δ(τ, x i ) Pressure: P(τ, x i ) = P(τ)+δP(τ, x i ) = P(τ) ( ) 1+Π(τ, x i ) Equation of state: P(τ) w(τ) ρ(τ), w < 1/3 for DE Sound speed: δp(τ, x i ) c s 2 (τ, x i )δρ(τ, x i ) c s 2 δ = wπ ( ) Rest frame of the fluid: c 2 s δ = ĉ 2 s δ + 3H(1+w) ĉ 2 2 s c a θ/k 2 0 ĉ s 2 1 for DE Adiabatic sound speed: c a 2 = P/ ρ, ẇ = 3(1+w) ( w c a 2 ) H
The smaller is ĉs 2 the largest is the relevance of dark energy perturbations. Matter (dust): ĉ 2 s = 0 Examples Radiation (photons, masless neutrinos): ĉ 2 s = 1/3 Quintessence (with a canonical kinetic term): ĉ 2 s = 1 K-essence (non-canonical kinetic term): ĉs 2 1 (even ) Two scalar fields (interacting or not): ĉs 2 1 String gas: 0 < ĉ 2 s << 1 Chaplygin gas (P 1/ρ): ĉ 2 s 0... Current status - ĉ s 2 cannot be detected with presently available data - Planck + LSST will be able to set a lower bound
Non linear equations Energy momentum tensor: T µν = (ρ+p)v µ v ν + Pg µν v µ =γ(1, u i ), γ = (1 u 2 ) 1/2 1 a u i = dx i /dτ, ah = da/dτ Scalar perturbations: } ds 2 = a 2 (τ) { (1+2φ)dτ 2 +(1 2φ)dx i dx i comoving conformal coordinates
Non linear equations in real space Continuity and Euler equations from T µν ;µ = 0 ( ρ+u 2 P ) + 3H(ρ+P)+ [(ρ+p) u] = 0 u+hu+(u ) u+ P + uṗ ρ+p + φ N = 0 Poisson equation from G µν = 8πG T µν 2 φ N = 3 2 H2 Ω α (δ α 3(1+w α )Hχ α ) α=m, x ( ) ρ(τ, x i ) = ρ(τ) 1+δ(τ, x i ) P(τ, x i ) = P(τ) ( ) 1+Π(τ, x i ) c s 2 δ wπ u = χ
Non linear perturbation equations in Fourier space Continuity ( ) ( ( ) ) δ(k)+3 ĉ 2 s w Hδ(k)+(1+w) 1+9 ĉ 2 H 2 s w θ(k) ( ) 2 + 1+ĉ s d 3 p d 3 qδ D (k p q)α(q, p)θ(q)δ(p) +O(2) = 0 Euler θ(k) + (1 3ĉ ) 2 s Hθ(k) ĉs 2 k 2 (1+w) δ(k) k2 φ N +(1 ĉ 2 s ) d 3 p d 3 qδ D (k p q)β(q, p)θ(q)θ(p)+o(2) = 0 k 2 Poisson k 2 φ N = 3 2 H2 α ( ) Ω α δ α (k)+3(1+w α ) H2 θ α (k) k 2 H
Correlation functions of matter and dark energy δ m ϕ = θ m /H δ x θ x /H e η, η = log a a in η ϕ a (k,η) = Ω ab (k, η)ϕ b (k,η) +e η d 3 p d 3 qδ D (k p q)γ abc (k, p, q)ϕ b (p,η)ϕ c (q,η) ϕ a (k)ϕ b (q) δ D (k+q)p ab (k) ϕ a (k)ϕ b (q)ϕ c (p) δ D (k+q+p)b abc (k, q, p) ϕ a (k)ϕ b (q)ϕ c (p)ϕ d (r) δ D (k+p+q+r) Q abcd (k,q,p,r) +δ D (k+q)δ D (p+r)p ab (k)p cd (p) +δ D (k+p)δ D (q+r)p ac (k)p bd (q) +δ D (k+r)δ D (q+p)p ad (k)p bc (q).
Time renormalization group (TRG) η ϕ a ϕ b = Ω ac ϕ c ϕ b Ω bc ϕ a ϕ c + e η γ acd ϕ c ϕ d ϕ b + e η γ bcd ϕ a ϕ c ϕ d η ϕ a ϕ b ϕ c = Ω ad ϕ d ϕ b ϕ c Ω bd ϕ a ϕ d ϕ c Ω cd ϕ a ϕ b ϕ d + e η γ ade ϕ d ϕ e ϕ b ϕ c + e η γ bde ϕ a ϕ d ϕ e ϕ c + e η γ cde ϕ a ϕ b ϕ d ϕ e η ϕ a ϕ b ϕ c ϕ d =.... Q abcd (k,q,p,r) = 0 η P ab (k) =... η B abc (k, q,q k) =...
η P ab (k) = Ω ac (k)p cb (k) Ω bc (k)p ac (k) + e η d 3 q [γ acd (k, q, q k) B bcd (k, q, q k) +B acd (k, q, q k)γ bcd (k, q, q k)] η B abc (k, q, q k) = Ω ad (k)b dbc (k, q, q k) Ω bd ( q)b adc (k, q, q k) Ω cd (q k)b abd (k, q, q k) + 2e η [γ ade (k, q, q k)p db (q)p ec (k q) +γ bde ( q, q k, k)p dc (k q)p ea (k) + γ cde (q k, k, q)p da (k)p eb (q)]
Power spectra of density perturbations 1. Matter: P m = δ m δ m 2. Total clustering density (matter & dark energy): ρ T = ρ m +δρ m + ρ x +δρ x δ T = δρ T / ρ T = Ω m δ m +Ω x δ x P T (k) = (Ω m δ m (k)+ω x δ x (k)) 2 = Ω 2 m P m(k)+2ω m Ω x δ m (k)δ x (k) +Ω 2 x δ x(k) 2
The case of zero sound speed (maximal clustering) For ĉ 2 s = 0 we can use δ T δ m + Ωx Ω m δ x and θ θ m δ T (k)+c(τ)θ(k)+ d 3 p d 3 qδ D (k p q)α(q, p)θ(q)δ (n) T (p) = 0 θ(k)+hθ(k)+ 3 2 Ω mh 2 δt (k) + d 3 p d 3 qδ D (k p q)β(q, p)θ(q)θ(p) = 0 C(τ) = 1+(1+w) Ω x Ω m
Linear DE approximation δ m (k)+θ m (k)+ d 3 p d 3 qδ D (k p q)α(q, p)θ m (q)δ m (p) = 0 θ m (k)+hθ m (k)+ 3 2 H2 (Ω m δ m (k)+ω x δ x (k)) + d 3 p d 3 qδ D (k p q)β(q, p)θ m (q)θ m (p) = 0,
Linear DE approximation δ m (k)+θ m (k)+ d 3 p d 3 qδ D (k p q)α(q, p)θ m (q)δ m (p) = 0 θ m (k)+hθ m (k)+ 3 ( 2 H2 Ω m δ m (k) 1+ Ω xδx L(k) ) Ω m δm L (k) + d 3 p d 3 qδ D (k p q)β(q, p)θ m (q)θ m (p) = 0 2 1 2ĉ s For Ω m 1, δ x = (1+w) 2 1 3w + ĉ δ m, ĉs 1 H k H s w 1 δ x δ m < 1 δ 2 x δ2 m
Linear DE approximation δ m (k)+θ m (k)+ d 3 p d 3 qδ D (k p q)α(q, p)θ m (q)δ m (p) = 0 θ m (k)+hθ m (k)+ 3 ( 2 H2 Ω m δ m (k) 1+ Ω xδx(k) L ) Ω m δ m(k) L + d 3 p d 3 qδ D (k p q)β(q, p)θ m (q)θ m (p) = 0 ( ) ( ( δ x(k) L = 3 ĉ 2 s w Hδx(k) (1+w) L 1+9 ĉ 2 s w θ x(k) L = (1 3ĉ ) 2 s Hθx(k)+ L ĉs 2 k 2 (1+w) δl x(k) ) H 2 k 2 ) θ L x(k) + 3 2 H2( Ω m δ L m (k)+ω x δ L x (k))
Linear DE approximation δ m (k)+θ m (k)+ d 3 p d 3 qδ D (k p q)α(q, p)θ m (q)δ m (p) = 0 θ m (k)+hθ m (k)+ 3 ( 2 H2 Ω m δ m (k) 1+ Ω xδx(k) L ) Ω m δ m(k) L + d 3 p d 3 qδ D (k p q)β(q, p)θ m (q)θ m (p) = 0 ( ) ( ( δ x(k) L = 3 ĉ 2 s w Hδx(k) (1+w) L 1+9 ĉ 2 s w θ x(k) L = (1 3ĉ ) 2 s Hθx(k)+ L ĉs 2 k 2 (1+w) δl x(k) ) H 2 k 2 + 3 2 H2( Ω m δ L m (k)+ω x δ L x (k)) ) θ L x(k) P T (k) Ω 2 m P [TRG] m (k)+2ω m Ω x δ L m(k)δ L x(k) +Ω 2 x δ L x(k) 2
0.10 w 0.8 z 0 0.08 0.06 Matter power spectrum lin P P sm lin P sm 0.04 0.02 0.00 0.01 0.015 0.02 0.03 0.05 0.07 0.1 0.15 0.2 k h Mpc
0.6 w 0.8 z 0 0.5 0.4 Total power spectrum lin P P sm lin P sm 0.3 0.2 0.1 0.0 0.05 0.10 0.15 0.20 k h Mpc
0.20 w 0.8 z 1 0.15 Total power spectrum lin P P sm lin P sm 0.10 0.05 0.00 0.05 0.10 0.15 0.20 k h Mpc
Matter growth factor and index δ m a g m(a) = g m (a i ) exp a a i (Ω c (ã) γm 1) dã ã for wcdm : γ m 0.55+0.05[1+w(z = 1)] δ x g m and γ m scale and redshift dependent δ m changes by max. 1% (linear) γ m changes by max. 5% (linear) ( ) γ m = (logω m ) 1 d logδm log dη
Matter growth factor and index δ m a g m(a) = g m (a i ) exp a a i (Ω c (ã) γm 1) dã ã for wcdm : γ m 0.55+0.05[1+w(z = 1)] δ x g m and γ m scale and redshift dependent δ m changes by max. 1% (linear)/ 3.5% (non linear) γ m changes by max. 5% (linear)/ 10 15% (non linear) ( ) γ m = (logω m ) 1 d logδm log dη Non linearities enhance the clustering and are nearly independent on ĉ s
0.55 Γ k 0.1 h Mpc 0.50 0.45 Matter growth index w 0.8 0.40 0.2 0.4 0.6 0.8 1.0 a
1.0 0.8 g k 0.1 h Mpc 0.6 0.4 Total growth 0.2 w 0.8 0.2 0.4 0.6 0.8 1.0 a
Summary of assumptions and approximations 1. constant w and 0 ĉ s 2 1 2. zero anisotropic stress 3. γ = 1 4. k H 5. u = 0 6. perturbations up to O(2) 7. terms ϕ a ϕ b neglected 8. zero trispectrum 9. k Ω ab = 0 10. linear treatment of DE perturbations And some references: - TRG. arxiv:0806.0971. Pietroni. - Growth index. arxiv:0807.3343. Ballesteros and Riotto - Dectectability of ĉ s 2. arxiv:1004.5509. Ballesteros & Lesgourgues - arxiv:1101.1026. Sefusatti and Vernizzi - Zero sound speed, TRG. arxiv:1106.0314. D Amico and Sefusatti - Most of this talk. arxiv:1106.0834. Anselmi, Ballesteros & Pietroni
Conclusions For DE (with constant w and ĉ s ) [and no anisotropic stress] we saw: DE fluctuations must be considered and ĉ s should be non adiabatic Planck + LSST: will set a lower bound on ĉ s Non linearities: needed for future precision observations (Euclid...) DE perturbations can accurately be treated linearly Corrections to Pm nearly independent on ĉ s (below 1%) Outlook: tomographic surveys, montecarlo sampling, N body...