Dark energy and non-linear power spectrum Jinn-Ouk Gong APCTP , - PowerPoint PPT Presentation

Dark energy and non-linear power spectrum Jinn-Ouk Gong APCTP , Pohang 790-784, Korea 2nd APCTP-TUS Joint Workshop Tokyo University of Science 3rd August, 2015 Based on S. G. Biern and JG, 1505.02972 [astro-ph.CO]

Introduction Formulation of perturbation theory Relativistic theory with homogeneous dark energy Geodesic approach Conclusions Outline Introduction 1 Formulation of perturbation theory 2 Newtonian theory Relativistic theory Relativistic theory with homogeneous dark energy 3 Effects of dark energy Non-linear power spectrum with dark energy Geodesic approach 4 Conclusions 5 Dark energy and non-linear power spectrum Jinn-Ouk Gong

Introduction Formulation of perturbation theory Relativistic theory with homogeneous dark energy Geodesic approach Conclusions Why GR in LSS? Planned galaxy surveys: DESI, HETDEX, LSST, Euclid, WFIRST... Larger and larger volumes, eventually accessing the scales comparable to the horizon: beyond Newtonian gravity, fully general relativistic approach (or any modification) is necessary Dark energy and non-linear power spectrum Jinn-Ouk Gong

Introduction Formulation of perturbation theory Relativistic theory with homogeneous dark energy Geodesic approach Conclusions Why dark energy in non-linear regime? DE was negligible at very early times DE becomes significant at later stage when non-linearities in cosmic structure are developed Naturally DE affects the evolution of gravitational instability, so that its effects emerge more prominently at non-linear level What are the effects of DE in non-linear regime of LSS? Dark energy and non-linear power spectrum Jinn-Ouk Gong

Introduction Formulation of perturbation theory Relativistic theory with homogeneous dark energy Geodesic approach Conclusions Newtonian theory 3 basic equations for density perturbation δ ≡ δρ / ¯ ρ , peculiar velocity u and gravitational potential Φ with a pressureless fluid δ + 1 a ∇· u = − 1 ˙ a ∇· ( δ u ) continuity eq u + H u + 1 a ∇ Φ = − 1 ˙ a ( u ·∇ ) u Euler eq ∆ a 2 Φ = 4 π G ¯ ρδ Poisson eq Newtonian system is closed at 2nd order ρδ = − 1 dt [ a ∇· ( δ u )] + 1 d δ + 2 H ˙ ¨ δ − 4 π G ¯ a 2 ∇· ( u ·∇ u ) a 2 → at linear order, δ + ∝ a (growing) and δ − ∝ a − 3/2 (decaying) − (Bernardeau et al. 2002) Dark energy and non-linear power spectrum Jinn-Ouk Gong

Introduction Formulation of perturbation theory Relativistic theory with homogeneous dark energy Geodesic approach Conclusions Basic non-linear equations Based on the ADM metric � N i dx 0 + dx i �� N j dx 0 + dx j � ds 2 = − N 2 ( dx 0 ) 2 + γ ij the fully non-linear equations are (Bardeen 1980) i + 2 R − K i j K j 3 K 2 − 16 π GE = 0 i ; j − 2 K j 3 K , i = 8 π GJ i K , i N i N ; i ; i K ,0 i − 1 − K i j K j 3 K 2 − 4 π G ( E + S ) = 0 N − + N N K i K i K i � � j ; k N k K jk N i ; k k N k ; j δ ij j − 1 j ,0 = KK i + R i j − 8 π GS i N ; i ; j − 3 N ; k − + − ; k j N N N N N � N 2 J i � E , i N i � � E ,0 E + S − K i j S j ; i N − − K i + = 0 N 2 N 3 J i ; j N j J j N j ; i S ji N , j J i ,0 EN , i + S ji ; j + − KJ i + = 0 N − − N N N N Fluid quantities: E ≡ n µ n ν T µν , J i ≡ − n µ T µ i , S ij ≡ T ij Dark energy and non-linear power spectrum Jinn-Ouk Gong

Introduction Formulation of perturbation theory Relativistic theory with homogeneous dark energy Geodesic approach Conclusions Einstein-de Sitter universe Usually, structure formation is described in EdS T µν = ρ m u µ u ν − → J i = S ij = 0 Linear growth factor is all: D 1 = a , D 2 = 3 D 2 1 /7 and so on Comoving gauge ( γ = 0 and T 0 i = 0) gives identical equations to the Newtonian counterparts up to 2nd order Pure GR contribution appears from 3rd order and is totally sub-dominant (Jeong, JG, Noh & Hwang 2011, Biern, JG & Jeong 2014) In e.g. synchronous gauge ( g 00 = − 1 and g 0 i = 0) we can have another Newtonian correspondence (Hwang, Noh, Jeong, JG & Biern 2015) Linear power spectrum is obtained by solving the Boltzmann eq (e.g. CAMB ) and is used iteratively to obtain non-linear contributions Dark energy and non-linear power spectrum Jinn-Ouk Gong

Introduction Formulation of perturbation theory Relativistic theory with homogeneous dark energy Geodesic approach Conclusions Putting dark energy on the table Previous strategy is not complete Λ CDM power spectrum in EdS background Matter domination all the way But we know the universe has been dominated by DE for a long time ρ = ρ m − → ρ = ρ m + ρ de with p de = w ρ de For simplicity No DE perturbation: ρ dm = ¯ ρ de (cf. Park, Hwang, Lee & Noh 2009) 1 Comoving gauge: T 0 i = 0 2 Dark energy and non-linear power spectrum Jinn-Ouk Gong

Introduction Formulation of perturbation theory Relativistic theory with homogeneous dark energy Geodesic approach Conclusions Dark energy changes the game DE provides different BG from both EdS and Λ CDM: H 2 = 8 π G H ′ = − 1 a 2 � � 2 H 2 (1 + 3 w ) ρ m + ¯ ¯ ρ de and 3 DE permeates all order in perturbation: e.g. energy conservation � � 1 − 1 δ ′ − κ (1 − λ ) = (non-linear terms) where λ ≡ (1 + w ) Ω m Thus away from EdS ( Ω m = 1) and Λ CDM ( w = − 1) the effects of general, dynamical DE are manifest : we use the parametrization (Chevallier & Polarski 2001, Linder 2003) w ( a ) = w 0 + (1 − a ) w a Dark energy and non-linear power spectrum Jinn-Ouk Gong

Introduction Formulation of perturbation theory Relativistic theory with homogeneous dark energy Geodesic approach Conclusions Non-linear solutions with DE Curvature perturbation is not conserved: from energy constraint � � ϕ = − H 2 f 1 + 3 2(1 − λ ) Ω m ∆ − 1 δ �= constant 1 − λ f Thus δ receives a) curvature evolution effects from 3rd order and b) general, dynamical DE effects from BG and linear order: � � λ ′ δ ′ − 3 δ ′′ + 2(1 − λ ) H 2 Ω m δ = N N + N ϕ + N ϕ ′ + N λ H + 1 − λ � �� � = non-linear source terms Newtonian EdS Λ CDM DE N N O O O O N ϕ X O O O N ϕ ′ X X X O X X X O N λ Dark energy and non-linear power spectrum Jinn-Ouk Gong

Introduction Formulation of perturbation theory Relativistic theory with homogeneous dark energy Geodesic approach Conclusions Relativistic kernels 2nd and 3rd order solutions are (Biern & JG 2015) � d 3 q 1 d 3 q 2 b � δ 2 ( k , a ) = D 2 δ (3) ( k − q 12 ) F 2 i ( q 1 , q 2 ) δ 1 ( q 1 ) δ 1 ( q 2 ) c 2 i ( a ) 1 (2 π ) 3 i = a f �� � c ni ≡ D ni � δ 3 ( k , a ) = D 3 ··· F 3 i ··· 3 δ ′ c 3 i ( a ) 1 s 1 D n i = a 1 D ϕ b �� � � c ϕ ··· F ϕ c ϕ 3 i + D 3 1 H 2 3 i ··· 3 δ 1 ’s 3 i ( a ) 3 i ≡ D 3 1 H 2 i = a In the EdS universe c ’s are fixed as certain numbers ( c 2 a = 3/7...) and (also in Λ CDM) c ni terms become purely Newtonian [Kamionkowski & Buchalter 1999 (2nd) and Takahashi 2008 (3rd)] and only c ϕ 3 i terms remain relativistic N. B. λ is completely entangled and cannot be separated like ϕ Dark energy and non-linear power spectrum Jinn-Ouk Gong

Introduction Formulation of perturbation theory Relativistic theory with homogeneous dark energy Geodesic approach Conclusions One-loop corrected power spectrum: versus Λ CDM 10 4 10 4 P total P total P 22 + P 13 P 22 + P 13 100 100 w o =- 1.2 P ( k )[ Mpc / h ] 3 P ( k )[ Mpc / h ] 3 Λ CDM w o =- 0.8 w a =- 1.0 1 1 w a =- 0.5 Λ CDM φ φ P 13 P 13 w a = 0.5 10 - 2 10 - 2 20 20 CDM - 1 [%] 10 CDM - 1 [%] 10 w a =- 1.0 0 w o =- 1.2 0 w a =- 0.5 w o =- 0.8 P / P Λ P / P Λ - 10 w a = 0.5 - 10 - 20 - 20 10 - 2 0.1 1 10 - 2 0.1 1 k [ h / Mpc ] k [ h / Mpc ] Overall almost constant deviation on large scales ( k � 0.1 h /Mpc) Deviation becomes significant on k � 0.1 h /Mpc, close to baryon acoustic oscillations w 0 > − 1 / w a > 0 ( w 0 < − 1 / w a < 0) give smaller (larger) P ( k ) Dark energy and non-linear power spectrum Jinn-Ouk Gong

Introduction Formulation of perturbation theory Relativistic theory with homogeneous dark energy Geodesic approach Conclusions One-loop corrected power spectrum: versus EdS In Newtonian studies, usually EdS power spectrum is transferred to an arbitrary DE model by replacing a → D 1 ( a ): P ( k , a ) = D 2 1 ( a ) P 11 ( k ) + D 4 1 ( a )[ P 22 ( k ) + P 13 ( k )] EdS P w a =- 1.0 P w o =- 1.2 P w a =- 0.5 2000 P Λ CDM 2000 P Λ CDM P w a = 0.5 P w o =- 0.8 P ( k )[ Mpc / h ] 3 P ( k )[ Mpc / h ] 3 w a =- 1.2 w a =- 1.0 P EdS P EdS w a =- 0.5 Λ CDM 1000 P EdS 1000 P EdS w a =- 0.8 Λ CDM P EdS P EdS w a = 0.5 P EdS 500 500 10 5 w a =- 1.0 P EdS / P - 1 [%] P EdS / P - 1 [%] 5 w a =- 0.5 w o =- 1.2 0 Λ CDM Λ CDM 0 w a = 0.5 w o =- 0.8 - 5 - 5 - 10 - 10 0.1 0.2 0.5 0.1 0.2 0.5 k [ h / Mpc ] k [ h / Mpc ] For Λ CDM, only ϕ drives difference so almost identical to EdS For general DE, the difference notably increases from k ≈ 0.1 h /Mpc Dark energy and non-linear power spectrum Jinn-Ouk Gong