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2008.11.11-14 7th RESCEU International Symposium Initial conditions based on Lagrangian perturbation theory for N-body simulations Takayuki Tatekawa (Ochanomizu University) e-mail: tatekawa@cosmos.phys.ocha.ac.jp Collaborator: Shuntaro Mizuno (RESCEU, Univ. of Tokyo) Ref: Tatekawa and Mizuno, JCAP 0712 (2007) 014. Tatekawa et al., in preparation. 1 Introduction Lagrangian perturbative theory (LPT) is widely used in observational cosmology. Nowadays, we have obtained up to third-order perturbative solutions (cf. Tatekawa, astro-ph/0412025). In cosmological N-body simulations, the initial condition is given by Lagrangian perturbative solution at high-z era (z~30). For a long time, we used linear-order Lagrangian perturbation (Zelʼdovich approximation, hereafter ZA) for the initial condition. In precision cosmology, this initial condition seems not reasonable. 2 Initial condition problem For many years, ZA has been used for the initial condition. Crocce, Pueblas, Scoccimarro (2006) Setting initial conditions in cosmological N-body simulations Lagrangian linear perturbation (ZA) Lagrangian second-order perturbation (2LPT) They noticed transients in the evolution of the statistical properties of density ﬁelds. They showed that 2LPT initial conditions reduce transients signiﬁcantly. -> 2LPT initial conditions are much more appropriate for numerical simulations. 3 Our work Crocce, Pueblas, Scoccimarro (2006) showed 2LPT initial conditions are much more appropriate for numerical simulations. We consider if 2LPT is more appropriate than ZA, 3LPT would be more appropriate than 2LPT. Should we consider 3LPT initial condition? Is 2LPT (or 3LPT) enough as the initial conditions? What is reasonable initial condition? (the order of the perturbative solutions and the initial era) 4 Set up the initial condition The initial condition is generated by COSMICS (by Bertschinger), which outputs Gaussian ﬂuctuation. The initial condition is based on ﬁrst-order (ZA). Then we improve the initial condition up to second-order (2LPT) and third-order (3LPT). At z>20, even if the dark energy is existed, the Universe is matter dominant. Therefore we apply the perturbative solutions in E-dS Universe model. x(q, t)=q+S(q, t) x: Comoving Eulerian coordinates q: Lagrangian coordinates S: Lagrangian displacement (perturbation) The linear-order solutions（ZA）are described as 2/3 −1 (1) t t S (q, t) = s+ (q) + s− (q) t0 t0 5 Higher-order perturbative solutions Catelan (1995) 4/3 (2) 3 t 2LPT: S (q, t) = − s(1) ∇q · s(1) − s(1) · ∇q s(1) 14 t0 (3) (3) 3LPT: S(3) (q, t) = sA (q, t) + sB (q, t) 2 (3) 1 t (1)C (1) sAα (q, t) =− sαβ sβ 9 t0 sC : adjoint matrix of sαβ αβ 2 (3) 5 t sB (q, t) = s(1) ∇q · s(2) − s(1) · ∇q s(2) 42 t0 + s(2) ∇q · s(1) − s(2) · ∇q s(1) Here we ignore divergence-free vector. 6 Third-order perturbative solutions If we consider only the longitudinal mode in linear order, third-order transverse mode appears (Buchert (1994), Sasaki and Kasai (1998)). ¨(3)T ˙ a ˙ (3)T εijk Sk,j + 2 Sk,j a (1) (1) (1) 1 (1) (2) = 4πGρb εijk D3 ψ,jl ψ,km ψ,lm − DEψ,jl ψ,kl , 2 Here we divide this solution to (1st)^3 and (1st)x(2nd). (3A)T Si A = F AT ζi , (3B)T Si B = F BT ζi , AT t2 3 2 F = , F BT = t , 7 98 (1) (1) (1) (1) (1) (1) A −∇2 ζi = ψ,il ψ,km ψ,lm − ψ,jl ψ,im ψ,lm , ,k ,j (1) (2) (1) (2) B −∇2 ζi = ψ,il ψ,kl − ψ,jl ψ,il , ,k ,j 7 Reconstruction of initial conditions We obtain linear-order initial condition (the peculiar velocity, the density ﬂuctuation, and the initial scale factor) by COSMICS. From the peculiar velocity, we compute initial displacement vector S(q). Using the initial scale factor and S(q), we compute second- and third- order Lagrangian perturbation (the displacement vector and the peculiar velocity). Finally we transfer these initial conditions to P3M code. In this poster, we set up the initial conditions based on ZA, 2LPT, 3LPTw/oT (2LPT+3LPT longitudinal mode), and 3LPT. 8 N-body simulation Numerical code: P3M code (by Bertschinger) Here we set cosmological parameters as follows. We regard the dark energy as cosmological constant. cosmological parameters (WMAP 5th-year) parameters in P3M code #particles : N = 1283 , Ωm = 0.26 , Box size : L = 128h−1 Mpc (at a = 1) ΩΛ = 0.74 , Softening length : H0 = 72[km/s/Mpc] , ε = 50h−1 kpc (at a = 1) . σ8 = 0.80 . We start the simulations at z=24, 50, 126. Smoothing scale: R= 2 Mpc/h 9 Non-Gaussianity in density ﬁeld We notice 1-point fn. of the density ﬂuctuaion P(δ). If the distribution function obeys Gaussian, the distribution is described as 1 2 2 P (δ) = e−δ /2σ (2πσ)1/2 2 σ 2 ≡ (δ − δ ) :density dispersion We set the initial conditions by Gaussian ﬂuctuation. Of course, if we consider only linear growth, the Gaussian distribution remains all time. By the eﬀect of non-linear evolution, the distribution function deviates from the Gaussian distribution. 10 Non-Gaussianity in density ﬁeld (cont.) We introduce higher-order statistical quantities. δ3 c skewness : γ = , < δ 3 >c ≡< δ 3 > , σ4 δ4 c kurtosis : η = 6 , < δ 4 >c ≡< δ 4 > −3σ 4 . σ In E-dS Universe model, these quantities were computed in weakly nonlinear stage with Eulerian second-order perturbation. 34 γ = + O(σ 2 ) , 7 60712 η = + O(σ 2 ) . 1323 We compute skewness and kurtosis in density ﬁeld. Because of nonlinear evolution, the quantities deviate from above values. 11 Started at z≒24 The diﬀerence of the density dispersion between ZA and 3LPT is less than 4%. 8 12 ZA ZA 2LPT 2LPT 7 3LPTw/oT 11 3LPTw/oT 3LPT 3LPT 6 10 5 9 dispersion skewness 4 8 3 7 2 6 1 5 0 5 4 3 2 1 0 4 z 5 4 3 2 1 0 However, the diﬀerence of the non-Gaussianity is large. z right-upper: skewness (a) right-lower: kurtosis 300 ZA 2LPT 3LPTw/oT The diﬀerence between 3LPT and ZA 250 3LPT skewness: less than 9% @ z=2 200 kurtosis: less than 13% @ z=2 kurtosis 150 The diﬀerence between ZA, 2LPT, and 3LPT appears. 100 We cannot decide that we should 50 consider until n-th order. 0 (3LPT is not enough to compute?) 5 4 3 z 2 1 0 The eﬀect of the transverse mode in 3LPT seems quite small. (b) 12 How to solve this problem? COSMICS code sets the initial time by maximum value of the density ﬂuctuation. (default conﬁguration: 1.0). We change the value and start N-body simulation at early era. -> We change the value to 0.5, 0.2. Of course, the nonlinear eﬀect becomes small when we set up the initial condition in early stage, we hope that the diﬀerence between ZA, 2LPT, 3LPTw/oT, and 3LPT becomes neglegible. 13 Started at z≒50, 126 z≒50 z≒126 12 12 ZA ZA 2LPT 2LPT 11 3LPTw/oT 11 3LPTw/oT 3LPT 3LPT 10 10 9 9 skewness skewness 8 8 7 7 6 6 5 5 4 4 5 4 3 2 1 0 5 4 3 2 1 0 z z (a) (a) 300 300 ZA ZA 2LPT 2LPT 3LPTw/oT 3LPTw/oT 250 3LPT 250 3LPT 200 200 kurtosis kurtosis 150 150 100 100 50 50 0 0 5 4 3 2 1 0 5 4 3 2 1 0 z z (b) (b) The diﬀerence between 3LPT and ZA skewness: less than 4% @ z=2 The diﬀerence between ZA and 3LPT kurtosis: less than 6% @ z=2 skewness: less than 1.5% @ z=2 The diﬀerence between 2LPT and 3LPT kurtosis: less than 2.5% @z=2 almost disappears. The diﬀerence between ZA and 2LPT remains. -> we need not apply 3LPT. -> ZA cannot set up reasonable initial condition. 14 Power spectrum We compare the power spectrum of the density ﬁeld. Initial condition: z=24, 50, 126 The diﬀerence of the power spectrum between the case of ZA and 3LPT is less than 5%. 100000 100000 ZA ZA 2LPT 2LPT 3LPTw/oT 3LPTw/oT 3LPT 3LPT 10000 10000 P(k) [h3 Mpc-3] P(k) [h3 Mpc-3] 1000 1000 100 100 0.01 0.1 0.01 0.1 k [h/Mpc] k [h/Mpc] In the case of z=24 In the case of z=50 15 The diﬀerence of the power spectrum Here we show the diﬀerence of the power spectrum. In the case of z=24, the diﬀerence of the power spectrum between 3LPT and ZA is less than 7%. 0.07 0.012 3LPT vs ZA 3LPT vs ZA 3LPT vs 2LPT 3LPT vs 2LPT 0.06 3LPT vs 3LPTw/oT 0.01 3LPT vs 3LPTw/oT Difference of the power spectrum Difference of the power spectrum 0.05 0.008 0.04 0.006 0.03 0.004 0.02 0.002 0.01 0 0 -0.002 -0.01 -0.004 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 k [h/Mpc] k [h/Mpc] z=24 z=126: The diﬀerence of the power spectrum: The diﬀerence of the power spectrum: 3LPT vs ZA : < 7% 3LPT vs ZA : < 1% 3LPT vs 2LPT : < 2% 3LPT vs 2LPT : < 0.2% 16 Summary For the initial coditions for cosmological N-body simulations, Lagrangian linear perturbation (Zelʼdovich approximation) is applied for long time. However the analyses of non-Gaussianity in the density ﬁeld, Lagrangian linear perturbation is insuﬃcient for the initial condition, i.e. it cannot describe nonlinear eﬀect well. In our analyses, we should apply at least 2LPT for the initial conditions before z=30. When we set up the initial conditions with Zelʼdovich approximation, even if we start the simulation at z > 100, the nonlinear eﬀect is not realized well. The eﬀect of third-order transverse mode seems small. 17 Future work Other eﬀects for the error in cosmological simulation. For example, discreteness eﬀect in N-body simulation has been considered. (Joyce and Marcos (2007)) Dispersion between the samples. In our analyses, even if we increase number of samples, the error bars for the skewness and the kurtosis still be large. 18