Initial conditions based on Lagrangian perturbation theory for N-body by znw65712

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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

								
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