nbody_iitk.pdf

Cosmological N-Body simulations
Technique, scope & limitations
Jayanti Prasad
Inter-University Centre for Astronomy & Astrophysics
Pune, India (411007)
January 13, 2011
Jayanti Prasad (IUCAA-PUNE) Cosmological N-Body simulations January 13, 2011 1 / 34

Plan of the Talk
Non-linear gravitational clustering
Cosmological N-body simulations
Role of substructure in gravitaional clustering
Finite volume effects in N-body simulations
Summary & Conclusions

Growth of perturbations
Density field
δ(x, t) =
ρ(x, t) − ρb(t)
ρb(t)
=
1
a3(t)
N
X
i=1
mi δD(x − xi (t)) (1)
and
δk(t) =
1
M
N
X
i=1
mi exp(−ik.xi ) − δ3
D(k); where M =
N
X
i=1
mi (2)
δ̈k + 2
ȧ
a
δk =
1
M
N
X
i=1
mi

−ik.

ẍi + 2
ä
a
ẋi

− (k.xi )2

exp(−ik.xi )
[Peebles, P. J. E. (1980)]

Dynamics of particles
ẍi + 2
ȧ
a
ẋi = −
∇x φ(x, t)
a2(t)
(3)
∇2
x φ(xi , t) = 4πGa2
(t)ρb(t)δ(xi , t) (4)
H2
(t) = H2
(t0)

Ωm
a3(t)
+ ΩΛ +
1 − Ωm − ΩΛ
a2(t)

(5)
where:
H(t) =
1
a(t)
da(t)
dt
; ΩX =
8πGρX
3H2(t)
(6)

Mode Coupling
δ̈k + 2
ȧ
a
δ̇k(t) = 4πGρbδk + Ak − Bk (7)
where
Ak = 2πGρb
X
k06=0,k6=k0

k0.k
|k0|2
+
k.(k − k0)
|k − k0|2

δk0 δk−k0 (8)
and
Bk =
1
M
N
X
i=1
mi (k.ẋi )2
exp(−ik.xi ) (9)
[Padmanabhan (1993)]

Cosmological N-body simulations
Mass dicreatization
mp =
(Llbox /Mpc)3
Np
2.78 × 1011
M Ωmh2
(10)
Periodic boundary conditions
x + Lbox −→ x (11)
Force softening
~
Fij = −
GMi Mj r̂ij
r2
ij + 2
(12)
[ Hockney Eastwood (1981); Efstathiou et al. (1985); Sellwood (1987);
Bertschinger (1998); Aarseth (2003)]

A. Setting-up initial conditions
The real and imaginary parts of the gravitational potential in Fourier
space are drawn from a Gaussian random distribution with zero mean
Pφ(k) variance
P(φk) =
1
p
2πPφ(k)
exp

−
1
2
φ2
k
Pφ(k)

(13)
or
φk = Gauss(0.0, Pφ(k)) (14)
Initial position/velocity of particles are computed using Zelodovich
Approximation
v = −∇φint (15)
Normalization of P(k) is given in the form of the scale of
non-linearity at the final epoch and the initial epoch is computed on
the basis of the maximum displacement in the first step.

B. Force computation
Particle-Particle (PP)
Direct pair-force computationally expensive i.e., O(N2
p ), however, accurate
at all scales.
Particle-Mesh (PM)
Force is computed by solving Poisson’s equation in Fourier space (on
a grid) by using Fast-Fourier-Techniques (FFT). Computational cost
grows as O(Np log Np) and force is inaccurate at sub-grid scales.
Periodic boundary conditions and force softening are automatically
incorporated.
[Efstathiou et al. (1985); Bagla Padmanabhan (1997)].

Force..
Tree
Tree structured hierarchical division of space into cubic cells, each of
which is recursively divided into eight sub-cells whenever more than
on particles are in the same cell. The tree is constructed at every time
step.
The force acting on a particle ”p” due to all the particles in a cell of
size l at distance d is approximated by the force acting between the
particle and the ”cell” if l/d θ, where θ is a parameter.
[Barnes Hut (1986)]
TreePM
Force is split into two parts: long range and the short range.
Long range force is computed using PM method and the short range
force is computed using Tree method.
[Bagla (2002); Bagla Ray (2003)]

C. Moving Particles
Time centered leap-from scheme
xn = xn−1 + vn−1/2∆t (16)
vn+1/2 = vn−1/2 + an∆t (17)

Power spectrum and two point correlation functions
The two point correlation function ξ(r) and density contrast δ(x) are
related as
ξ(r) = hδ(x + r)δ(x)i (18)
Here averaging is done over an ensemble of large spatial regions of the
universe that are statistically independent from each other. If we write
density contrasts δ(x + r) and δ(x) in terms of Fourier components and
simplify equation (18) we get
ξ(r) =
Z
d3k
(2π)3
P(k) exp(ik.r) (19)
Here P(k) is called the power spectrum and is defined as
hδkδ?
k0 i = (2π)3
δ
(3)
D (k − k0
)P(k) (20)

Mass variance
Variance of the mass inside cells of various sizes, after smoothing the
density by a window function W .
σ2
(r) =
(M− M )2
M 2
=
1
2π2
Z
dkk2
P(k)|W (k, r)|2
(21)
where M = 4πρr3/3. For a spherical-top hat window W (k, r) is given by
W (k, r) = 3

sin kr − kr cos kr
k3r3

(22)
It is more useful to express σ2(r) in the following form
σ2
(r) =
Z
dk
k

k3P(k)
2π2

|W (k, r)|2
=
Z
dk
k
∆2
(k)|W (k, r)|2
(23)

Press-Schecter mass function
If we consider that the initial density field is Gaussian then the fraction of
mass F(M) in the collapsed objects with mass greater than M can be
identified with the fraction of the initial volume for which δ δc.
F(M) =
Z ∞
δc
P(δ, r)dδ = Erfc

ν
√
2

(24)
f (M)dM =
∂F(M)
∂M
=
r
2
π
dlnσ
dM
ν exp(−ν2
/2)dM (25)
and comoving number density of objects in mass range [M, M + dM] is
N(M)dM =
ρ
M
× f (M)dM =
r
2
π
ρ
M2
dlnσ
dlnM
ν exp(−ν2
/2)dM (26)
where ν = δc/σ
[Press Schechter (1974)]

Mode coupling in Non-linear gravitational clustering
Structure formation in a cold dark matter dominated universe takes
place hierarchically. In this situation it becomes relevant to ask:
What role perturbations at small scales (substructure) play in the collapse
of perturbations at larger scales ? [Bagla et al. (2005); Bagla Prasad
(2009)]
It is well knows that there are strong effects of perturbations at large
scales on the growth of perturbations at small scales. In cosmological
N-body simulations perturbations at scales larger than the size of
simulation volume are ignored.
How the size of simulation volume affect various physical quantities in
N-body simulations ? [Bagla Prasad (2006); Prasad (2007); Bagla et al.
(2009)]

1. Role of substructure
Name Method α Plane wave IC
PM 00L PM 0.0 Yes Grid
T 00L TreePM 0.0 Yes Perturbed grid
T 05L TreePM 0.5 Yes Grid
T 10P TreePM 1.0 Yes Perturbed grid
T 40P TreePM 4.0 Yes Perturbed grid
T 05 TreePM 0.5 No Grid
Table: Cosmological simulations with power at two scales: large scale (plane
wave) and small scales (power non-zero only for a narrow range on scales
isotropic and anisotropic). We found that the presence of isotropic power
speed-up the collapse of plane wave [Bagla et al. (2005)].

A. Planar Collapse
[Bagla et al. (2005)]

Planar Collapse
[Bagla et al. (2005)]

B. Features in Power spectrum
Models
P(k) =









Ak−1, for Model I
Ak−1e
− k2
2k2
c , for Model II
Ak−1 + Aαk−1
c e
(k−kc )
2σ2
k for Model III
(27)
Parameters
σk = 2π/Lbox , kc = 4kf = 4(2π/Lbpx ), α = 4
rnl (a = 1) = 12, Npart = 2003, Lbox = 200

[Bagla Prasad (2009)]

Correlation function
Figure: The left and right panels show the average two point correlation functions
for all the models at an early and later epoch respectively.

Skewness
Figure: The left and right panels show the Skewness for all the models at an early
and later epoch respectively.

Number density
Figure: The left and right panels show the number density of haloes for all the
models at an early and later epoch respectively.

2. Finite volume effects
In cosmological N-body simulations the initial power spectrum is
sampled at discrete points in k space in the range [2π/Lbox , 2π/Lgrid ].
Perturbations k 2π/Lbox can be ignored since there are no
significant effects or perturbations at small scales on large scale.
However, that is not the case of k 2π/Lbox perturbations at scales
larger than the simulation volume can affect physical quantities
significantly.
[Bagla Prasad (2006); Prasad (2007); Bagla et al. (2009)]

Our formalism [Bagla Prasad (2006)]
Mass variance in theory
σ2
0(r) =
∞
Z
0
dk
k
W 2
(kr) where ∆2
(k) =
k3P(k)
2π2
(28)
In simulations
σ2
(r, Lbox ) =
2π/Lgrid
Z
2π/Lbox
dk
k
∆2
(k)W 2
(kr) (29)
From the above two equations:
σ2
(r, Lbox ) =
∞
Z
0
dk
k
∆2
(k)W 2
(kr)−
2π/Lbox
Z
0
dk
k
∆2
(k)W 2
(kr) = σ2
0(r)−σ2
1(r)
(30)

Correction in mass variance
σ2
1(r, Lbox ) =
2π/Lbox
Z
0
dk
k
∆2
(k)W 2
(kr)
≈ C1 −
1
5
C2r2
+
3
175
C3r4
+ O(r6
) (31)
where
Cn =
2π/Lbox
Z
0
dk
k
k2(n−1)
∆2
(k) (32)
Power law model:P(k) = Akn
C1 =
A
2π2(n + 3)

2π
Lbox
(n+3)
(33)

Correction in Mass function
The Press-Schecter mass function (theory)
F(M) = F0(M) = 2
Z ∞
δc
1
q
2πσ2
0(M)
exp

−
δ2
2σ2
0(M)

dδ (34)
N-body simulations
F(M, Lbox ) = 2
Z ∞
δc
1
p
2πσ2(M, Lbox )
exp

−
δ2
2σ2(M, Lbox )

dδ (35)
Correction :
F1(M, Lbox ) = F0 − F =
2
√
π
Z δc
σ(M,Lbox )
√
2
δc
σ0(M)
√
2
exp(−x2
)
≈
δc
√
2π
σ2
1
σ3
0
exp

−
δ2
c
2σ2
0

(36)

Summary Conclusions
In the models of structure formation in which cold dark matter is the
dominating component, gravitational clustering takes place
hierarchically i.e., small structures form first and bigger structures
follow later.
It is important to understand what role substructures play in the
collapse of perturbations at larger scales.
We have found that in most cases the role of structure can be
neglected which follows the paradigm that transfer of power in
non-linear gravitational clustering is mostly from large to small scales.
In cosmological N-body simulations perturbations at sub-grid scales
are assumed to be zero. Our study can help to understand the effects
of pre-initial condition in cosmological N-body simulations.

Summary cont...
We have developed a prescription for estimating corrections in various
physical quantities (related to gravitational clustering) due to the
finite size of the simulation box.
We showed that the amplitude of clustering is underestimated at all
scales when the size of simulation volume is reduced.
We presented explicit expressions for corrections in mass function,
multiplicity function, number density and formation and destruction
rates of haloes.
We showed that the correction in mass function is maximum at the
scales of non-linearity and it decreases at large as well as small scales.
We showed that the number density of low mass haloes is
overestimated and that of large mass haloes is underestimated when
the size of the simulation box is reduced.

Thank You !

References
Aarseth, S. J. 2003, Gravitational N-Body Simulations ( Cambridge, UK:
Cambridge University Press, November 2003.)
Bagla, J. S. 2002, Journal of Astrophysics and Astronomy, 23, 185
Bagla, J. S., Padmanabhan, T. 1997, Pramana, 49, 161
Bagla, J. S., Prasad, J. 2006, Monthly Notices of Royal Aastron. Society, 370,
993
—. 2009, Monthly Notices of Royal Aastron. Society, 393, 607
Bagla, J. S., Prasad, J., Khandai, N. 2009, Monthly Notices of Royal Aastron.
Society, 395, 918
Bagla, J. S., Prasad, J., Ray, S. 2005, Monthly Notices of Royal Aastron.
Society, 360, 194
Bagla, J. S., Ray, S. 2003, New Astronomy, 8, 665
Barnes, J., Hut, P. 1986, Nature, 324, 446
Bertschinger, E. 1998, Ann. Rev. Astron. Astrophys., 36, 599
Efstathiou, G., Davis, M., White, S. D. M., Frenk, C. S. 1985, Astrophical
Journal Suppliment, 57, 241
Hockney, R. W., Eastwood, J. W. 1981, Computer Simulation Using Particles
(New York: McGraw-Hill, 1981)

Padmanabhan, T. 1993, Structure Formation in the Universe (Cambridge, UK:
Cambridge University Press, June 1993.)
Peebles, P. J. E. 1980, The large-scale structure of the universe (Princeton, N.J.,
Princeton University Press, 1980. 435 p.)
Prasad, J. 2007, Journal of Astrophysics and Astronomy, 28, 117
Press, W. H., Schechter, P. 1974, Astrophical Journal, 187, 425
Sellwood, J. A. 1987, Ann. Rev. Astron. Astrophys., 25, 151

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