1. Simulating the Universe
Andreas Faltenbacher
Cape Town International Cosmology Summer School, 23. January 2012

2. Outline:
• Celestial peace
• Gravitation
• Initial conditions
• Background cosmology
• From dark to light

3. Celestial peace

4. Dunhuang Star Chart from Tang Dynasty (618 - 907)
Orion didn’t move much the last 1200 years

5. Dunhuang Star Chart from Tang Dynasty (618 - 907)
Orion didn’t move much the last 1200 years

6. We hardly see motion in astronomical observations
but if Big Bang Theory is correct
objects must have formed at some point
Simulations are the only tool to
directly investigate the evolution
of the Universe and it’s constituents

7. Gravitation

8. Erik Holmberg 1941: Replacing gravitation by light
P
Intensity (Power per unit Area): I = 2πr2

9. The 1/r2 law for ∼homogeneous distributions
Impact of individual spheres is ≈equal

10. The 1/r2 law for ∼homogeneous distributions
Impact of individual spheres is ≈equal

11. Simulations of gravitationally interacting N-body systems:
The long range nature of gravity requires a double sum over all
interacting objects ⇒ N 2 problem

12. Can energy loss due to tides cause capture ?
hyperbolic orbits & tidal friction ⇒ capture

13. Initial conditions
how to get the fluctuation spectrum right

14. Aarseth 1963
How to simulate an irregular cluster ?

15. Peebles 1970: Top hat collapse
density profiles of clusters too steep

16. White 1976: expanding initial conditions
700 particles representing galaxies with different masses

17. White 1976: expanding initial conditions
O > M > H > ∗, too much mass segregation

18. Poisson (P (k) = const.) observed power spectrum

19. Aarseth, Gott & Turner 1979: Cosmic density field
In order to generate fluctuations with power spectrum,
P (k) ∝ k−1, particles are placed along rods

20. Aarseth, Gott & Turner 1979: Cosmic density field
In order to generate fluctuations with power spectrum,
P (k) ∝ k−1, particles are placed along rods

21. Aarseth, Gott & Turner 1979: Cosmic density field
In order to generate fluctuations with power spectrum,
P (k) ∝ k−1, particles are placed along rods

22. Klypin & Shandarin 1983:
323 particle, 160 Mpc/h box, Zel’dovich approximation, FFT

23. Klypin & Shandarin 1983:
323 particle, 160 Mpc/h box, Zel’dovich approximation, FFT

24. Klypin & Shandarin 1983:
323 particle, 160 Mpc/h box, Zel’dovich approximation, FFT

25. Current approach:
• Compute initial power spectrum CMB-
FAST, CAMB, CMBeasy, ...
• Generate a random realization of the
density field in k-space
• Do Fourier transform to get real space
density fluctuations
• Apply Zel’dovich approximation to obtain
initial positions and velocities of simula-
tion particles

26. Initial power spectrum & transfer function
P (k) = |δ(k)|2
δ(r) = δ(k) exp(−ikr)dk
ρ(r) − ρ
¯
δ(r) =
ρ
¯
Bardeen, Bond, Kaiser & Szalay 1986

27. Zel’dovich approximation: r(q, t) = a(t)[q + b(t)s(q)]
s(q) = Φ0(q)
Edmund Bertschinger’s COSMICS package (http://web.mit.edu/edbert/)

28. Springel at al. 2005 :
as time went by ...

29. Background cosmology
Newtonian gravity on expanding background

30. The collosionless Boltzmann equation (Vlasov equation) for the
dark matter distribution function, f , in comoving coordinates x:
f = f (x, x, t)
˙
∂f ∂f ∂f
+ x
˙ − φ = 0, p = a2x,
˙
∂t ∂x ∂p
2 φ = 4πGa2 (ρ(x, t) − ρ) = 4πGa2 Ω
¯ dm δρcr

31. The solution of the Vlasov equation can be written in terms of
equations for characteristics, which look like equations of parti-
cle motion:
dp φ dv a
˙ φ
= − , +2 v = − 3
da a
˙ dt a a
dx p dx
= 2
, = v
da aa
˙ dt
2 φ = 4πGΩ δρ
0 cr,0 /a, φ = aφ
1
a = H0 1 + Ω 0
˙ − 1 + ΩΛ a2 − 1
a

32. Mare Nostrum Universe: 100 Mpc/h
10243 particles, 500 Mpc/h, mDM = 8.24 × 109h−1M
credit: Arman Khalatyan et al.

33. Mare Nostrum Universe: 20 Mpc/h
10243 particles, 500 Mpc/h, mDM = 8.24 × 109h−1M
credit: Arman Khalatyan et al.

34. Mare Nostrum Universe:
10243 particles, 500 Mpc/h, mDM = 8.24 × 109h−1M
credit: Arman Khalatyan et al.

35. From dark to light
adding baryons

36. Mare Nostrum Universe: Adiabatic Hydrodynamics
10243 particles, 500 Mpc/h, mgas = 1.45 × 109h−1M
credit: Arman Khalatyan et al.

37. Mare Nostrum Universe: Adiabatic Hydrodynamics
10243 particles, 500 Mpc/h, mgas = 1.45 × 109h−1M
credit: Arman Khalatyan et al.

38. Mare Nostrum Universe: Adiabatic Hydrodynamics
10243 particles, 500 Mpc/h, mgas = 1.45 × 109h−1M
credit: Arman Khalatyan et al.

39. Other recipes to take baryons into account:
• Full astro-hydrodynamics, including
cooling, feed back, etc.
• Semi-analytical approach
• Halo occupation distribution, abundance
matching

40. Guedes 2011: Succeeded to simulate a realistic disk
15 kpc 0.3 0.7

41. ... how far are we from ...