Cosmological n-body simulations

Laszlo DobosDept. of Physics of Complex Systems

dobos@complex.elte.huE 5.60

April 30, 2018

Inhomogeneities in the universe

At the time of decoupling of cosmic mw background radiation(z ≈ 1100)

I density fluctuations are very small ∆ρρ ∼ 10−5

I how did the the galaxies of today form? ∆ρρ ∼ 106

Gravity on small scales quickly becomes non-linear

I the more matter gathers around the same spot, the deeper thepotential gets

I it can gather even more matter

Follow the evolution of structure theoretically

I gravity is still linear in the early universe

I perturbations of the FLRW metric to first order

I later: ?? not exact general relativistic method

Description of the non-linear evolution

Einstein’s equations → Newtonian gravity

I can be linearized: gµν = ηµν + hµνI hµν is a small perturbation

I weak gravitational force approximation

I works only if peculiar velocities are small

Non-linear structure formation is followed in Newtonianapproximation

I cannot be handled in detail analytically

I numerical approximation: N-body simulations

Cosmological n-body simulations

Non-linear structure formation cannot be described analytically

I gravitating “particles” in an expanding universe

I pressureless liquid

I core simulation dark matter only, galaxies “need to be put in”

Can be connected with hydrodinamical codes

I take effect of baryonic matter, pressure into account

I star formation, galaxy evolution, supernovae

I AGN feedback

A few important simulations

I Millennium, Millennium II, Millennium XXL (Springel et al.)

I Bolshoi (Klipyn et al.)

I Indra (Szalay et al.)

I Illustris

Simulations have to be very large

Typical sizes required to simulate a single supercluster:

I 1010 ,,particles”

I particle mass in 105-1010 M� range

I simulation box size in 75-500 Mpc range

I periodic boundary conditions

I number of dark matter halos forming by z = 0 is ∼ 20 million

Computation cost requires computing clusters

I 10000 CPU core running for months

I millions of CPU hours, 100s of CPU years

I 5-20 TB memory

I very fast network

I simulation output is in the 10–100 TB range

Computing the gravitational force

Number of particles is very high: ∼ 1010

I direct force computation would require 1020 distances

I particles don’t fit into a single machine’s memory

I all coordinates would need to be communicated over thenetwork

I not feasible, would be extremely slow

Idea: average out force from far away particles

I divide simulation box into subsets

I nearby particles are almost always in the same machine’smemory

I compute force from nearby particles exactly

I averaged out far away force is communicated over the network

Simulation at different time steps

Generating initial conditions

Mass points are randomly distributed with prescribed correlations

I P(k) ∼ kn power spectrum is given

I phase of planar waves is chosen randomly

I need a relaxed configuration at start

Universe doesn’t have an edge

I simulation with periodic boundary conditions

I what goes out on one side comes in the opposite side

I box size affects small wave numbers (corner modes)

The universe expands

I particle coordinates and velocities are rescaled

I must put in cosmology explicitly

Motion of particlesDark matter is never turbulent, no pressure, no viscosity

I particle trajectories never cross on large scales

Aragon-Calvo et al. (2011)

Identifying dark matter halosSimulation provides the phase space parameters at a few timesteps

I position and velocity for each particle

I only a few time steps can be saved to disk

Need to identify dark matter halos in simulation output

I use same friend-of-friend algorithm as for cluster finding

I two particles closer than dcut are in same cluster

I match merging halos from time step to time step

Can be determined

I density profile of halos: NFW profile

I mass distribution of halos, sub halos

I velocity dispersion of subhalos

I evolution of mass distribution

I merger hierarchy of halos

Can only be investigated from simulations

Spatial shape and 3D velocity field of halos

I observations provide line of sight Doppler shifts only

I all galaxies of a cluster are at the same z , no third coordinate

Number and mass distribution of sub halos

I only a handful of satellites around the Milky Way

I we can’t observe them around extragalaxies

I simulations provide an estimate on their number andimportance

Merger tree of halos

I can follow merging of halos in time

I detailed view on the hierarchical formation of structure

I can follow cluster formation back in time

Merger tree of halos

Modelling galaxies in DM simulations

Simpler simulations compute dark matter only

I we can put galaxies inside halo centers

I amount of baryonic matter depends on a chosen model

I just like total mass of stars

Halo occupation distribution (HOD)

I what is the distribution of halo-galaxy mass ratio

I what mass and how many galaxies form in a halo of givenmass

I how is baryonic matter distributed wrt. dark matter(baryonic matter has pressure, can decouple from DM)

I velocity of baryonic matter wrt. DM

Semi-analytic modelling of galaxies

We know the merger tree of halos

I when two halos merge, the galaxies inside them do too

I star-burst period followed by AGN activity

Simulate the stellar population of the galaxy

I merger events determine star formation history Ψ(t)[M� yr−1]

I star-formation rate as a function of cosmic time

I we can model the spectrum of the galaxy from Ψ(t)

Simulated catalogs and light cones

Result is a semi-analytic mock catalog

I as if it was a real sky survey

I magnitudes, colors, redshifts

Simulation is only saved at predefined values of z (snapshots)

I farther galaxies need to be taken from earlier time steps

I redshifting and evolution can also be simulated this way

Semi-analytic models for supermassive black holes

I also start from the merger tree

I mass is summed together at every merger + accretion

Light cones: stitching different redshifts together