Cosmological N-body Simulations

Cosmological N-body Simulations

Julian Adamek

The IFT School on Cosmology Tools

Madrid, 16/03/2017

Beyond the Linear Frontier

At late time (t & 1 Gyr) and on “not too large” scales (r . 100Mpc) the Universe is clumpy. A linear treatment is insufficient.

This so-called large-scale structure (LSS) contains a hugeamount of information that we want to harness with the nextgeneration of telescopic surveys.

Julian Adamek IFT School on Cosmology Tools 2017 1 / 16

Beyond the Linear Frontier

At late time (t & 1 Gyr) and on “not too large” scales (r . 100Mpc) the Universe is clumpy. A linear treatment is insufficient.

This so-called large-scale structure (LSS) contains a hugeamount of information that we want to harness with the nextgeneration of telescopic surveys.

Things we may be interested in:

• Non-linear power spectra / correlation functions

• Structure and evolution of dark matter halos

• Local and integrated projection effects (Doppler RSD,gravitational lensing . . . )


N-body Challenges

Cosmological N-body Simulation = full simulation of thenon-linear (gravitational) evolution of a N-body system

Challenges:

• Computationally expensive → parallelization

• Too much information → data reduction

• Competition between finite volume and finite resolution

• Validation of results → convergence studies, codecomparison. . .

• Systematics due to unmodelled astrophysics


Choose the Right Tool

gevolution Gadget-2 RAMSESparadigm particle-mesh tree / tree-PM particle-meshresolution fixed adaptive (tree) adaptive (AMR)

hydro no SPH Cartesian FVMgravity metric (GR) Newtonian F Newtonian ψ

neutrinos yes (no) nolanguage C++ ANSI C Fortran 90

release date 2016 2005 2008

There are many more N-body codes on the market (Enzo,pkdgrav, CubePM. . . )


Particle-Mesh (PM) Scheme










Adaptive Mesh Refinement (AMR)

Idea: by subdividing cells, increaseresolution of PM grid in “interesting”regions

• choose refinement criterion(e.g. density threshold)

• work out boundary conditionsat coarse/fine transition

• implement appropriatenumerical solvers (no FFT!)

• worry about load balance! credit: R. Teyssier


Tree Algorithm

Idea: speed up computation of two-body forces by “lumping together”clouds of particles

• choose “tree opening angle”

• still needs “softening length”

• worry about load balance!

credit: University of Texas / Austin


Initial DataSimulations are initialized at early time (typically redshift50–100) where perturbation theory is still valid

• initial fluctuation amplitudes can be computed with aBoltzmann code (e.g. CAMB or CLASS)




Procedure:• set up homogeneous particle ensemble• generate random realization of the perturbation field in

Fourier space• Fourier transform to obtain displacement• displace particles




Procedure:• set up homogeneous particle ensemble• generate random realization of the perturbation field in

Fourier space• Fourier transform to obtain displacement• displace particles

Radiation is usually ignored (often even in the background!)• compute linear transfer function at redshift z=0 and scale

back using the appropriate growth function• can use Newtonian 2LPTJulian Adamek IFT School on Cosmology Tools 2017 7 / 16

Post-processing: Power Spectra

Pδ cdm

+b(k)[M

pc3/h3]

Pδ ν(k)[M

pc3/h3]

gevolutionCLASS

z = 63z = 31z = 15z = 7z = 3z = 1z = 0

100

100

10

10

1

1

1

1

1

1

0.1

0.1

0.1

0.1

0.1

0.1

0.01

0.01

0.01

0.01

0.01

0.01

0.001

0.001

∑mν = 0 meV

∑mν = 200 meV

∑mν = 200 meV

∑mν = 200 meV

mν = 60 meV mν = 80 meV


Post-processing: Halo Finder

Break down particle ensembleinto halos using halo finder algo-rithm

• friends-of-friends

• spherical overdensity

Halo catalog = huge data reduc-tion

• halo distribution (e.g.two-point statistics)

• individual halo properties(e.g. density profiles)


Post-processing: Halo Finder

Break down particle ensembleinto halos using halo finder algo-rithm

• friends-of-friends

• spherical overdensity

Halo catalog = huge data reduc-tion

• halo distribution (e.g.two-point statistics)

• individual halo properties(e.g. density profiles)


Post-processing: Ray Tracing

Write output in form of a lightcone (as opposed to equal-timesnapshot) → one can constructobservables using ray tracing


A Brief Overview of gevolution

spin-1 metric perturbationwith gevolution

gevolution, a general relativistic N-body code

• based on weak-field expansion (inPoisson gauge)

• for any given T µν computes the six

metric d.o.f. (Φ, Ψ, Bi, hij)

• N-body particle ensemble evolved usingrelativistic geodesic equation









Models beyond ΛCDM may have relativistic sources ofstress-energy perturbations

• Newtonian limit not always a good approximation











Increasing data quality imposes new challenge to take intoaccount relativistic effects (e.g. in modelling RSD, WL. . . )

• perturbations of spacetime geometry are signal, not noise!











Increasing data quality imposes new challenge to take intoaccount relativistic effects (e.g. in modelling RSD, WL. . . )

• perturbations of spacetime geometry are signal, not noise!

https://github.com/gevolution-code/gevolution-1.1.git


Strategy

• choose ansatz for the metric (perturbed FLRW)

ds2=a2(τ)[

−e2Ψdτ2+ e−2Φδijdxidxj+ hijdx

idxj− 2Bidxidτ

]


Strategy


ds2=a2(τ)[


idxj− 2Bidxidτ

]

• metric components are evolved with Einstein’s equations

Gµν = 8πGT µ

ν


Strategy


ds2=a2(τ)[


idxj− 2Bidxidτ

]

• metric components are evolved with Einstein’s equations

Gµν = 8πGT µ

ν

• stress-energy tensor is determined by solving the EOM’s ofall sources of stress-energy

Tµνm =

∑

nm(n)

δ(3)(x−x(n))√

−g

(

−gαβdxα

(n)

dτ

dxβ

(n)

dτ

)

−12 dxµ

(n)

dτ

dxν(n)

dτ


Design Principles

We use the LATField2 libraryas data handling / parallelizationback end.

• metric field represented ona regular lattice

• Fourier analysis possible(LATfield2 provides FFT)

dim=0

dim=1

dim=2


Design Principles

We use the LATField2 libraryas data handling / parallelizationback end.

• metric field represented ona regular lattice

• Fourier analysis possible(LATfield2 provides FFT)

dim=0

dim=1

dim=2

The front end / user interface borrows a lot from CLASS

• code can be directly interfaced with CLASS!

• use unified notation!


k [h/Mpc] k [h/Mpc] k [h/Mpc]

∆(k)

Φ

Φ-Ψ

hij

B

z = 3 z = 1 z = 0

10

10

10

10

10

10

10

10

1 1 10.1 0.1 0.10.01 0.01 0.01

-10

-12

-14

-16

-18

-20

-22

-24

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Cosmological N-body Simulations