Cosmological N-body Simulations
Transcript of Cosmological N-body Simulations
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Cosmological N-body Simulations
Julian Adamek
The IFT School on Cosmology Tools
Madrid, 16/03/2017
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Beyond the Linear Frontier
At late time (t & 1 Gyr) and on ānot too largeā 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.
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Beyond the Linear Frontier
At late time (t & 1 Gyr) and on ānot too largeā 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:
ā¢ Non-linear power spectra / correlation functions
ā¢ Structure and evolution of dark matter halos
ā¢ Local and integrated projection effects (Doppler RSD,gravitational lensing . . . )
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N-body Challenges
Cosmological N-body Simulation = full simulation of thenon-linear (gravitational) evolution of a N-body system
Challenges:
ā¢ Computationally expensive ā parallelization
ā¢ Too much information ā data reduction
ā¢ Competition between finite volume and finite resolution
ā¢ Validation of results ā convergence studies, codecomparison. . .
ā¢ Systematics due to unmodelled astrophysics
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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 Ļ
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. . . )
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Particle-Mesh (PM) Scheme
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Particle-Mesh (PM) Scheme
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Particle-Mesh (PM) Scheme
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Particle-Mesh (PM) Scheme
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Particle-Mesh (PM) Scheme
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Adaptive Mesh Refinement (AMR)
Idea: by subdividing cells, increaseresolution of PM grid in āinterestingāregions
ā¢ choose refinement criterion(e.g. density threshold)
ā¢ work out boundary conditionsat coarse/fine transition
ā¢ implement appropriatenumerical solvers (no FFT!)
ā¢ worry about load balance! credit: R. Teyssier
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Tree Algorithm
Idea: speed up computation of two-body forces by ālumping togetherāclouds of particles
ā¢ choose ātree opening angleā
ā¢ still needs āsoftening lengthā
ā¢ worry about load balance!
credit: University of Texas / Austin
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Initial DataSimulations are initialized at early time (typically redshift50ā100) where perturbation theory is still valid
ā¢ initial fluctuation amplitudes can be computed with aBoltzmann code (e.g. CAMB or CLASS)
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Initial DataSimulations are initialized at early time (typically redshift50ā100) where perturbation theory is still valid
ā¢ initial fluctuation amplitudes can be computed with aBoltzmann code (e.g. CAMB or CLASS)
Procedure:ā¢ set up homogeneous particle ensembleā¢ generate random realization of the perturbation field in
Fourier spaceā¢ Fourier transform to obtain displacementā¢ displace particles
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Initial DataSimulations are initialized at early time (typically redshift50ā100) where perturbation theory is still valid
ā¢ initial fluctuation amplitudes can be computed with aBoltzmann code (e.g. CAMB or CLASS)
Procedure:ā¢ set up homogeneous particle ensembleā¢ generate random realization of the perturbation field in
Fourier spaceā¢ Fourier transform to obtain displacementā¢ displace particles
Radiation is usually ignored (often even in the background!)ā¢ compute linear transfer function at redshift z=0 and scale
back using the appropriate growth functionā¢ can use Newtonian 2LPTJulian Adamek IFT School on Cosmology Tools 2017 7 / 16
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Post-processing: Power Spectra
PĪ“ cdm
+b(k)[M
pc3/h3]
PĪ“ Ī½(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
āmĪ½ = 0 meV
āmĪ½ = 200 meV
āmĪ½ = 200 meV
āmĪ½ = 200 meV
mĪ½ = 60 meV mĪ½ = 80 meV
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Post-processing: Halo Finder
Break down particle ensembleinto halos using halo finder algo-rithm
ā¢ friends-of-friends
ā¢ spherical overdensity
Halo catalog = huge data reduc-tion
ā¢ halo distribution (e.g.two-point statistics)
ā¢ individual halo properties(e.g. density profiles)
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Post-processing: Halo Finder
Break down particle ensembleinto halos using halo finder algo-rithm
ā¢ friends-of-friends
ā¢ spherical overdensity
Halo catalog = huge data reduc-tion
ā¢ halo distribution (e.g.two-point statistics)
ā¢ individual halo properties(e.g. density profiles)
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Post-processing: Ray Tracing
Write output in form of a lightcone (as opposed to equal-timesnapshot) ā one can constructobservables using ray tracing
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A Brief Overview of gevolution
spin-1 metric perturbationwith gevolution
gevolution, a general relativistic N-body code
ā¢ based on weak-field expansion (inPoisson gauge)
ā¢ for any given T ĀµĪ½ computes the six
metric d.o.f. (Ī¦, ĪØ, Bi, hij)
ā¢ N-body particle ensemble evolved usingrelativistic geodesic equation
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A Brief Overview of gevolution
spin-1 metric perturbationwith gevolution
gevolution, a general relativistic N-body code
ā¢ based on weak-field expansion (inPoisson gauge)
ā¢ for any given T ĀµĪ½ computes the six
metric d.o.f. (Ī¦, ĪØ, Bi, hij)
ā¢ N-body particle ensemble evolved usingrelativistic geodesic equation
Models beyond ĪCDM may have relativistic sources ofstress-energy perturbations
ā¢ Newtonian limit not always a good approximation
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A Brief Overview of gevolution
spin-1 metric perturbationwith gevolution
gevolution, a general relativistic N-body code
ā¢ based on weak-field expansion (inPoisson gauge)
ā¢ for any given T ĀµĪ½ computes the six
metric d.o.f. (Ī¦, ĪØ, Bi, hij)
ā¢ N-body particle ensemble evolved usingrelativistic geodesic equation
Models beyond ĪCDM may have relativistic sources ofstress-energy perturbations
ā¢ Newtonian limit not always a good approximation
Increasing data quality imposes new challenge to take intoaccount relativistic effects (e.g. in modelling RSD, WL. . . )
ā¢ perturbations of spacetime geometry are signal, not noise!
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A Brief Overview of gevolution
spin-1 metric perturbationwith gevolution
gevolution, a general relativistic N-body code
ā¢ based on weak-field expansion (inPoisson gauge)
ā¢ for any given T ĀµĪ½ computes the six
metric d.o.f. (Ī¦, ĪØ, Bi, hij)
ā¢ N-body particle ensemble evolved usingrelativistic geodesic equation
Models beyond ĪCDM may have relativistic sources ofstress-energy perturbations
ā¢ Newtonian limit not always a good approximation
Increasing data quality imposes new challenge to take intoaccount relativistic effects (e.g. in modelling RSD, WL. . . )
ā¢ perturbations of spacetime geometry are signal, not noise!
https://github.com/gevolution-code/gevolution-1.1.git
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Strategy
ā¢ choose ansatz for the metric (perturbed FLRW)
ds2=a2(Ļ)[
āe2ĪØdĻ2+ eā2Ī¦Ī“ijdxidxj+ hijdx
idxjā 2BidxidĻ
]
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Strategy
ā¢ choose ansatz for the metric (perturbed FLRW)
ds2=a2(Ļ)[
āe2ĪØdĻ2+ eā2Ī¦Ī“ijdxidxj+ hijdx
idxjā 2BidxidĻ
]
ā¢ metric components are evolved with Einsteinās equations
GĀµĪ½ = 8ĻGT Āµ
Ī½
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Strategy
ā¢ choose ansatz for the metric (perturbed FLRW)
ds2=a2(Ļ)[
āe2ĪØdĻ2+ eā2Ī¦Ī“ijdxidxj+ hijdx
idxjā 2BidxidĻ
]
ā¢ metric components are evolved with Einsteinās equations
GĀµĪ½ = 8ĻGT Āµ
Ī½
ā¢ stress-energy tensor is determined by solving the EOMās ofall sources of stress-energy
TĀµĪ½m =
ā
nm(n)
Ī“(3)(xāx(n))ā
āg
(
āgĪ±Ī²dxĪ±
(n)
dĻ
dxĪ²
(n)
dĻ
)
ā12 dxĀµ
(n)
dĻ
dxĪ½(n)
dĻ
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Design Principles
We use the LATField2 libraryas data handling / parallelizationback end.
ā¢ metric field represented ona regular lattice
ā¢ Fourier analysis possible(LATfield2 provides FFT)
dim=0
dim=1
dim=2
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Design Principles
We use the LATField2 libraryas data handling / parallelizationback end.
ā¢ metric field represented ona regular lattice
ā¢ Fourier analysis possible(LATfield2 provides FFT)
dim=0
dim=1
dim=2
The front end / user interface borrows a lot from CLASS
ā¢ code can be directly interfaced with CLASS!
ā¢ use unified notation!
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k [h/Mpc] k [h/Mpc] k [h/Mpc]
ā(k)
Ī¦
Ī¦-ĪØ
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