How accurately do N body simulations reproduce the clustering of CDM?

Michael Joyce

LPNHE, Université Paris VI

Work in collaboration with:

T. Baertschiger (La Sapienza, Rome)A. Gabrielli (Istituto dei Sistemi Complessi-CNR , Rome)

B. Marcos & F. Sylos Labini (Centro E. Fermi, Rome)

Outline

Intro:Theory vs. simulation or what is the problem?Qualitative expectations or is it a real problem?

Systematic analytical approaches: (Initial conditions)Perturbative regime

Towards control of the non-linear regimeComments on numerical testingOther approaches

What is the problem?

N body simulations

are not a direct discretization

of the theoretical equations of motion

a “numerically perfect” simulation ≠ theory

What is the problem?

Theory (What one would like to simulate)

Purely self-gravitating microscopic particles (typically ~1070/[Mpc]3)

Treated statistically ---> Vlasov-Poisson equations for f(v,x,t)

“Collisionless” (mean field) limit

Fluid/continuum limit (appropriate N --> )

Physics: separation of scales

scales of discrete “graininess” << scales of clustering

What is the problem?

N body systems (What is in fact simulated)

Purely self-gravitating macroscopic particles (typically ~(1-100)/[Mpc]3)

Direct evolution under Newtonian self-gravity

Expanding background + small scale smoothing on force

What is the problem? The discreteness (finite N) problem

What is the relation between e.g. a correlation function or power spectrum

calculated from output of an NBS and the same quantity in the theory?

Answer: we don’t know !

Since theory is an appropriate N --> limit, the problem may be stated:

what are the corrections due to the use of finite N ?

Is it really a problem?

Is N ~ 1010 (e.g. “Millenium”) not enough?

Answer: it depends on what you want to resolve.

Simulators systematically make very optimistic assumptions

Surely simulators understand and control this?

Answer: No!

There are some (but very few) numerical studies.

In general only qualitative arguments for trusting results are given.

Is it really a problem? The issue of resolution

Unphysical characteristic scales are introduced by the “discretization”:

Interparticle separation l, force smoothing [and box size L, with N=(L/l)3]

Naively: fluid continuum limit for scales >> l

In practice: results are taken as physical (usually) down to , where << l

Why? This is the “interesting” regime (strongly non-linear)…

e.g. “Millenium” simulation: l ≈ 0,25 h-1 Mpc, ≈ 5 h-1 kpc

Is it justified? If so, what are errors?

Is it really a problem? Some common wisdom justifying this practice

Numerical tests show that results are robust to changes in N (---> l)

Some analytical “predictions” work well: notably

Press-Schecter formalism

Self-similar scaling for power law initial power spectra

Physics: “transfer of power to small scales is very efficient”

Is it really a problem? Caveats to this common wisdom

Numerical studies in the literature are

few and unsystematic (other parameters varied --- see below),

very limited range of l (at very most factor of 10, typically by 2)

do not agree (e.g. Melott et al.conclude that extrapolation is not justified)

Physics: PS, self-similarity --> structures form predominantly by collapse, with

linear theory setting the appropriate mass/time scales.

This does not establish validity of Vlasov/fluid description in non-linear regime.

Important: N independence does not imply Vlasov!

Is it really a problem? So..

Our understanding of this fundamental issue about NBS is, at best, qualitative

We need a “theory of discreteness errors” leading to:

A physical understanding of these effects

Methods for quantifying these effects (analytically or numerically)

Rest of talk: A problem in three parts

Initial conditions of simulations

The perturbative regime (up to “shell-crossing”)

The non-linear regime

Analytical approaches I Discreteness effects in initial conditions (IC)

IC are generated by displacing particles off a lattice (or “glass”) using

Zeldovich Approximation.

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QuickTime™ and aTIFF (LZW) decompressorare needed to see this picture.Input theoretical power spectrum

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power spectrum of lattice (or glass)

Analytical approaches I Full power spectrum of discrete IC

Theoretical correlation properties very well represented in reciprocal space for k<kN with discrete contribution at k>kN

In real space (e.g. mass variance) the relation is more complicated (discreteness terms are delocalized)--->In the limit of low amplitude (i.e. high initial red-shift), atfixed N, the real space properties are not represented accurately

Is this of dynamical importance?

Analytical approaches I Conclusions on discreteness in IC

Evolution of N body system can be solved perturbatively in displacements off the lattice

Gives discrete generalisation of Lagrangian perturbative theory for fluid.

---> Recover the fluid limit and study N dependent corrections to it

Analytical approaches II

Perturbative treatment of the N body problem

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Analytical approaches II

Linearisation of the N body problem

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Analytical approaches II

Linear evolution of displacement fields

Analytical approaches II

Eigenvalues for a simple cubic lattice

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Analytical approaches II

Growth of power in “particle linear theory”

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Analytical approaches II

Corrections in amplification due to discreteness

• Simulation begins at a=1 • Deviation from unity is the discreteness effect

Analytical approaches II

What we learn from this perturbative regime

Fluid evolution for a mode k recovered for kl << 1 i.e. as naively expected.

Exact fluid evolution is thus recovered by imposing a cut-off kC in the input

power spectrum, and taking kC l --> 0

Discreteness effects in this regime accumulate in time.

Taking initial red-shift zI -->, at fixed l, the simulation diverges from fluid

(--> zI is a relevant parameter for discreteness!)

These dynamical effects of discreteness are not two-body collision effects

Not analytically tractable (that’s why we use simulations!)

Need at least well defined numerical procedures to quantify discreteness

Some approaches towards understanding physics:

Detailed study of “simplified” simulations (e.g. “shuffled lattice”)

Rigorous studies of simplified toy models

(--> statistical physics of long range interactions)

Towards control on the non-linear regime

Increasing N to test for discreteness effects we should extrapolate towards the correct continuum limit.

Formally it is N --> i.e. l --> 0 (in units of box size)

What do we do with other relevant parameters: , zI, kC ?

(Non-unique) answer: keep them fixed (in units of box size for , kC )

Note: For robust conclusions on NBS we need to extrapolate to l << kC-1

l<< ---> large PM type simulations

Towards control on the non-linear regime

The continuum limit

Lattice with uncorrelated perturbations (<--> random error on

positions)

Power spectrum k2 at small k

Non-expanding space

Findings:

Self-similarity with temporal behaviour of fluid limit

Form of non-linear correlation function already defined in nearest

neighbour dominated (i.e. non-Vlasov) phase.

N body “coarse-grainings” only converge in continuum limit (as above)

Towards control on the non-linear regime

Study of “shuffled lattices”

N body simulators make very optimistic and rigorously unjustified

assumptions about extrapolation to theory

New formalism resolving the problem in the perturbative regime

(--> defined continuum limit, quantifiable error, “correction” of IC)

Physical effects of discreteness are more complex than two body

collisionality + sampling in IC

Numerical tests should extrapolate to continuum limit as defined.

Other numerical and analytical approaches necessary.

Conclusions

References

M. Joyce, B. Marcos, A. Gabrielli, T. Baertschiger, F. Sylos Labini Gravitational evolution of a perturbed lattice and its fluid limitPhys. Rev. Lett. 95:011334(2005)

B. Marcos, T. Baertschiger, M. Joyce, A. Gabrielli, F. Sylos Labini Linear perturbative theory of the discrete cosmological N body problemPhys.Rev. D73:103507(2006)

M. Joyce and B. Marcos, Quantification of discreteness effects in cosmological N body simulations. I: Initial conditionsPhys. Rev. D, in press,(2007)

M. Joyce and B. Marcos, Quantification of discreteness effects in cosmological N body simulations. II: Early time evolution.In preparation (astro-ph soon)

T. Baertschiger M. Joyce, A. Gabrielli, F. Sylos Labini Gravitational Dynamics of an Infinite Shuffled Lattice of Particles Phys.Rev. E , in press (2007)

T. Baertschiger M. Joyce, A. Gabrielli, F. Sylos Labini Gravitational Dynamics of an Infinite Shuffled Lattice: Particle Coarse-grainings, Non-linear Clustering and the Continuum Limit , cond-mat/0612594