1. 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)

2. 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 regime  Comments on numerical testing  Other approaches

3. What is the problem? N body simulations are not a direct discretization of the theoretical equations of motion a “numerically perfect” simulation ≠ theory

4. What is the problem? Theory (What one would like to simulate)  Purely self-gravitating microscopic particles (typically ~10 70 /[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

5. 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

6. 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 ?

7. Is it really a problem? Is N ~ 10 10 (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.

8. 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?

9. 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”

10. 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!

11. 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)

12. Rest of talk: A problem in three parts  Initial conditions of simulations  The perturbative regime (up to “shell-crossing”)  The non-linear regime

13. Analytical approaches I Discreteness effects in initial conditions (IC) IC are generated by displacing particles off a lattice (or “glass”) using Zeldovich Approximation.

14. Input theoretical power spectrum Convolution term (linear in P th ) power spectrum of lattice (or glass) Analytical approaches I Full power spectrum of discrete IC

16. Theoretical correlation properties very well represented in reciprocal space for k k N 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), at fixed N, the real space properties are not represented accurately Is this of dynamical importance? Analytical approaches I Conclusions on discreteness in IC

17. 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

18. Analytical approaches II Linearisation of the N body problem

19. Analytical approaches II Linear evolution of displacement fields

20. Analytical approaches II Eigenvalues for a simple cubic lattice

21. Analytical approaches II Growth of power in “particle linear theory”

22. Analytical approaches II Corrections in amplification due to discreteness Simulation begins at a=1 Deviation from unity is the discreteness effect

23. 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 k C in the input power spectrum, and taking k C l --> 0  Discreteness effects in this regime accumulate in time.  Taking initial red-shift z I --> , at fixed l, the simulation diverges from fluid (--> z I is a relevant parameter for discreteness!)  These dynamical effects of discreteness are not two-body collision effects

24. 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

25. 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: , z I, k C ? (Non-unique) answer: keep them fixed (in units of box size for , k C ) Note: For robust conclusions on NBS we need to extrapolate to l << k C -1 l large PM type simulations Towards control on the non-linear regime The continuum limit

26.  Lattice with uncorrelated perturbations ( random error on positions)  Power spectrum  k 2 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”

27.  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

28. References  M. Joyce, B. Marcos, A. Gabrielli, T. Baertschiger, F. Sylos Labini Gravitational evolution of a perturbed lattice and its fluid limit Phys. 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 problem Phys.Rev. D73:103507(2006)  M. Joyce and B. Marcos, Quantification of discreteness effects in cosmological N body simulations. I: Initial conditions Phys. 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