Prsentation PowerPoint

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Cosmological Simulations of Structure Formation
and the Vlasov Equation
Michael Joyce LPNHE Université de Paris VI
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Work in collaboration with
T. Baertschiger (La Sapienza, Roma) A.
Gabrielli (ISC, Roma), B. Marcos (La Sapienza,
Roma) F. Sylos Labini (Centro Fermi, Roma)
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Outline

• The problem of structure formation in cosmology
• Cosmological N body simulation (NBS)
• The problem of discreteness in NBS
• Conclusion

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I. Structure formation in cosmology

• There is a now widely accepted and highly
  successful  standard  model in cosmology
  Lambda Cold Dark Matter
• Uniformly expanding universe perturbations
• Perturbations at early times strongly constrained
  by measurements of fluctuations in Cosmic
  Microwave Background,
• initial small fluctuations ---gt very
  structured distribution of galaxies (and other
  objects) today by gravitational instability

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CMB t105yrs
galaxies t1010 yrs
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CDM and the Vlasov Equation

• Dark Matter particle-like matter
  interacting only by gravity
• Cold very low velocity dispersion --gt
  non-relativistic
• Length scales are those at which gravity is
  Newtonian
• (expansion gives a simple modification of eom)
• ---gt
• Dynamics of Newtonian particles in an infinite
  space starting from quasi-uniform initial
  conditions.
• Particles are microscopic i.e. very many
  particles in regions of size we are interested in
• ---gt (??)
• CDM described by Vlasov-Poisson equations

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Derivations of Vlasov-Poisson system

• Qualitatively using estimates of collision times
  (e.g. Binney Tremaine)
• Truncation of BBGKY hierarchy (e.g. Peebles,
  Saslaw)
• Coarse-graining of spiky one particle particle
  distribution function (cf. Buchert Dominguez)
• None of these derivations are rigorous in
  particular 2 3 do not establish whether (and
  when) the additional terms can be neglected.
• Let us assume it is valid.

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II. Cosmological N body Simulation

• Aim to solve evolution of CDM
• But direct solution of Vlasov-Poisson system is
  not numerically feasible gravity forms
  structures by collapse, and we thus need to
  follow a range of scale in 6-d phase space
• Instead solve the N-body problem directly but for
• N ltlt real number of CDM
  particles
• N 105 in 1980s ----gt 1010 currently ltlt
  1080 real

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Cosmological N body Simulation methods

• N particles in a cubic box periodic boundary
  conditions
• Newtonian potential with regularization of r0
  singularity
• Simple change of coordinates to include expansion
  of background
• Initial conditions particles slightly displaced
  off a perfect lattice (see below)
• Many different numerical implementations of force
  calculation (PM, P3M, tree codes.)

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III. The problem of discreteness in NBS

• Finite N simulation lt----gt Vlasov-Poisson(VP)
  equations
• In what limit do NBS reproduce VP ?
• What are the corrections at finite N to VP, at a
  given spatial scale and time ?
• These questions are important both conceptually
  and practically observations in the coming years
  will require great precision on the results of
  NBS.
• There are currently only very qualitative and
  ambiguous answers to them in the literature on
  NBS

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• (The problem of discreteness in NBS)
• The problem is naturally approached in three
  parts
• Initial conditions
• Early time evolution
• C. Non-linear evolution

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III.A Discreteness in Initial Conditions
Theoretical IC specified entirely by power
spectrum of density fluctuations P(k)
Zeldovich approximation gives displacements
and velocities of fluid elements in a perturbed
self-gravitating fluid. N-body
discretisation Uniform fluid elements ----gt N
particles on a lattice The NBS then has IC whose
correlation properties are a convolution of
those of theoretical model and those oflattice.

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(Discreteness in Initial Conditions) A
characteristic length scale l ( lattice
spacing) is introduced. Expect to recover
theoretical correlation properties for length
scales gtgtl, and for wavenumbers ltlt l-1 Result
Algorithm produces reciprocal space properties
very accurately, but not necessarily real space
ones. In the limit of low amplitude of the input
fluctuations real space properties are not
represented accurately. Is this of dynamical
importance?
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III.B Discreteness at Early Times

• Early times ? Displacements to lattice are
  small
• ---gt perturbative treatment completely analogous
  to that used in studying crystals (eigenmodes of
  displacements etc.)
• Main results (cf. Talk of B. Marcos)
• Known perturbative results in fluid
  limit(truncated VP) recovered when kl --gt 0
  (kwave number, llattice spacing)
• Temporal evolution of finite fixed N system
  diverges from that of Vlasov-Poisson system.
• Divergence is exponential, with an exponent which
  depends on length scale.
• Discreteness effect is due to sparseness, not
  collisionality.

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III.C Discreteness in the non-linear regime

• Perturbative approach breaks downthe full
  non-linear problem is very difficultmore
  difficult than the Vlasov-Poisson system which we
  cant solve to start with!
• Aim to
• give numerical procedures for testing for
  discreteness
• (cf. talk of B. Marcos)
• define the convergence problem clearly (i.e. what
  to keep fixed when one changes N)
• test for specific effects which should be
  negligible in the Vlasov limit e.g. the role
  played by two body interactions

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(Discreteness in non-linear regime) Some
results Immediately after perturbative regime,
forces on particles in a perturbed lattice become
strongly dominated by a single nearest
neighbour(NN). Correlations which develop very
well approximated for a time taking into account
only this NN interaction ---gt system is thus not
Vlasov-like in the relevant range of temporal and
spatial scales. Vlasov-like at longer times?
Perhaps, but we have found that the correlation
function which develops with NN interactions
strikingly resembles that at longer times.
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Conclusion

• Summary
• Cosmologists wish to solve a coupled set of VP
  equations
• They attempt to do so indirectly through finite N
  simulation.
• Issue of the relation between the two is poorly
  understood, but some progress has been made.
• Important not just conceptually but also
  practically cosmologists need to know now how
  precise their simulations are.
• Remarks/questions
• Rigorous derivation of Vlasov-Poisson equation?
• Study of other systems/toy models
• Simulate Vlasov-Poisson directly?

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Common wisdom on discreteness

• Varying N gives approximately same results. But
  (i) modest range of N, and (ii) not all studies
  agree
• Gravity gives a top-down dynamics which tends
  to wipe out effects of discreteness.
• Dynamics manifest self-similar (scaling)
  solutions from scale-free initial conditions,
  which are manifestly independent of discreteness
  scale.