How smooth are particle trajectories in a ΛCDM universe?

Cornelius Rampf, Barbara Villone, Uriel Frisch

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It is shown here that in a flat, cold dark matter (CDM)-dominated Universe with positive cosmological constant (Λ), modelled in terms of a Newtonian and collisionless fluid, particle trajectories are analytical in time (representable by a convergent Taylor series) until at least a finite time after decoupling. The time variable used for this statement is the cosmic scale factor, i.e. the 'a-time', and not the cosmic time. For this, a Lagrangian-coordinate formulation of the Euler-Poisson equations is employed, originally used by Cauchy for 3D incompressible flow. Temporal analyticity for ΛCDM is found to be a consequence of novel explicit all-order recursion relations for the a-time Taylor coefficients of the Lagrangian displacement field, from which we derive the convergence of the a-time Taylor series. A lower bound for the a-time where analyticity is guaranteed and shell-crossing is ruled out is obtained, whose value depends only on Λ and on the initial spatial smoothness of the density field. The largest time interval is achieved when Λ vanishes, i.e. for an Einstein-de Sitter universe. Analyticity holds also if, instead of the a-time, one uses the linear structure growth D-time, but no simple recursion relations are then obtained. The analyticity result also holds when a curvature term is included in the Friedmann equation for the background, but inclusion of a radiation term arising from the primordial era spoils analyticity.

Original languageEnglish
Pages (from-to)1421-1436
Number of pages16
JournalMonthly Notices of the Royal Astronomical Society
Issue number2
Early online date14 Jul 2015
Publication statusPublished - 11 Sept 2015


  • cosmology: theory
  • dark energy
  • dark matter
  • large scale structure of universe


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