How smooth are particle trajectories in a $\Lambda$CDM Universe?

Abstract

It is shown here that in a flat, cold dark matter (CDM) dominated Universe with positive cosmological constant ($\Lambda$), 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-coordinates formulation of the Euler-Poisson equations is employed, originally used by Cauchy for 3-D incompressible flow. Temporal analyticity for $\Lambda$CDM 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 $\Lambda$ and on the initial spatial smoothness of the density field. The largest time interval is achieved when $\Lambda$ 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.

arXiv comment16 pages, 4 figures, published in MNRAS, this paper introduces a convergent formulation of Lagrangian perturbation theory for LCDM

References in corpus (30)

• Planck 2013 results. XVI. Cosmological parameters
• Large-Scale Structure of the Universe and Cosmological Perturbation Theory
• An analytic model for the spatial clustering of dark matter haloes
• A measurement by BOOMERANG of multiple peaks in the angular power spectrum of the cosmic microwave background
• Transients from Initial Conditions in Cosmological Simulations
• Resumming Cosmological Perturbations via the Lagrangian Picture: One-loop Results in Real Space and in Redshift Space
• Nonlinear perturbation theory with halo bias and redshift-space distortions via the Lagrangian picture
• Reconstruction of the early Universe as a convex optimization problem
• A reconstruction of the initial conditions of the Universe by optimal mass transportation
• Newtonian Cosmology in Lagrangian Formulation: Foundations and Perturbation Theory
• Lagrangian Dynamics in Non-Flat Universes and Non-Linear Gravitational Evolution
• Post-Newtonian Cosmological Dynamics in Lagrangian coordinates
• Lagrangian perturbations and the matter bispectrum I: fourth-order model for non-linear clustering
• Lagrangian perturbation theory at one loop order: successes, failures, and improvements
• Time-analyticity of Lagrangian particle trajectories in ideal fluid flow
• Behaviour of Lagrangian Approximations in Spherical Voids
• Lagrangian theory of structure formation in relativistic cosmology II: average properties of a generic evolution model
• The recursion relation in Lagrangian perturbation theory
• Cauchy's almost forgotten Lagrangian formulation of the Euler equation for 3D incompressible flow
• Lagrangian theory of structure formation in relativistic cosmology I: Lagrangian framework and definition of a nonperturbative approximation
• Lagrangian perturbations and the matter bispectrum II: the resummed one-loop correction to the matter bispectrum
• Lagrangian theory of structure formation in relativistic cosmology III: gravitoelectric perturbation and solution schemes at any order
• Zel'dovich approximation and General Relativity
• The Monge-Ampere equation: various forms and numerical methods
• On Second-order Perturbation Theories of Gravitational Instability in Friedmann-Lemaitre Models
• Relativistic Lagrangian displacement field and tensor perturbations
• Frame dragging and Eulerian frames in General Relativity
• Cosmological Density Perturbations with a Scale-Dependent Newton's G
• Evolution of density perturbations in a realistic universe
• Lagrangian Perturbation Theory : Exact One-Loop Power Spectrum in General Dark Energy Models

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• Redshift-Space Distortions in Lagrangian Perturbation Theory
• Accurate initial conditions for cosmological N-body simulations: Minimizing truncation and discreteness errors
• Higher-order initial conditions for mixed baryon-CDM simulations
• Geometric formulation of the Cauchy invariants for incompressible Euler flow in flat and curved spaces
• An $n$-th order Lagrangian Forward Model for Large-Scale Structure
• Non-linear Fields in Generalized Cosmologies
• Higher-order initial conditions with massive neutrinos
• Shell-crossing in quasi-one-dimensional flow
• Cosmological $N$-body simulations including radiation perturbations
• Lagrangian cosmological perturbation theory at shell-crossing
• Cosmological perturbations for two cold fluids in $\Lambda$CDM
• Semiclassical path to cosmic large-scale structure
• The Cauchy-Lagrangian method for numerical analysis of Euler flow
• Shell-crossing in a $\Lambda$CDM Universe
• Unveiling the singular dynamics in the cosmic large-scale structure
• Cosmological Vlasov-Poisson equations for dark matter: Recent developments and connections to selected plasma problems
• Lagrangian theory for cosmic structure formation with vorticity: Newtonian and post-Friedmann approximations
• Perturbation theory of LSS in the $\Lambda$CDM Universe: exact time evolution and the two-loop power spectrum
• Integrated perturbation theory for cosmological tensor fields. I. Basic formulation
• Quasi-spherical collapse of matter in $\Lambda$CDM
• Eye of the Tyger: early-time resonances and singularities in the inviscid Burgers equation
• Cold dark matter protohalo structure around collapse: Lagrangian cosmological perturbation theory versus Vlasov simulations
• A constructive approach to regularity of Lagrangian trajectories for incompressible Euler flow in a bounded domain
• Analytical growth functions for cosmic structures in a $\Lambda$CDM Universe
• Renormalization group and UV completion of cosmological perturbations: Gravitational collapse as a critical phenomenon
• On general-relativistic Lagrangian perturbation theory and its non-perturbative generalization
• The gravitational force field of proto-pancakes
• Fast and accurate collapse-time predictions for collisionless matter
• Stability analysis of fluid flows using Lagrangian Perturbation Theory (LPT): application to the plane Couette flow
• Post-collapse Lagrangian perturbation theory in three dimensions