Lagrangian Cosmological Perturbation Theory at Shell Crossing

Abstract

We consider the growth of primordial dark matter halos seeded by three crossed initial sine waves of various amplitudes. Using a Lagrangian treatment of cosmological gravitational dynamics, we examine the convergence properties of a high-order perturbative expansion in the vicinity of shellcrossing, by comparing the analytical results with state-of-the-art high resolution Vlasov-Poisson simulations. Based on a quantitative exploration of parameter space, we study explicitly for the first time the convergence speed of the perturbative series, and find, in agreement with intuition, that it slows down when going from quasi one-dimensional initial conditions (one sine wave dominating) to quasi triaxial symmetry (three sine waves with same amplitude). In most cases, the system structure at collapse time is, as expected, very similar to what is obtained with simple one-dimensional dynamics, except in the quasi-triaxial regime, where the phase-space sheet presents a velocity spike. In all cases, the perturbative series exhibits a generic convergence behavior as fast as an exponential of a power-law of the order of the expansion, allowing one to numerically extrapolate it to infinite order. The results of such an extrapolation agree remarkably well with the simulations, even at shell-crossing. arXiv:1805.08787v2 [astro-ph.CO]

Semiclassical path to cosmic large-scale structure

“…Deviating from 1D, we have generically V eff = 0, and, as a result, the ZA performs rather poorly as gravitational tidal effects become non-negligible. In particular, this reflects in inaccurate predictions of the ZA for the shell-crossing time that worsen successively when deviating more and more from the 1D collapse [15][16][17]. Very similar performance issues are expected for the propagator method.…”

“…The central quantity is the Lagrangian displacement field that encodes how fluid elements are displaced as a function of time and initial (Lagrangian) position. The corresponding perturbative framework is usually called Lagrangian perturbation theory (LPT), and, although a challenge, allows to investigate the instance of shell-crossing by analytic means [12][13][14][15][16][17]. However, to update the gravitational potential that is responsible for displacing the fluid elements, one still requires, effectively, the Eulerian density.…”

The generation of vorticity in cosmological N-body simulations

“…Furthermore, it has the drawback to require the introduction of a free parameter for the velocity dispersion which regulates the amplitude of the vorticity spectrum. Several studies of shell-crossing in the context of perturbation theory have been recently presented (see [8][9][10][11] and references therein). However, the application of these techniques to the full 3D problem is still lacking.…”

Cosmological Vlasov–Poisson equations for dark matter

“…See Fig. 5 for comparisons of various analytical solutions against numerical simulations (black dotted line): the blue (dot-dashed) line denotes the result using the Ansatz (26b) up to second order in (derived in Saga et al 2018), while the red-dashed line is obtained using (20a) up to the 10th order in the series. The left panel in Fig.…”

“…The left panel in Fig. 5 denotes the stated quasi-one-dimensional problem, while the right panel shows the collapse for tri-axial sine-wave initial conditions (see Saga et al 2018 for the specific setup). Note that due to the use of trigonometric functions in the initial data, the theoretical solutions for the displacement are combinations of trigonometric functions as well-thus, no numerics are required for the shown theoretical solutions.…”