The gravitational force field of proto-pancakes

¹ Sorbonne Université, CNRS, UMR7095, Institut d’Astrophysique de Paris, 98bis boulevard Arago, 75014 Paris, France
² Institute for Advanced Research, Nagoya University, Furo-cho Chikusa-ku, Nagoya 464-8601, Japan
e-mail: shohei.saga@yukawa.kyoto-u.ac.jp
³ Kobayashi-Maskawa Institute for the Origin of Particles and the Universe, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan
⁴ Laboratoire Univers et Théories, Observatoire de Paris, Université PSL, Université de Paris, CNRS, 5 place Jules Janssen, 92190 Meudon, France
⁵ Center for Gravitational Physics and Quantum Information, Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
⁶ Kavli Institute for the Physics and Mathematics of the Universe (WPI), Todai institute for Advanced Study, University of Tokyo, Kashiwa, Chiba 277-8568, Japan

Received: 22 May 2023
Accepted: 9 September 2023

Abstract

It is well known that the first structures that form from small fluctuations in a self-gravitating, collisionless, and initially smooth cold dark matter (CDM) fluid are pancakes. We studied the gravitational force generated by such pancakes just after shell crossing and have found a simple analytical formula for the force along the collapse direction, which can be applied to both the single- and multi-stream regimes. We tested the formula on the early growth of CDM proto-haloes seeded by two or three crossed sine waves. Adopting the high-order Lagrangian perturbation theory (LPT) solution as a proxy for the dynamics, we confirm that our analytical prediction agrees well with the exact solution computed via a direct resolution of the Poisson equation, as long as the local caustic structure remains sufficiently one-dimensional. These results are further confirmed by comparisons of the LPT predictions performed this way to measurements in Vlasov simulations performed with the public code ColDICE. We also show that the component of the force orthogonal to the collapse direction preserves its single-stream nature – it does not change qualitatively before or after the collapse – allowing sufficiently high-order LPT acceleration to be used to approximate it accurately as long as the LPT series converges. As expected, solving the Poisson equation on the density field generated with LPT displacement provides a more accurate force than the LPT acceleration itself, as a direct consequence of the faster convergence of the LPT series for the positions than for the accelerations. This may provide a clue as to how we can improve standard LPT predictions. Our investigations represent a very needed first step in the study of gravitational dynamics in the multi-stream regime analytically: we estimate, at the leading order in time and space, the proper backreaction on the gravitational field inside the pancakes.