Self-consistent initial conditions for primordial black hole formation

TL;DR

This paper develops an analytical method to generate precise initial conditions for simulating primordial black hole formation, improving accuracy and stability in numerical computations.

Contribution

It introduces a recursive quasi-linearization approach to solve Einstein equations for super-horizon curvature perturbations, enabling optimal initial condition setting.

Findings

Analytical solutions up to eighth order for curvature perturbations

Method to determine when to set initial conditions for simulations

Enhanced accuracy in numerical primordial black hole formation models

Abstract

For an arbitrary strong, spherically symmetric super-horizon curvature perturbation, we present analytical solutions of the Einstein equations in terms of asymptotic expansion over the ratio of the Hubble radius to the length-scale of the curvature perturbation under consideration. To obtain this solution we develop a recursive method of quasi-linearization which reduces the problem to a system of coupled ordinary differential equations for the $N$-th order terms in the asymptotic expansion with sources consisting of a non-linear combination of the lower order terms. We use this solution for setting initial conditions for subsequent numerical computations. For an arbitrary precision requirement predetermined by the intended accuracy and stability of the computer code, our analytical solution yields optimal truncated asymptotic expansion which can be used to find the upper limit on the moment of time when the initial conditions expressed in terms of such truncated expansion should be set. Examples of how these truncated (up to eighth order) solutions provide initial conditions with given accuracy for different radial profiles of curvature perturbations are presented.