Restrictions on Initial Conditions in Cosmological Scenarios and Implications for Simulations of Primordial Black Holes and Inflation

TL;DR

This work analyzes how nonperturbative inhomogeneities constrain cosmological initial data by applying a cosmological adaptation of a toy model for the Hamiltonian constraint. It shows that, for given overdensity $\delta \rho$ and mean-curvature perturbation $\delta K$, solutions exist only if $\delta \rho - \delta K(2+\delta K)$ lies below a critical bound, and that, when solutions do exist, they are not unique, yielding two branches (strong- and weak-field) that merge at a maximum. The two branches share the same cosmological-horizon entropy, but differ in interior structure and possible horizon content, with strong-field data capable of producing pronounced compactness or horizons depending on parameters. The results have implications for constructing cosmological initial data in simulations of primordial black hole formation and inflation robustness, highlighting potential ambiguities in iterative solvers and the need to account for spatial variations in $K$ to balance large inhomogeneities.

Abstract

Numerical relativity simulations provide a means by which to study the evolution and end point of strong over-densities in cosmological spacetimes. Specific applications include studies of primordial black hole formation and the robustness of inflation. Here we adopt a toy model previously used in asymptotically flat spacetimes to show that, for given values of the over-density and the mean curvature, solutions to the Hamiltonian constraint need not exist, and if they do exist they are not unique. Specifically, pairs of solutions exist on two branches, corresponding to strong-field and weak-field solutions, that join at a maximum beyond which solutions cease to exist. As a result, there is a limit to the extent to which an over-density can be balanced by intrinsic rather than extrinsic curvature on the initial slice. Even below this limit, iterative methods to construct initial data may converge to solutions on either one of the two branches, depending on the starting guess, leading to potentially inconsistent physical results in the evolution.

Figures (4)

• Figure 1: The function $f(\alpha)$ defined in Eq. (\ref{['eq:f_alpha']}). There are no solutions with $f(\alpha)$ greater than the value at the maximum of the function $f(\alpha_c)$. For smaller values of $f(\alpha)$ there are two solutions for $\alpha$, identified with the strong, $\alpha < \alpha_c$, and weak, $\alpha > \alpha_c$, branches of the solution.
• Figure 2: Top panel: Representative solutions $\psi(r)$ for a fixed value of the additional expansion $\delta K = 0.1$, a Hubble size region $n=0.5$ and the two values of $\alpha$ from the strong and weak branches as marked on Fig. \ref{['fig-falpha']}, namely $\alpha=0.7$ and $\alpha = 19.8$. Both values give $f(\alpha)=0.05$ and so the same profile for $\rho$, but the profiles for $\psi$ differ. The vertical lines show the coordinate locations of the cosmological horizons for the two solutions. Bottom panel: The areal radius $R = \psi^2 r$ as a function of the coordinate radius $r$ for the two solutions. Although the coordinate radii of the cosmological horizons are different in each solution, both horizons have the same areal radius $R(r_{\rm cos})=1$ and therefore the same entropy. They are, however, physically distinct, with those on the strong branch exhibiting a non-monotonic behavior in the areal radius $R$ as a function of coordinate radius $r$ in cases with $\alpha<1$.
• Figure 3: Values of the compaction \ref{['eq:compaction']} versus $\delta \rho - \delta K(2 + \delta K)$ for different values of $n$. For a given value of $n$, solutions to the Hamiltonian constraint \ref{['ham1']} exist only for suitable combinations of the inhomogeneities $\delta \rho$ and $\delta K$. If such a solution exists, it is not unique: instead, the Hamiltonian constraint allows both a weak-field solution (the lower branch) and a strong-field solution (the upper branch). The two branches meet at the critical solution (corresponding to $\alpha = \alpha_{\rm crit}$), marked by the filled dots.
• Figure 4: Illustration of the coordinate radius of the horizons for a small $n$ (top panel) and large $n$ (bottom panel) case as a function of the parameter $\alpha$. For a given value of $\alpha$, there is always an even number of black-hole horizons and an odd number of cosmological horizons. In the small $n$ case we have two black hole horizons and three cosmological ones in the region $6\sqrt{3}n < \alpha^2 < 1$ (see Eq. \ref{['eq:alpha_condition']}). For the large $n$ case no BH horizons exist for any value of $\alpha$, and there is only one cosmological horizon. As discussed in the text, whilst the horizons on the strong and weak branches have different coordinate radii, their proper radii are the same for corresponding solutions (see Eq. \ref{['eq:cos_hor_proper']}). The dotted line in the lower panel represents the asymptotic solution \ref{['eq:r_cos_limit']}.