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

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.

💡 Research Summary

This paper investigates fundamental limitations in constructing valid initial data for numerical relativity simulations of strong inhomogeneities in cosmological spacetimes, with applications to primordial black hole (PBH) formation and the robustness of inflation. The core finding is that the nonlinear Hamiltonian constraint imposes unexpected restrictions: for given values of an over-density and the mean extrinsic curvature, solutions may not exist, and if they do, they are not unique.

The authors employ a simple, analytically tractable toy model: a spherically symmetric, conformally flat overdense region (constant density ρ₀ + Δρ within a coordinate radius r_od) embedded in a homogeneous Friedmann-Lemaître-Robertson-Walker (FLRW) background (density ρ₀, mean curvature K₀). They allow the mean curvature inside the overdensity to differ from the background value by ΔK. Under these assumptions, and by momentarily ignoring the momentum constraint via a specific matter ansatz, the Hamiltonian constraint reduces to a nonlinear equation for the conformal factor ψ.

Analysis reveals a critical constraint linking the overdensity amplitude (δρ = Δρ/ρ₀), the curvature deviation (δK = ΔK/K₀), and the size of the overdense region (scaled to the Hubble length, n = r_od / L_H). Solutions exist only if δρ ≤ δρ_crit, where δρ_crit = (5⁵/(6⁶)) * (4/n²) + δK(2+δK). This demonstrates that intrinsic curvature (represented by ψ) alone cannot compensate for an arbitrarily large overdensity, especially for horizon-scale perturbations (n ~ 1). The term δK(2+δK) shows how adjusting the extrinsic curvature inside the lump can alter this limit.

Furthermore, within the allowed parameter space, solutions are not unique. The matching conditions lead to an equation, f(α) = constant, where the function f(α) has a maximum. For values below this maximum, two distinct solutions for the parameter α exist, corresponding to two continuous branches of solutions for ψ:

1. The Strong-Field Branch (α < α_c): Characterized by significant intrinsic curvature. The conformal factor ψ is smaller at the center, potentially indicating the presence of a black hole apparent horizon or a throat-like geometry on the initial slice.
2. The Weak-Field Branch (α > α_c): Characterized by mild intrinsic curvature. As the overdensity δρ approaches zero, this branch smoothly connects to the unperturbed FLRW background (ψ → 1).

This duality has direct implications for numerical simulations. Iterative methods used to solve the constraint equations for initial data can converge to either branch depending on the initial guess for ψ. Consequently, for the same physical parameters (δρ, δK, r_od), one could generate two geometrically distinct initial datasets, leading to potentially different evolutionary outcomes (e.g., collapse vs. dispersion). This ambiguity complicates the interpretation of simulations aimed at finding PBH formation thresholds or testing inflationary robustness against “generic” large inhomogeneities.

The paper attributes these phenomena to the “wrong-sign” of the source term in the Hamiltonian constraint, which prevents standard uniqueness proofs. While the toy model is highly simplified, the authors argue that the qualitative conclusions—existence limits and solution non-uniqueness—are expected to hold in more general, non-spherical, and non-conformally-flat scenarios due to the persistent nonlinear structure of the constraints. The work serves as a crucial cautionary note for setting up cosmological numerical relativity simulations, emphasizing that the initial geometry is not a free parameter but is tightly constrained and ambiguous, even for seemingly simple density profiles.