Primordial Black Hole Formation in $f(R)=R+αR^2$ Gravity: Perturbative and Non-Perturbative Analysis

TL;DR

The paper tackles PBH formation in the quadratic $f(R)$ gravity model $f(R)=R+αR^2$ by combining a perturbative expansion around GR with a non-perturbative Einstein-frame treatment. The perturbative analysis yields explicit first-order corrections to the collapse dynamics in a flat FLRW dust interior, showing that $α>0$ accelerates collapse and lowers the horizon-formation time, hence reducing the PBH threshold $δ_c$. The non-perturbative regime is handled by mapping to GR plus a scalaron with the Starobinsky potential, and by constructing a complete ODE system for a closed overdense patch to compute $δ_c^{(α)}(k)$. The results indicate a potentially dramatic enhancement of PBH production in high-curvature phases, enabling robust observational constraints on the curvature parameter $α$ from PBH abundances, with current bounds suggesting $α oxed{ ext{not too large}}$. Overall, the work provides a unified framework linking high-curvature gravity corrections to PBH formation and associated observational signatures.

Abstract

We present a complete analytic and semi-analytic study of gravitational collapse and primordial black hole (PBH) formation in the quadratic $f(R)$ model $f(R)=R+αR^2$. We first derive the perturbative expansion around General Relativity (GR), working to first order in the small parameter $α$. For a collapsing flat FLRW dust interior we compute the explicit first-order corrections to the scale factor, the stellar radius, and the horizon formation time. We then use these results to obtain the shift in the PBH formation threshold $δ_c$. The perturbative effect is small for PBHs forming in the deep radiation era, but becomes important when the background curvature is high. To access this early regime we reformulate the theory in the Einstein frame, where the model becomes GR plus the scalaron field $φ$ with the Starobinsky potential. We provide the complete ODE system governing both the cosmological background and the evolution of an overdense closed FLRW patch. This system can be numerically integrated to obtain the critical overdensity $δ_c(k)$ for PBH formation near the end of inflation.

Figures (1)

• Figure 1: Schematic behaviour of the collapsing scale factor in GR (solid line) versus the $f(R)=R+\alpha R^2$ model (dashed line) for $\alpha>0$. Panel \ref{['fig:dust']} shows the dust-dominated case, while panel \ref{['fig:rad']} corresponds to the radiation-dominated case. In both situations the $R^2$ correction accelerates the collapse, causing the scale factor to deviate from the GR solution as $t\to t_0$.