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

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.