arXiv:2601.02416v1 Announce Type: new
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 $f(R)=R+alpha R^2$. We first derive the perturbative expansion around General Relativity (GR), working to first order in the small parameter $alpha$. 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 $delta_c$. 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 $phi$ 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 $delta_c(k)$ for PBH formation near the end of inflation.
Conclusions
The study of gravitational collapse and primordial black hole formation in the quadratic $f(R)$ model $f(R)=R+alpha R^2$ provides valuable insights into the early universe dynamics. The perturbative expansion around General Relativity allows for a detailed analysis of corrections to scale factors, stellar radii, and horizon formation times. The shift in the PBH formation threshold $delta_c$ is found to be significant in high-curvature backgrounds, highlighting the importance of considering alternative gravity models in extreme conditions.
Reformulating the theory in the Einstein frame offers a new perspective, transforming the model into GR plus the scalaron field $phi$ with the Starobinsky potential. By solving the complete ODE system governing cosmological backgrounds and the evolution of overdense closed FLRW patches, critical overdensities $delta_c(k)$ for PBH formation near the end of inflation can be obtained.
Future Roadmap
Challenges:
- Validation of perturbative results against simulations and observational data.
- Understanding the implications of high-curvature backgrounds on early universe dynamics.
- Numerical integration of the ODE system for accurate determination of critical overdensities.
Opportunities:
- Exploration of the role of alternative gravity models in cosmological evolution.
- Investigation of PBH formation in the early universe as a probe of fundamental physics.
- Potential implications for the nature of dark matter and the cosmological constant.