Gravitational Bounce from the Quantum Exclusion Principle

We investigate the fully relativistic spherical collapse model of a uniform distribution of mass $M$ with initial comoving radius $\chi_*$ and spatial curvature $k \equiv 1/\chi_k^2 \le 1/\chi_*^2$ representing an over-density or bounded perturbation within a larger background. Our model incorporates a perfect fluid with an evolving equation of state, $P = P[\rho]$, which asymptotically transitions from pressureless dust ($P = 0$) to a ground state characterized by a uniform, time-independent energy density $\rho_{\rm G}$. This transition is motivated by the quantum exclusion principle, which prevents singular collapse, as observed in supernova core-collapse explosions. We analytically demonstrate that this transition induces a gravitational bounce at a radius $R_{\rm B} = (8 \pi G \rho_{\rm G}/3)^{-1/2}$. The bounce leads to an exponential expansion phase, where $P[\rho]$ behaves effectively as an inflation potential. This model provides novel insights into black hole interiors and, when extended to a cosmological setting, predicts a small but non-zero closed spatial curvature: $ -0.07 \pm 0.02 \le \Omega_k < 0$. This lower bound follows from the requirement of $\chi_k \ge \chi_* \simeq 15.9$ Gpc to address the cosmic microwave background low quadrupole anomaly. The bounce remains confined within the initial gravitational radius $r_{\rm S} = 2GM$, which effectively acts as a cosmological constant $\Lambda$ inside $r_{\rm S}=\sqrt{3/\Lambda}$ while still appearing as a Schwarzschild black hole from an external perspective. This framework unifies the origin of inflation and dark energy, with its key observational signature being the presence of a small, but nonzero, spatial curvature, a testable prediction for upcoming cosmological surveys.