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 ${\ensuremath{\chi}}_{*}$ and spatial curvature $k\ensuremath{\equiv}1/{\ensuremath{\chi}}_{k}^{2}\ensuremath{\le}1/{\ensuremath{\chi}}_{*}^{2}$ representing an overdensity or bounded perturbation within a larger background. Our model incorporates a perfect fluid with an evolving equation of state, $P=P(\ensuremath{\rho})$, which asymptotically transitions from pressureless dust ($P=0$) to a ground state characterized by a uniform, time-independent energy density ${\ensuremath{\rho}}_{\mathrm{G}}$. 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 ${R}_{\mathrm{B}}=(8\ensuremath{\pi}G{\ensuremath{\rho}}_{\mathrm{G}}/3{)}^{\ensuremath{-}1/2}$. The bounce leads to an exponential expansion phase, where $P(\ensuremath{\rho})$ 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 nonzero closed spatial curvature: $\ensuremath{-}0.07\ifmmode\pm\else\textpm\fi{}0.02\ensuremath{\le}{\mathrm{\ensuremath{\Omega}}}_{k}<0$. This lower bound follows from the requirement of ${\ensuremath{\chi}}_{k}\ensuremath{\ge}{\ensuremath{\chi}}_{*}\ensuremath{\simeq}15.9\text{ }\text{ }\mathrm{Gpc}$ to address the cosmic microwave background low quadrupole anomaly. The bounce remains confined within the initial gravitational radius ${r}_{\mathrm{S}}=2GM$, which effectively acts as a cosmological constant $\mathrm{\ensuremath{\Lambda}}$ inside ${r}_{\mathrm{S}}=\sqrt{3/\mathrm{\ensuremath{\Lambda}}}$ 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 small but nonzero spatial curvature, a testable prediction for upcoming cosmological surveys.