Robust Non-Singular Bouncing Cosmology from Regularized Hyperbolic Field Space

Abstract

We present a framework for non-singular bouncing cosmology in a closed ($k=+1$) universe with a two-field sigma model whose regularized hyperbolic field-space metric $g^S_{\chi\chi}(\phi) = (1 + e^{-2\alpha\phi/M_{\mathrm{Pl}}})^{-1}$ is derived from three physical boundary conditions: (i) kinetic suppression during contraction enabling the bounce, (ii) canonical normalization during inflation preserving perturbative unitarity, and (iii) positive-definiteness ensuring ghost-freedom. The bounce preserves the Null Energy Condition and is BKL-stable. The full two-field perturbation system $(\delta\phi, \delta\chi, \Phi)$ is integrated in the Newtonian gauge through the bounce over 65 e-folds, circumventing the comoving-gauge $H=0$ singularity, with both Einstein constraints verified a posteriori. Scalar sound speeds numerically measured adjacent to $H=0$ satisfy $|c_\phi^2-1|, |c_\chi^2-1| \leq 8 \times 10^{-16}$, establishing strict hyperbolicity (no ghost or gradient instability); $\mathcal{R}$ is conserved on super-Hubble scales to $|\Delta\mathcal{R}^2/\mathcal{R}^2| = 4.43 \times 10^{-3}$. An independent CMB-scale Mukhanov-Sasaki integration confirms $n_s = 0.9683$, matching exact Starobinsky slow-roll to $|\Delta n_s| = 5.47 \times 10^{-4}$. $\delta N$ non-Gaussianity yields $f_{\mathrm{NL}}^{\mathrm{local}} = +0.0133$, consistent with Maldacena's relation to $|\Delta f_{\mathrm{NL}}| = 1.54 \times 10^{-4}$. A bounce-scale spectral feature is pushed by $\sim 2630$ post-bounce e-folds to $k_{\mathrm{CMB}}/k_H \sim 10^{1116}$, far beyond the observable universe, so the model recovers Starobinsky predictions on all observable scales while resolving the initial singularity. Predictions ($n_s \approx 0.967$, $r \approx 0.003$, $f_{\mathrm{NL}}^{\mathrm{local}} \approx +0.013$, independent of $\alpha$) are consistent with Planck 2018 and testable by next-generation CMB experiments.