Covariant Two-Field Perturbations Through a Curvature-Driven Bounce in Closed FRW Cosmology

Abstract

We study perturbations through a locally non-singular bounce in a closed ($K=+1$) FRW universe with a regularised two-field sigma model. Positive curvature drives the bounce in General Relativity without NEC violation; the expanding branch approaches Starobinsky inflation and yields the standard leading slow-roll observables at $N=60$. In the exact fiducial reduced-scalar calculation, kinetic and gradient coefficients remain positive on the declared harmonic-time grid and tensor characteristics are luminal; a wall-free homogeneous-shear scan establishes only sampled robustness. We integrate the scalar sector in two field-space representations of the comoving curvature perturbation, both in Newtonian gauge, with an algebraic Gordon-Wands projection as a third reconstruction: across 64 turning cells the maximum cross-representation residual is $5.6\times 10^{-10}$, and across $H=0$ the Einstein-constraint, tolerance-refinement and DOP853-Radau diagnostics stay at the $10^{-8}$-$10^{-12}$ level. On the fiducial no-turn trajectory we derive and integrate the exact finite-harmonic closed-$S^3$ reduced scalar action, propagating two canonically normalised finite-time states without claiming a unique asymptotic vacuum. This gives an early-plateau tensor-to-scalar ratio $r_{\ell =60}=1.73\times 10^{-6}$ (state range $[1.62,1.73]\times 10^{-6}$), distinct from the observable-pivot prediction $r\simeq 3.1\times 10^{-3}$, and an entropy-to-curvature power ratio $0.983$ at $\ell =60$ (state range $[0.896,0.983]$), which is spectator power rather than a transfer coefficient or an isocurvature fraction. At the lowest harmonics the two states span $n_s^{\mathrm{curv}}\in [0.86,4.17]$; state sensitivity and cosmic variance preclude a likelihood claim. The work establishes a fully specified finite-grid perturbative continuation through a curvature-driven bounce, not a proof of past geodesic completeness.

v5: rescoped replacement of v1-v4 under a new title; 28 itemised corrections in Appendix C withdraw or rescope earlier uniqueness, unitarity, BKL-stability, basin, trans-Planckian, late-time-transfer and completeness claims. 51 pages, 6 figures. Code, artefacts and verification report: GitHub release 5.0.0, DOI 10.5281/zenodo.22819371. v1-v4 remain accessible and are superseded