Robust Non-Singular Bouncing Cosmology from Regularized Hyperbolic Field Space

TL;DR

This work develops a complete, non-singular bouncing cosmology in a closed universe by introducing a regularized hyperbolic field space with the sigmoid metric $g^S_{\chi\chi} = \left(1 + e^{-2\alpha \phi / M_{\rm Pl}}\right)^{-1}$. The three physical boundary conditions—bounce, perturbative unitarity, and ghost-freedom—impose a unique, minimal-complexity solution solved by the logistic equation, linking hyperbolic geometry with $\alpha$-attractors. The resulting two-field model achieves a 60+ e-folds inflation after a non-singular bounce, preserves the NEC, and expands the basin of attraction by about $10^{21}$ relative to the pure exponential metric; curvature is efficiently diluted so that $|\Omega_k| < 0.001$ after inflation. Perturbations remain regular through the bounce and $\mathcal{R}$ is conserved on super-Hubble scales, with universal observational predictions $n_s \approx 0.967$, $r \approx 0.003$, independent of $\alpha$, consistent with Planck 2018 and testable by future CMB experiments.

Abstract

We present a complete and robust framework for non-singular bouncing cosmology in a closed universe, where the scalar field space is endowed with a regularized hyperbolic geometry. The field space metric $g^S_{χχ} = (1 + e^{-2αφ/M_{\mathrm{Pl}}})^{-1}$ is rigorously derived from fundamental physical principles: (i) exponential suppression during contraction enabling the bounce mechanism, (ii) asymptotic saturation to unity during inflation preserving perturbative unitarity, and (iii) positive-definiteness ensuring ghost-freedom. We prove that the sigmoid function emerges uniquely as the minimal-complexity solution satisfying these boundary conditions, with deep connections to hyperbolic geometry and cosmological $α$-attractors. Through comprehensive numerical validation, we demonstrate that the model achieves 60+ e-folds of inflation following a non-singular bounce without violating the Null Energy Condition. The sigmoid regularization expands the basin of attraction by $\sim 10^{21}$ compared to the pure exponential metric, achieving 100\% success rate across 16 orders of magnitude in initial velocities. The model is compatible with the BKL conjecture (anisotropic shear suppressed by spatial curvature) and solves the flatness problem (only $\sim 3.5$ e-folds needed for $|Ω_k| < 0.001$). Perturbation equations remain regular throughout the bounce, and the comoving curvature perturbation $\mathcal{R}$ is conserved on super-Hubble scales. Observable predictions ($n_s \approx 0.967$, $r \approx 0.003$) are independent of the regularization parameter $α$, in excellent agreement with Planck 2018 data and testable by next-generation CMB experiments.

Figures (9)

• Figure 1: Field space geometry and regularization. (a) The sigmoid metric $g^S_{\chi\chi}$ (blue solid) regularizes both boundaries compared to the exponential metric (red dashed), remaining finite during inflation while providing exponential suppression during contraction. (b) Field space curvature transitions smoothly from hyperbolic ($K = -\alpha^2/2$) during contraction to flat ($K = 0$) during inflation. (c) Christoffel symbols and the geometric decoupling factor $(1-g_{\chi\chi})$ automatically suppress kinetic coupling during inflation.
• Figure 2: Potential and inflationary dynamics. (a) Starobinsky-type potential providing plateau inflation. (b) Slow-roll parameters remain small during inflation, ensuring sufficient e-folds. (c) Number of e-folds as function of field value, with CMB scales corresponding to $\phi \approx 5.4 M_{\rm Pl}$. (d) Observable predictions for spectral index $n_s$ and tensor-to-scalar ratio $r$ as function of e-folds $N$, showing excellent agreement with Planck constraints (red band).
• Figure 3: Complete background evolution through bounce and inflation. (a) Scale factor evolution showing non-singular bounce at finite $a_{\rm min}$. (b) Hubble parameter smoothly transitioning from contraction ($H<0$) to expansion ($H>0$). (c) Inflaton field evolution from negative values (bounce regime) to positive values (inflation). (d) Field space metric saturating to unity during inflation, ensuring canonical dynamics.
• Figure 4: Basin of attraction analysis demonstrating robustness. (a) Success rate across 16 orders of magnitude in initial conditions, compared to exponential metric requiring extreme fine-tuning (red shaded region). (b) Post-bounce inflationary e-folds exceeding the required $N=60$ for CMB across the entire basin.
• Figure 5: Exponential dilution of spatial curvature after bounce. (a) Evolution of curvature density parameter $|\Omega_k|$ as function of e-folds after bounce. Only $\sim 3.5$ e-folds are required to satisfy current observational bounds ($|\Omega_k| < 0.001$), while the model naturally produces $N > 60$ e-folds. (b) Comparison of required vs achieved curvature suppression, demonstrating that flatness is easily achieved without fine-tuning.