Torsional pseudo-inflation beyond Einstein-Cartan

Abstract

Torsion entered early-universe cosmology twice: through the nonsingular Einstein--Cartan bounce, where the quantum spin of fermions halts the contraction, and through the proposal that the expansion following that bounce could take over the duties of cosmic inflation. We revisit the second idea in a constructive spirit. First we compute its budget: for a Weyssenhoff spin fluid with barotropic source, $w\in\left[0,1\right]$, the accelerated window that follows the bounce spans $N_e=\ln\left[4/\left(1+3w\right)\right]/3\left(1-w\right)\leq\tfrac{1}{3}\ln4\simeq0.46$ e-folds, and positive spatial curvature only shrinks it. The mechanism is sound; its torsional ``fuel'' dilutes as $a^{-6}$, and the engine stops two orders of magnitude short of inflationary needs. We then ask what torsion would need in order to sustain a quasi-de Sitter phase, and give a two-branch answer in Riemann--Cartan cosmology with nonminimal couplings, where torsion is fed by the couplings rather than by spin. The vectorial branch reproduces Palatini inflation, observationally alive. On the axial branch we exhibit an exact de Sitter solution sustained by a torsion condensate: constant axial torsion $f_{0}$, a linear Gauss--Bonnet coupling, and a potential fixed to $V_{0}=6f_{0}^{2}+\tfrac{5}{2}v^{2}$ by the Friedmann constraint. The solution is an attractor of the homogeneous dynamics, with eigenvalue exactly $-3H$. The phase ends when the slope $U^{\prime}$, a decreasing function of the roll speed at fixed flat potential, drifts down to the floor $5/16V_{0}$ and the condensate switches itself off; consistency of the effective theory pins the whole episode near the Planck scale, $H\gtrsim0.3$ in reduced Planck units. We studied this branch at background level; its perturbation spectrum remains open.