Torsion, Frames and Quantum Bounces: Inflation and Ekpyrotic Contraction in Quantum Trajectory Cosmology with a Non-Minimally Coupled Scalar

Abstract

We study the de Broglie-Bohm quantum cosmology of a scalar field non-minimally coupled to a metric-compatible Riemann-Cartan geometry, comparing inflation with ekpyrotic contraction. The torsion trace is auxiliary and is eliminated algebraically before quantisation. The minisuperspace metric depends on the coupling $F(ϕ)$ and a constant $κ$ for an independent quadratic torsion term ($κ=0$ Einstein-Cartan, $κ\to\infty$ torsion-free). For every $F$, $κ$ and potential this metric is conformally flat, so torsion acts only through the Einstein-frame kinetic function $K(ϕ)$. Its curvature is therefore not a diagnostic of torsion, and the Laplace-Beltrami Wheeler-DeWitt equation, quantum potential and Bohmian trajectories are frame covariant. For a massless scalar the equation is solved exactly; in a two-packet state all 41 flux-weighted Bohmian trajectories bounce in the Einstein frame. In the Jordan frame the bounce at the symmetric event is independent of torsion and persists only for $ξ<ξ_c$ (closed form); torsion changes the late-time exponent from $1-\sqrt{6ξ/(1+6ξ)}$ to $1-\sqrt{6ξ}$. The bounce comes from interference in the state, not from torsion. For exponential potentials, exact Wheeler-DeWitt solutions show that in the sector containing the ekpyrotic attractor no Bohmian trajectory bounces, and single-momentum states bounce only where the potential is negligible and end in the crunch. For quadratic inflation the torsion-specific shift of $r$ is below one per cent. A single ekpyrotic field gives blue spectra, $n_s=3+2/(ε-1)$ and $r=16ε$, whatever the torsion, also for running $ε$. In an entropic two-field version the tilt shifts at order $\sqrtξ$ (non-minimal coupling) and $ξ^{3/2}$ (torsion). Tensor modes cross the massless Bohmian bounce almost unchanged and stay blue.