Finite-Time Bounds on Conformal Vacuum Energy Relaxation in Scalar-Tensor Gravity

Abstract

We investigate finite time vacuum energy relaxation in conformally coupled scalar-tensor cosmology distinguishing the suppression accumulated along a physical trajectory from its asymptotic rate. For a canonical scalar with an exponential potential and constant conformal coupling, the autonomous system exhibits a stability exchange between scalar dominated and vacuum kinetic scaling solutions. In the power-law metric $f(R)=f_0R^n$ realization with $n>2$, the stable scalar dominated branch has a sustained relaxation rate below $0.579$ decades per Einstein frame e-fold. We additionally derive a finite time upper bound that accounts for depletion of the vacuum relative to the scalar potential. For the specified reference initial state $Ω_{Λ,i}=0.99$ and $Ω_{{\rm kin},i}=10^{-4}$, the bound excludes $60$ decades of suppression within $60$ e-folds throughout the $n>2$ family: $D_Λ(60)<47.7$. Numerical trajectories illustrate why this bound is conservative and why the transient contribution cannot be inferred from the final attractor alone. For the reference $f(R)\propto R^{2.14}$ solution, $D_Λ(60)\simeq7.53$ decades including a transient excess of about $3.27$ decades relative to uniform evolution at the asymptotic rate. The relaxed quantity is the dimensionless vacuum to gravitational scale hierarchy, its evolution also entails variation of the conformal frame Planck mass. The bounds are conditional on the homogeneous model and specified initial state, not a demonstration of complete cosmological viability.