Inflationary Cosmology in Scalar-Tensor Gravity: Reconstructing Higgs-Like Potentials
We study cosmology in scalar-tensor (Bergmann Wagoner) gravity, restricting the coupling function, $ω(ϕ)$ to be constant. Rather than specify the form of the cosmological function, $λ(ϕ)$, the scalar field is modelled as a function which decays to its present value $ϕ=1$. Solutions of the field equations are found for which $λ(ϕ)$ evolves from a large value (approximately 1) near the singularity to a small, non-zero value at later times, avoiding the problem of pre-inflationary collapse in standard general relativistic cosmology. Interpreting the model within the framework of Brans-Dicke theory with a scalar potential, we find that for suitable initial conditions, the reconstructed potential at late times for flat, open and closed universes is well described by a Higgs-like Mexican hat potential, quartic in $ϕ$, though this was not built in to the initial assumptions.