Strong coupling and instabilities in singularity-free inflation from an infinite sum of curvature corrections

Four-dimensional gravitational theories derived from an infinite sum of Lovelock curvature invariants, combined with a conformal rescaling of the metric, are equivalent to a subclass of shift-symmetric Horndeski theories that possess a single scalar degree of freedom. Under the assumption of a homogeneous and isotropic cosmological background, the theory admits an inflationary solution that replaces the Big Bang singularity. This can be achieved by a solution where the Hubble expansion rate $H$ is equal to the time derivative of the scalar field $\dotϕ$. We show that the solution $H=\dotϕ$ suffers from a strong coupling problem, characterized by the vanishing kinetic term of linear scalar perturbations at all times. Consequently, nonlinear scalar perturbations remain uncontrolled from the onset of inflation throughout the subsequent cosmological evolution. Moreover, tensor perturbations are generally subject to Laplacian instabilities during inflation. This instability in the tensor sector also persists under background initial conditions where $H \neq \dotϕ$. In the latter case, both the coefficient of the kinetic term for scalar perturbations and the scalar sound speed diverge at the onset of inflation. Thus, the dominance of inhomogeneities in this theory renders the homogeneous background solution illegitimate.