Cosmology in sixth-order higher-derivative theories of gravity

We analyze the cosmological implications of theories of gravity with quadratic curvature invariants R squared and C μνρσC μνρσ, and six-derivative operators R R and C μνρσ C μνρσ. We first analyze homogeneous and isotropic cosmological solutions, showing that the theory admits expanding, recollapsing and non-singular bouncing solutions. We clarify in the Einstein-frame formulation the mechanism that circumvents the Hawking--Penrose singularity theorems, and identify the role played by the additional higher-derivative degrees of freedom in that mechanism. We show that the complete set of linear equations governing tensor perturbations on standard backgrounds admit analytically stable solutions. The analysis reveals that the stability properties are determined by the pole structure of the propagator and by specific relations among the higher-derivative couplings, while the singularity-free backgrounds are found to be unstable against tensor perturbations. For scalar cosmological perturbations, the higher-derivative theories presents super-horizon scalar instabilities in de Sitter spacetimes for both quadratic and sixth-order gravity. To extract inflationary observables, we adopt an EFT approach to the sixth-order theory, treating these terms as perturbative corrections around the Starobinsky background. Within this framework, we compute the modification of the primordial scalar and tensor power spectra, obtaining predictions for the scalar spectral index and tensor-to-scalar ratio. We find that sixth-order operators produce controlled deviations from the Starobinsky model while preserving the EFT expansion, thereby providing a consistent phenomenological framework to probe ultraviolet gravitational corrections through cosmological observables.