Lower tensor-to-scalar ratio as possible signature of modified gravity

This paper simplifies the induced four-dimensional gravitational equations originating from a five-dimensional bulk within the framework of Nash’s embeddings, incorporating them into a well-known $\mu -\mathrm{\Sigma }$ modified gravity (MG) parametrization. By leveraging data from Planck Public Release 4 (PR4), BICEP/Keck Array 2018, Planck cosmic microwave background lensing, and baryon acoustic oscillation observations, we establish a stringent lower limit for the tensor-to-scalar ratio parameter: $r<0.0303$ at a confidence level (CL) of 95%. This finding suggests the presence of extrinsic dynamics influencing standard four-dimensional cosmology. Notably, this limit surpasses those typically obtained through Bayesian analysis using Markov Chain Monte Carlo (MCMC) techniques, which yield $r<0.038$, or through the frequentist profile likelihood method, which yields $r<0.037$ at 95% CL.