Probing dynamical embeddings in a five-dimensional spacetime in light of DESI BAO

We here investigate the observational viability of Nash gravity as an alternative to the standard $\Lambda$CDM cosmology. Based on Nash's embedding theorem, the model introduces orthogonal perturbations via variations in the extrinsic curvature, generating scalar-type metric perturbations directly from geometry, without the need to introduce additional fields. We confront the model with current observational data, including Cosmic Microwave Background (CMB) measurements from Planck, Baryon Acoustic Oscillations (BAO) from DESI DR2, and recent Type Ia supernova (SN Ia) compilations. Our analysis shows that Nash gravity provides a good fit to the data, yielding a slightly higher value for the Hubble constant, $H_0 = 69.32 \pm 0.72$ km/s/Mpc, compared to the $\Lambda$CDM model, thus offering a potential alleviation of the $H_0$ tension. Furthermore, the model naturally predicts a suppressed growth of structure, with $S_8 \approx 0.76$ across various joint analyses, potentially alleviating the so-called $S_8$ tension, assuming that this discrepancy is not solely due to systematic effects in other independent measurements. In some cases, Nash gravity achieves a better fit to the data than the $\Lambda$CDM paradigm at the $2\sigma$ level.