Can the NANOGrav observations constrain the geometry of the universe?

The theory of inflation provides an elegant explanation for the nearly flat universe observed today, which represents one of the pillars of the standard cosmological model. However, recent studies have reported some deviations from a flat geometry, arguing that a closed universe would be instead favored by observations. Given its central role played in the cosmological context, this paper revisits the issue of spatial curvature in light of the stochastic gravitational wave background signal recently detected by the NANOGrav collaboration. For this purpose, we investigate the primordial gravitational waves generated during inflation and their propagation in the post-inflationary universe. We propose a new parametrization of the gravitational wave power spectrum, taking into account spatial curvature, the tensor-to-scalar ratio and the spectral index of tensor perturbations. Therefore, we compare the theoretical predictions with NANOGrav data to possibly constrain the geometry of the universe. We find that the choice of the priors has a significant effect on the computed posterior distributions. In particular, using flat uniform priors results in $\Omega_{\mathcal{K},0}= 0.00 \pm 0.67$ at the 68\% confidence level. On the other hand, imposing a Planck prior, we obtain $\Omega_{\mathcal{K},0}= -0.05 \pm 0.17$ at the 68\% confidence level. This result aligns with the analysis of the cosmic microwave background radiation, and no deviations from a flat universe are found.