Primordial observables of explicit diffeomorphism violation in gravity

We investigate the potential for current and future gravitational-wave detectors to observe imprints of explicit diffeomorphism violation in primordial signals. Starting from a simple model with known effects, we derive the strain amplitude and power spectrum for primordial gravitational waves, both of which are affected by the symmetry breaking. Through this, we directly find predictions for the tensor spectral index and tensor-to-scalar which are different from general relativity. By considering the known sensitivity curves for NANOGrav, SKA, THEIA, $\mu$-ARES, ASTROD-GW, LISA, BBO, DECIGO, CE, AION-km, AEDGE, ET, and aLIGO, we place observability limits on the parameters controlling the diffeomorphism violation. For instance, we find that aLIGO could observe signals for $s_{00}\lesssim -0.1$, while more sensitive future detectors like LISA and DECIGO could probe violations as small as $s_{00}\approx -5\times {10}^{-4}$ and $-3\times {10}^{-3}$, respectively. Finally, we consider the existing constraints on the number of relativistic degrees of freedom $\mathrm{\Delta }{N}_{\mathrm{e}\mathrm{f}\mathrm{f}}$ which is tightly constrained by Big-Bang Nucleosynthesis (BBN) and we find that $\mathrm{\Delta }{N}_{\mathrm{e}\mathrm{f}\mathrm{f}}$ only weakly depends on the symmetry breaking but places a lower bound on the coefficients which is consistent with available bounds from the speed of gravitational waves.