Primordial observables of explicit diffeomorphism violation in gravity

TL;DR

This work investigates explicit diffeomorphism violation in gravity via a fixed background tensor $s_{\mu\nu}$, emphasizing the simple case with $s_{00}$ nonzero and constant. It shows how this breaking modifies the cosmological background and the propagation of primordial tensor modes, leading to a modified PGW spectrum with a tunable tilt $n_T$ and altered horizon crossing conditions. The authors derive observables such as the modified transfer function, the present-day GW energy density, and detector-thresholds, revealing a viable early-Universe window for $s_{00}$ (roughly $-0.08\lesssim s_{00}\le 0$) that could be probed by upcoming GW detectors and constrained by BBN through $\Delta N_{\rm eff}$. While late-time constraints from GW170817 dominate the bound on $s_{00}$ today, the primordial era remains open to testing diffeomorphism-violating physics, with time-dependent extensions and more general coefficients as promising directions for future work.

Abstract

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, $μ$-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 $ΔN_{\rm eff}$ which is tightly constrained by Big-Bang Nucleosynthesis, where we find that $ΔN_{\rm eff}$ 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.

Figures (4)

• Figure 1: Plot of the strain amplitude of $\Omega_{GW}h^2$ versus frequency $f$ (range of $(10^{-10}~\mathrm{Hz}, 10^4~\mathrm{Hz})$) for the several values of $s_{00}$ within both negative and positive ranges. The colored shaded areas denote different sensitivity regions for upcoming GW detectors: NANOGrav (indigo curve), SKA (orange curve), THEIA (yellow-dark curve), $\mu$--Ares (blue curve), ASTROD--GW (red curve), LISA (purple curve), BBO (magenta curve), DECIGO (green curve), CE (brown curve), AION--km (pink curve), AEDGE (gray curve), ET (black curve), aLIGO (cyan curve).
• Figure 2: The plot of tensor spectral index $n_T$ in terms of frequency $f$ for some selected values of $r$ with fixed value of $s_{00}=-10^{-4}$ (left) and some selected values of $s_{00}$ with fixed value of $r=0.03$ (right).
• Figure 3: The plot of tensor-to-scalar ratio $r$ in terms of frequency $f$ for some selected values of $n_T$ with fixed values of $s_{00}=-10^{-3}$, and $-10^{-2}$ for left and right panels, respectively. The horizontal black line shows the upper bound $r\lesssim 0.032$.
• Figure 4: The behavior of $\Delta N_{\rm eff}$ in terms of $s_{00}$. The horizonal dashed line shows the upper bound of $\Delta N_{\rm eff}\lesssim 0.4$.