Exorcising ghosts with gravitational waves: cases of ghostful and ghost-free fourth-order gravity

General Relativity (GR) is an effective field theory valid in the infrared regime. Quadratic curvature extensions intended to probe ultraviolet physics generically propagate a massive spin-$2$ ghost and are therefore non-unitary. One route to remove ghost is by enlarging the geometric sector (torsion, non-metricity). We investigate the infrared phenomenology of both the standard (ghostful) and ghost-free fourth-order gravity theories by computing Gravitational Wave (GW) emission and confronting the results with observations such as the orbital-period decay of quasi-stable binaries such as PSR B1913+16 and PSR J1738+0333 and the chirp-mass evolution of GW170817. In the ghostful theory, besides the theoretical inconsistency due to non-unitarity, there are also phenomenological problems: the massless spin-$2$ GW flux cancels the combined GW fluxes of the massive spin-$2$ ghost and massive spin-$0$ scalar in the vanishing-mass limit, so the GR quadrupole formula is not recovered at the leading order. As a result, we obtain the GW constraint on the ghostful theory as $m\gtrsim 10^{-11}~\mathrm{eV}$, where $m$ is the mass of the massive modes. By contrast, the ghost-free theory smoothly reproduces the Newtonian potential and GR quadrupole formulae when the two coupling constants $\alpha_1$ and $\alpha_2$ vanish, independently of the mass $m$. Therefore, GW observations put mass-dependent upper bounds on the size of the coupling constants. For example, if we assume $\alpha_1\simeq\alpha_2$ for simplicity, then we obtain $\alpha_{1,2}\lesssim 4.2\times 10^{83}$ for $m\sim 3\times 10^{-16}\,\mathrm{eV}$ and $\alpha_{1,2}\lesssim 1.3\times 10^{75}$ for $m\sim 10^{-11}\,\mathrm{eV}$. To our knowledge, these are the first astrophysical-scale bounds reported for ghostful and ghost-free fourth-order gravity.