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

TL;DR

The paper systematically contrasts ghostful and ghost-free fourth-order gravity by computing gravitational-wave emission and orbital-energy loss in both frameworks. Using a QFT weak-field approach, it derives the modified gravitational potential with Yukawa corrections and evaluates GW energy loss from massless gravitons and from additional massive modes, for both quasi-stable and coalescing binaries. Observational data from the Hulse–Taylor binary, PSR J1738+0333, and GW170817 are then used to place bounds on the masses of the extra modes and on the couplings α1, α2, with ghostful gravity requiring m ≳ 10^−11 eV and ghost-free gravity yielding scale-dependent upper bounds on the couplings that become tighter as the mode masses increase. The results show that ghostful theories fail to reproduce GR at leading order in the weak-field limit and face vacuum-instability concerns, while ghost-free constructions recover GR in the appropriate limit and are constrained mainly by multi-band GW observations, highlighting the potential of GW astronomy to probe UV extensions of gravity.

Abstract

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 $α_1$ and $α_2$ 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 $α_1\simeqα_2$ for simplicity, then we obtain $α_{1,2}\lesssim 4.2\times 10^{83}$ for $m\sim 3\times 10^{-16}\,\mathrm{eV}$ and $α_{1,2}\lesssim 1.3\times 10^{75}$ 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.

Figures (8)

• Figure 1: Contributions of different massless (spin-$2$) and massive (spin-$2$ ghost and spin-$0$) modes, their total contribution and the behaviour in the limit of larger mode masses in standard fourth-order gravity for the rate of orbital period loss of Hulse-Taylor compact binary system: (a) Rates of orbital period loss for the individual modes, (b) Total rate of orbital period loss compared to the GR value, (c) Total rate of orbital period loss in the limit $m\gtrsim \Omega$, and (d) Total rate of orbital period loss in the limit $m\gtrsim 1/a$. See texts for details.
• Figure 2: Contributions of different massless (spin-$2$) and massive (spin-$2$ ghost and spin-$0$) modes and their total contribution in standard fourth-order gravity for the rate of orbital period loss of PSR J1738+0333 compact binary system: (a) Rates of orbital period loss for the individual modes, (b) Total rate of orbital period loss compared to the GR value, (c) Total rate of orbital period loss in the limit $m\gtrsim \Omega$, and (d) Total rate of orbital period loss in the limit $m\gtrsim 1/a$. See texts for details.
• Figure 3: Orbital period loss in different limiting cases for (a) Hulse-Taylor binary and (b) PSR J1738+0333 in standard fourth-order gravity. See texts for details.
• Figure 4: Constraints on the couplings $\alpha_i$ for ghost-free fourth-order gravity from orbital period decay measurements of the Hulse-Taylor binary and PSR J1738+0333, considering (a) the combined effects of the modified force and additional radiation, and (b) radiation effects only. See texts for details.
• Figure 5: Constraints on the mass of the modes from GW170817 in standard fourth-order gravity. See texts for details.