Abstract
Recent gravitational wave observations of binary black hole mergers and a binary neutron star merger by LIGO and Virgo Collaborations associated with its optical counterpart constrain deviation from General Relativity (GR) both on strong-field regime and cosmological scales with high accuracy, and further strong constraints are expected by near-future observations. Thus, it is important to identify theories of modified gravity that intrinsically possess the same solutions as in GR among a huge number of theories. We clarify the three conditions for theories of modified gravity to allow GR solutions, i.e., solutions with the metric satisfying the Einstein equations in GR and the constant profile of the scalar fields. Our analysis is quite general, as it applies a wide class of single-/multi-field scalar–tensor theories of modified gravity in the presence of matter component, and any spacetime geometry including cosmological background as well as spacetime around black hole and neutron star, for the latter of which these conditions provide a necessary condition for no-hair theorem. The three conditions will be useful for further constraints on modified gravity theories as they classify general theories of modified gravity into three classes, each of which possesses i) unique GR solutions (i.e., no-hair cases), ii) only hairy solutions (except the cases that GR solutions are realized by cancellation between singular coupling functions in the Euler–Lagrange equations), and iii) both GR and hairy solutions, for the last of which one of the two solutions may be selected dynamically.
Highlights
Recent measurements of gravitational waves (GWs) from binary black hole (BH) mergers by LIGO and Virgo Collaborations [1,2]clarified that the observed GWs are consistent with the prediction of General Relativity (GR) for binary coalescence waveforms.the almost simultaneous detection of GWs from a neutron star (NS) merger [3], and the short gamma-ray burst [4] has significantly constrained a deviation of propagation speed of GWs over cosmological distance from the speed of light down order10−15 [5]
It should be emphasized that our analysis focuses on GR solutions with the constant profile of the scalar fields, and there are several theories that do not fit our analysis, e.g., theories with self-gravitating media such as Lorentz-violating massive gravity [73–78], and theories where the small-scale behavior such as breaking of the Vainshtein screening is sensitive to the asymptotic time-dependence of the scalar fields [79–81]
From the fact that GW observations are consistent with GR, we are interested in identifying a class of modified gravity that allows GR solutions by satisfying the conditions 1–3
Summary
Recent measurements of gravitational waves (GWs) from binary black hole (BH) mergers by LIGO and Virgo Collaborations [1,2]. There exist some class of modified gravity theories allowing only the BH metric solutions in GR with constant scalar field as the unique solutions [59–68]. This is the extension of no-hair theorems, and implies that these classes evade constraints on deviation of BH spacetime from GR one. In order n for Eq (1) to be covariant with respect to g μν , the dependence μμμμμμ of C 1I1 , C 2I1 2 , C 3I1 2 3 , · · · on the metric should be through metric itself, curvature tensors associated with it, and their covariant derivatives This action is very generic and covers a lot of single-/multiμν field models of scalar–tensor theories. The following analysis for (1) to allow GR solutions is powerful and exhausts almost all the known scalar–tensor theories of modified gravity
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