One century after its formulation, Einsteinʼs general relativity (GR) has made remarkable predictions and turned out to be compatible with all experimental tests. Most of these tests probe the theory in the weak-field regime, and there are theoretical and experimental reasons to believe that GR should be modified when Class. Quantum Grav. 32 (2015) 243001Topical Review physical laboratories to probe strong-field gravity are black holes and neutron stars, whether isolated or in binary systems. We review the motivations to consider extensions of GR. We present a (necessarily incomplete) catalog of modified theories of gravity for which strong-field predictions have been computed and contrasted to Einsteinʼs theory, and we summarize our current understanding of the structure and dynamics of compact objects in these theories. We discuss current bounds on modified gravity from binary pulsar and cosmological observations, and we highlight the potential of future gravitational wave measurements to inform us on the behavior of gravity in the strong-field regime.

Scalar-tensor theories are a compelling alternative to general relativity and one of the most accepted extensions of Einstein's theory. Black holes in these theories have no hair, but could grow "wigs" supported by time-dependent boundary conditions or spatial gradients. Time-dependent or spatially varying fields lead in general to nontrivial black hole dynamics, with potentially interesting experimental consequences. We carry out a numerical investigation of the dynamics of single and binary black holes in the presence of scalar fields. In particular we study gravitational and scalar radiation from black-hole binaries in a constant scalar-field gradient, and we compare our numerical findings to analytical models. In the single black hole case we find that, after a short transient, the scalar field relaxes to static configurations, in agreement with perturbative calculations. Furthermore we predict analytically (and verify numerically) that accelerated black holes in a scalar-field gradient emit scalar radiation. For a quasicircular black-hole binary, our analytical and numerical calculations show that the dominant component of the scalar radiation is emitted at twice the binary's orbital frequency.

An old result (astro-ph/9905303) by Jacobson implies that a black hole with Schwarzschild radius r s acquires scalar hair, Q ∝ r 2 s µ, when the (canonically normalized) scalar field in question is slowly time-dependent far from the black hole, ∂ t φ ≃ µM p with µr s ≪ 1 time-independent. Such a time dependence could arise in scalar-tensor theories either from cosmological evolution, or due to the slow motion of the black hole within an asymptotic spatial gradient in the scalar field. Most remarkably, the amount of scalar hair so induced is independent of the strength with which the scalar couples to matter. We argue that Jacobson's Miracle Hair-Growth Formula c implies, in particular, that an orbiting pair of black holes can radiate dipole radiation, provided only that the two black holes have different masses. Quasar OJ 287, situated at redshift z ≃ 0.306, has been argued to be a double blackhole binary system of this type, whose orbital decay recently has been indirectly measured and found to agree with the predictions of General Relativity to within 6%. We argue that the absence of observable scalar dipole radiation in this system yields the remarkable bound | µ| < (16 days) −1 on the instantaneous time derivative at this redshift (as opposed to constraining an average field difference, ∆φ, over cosmological times), provided only that the scalar is light enough to be radiated -i.e. m < ∼ 10 −23 eV -independent of how the scalar couples to matter. This can also be interpreted as constraining (in a more modeldependent way) the binary's motion relative to any spatial variation of the scalar field within its immediate vicinity within its host galaxy.

We examine the motion of light fields near the bottom of a potential valley in a multi-dimensional field space. In the case of two fields we identify three general scales, all of which must be large in order to justify an effective low-energy approximation involving only the light field, ℓ. (Typically only one of these -the mass of the heavy field transverse to the trough -is used in the literature when justifying the truncation of heavy fields.)We explicitly compute the resulting effective field theory, which has the form of a P (ℓ, X) model, with X = − 1 2 (∂ℓ) 2 , as a function of these scales. This gives the leading ways each scale contributes to any low-energy dynamics, including (but not restricted to) those relevant for cosmology. We check our results with the special case of a homogeneous roll near the valley floor, placing into a broader context recent cosmological calculations that show how the truncation approximation can fail. By casting our results covariantly in field space, we provide a geometrical criterion for model-builders to decide whether or not the singlefield and/or the truncation approximation is justified, identify its leading deviations, and to efficiently extract cosmological predictions.