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The recent detection of coalescing black holes by the Laser Interferometer Gravitational-wave Observatory has brought forth the era of gravitational wave astronomy. Physicists are only now beginning to probe the mergers of compact objects that send ripples through space and time. These distortions carry with them the information from the system where they originated. The dynamics of black hole collisions and neutron star mergers are new and exciting events which were undetectable just a few years ago. Einstein's theory of General Relativity has done an excellent job of describing gravity and the information that can be extracted from gravitational systems. However, his theory contains several anomalies such as the inability to explain the inflation of the universe, the effects of dark matter and energy, the presence of singularities, as well as a failure to reconcile with quantum mechanics. Modified theories of gravity have been proposed to answer any remaining questions about gravitation while prescribing solutions to the problems General Relativity still has. The work within this thesis describes how we may study modified theories of gravity in the strong field regime through two different means. The first, is through the calculation of the rate of gravitational radiation from binary systems. This rate varies depending on the theory of gravity being studied. Comparing the theoretical predictions of these rates from alternative theories to astronomical observation will allow us to place better constraints on modified gravity and test General Relativity like never before. The second way is through the investigation of the spacetime surrounding a neutron star. Unlike black holes which emit no light, we are able to see neutron stars (more specifically pulsars) through their light curve as they rotate. The shape of the light curve is dictated by the theory of gravitation used to describe the spacetime around the neutron star. My goal of constructing such a spacetime for neutron stars in modified gravity allows for future scientists to study the light curves to be detected and place constraints on the particular theory.

The spacetime surrounding compact objects such as neutron stars and black holes provides an excellent place to study gravity in the strong, non-linear, dynamical regime. Here, the effects of strong curvature can leave their imprint on observables which we may use to study gravity. Recently, NICER provided a mass and radius measurement of an isolated neutron star using x-rays, while LIGO/Virgo measured the tidal deformability of neutron stars through gravitational waves. These measurements can be used to test the relation between the tidal deformability and compactness of neutron stars that are known to be universal in general relativity. Here, we take (shift-symmetric) scalar-Gauss-Bonnet gravity (motivated by a low-energy effective theory of a string theory) as an example and study whether one can apply the NICER and LIGO/Virgo measurements to the universal relation to test the theory. To do so, this paper is mostly devoted on theoretically constructing tidally-deformed neutron star solutions in this theory perturbatively and calculate the tidal deformability for the first time. We find that the relation between the tidal deformability and compactness remains to be mostly universal for a fixed dimensionless coupling constant of the theory though the relation is different from the one in general relativity. We also present a universal relation between the tidal deformability \\textbf{of} one neutron star and the compactness for another neutron star that can be directly applied to observations by LIGO/Virgo and NICER. For the equations of state considered in this paper, it is still inconclusive whether one can place a meaningful bounds on scalar Gauss-Bonnet gravity with the new universal relations. However, we found a new bound from the mass measurement of the pulsar J0740+6620 that is comparable to other existing bounds from black hole observations.

Recent observations of gravitational waves from binary black holes and neutron stars allow us to probe the strong and dynamical field regime of gravity. On the other hand, a collective signal from many individual, unresolved sources results in what is known as a stochastic background. We here consider probing gravity with such a background from stellar-mass binary black hole mergers. We adopt a simple power-law spectrum and carry out a parameter estimation study with a network of current and future ground-based detectors by including both general relativistic and beyond general relativistic variables. For a network of second-generation detectors, we find that one can place meaningful bounds on the deviation parameter in the gravitational-wave amplitude if it enters at a sufficiently negative post-Newtonian order. However, such future bounds from a stochastic background are weaker than existing bounds from individual sources, such as GW150914 and GW151226. We also find that systematic errors due to mismodeling of the spectrum is much smaller than statistical errors, which justifies our use of the power-law model. Regarding a network of third-generation detectors, we find that the bounds on the deviation parameter from statistical errors improve upon the second-generation case, though systematic errors now dominate the error budget and thus one needs to use a more realistic spectrum model. We conclude that individual sources seem to be more powerful in probing general relativity than the astrophysical stochastic background.

One of the key ingredients for making binary waveform predictions in a beyond-GR theory of gravity is understanding the energy and angular momentum carried by gravitational waves and any other radiated fields. Identifying the appropriate energy functional is unclear in Hassan-Rosen bigravity, a ghost-free theory with one massive and one massless graviton. The difficulty arises from the new degrees of freedom and length scales which are not present in GR, rendering an Isaacson-style averaging calculation ambiguous. In this article we compute the energy carried by gravitational waves in bigravity starting from the action, using the canonical current formalism. The canonical current agrees with other common energy calculations in GR, and is unambiguous (modulo boundary terms), making it a convenient choice for quantifying the energy of gravitational waves in bigravity or any diffeomorphism-invariant theories of gravity. This calculation opens the door for future waveform modeling in bigravity to correctly include backreaction due to emission of gravitational waves.

The recent gravitational wave observations provide insight into the extreme gravity regime of coalescing binaries, where gravity is strong, dynamical and non-linear. The interpretation of these observations relies on the comparison of the data to a gravitational wave model, which in turn depends on the orbital evolution of the binary, and in particular on its orbital energy and angular momentum decay. In this paper, we calculate the latter in the inspiral of a non-spinning compact binary system within Einstein-\\AE{}ther theory. From the theory's gravitational wave stress energy tensor and a balance law, we compute the angular momentum decay both as a function of the fields in the theory and as a function of the multipole moments of the binary. We then specialize to a Keplerian parameterization of the orbit to express the angular momentum decay as a function of the binary's orbital elements. We conclude by combining this with the orbital energy decay to find expressions for the decay of the semi-major axis and the orbital eccentricity of the binary. We find that these rates of decay are typically faster in Einstein-\\AE{}ther theory than in General Relativity due to the presence of dipole radiation. Such modifications will imprint onto the chirp rate of gravitational waves, leaving a signature of Einstein-\\AE{}ther theory that if absent in the data could be used to stringently constrain it.

The observation of the inspiral and merger of compact binaries by the LIGO-Virgo collaboration has allowed for new tests of Einstein's theory in the extreme gravity regime, where gravitational interactions are simultaneously strong, non-linear, and dynamical. Theories beyond Einstein's can also be constrained by detecting the polarization modes of gravitational waves. In this paper, we show that dynamical Chern-Simons and Einstein-dilaton-Gauss-Bonnet gravity cannot be differentiated from general relativity based on the detection of polarization modes alone. To prove this result, we use the Newman-Penrose method and an irreducible decomposition method to find that only the tensorial modes can be detected in both these theories.

The spacetime around compact objects is an excellent place to study gravity in the strong, nonlinear, dynamical regime where solar system tests cannot account for the effects of large curvature. Understanding the dynamics of this spacetime is important for testing theories of gravity and probing a regime which has not yet been studied with observations. In this paper, we construct an analytical solution for the exterior spacetime of a neutron star in scalar-Gauss-Bonnet gravity that is independent of the equation of state chosen. The aim is to provide a metric that can be used to probe the strong-field regime near a neutron star and create predictions that can be compared with future observations to place possible constraints on the theory. In addition to constructing the metric, we examine a number of physical systems in order to see what deviations exist between our spacetime and that of general relativity. We find these deviations to be small and of higher post-Newtonian order than previous results using black hole solutions. The metric derived here can be used to further the study of scalar-Gauss-Bonnet gravity in the strong field, and allow for constraints on corrections to general relativity with future observations.

Vector-tensor theories beyond General Relativity have widely been studied in the context of ultraviolet completion of gravity, endowing a mass to the graviton and explaining dark energy phenomena. We here construct rotating black hole solutions in vector-tensor theories valid after the binary neutron star merger event GW170817 that placed very stringent bound on the propagation speed of gravitational waves away from the speed of light. Such valid vector-tensor theories are constructed by performing a generic conformal transformation to Einstein-Maxwell theory, and the new rotating black hole solutions are constructed by applying the same conformal transformation to the Kerr-Newman solution. These theories fall outside of beyond generalized Proca theories but are within an extended class of vector-tensor theories that satisfy a degenerate condition to eliminate instability modes and are thus healthy. We find that such conformal Kerr-Newman solutions preserve the location of the singularities, event horizons and ergoregion boundary from Kerr-Newman, as well as the multipole moments and the Petrov type. On the other hand, the Hamilton-Jacobi equation is no longer separable, suggesting that the Carter-like constant does not exist in this solution. The standard Newman-Janis algorithm also does not work to construct the new solutions. We also compute the epicyclic frequencies, the location of the innermost stable circular orbits, and the Schwarzschild precession and apply the latter to the recent GRAVITY measurement to place bounds on the deviations away from Kerr-Newman for Sgr A\\(^*\\).

Gravitational-wave memory effects are identified by their distinctive effects on families of freely falling observers: after a burst of waves pass by their locations, memory effects can cause lasting relative displacements of the observers. These effects are closely related to the infrared properties of gravity and other massless field theories, including their asymptotic symmetries and conserved quantities. In this paper, we investigate the connection between memory effects, symmetries, and conserved quantities in Brans-Dicke theory. We compute the field equations in Bondi coordinates, and we define a set of boundary conditions that represent asymptotically flat solutions in this context. Next, we derive the asymptotic symmetry group of these spacetimes, and we find that it is the same as the Bondi-Metzner-Sachs group in general relativity. Because there is an additional polarization of gravitational waves in Brans-Dicke theory, we compute the memory effects associated with this extra polarization (the so-called \"breathing\" mode). This breathing mode produces a uniform expansion (or contraction) of a ring of freely falling observers. After these breathing gravitational waves pass by the observers' locations, there are two additional memory effects that depend on their initial displacements and relative velocities. Neither of these additional memory effects seems to be related to asymptotic symmetries or conserved quantities; rather, they are determined by the properties of the nonradiative region before and after the bursts of the scalar field and the gravitational waves. We discuss the properties of these regions necessary to support nontrivial breathing-mode-type memory effects.

The recent detections of gravitational waves by the advanced LIGO and Virgo detectors open up new tests of modified gravity theories in the strong-field and dynamical, extreme gravity regime. Such tests rely sensitively on the phase evolution of the gravitational waves, which is controlled by the energy-momentum carried by such waves out of the system. We here study four different methods for finding the gravitational wave stress-energy pseudo-tensor in gravity theories with any combination of scalar, vector, or tensor degrees of freedom. These methods rely on the second variation of the action under short-wavelength averaging, the second perturbation of the field equations in the short-wavelength approximation, the construction of an energy complex leading to a Landau-Lifshitz tensor, and the use of Noether's theorem in field theories about a flat background. We apply these methods in General Relativity, scalar-tensor theories and Einstein-\\AE{}ther theory to find the gravitational wave stress-energy pseudo-tensor and calculate the rate at which energy and linear momentum is carried away from the system. The stress-energy tensor and the rate of linear momentum loss in Einstein-\\AE{}ther theory are presented here for the first time. We find that all methods yield the same rate of energy loss, although the stress-energy pseudo-tensor can be functionally different. We also find that the Noether method yields a stress-energy tensor that is not symmetric or gauge-invariant, and symmetrization via the Belinfante procedure does not fix these problems because this procedure relies on Lorentz invariance, which is spontaneously broken in Einstein-\\AE{}ther theory. The methods and results found here will be useful for the calculation of predictions in modified gravity theories that can then be contrasted with observations.