What one would like to believe is that gravitational waves will be detected by the end of this decade, either through ground-based detectors or through pulsar timing arrays. Given this, there is a concrete effort to develop the proper formalism and implementation pipelines to test Einstein’s theory once data becomes available. Currently, the research groups separate into two distinct classes: theory and implementation. The theory part of the research load is being carried out at a variety of institutions without a given focal point. The implementation part is being done mostly within the LIGO Scientific Collaboration and the pulsar timing consortia. Cross-communication between the theory and implementation groups has recently flourished and one expects more interdisciplinary work in the future.
So many accomplishments have been made in the past 50 years that it is almost impossible to list them here. From the implementation side, perhaps one of the most important is the actual construction and operation of the initial LIGO instruments at design sensitivity in all of their frequency domain. This is a tremendously important engineering and physics challenge. Similarly, the construction of impressive pulsar timing arrays, and the timing of these pulses to nano-second precision is an instrumental and data analysis feat to be admired. Without these observatories no waves would be detectable in the future, and of course, no tests of Einstein’s theory would be feasible. On the theory side, perhaps the most important accomplishment has been the understanding of the inspiral phase to really-high post-Newtonian order and the merger phase with numerical simulations. The latter, in particular, had been an unsolved problem for over 50 years, until very recently. It is these accomplishments that then allow us to postulate modified inspiral template families, since we understand what the GR expectation is. This is particularly true if one is considering small deformations away from Einstein’s theory, as it would be impossible to perturb about an unknown solution.
The main questions that are currently at the forefront are the following. On the theory side of things, one would wish to understand the inspiral to high post-Newtonian order in certain strong-field modifications to GR, like dynamical Chern–Simons gravity or Einstein-Dilaton-Gauss–Bonnet theory. One would also like to investigate theories with preferred frames, such as Einstein-Aether theory or Hořava–Lifshitz gravity, which will lead to Lorentz violating observables. Understanding these theories to high post-Newtonian order is particularly important for those that predict dipolar gravitational emission, such as Einstein-Dilaton-Gauss–Bonnet theory. Such corrections dominate over Einstein’s quadrupole emission at sufficiently low velocities.
Of course, a full inspiral-merger-ringdown template is not complete unless we also understand the merger. This would require full numerical simulations, which are very taxing even within GR. Once one modifies the Einstein field equations, the characteristic structure of the evolution equations will also likely change, and it is unclear whether the standard evolution methods will continue to work. Moreover, when dealing with the merger phase, one is usually forced to treat the modified theory as exact, instead of as an effective theory. Without the latter, it is likely that certain modified theories will not have a well-posed initial value problem, which would force any numerical evolution to fail. Of course, one could order-reduce these equations and then use these to evolve black-hole spacetimes. Much work still remains to be done to understand whether this is feasible.
On the implementation side of things, there is also much work that remains to be done. Currently, efforts are only beginning on the implementation of Bayesian frameworks for hypothesis testing. This seems today like one of the most promising approaches to testing Einstein’s theory with gravitational waves. Current studies concentrate mostly on single-detectors, but by the beginning of the next decade we expect four or five detectors to be online, and thus, one would like to see these implementations extended. The use of multiple detectors also opens the door to the extraction of new information, such as multiple polarization modes, a precise location of the source in the sky, etc. Moreover, the evidence for a given model increases dramatically if the event is observed in several detectors. One therefore expects that the strongest tests of GR will come from leveraging the data from all detectors in a multiply-coincident event.
Ultimately, research is moving toward the construction of robust techniques to test Einstein’s theory. A general push is currently observed toward the testing of general principles that serve as foundations of GR. This allows one to answer general questions, such as: Does the graviton have a mass? Are compact objects represented by the Kerr metric and the no-hair theorems satisfied? Does the propagating metric perturbation possess only two transverse-traceless polarization modes? What is the rate of change of a binary’s binding energy? Do naked singularities exist in nature and are orbits chaotic? Is Lorentz-violation present in the propagation of gravitons? These are examples of questions that can be answered once gravitational waves are detected. The more questions of this type that are generated and the more robust the methods to answer them are, the more stringent the test of Einstein’s theories and the more information we will obtain about the gravitational interaction in a previously unexplored regime.