1 Introduction

The first direct detection of gravitational waves (GWs), widely expected in the mid 2010s with advanced ground-based interferometers [219, 2], will represent the culmination of a fifty-year experimental quest [124]. Soon thereafter, newly plentiful GW observations will begin to shed light on the structure, populations, and astrophysics of mostly dark, highly relativistic objects such as black holes and neutron stars. In the low-frequency band that will be targeted by space-based detectors (roughly 10^–5 to 1 Hz), GW observations will provide a census of the massive black-hole binaries at the center of galaxies, and characterize their merger histories; probe the galactic population of binaries that include highly evolved, degenerate stars; study the stellar-mass objects that spiral into the central black holes in galactic nuclei; and possibly detect stochastic GW backgrounds from the dynamical evolution of the very early universe.

Thus, there are very strong astrophysical motivations to observe the universe in GWs, especially because the systems and phenomena that can be observed in this fashion are largely orthogonal to those accessible to traditional electromagnetic (EM) and astroparticle astronomy. The promise of GWs appears just as great for fundamental physics. Einstein’s theory of gravity, general relativity (GR), has been confirmed by extensive experimental tests; but these have largely been confined to the solar system, where gravity is well approximated by Newtonian gravity with small corrections. A few tests, based on observations of binary compact-object systems, have confirmed the weakest (leading-order) effects of GW generation. By contrast, observation of strong GWs will provide the first direct observational probe of the dynamical, strong-field regime of GR, where the nature and behavior of gravity can be significantly different from the Newtonian picture. GWs are prima facie the perfect probe to investigate gravitation, since they originate directly from the bulk motion of gravitating matter, relieving the need to understand and model the physics of other intermediate messengers, typically photons from stellar surfaces or black-hole surroundings.

Already today we can rely on a very sophisticated understanding of the analytical and numerical techniques required to model GW sources and their GW emission, including the post-Newtonian expansion [84, 190], black-hole perturbation theory [119], numerical relativity for vacuum spacetimes [368], spacetimes with gases or magnetized plasmas [184], and much more. That these techniques should have been developed so much in the absence of a dialogue with experimental data (except for the binary pulsar [293]) is witness to the great perceived promise of GW astronomy. For a bird’s-eye view of the field, see the Living Review by Sathyaprakash and Schutz [395], who cover the physics of GWs, the principles of operation of GW detectors, the nature of GW sources, the data analysis of GW signals, and the science payoffs of GW observations for physics, astrophysics, and cosmology.

This review focuses on the opportunities to challenge or confirm our understanding of gravitational physics that will be offered by forthcoming space-based missions to observe GWs in the low-frequency band between 10^–5 and 1 Hz. Most of the literature on this subject has focused on one mission design, LISA (the Laser Interferometer Space Antenna [64, 252, 370]), which was studied jointly by NASA and ESA between 2001 and 2011. In 2011, budgetary and programmatic reasons led the two space agencies to end this partnership, and to pursue space-based GW detection separately, studying cheaper, rescoped LISA-like missions.

ESA’s proposed eLISA/NGO [20] would be smaller than LISA, fly on orbits closer to Earth, and operate interferometric links only along two arms. In 2012 eLISA was considered for implementation as ESA’s first large mission (“L1”) in the Cosmic Vision program. A planetary mission was selected instead, but eLISA will be in the running for the next launch slot (“L2”), with a decision coming as soon as 2014. NASA ran studies on a broader range of missions [215], including several variants of LISA to be implemented by NASA alone, as well as options with a geocentric orbit (OMEGA [229]), and without drag-free control (LAGRANGE [304]). The final study report concludes that scientifically compelling missions can be carried out for less, but not substantially less, than the full LISA cost; that scientific performance decreases far more rapidly than cost; and that no design choice or technology can make a dramatic reduction in cost without much greater risks. The NASA study noted the possibility of participation in the ESA-led eLISA mission (if selected by ESA) as a minority partner.

Whatever specific design is eventually selected, it is likely that its architecture, technology, and scientific reach will bear a strong resemblance to LISA’s (with the appropriate scalings in sensitivity, mission duration, and so on). Thus, the research reviewed in this article, which was targeted in large part to LISA, is still broadly relevant to future missions. Such LISA-like observatories are characterized by a few common elements: a set of three spacecraft in long-baseline (Mkm) orbits, monitoring their relative displacements using laser interferometry; drag-free operation (except for LAGRANGE [304]), whereby displacement measurements are referenced to freely falling test masses protected by the spacecraft, which hover around the masses using precise micro-Newton thrusters; frequency correction of laser noise using a variety of means, including onboard cavities and interferometers, arm locking, and a LISA-specific technique known as Time-Delay Interferometry.

The predictions of GR that can be tested by space-based GW observatories include the absence of gravitational fields other than the metric tensor; the number and character of GW polarization states; the speed of GW propagation; the detailed progress of binary inspiral, as driven by nonlinear gravitational dynamics and loss of energy to GWs; the strength and shape of the GWs from binary merger and ringdown; the true nature of astrophysical black holes; and more.

Some of these tests will also be performed with ground-based GW detectors and pulsar-timing observations [500], but space-based tests will almost always have superior accuracy and significance, because low-frequency sources are intrinsically stronger, and will spend a larger time within the band of good detector sensitivity. For binary systems with very asymmetric mass ratios, such as extreme mass-ratio inspirals (EMRIs), LISA-like missions will measure hundreds of thousands of orbital cycles; because successful detections require matching the phase of signals throughout their evolution, it follows that these observations will be exquisitely sensitive to source parameters. The data-analysis detection problem will be correspondingly delicate, but has been tackled both theoretically [38], and in a practical program of mock data challenges for LISA [37, 39, 450].

The rest of this review is organized as follows. Section 2 provides the briefest overview of Einstein’s GR, of the theoretical framework in which it can be tested, and of a few leading alternative theories. It also introduces the “black-hole paradigm,” which augments Einstein’s equations with a few assumptions of physicality that lead to the prediction that the end result of gravitational collapse are black holes described by the Kerr metric. Section 3 reviews the “classic” LISA architecture, as well as possible options for LISA-like variants. Section 4 summarizes the main classes of GW sources that would be observed by LISA-like detectors, and that can be used to test GR. Section 5 examines the tests of gravitational dynamics that can be performed with these sources, while Section 6 discusses the tests of the black-hole nature and structure. (A conspicuous omission are possible stochastic GW backgrounds of cosmological origin [82]; indeed, in this article we do not discuss the role of space-based detectors as probes of cosmology and early-universe physics.) Last, Section 7 presents our conclusions and speculations.