List of Figures

	Figure 1: Time-Delay Interferometry (TDI). LISA-like detectors measure GWs by transmitting laser light between three spacecraft in triangular configuration, and comparing the optical phase of the incident lasers against reference lasers on each spacecraft. To avoid extreme requirements on laser-frequency stability over the course of the many seconds required for transmission around the triangle, data analysts will generate time-delayed linear combinations of the phase comparisons; the combinations simulate nearly equal-delay optical paths around the sides of the triangle, and (much like an equal-arm Michelson interferometer) they suppress laser frequency noise. Many such combinations, including those depicted here, are possible, but altogether they comprise at most three independent gravitational-wave observables. Image reproduced by permission from [447], copyright by APS.
	Figure 2: The discovery space for space-based GW detectors, covering the low-frequency region of the GW spectrum, $−5 10 Hz ≲ f ≲ 0.1 Hz$ . The discovery space is delineated by the LISA threshold sensitivity curve [277] in black, and by the eLISA sensitivity curve in red [21] (the curves were produced using the online sensitivity curve and source plotting website [321]). This region is populated by a wealth of strong sources, often in large numbers, including mergers of MBHs, EMRIs of stellar-scale compact objects into MBHs, and millions of close-orbiting binary systems in the galaxy. Thousands of the strongest signals from these galactic binary systems should be individually resolvable, while the combined signals of millions of them produce a stochastic background at low frequencies. These systems provide ample opportunities for astrophysical tests of GR for gravitational-field strengths that are not well characterized and studied in conventional astronomy.
	Figure 3: Contours of constant SNR for MBH binaries observed with eLISA. The left-hand panel shows contours in the total-mass–redshift plane for equal-mass binaries, while the right-hand panel shows contours in the total-mass–mass-ratio plane for sources at redshift $z = 4$ . Image reproduced by permission from [21].
	Figure 4: Effect of the six possible GW polarization modes on a ring of test particles. The GW propagates in the z-direction for the upper three transverse modes, and in the x-direction for the lower three longitudinal modes. Only modes (a) and (b) are possible in GR. Image reproduced by permission from [471].
	Figure 5: Estimating all the binary-inspiral phasing coefficients $ψk$ of Eq. (51 ) yields differently shaped regions in the $m1$ – $m2$ plane, which must intersect near true mass values if GR is correct. Image reproduced by permission from [27], copyright by APS.
	Figure 6: Constraints on phasing corrections in the ppE framework, as determined from LISA observations of $6 ∼ 10 M ⊙$ massive–black-hole inspirals at $z = 1$ and $z = 3$ . The figure also includes the $β$ bounds derived from pulsar PSR J0737–3039 [492], the solar-system bound on the graviton mass [435], and PN-coefficient bounds derived as described Section 5.2.1. The spike at $b = 0$ corresponds to the degeneracy between the ppE correction and the initial GW-phase parameter. (Adapted from [134].)
	Figure 7: Poincaré map for a regular orbit (left panel) and a chaotic orbit (right panel) in the Manko–Novikov spacetime. Image reproduced by permission from [195], copyright by APS.
	Figure 8: Comparative SNRs, as a function of redshifted black-hole mass $(1 + z)M$ , for the last year of inspiral of an equal-mass MBH binary and for the ringdown after the merger of the system. The method used to generate this figure follows that of [182], updated to use a modern LISA sensitivity curve [277] with a low-frequency cutoff of $f = 1 × 10− 4 Hz$ . The redshift is set to $z = 1$ , at which the luminosity distance is $D = 6.6 Gpc L$ using WMAP 7-year parameters [271].

	Abstract
1	Introduction
2	The Theory of Gravitation
	2.1	Will’s “standard model” of gravitational theories
	2.2	Alternative theories
	2.3	The black-hole paradigm
3	Space-Based Missions to Detect Gravitational Waves
	3.1	The classic LISA architecture
	3.2	LISA-like observatories
	3.3	Mid-frequency space-based observatories
4	Summary of Low-Frequency Gravitational-Wave Sources
	4.1	Massive black-hole coalescences
	4.2	Extreme-mass-ratio inspirals
	4.3	Galactic binaries
5	Gravitational-Wave Tests of Gravitational Physics
	5.1	The “classic tests” of general relativity with gravitational waves
	5.2	Tests of general relativity with phenomenological inspiral template families
	5.3	Beyond the binary inspiral
6	Tests of the Nature and Structure of Black Holes
	6.1	Current observational status
	6.2	Tests of black-hole structure using EMRIs
	6.3	Tests of black-hole structure using ringdown radiation: black-hole spectroscopy
	6.4	Prospects from gravitational-wave and other observations
7	Discussion
	Acknowledgements
	References
	Footnotes
	Figures
	Tables