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Gravitational wave

Gravitational waves are ripples in the fabric of spacetime produced by the acceleration of massive objects, such as the inspiral and merger of binary black holes or neutron stars, and they propagate outward from their sources at the speed of light.^[1] Predicted by Albert Einstein in 1916 as a direct consequence of his general theory of relativity, these waves carry energy as gravitational radiation and cause tiny distortions in spacetime that stretch and squeeze objects in their path by fractions smaller than the width of a proton.^[1] Unlike electromagnetic waves, gravitational waves interact very weakly with matter, allowing them to pass through the universe unimpeded and provide information about events obscured from optical or radio telescopes.^[2] The existence of gravitational waves was first indirectly confirmed in 1974 through observations of the binary pulsar PSR B1913+16, where the orbital decay matched general relativity's predictions for energy loss due to gravitational radiation, earning Russell Hulse and Joseph Taylor the 1993 Nobel Prize in Physics.^[3] Direct detection came a century after Einstein's prediction, with the Laser Interferometer Gravitational-Wave Observatory (LIGO) announcing on February 11, 2016, the observation of GW150914 on September 14, 2015—a signal from two black holes, each about 30 times the mass of the Sun, merging 1.3 billion light-years away to form a 62-solar-mass black hole. This breakthrough, involving LIGO's twin detectors in Louisiana and Washington, validated general relativity in the strong-field regime and initiated the era of gravitational-wave astronomy.^[1] Since 2015, more than 300 gravitational-wave events have been detected (as of September 2025) by the global network of ground-based observatories, including LIGO, Virgo in Italy, and KAGRA in Japan, primarily from binary black hole mergers but also from neutron star systems, during observing runs including the ongoing fourth run (O4, 2023–2025).^[4] ^[5] A landmark multi-messenger event, GW170817 in 2017, involved a binary neutron star merger observed in gravitational waves and across the electromagnetic spectrum, from gamma rays to radio, revealing connections between short gamma-ray bursts, kilonovae, and the production of heavy elements like gold via rapid neutron capture.^[6] These observations have tested fundamental physics, including the nature of black holes, the equation of state of nuclear matter, and constraints on deviations from general relativity, while population studies of mergers probe stellar evolution, binary formation, and the expansion rate of the universe.^[6] Looking ahead, gravitational-wave astronomy is expanding with upgrades to existing detectors and new facilities, such as the space-based Laser Interferometer Space Antenna (LISA), scheduled for launch in 2035, which will observe low-frequency waves (millihertz range) from supermassive black hole binaries and extreme mass-ratio inspirals.^[7] Ground-based third-generation observatories like the Einstein Telescope and Cosmic Explorer aim to increase sensitivity by an order of magnitude, enabling detections up to cosmological distances (redshift z ~ 20) and precision tests of gravity in uncharted regimes.^[6] Pulsar timing arrays, using millisecond pulsars as galactic interferometers, reported evidence in 2023 of a low-frequency stochastic gravitational-wave background, likely from supermassive black hole binaries, heralding nanohertz astronomy.^[8]

Overview and History

Definition and basic principles

Gravitational waves are transverse disturbances in the metric tensor of spacetime, arising from the acceleration of massive objects within the framework of general relativity. These waves manifest as rhythmic oscillations that cause the distances between freely falling test masses to alternately compress and stretch in directions perpendicular to the direction of propagation. Predicted by Albert Einstein in 1916, they represent a dynamic aspect of gravity, where spacetime itself acts as the medium for the propagation of these ripples.^[9] Analogous to electromagnetic waves generated by accelerating electric charges, gravitational waves are produced by the time-varying gravitational fields of accelerating masses, but they differ fundamentally as distortions of spacetime geometry rather than oscillations in a field permeating space. However, due to the tensor nature of gravity and conservation of momentum, gravitational radiation requires a non-spherical (quadrupole or higher) distribution of accelerating masses; symmetric motions, such as those of a uniformly expanding or contracting sphere, do not emit detectable waves. In the weak-field limit, where the dimensionless strain h (the fractional change in distance) satisfies |h| \ll 1, the effects of these waves can be analyzed using the linear approximation to general relativity. This approximation treats spacetime perturbations as small deviations from flat Minkowski spacetime, simplifying the Einstein field equations to a form amenable to wave solutions.^[10]^[11] The linearized Einstein field equations in this regime, under the Lorentz gauge condition \partial^\alpha \bar{h}_{\mu\alpha} = 0, yield the wave equation for the trace-reversed perturbation \bar{h}_{\mu\nu} = h_{\mu\nu} - \frac{1}{2} h \eta_{\mu\nu} (where h = h^\alpha_\alpha and \eta_{\mu\nu} is the Minkowski metric):

\Box \bar{h}_{\mu\nu} = -\frac{16\pi G}{c^4} T_{\mu\nu},

with \Box = \eta^{\alpha\beta} \partial_\alpha \partial_\beta the d'Alembertian operator and T_{\mu\nu} the stress-energy tensor. In vacuum regions far from sources (T_{\mu\nu} = 0), this reduces to \Box \bar{h}_{\mu\nu} = 0, describing plane waves propagating at the speed of light c. These solutions exhibit two independent polarization states, often denoted as the plus (+) and cross (×) modes, which shear spacetime in orthogonal planes.^[11] Unlike sound waves, which require a material medium and dissipate through interactions, or electromagnetic waves, which can be absorbed or scattered by charged particles, gravitational waves interact only feebly with matter due to the extreme weakness of gravity compared to other forces. Consequently, they traverse the universe virtually unimpeded, carrying pristine information from their distant sources over cosmological distances.^[1]

Historical predictions and early concepts

The concept of gravitational waves emerged in the early 20th century as physicists sought to extend wave propagation ideas from electromagnetism to gravity. In 1905, Henri Poincaré suggested that gravitational effects might travel as waves at the speed of light, analogous to electromagnetic waves, in his presentation to the French Academy of Sciences on the dynamics of the electron. This idea anticipated the need for a relativistic theory of gravity where disturbances propagate finitely rather than instantaneously as in Newtonian mechanics.^[12] Albert Einstein provided the first rigorous theoretical foundation for gravitational waves in 1916, shortly after completing general relativity. In his paper on the approximate integration of the gravitational field equations, Einstein derived linearized solutions to the Einstein field equations that describe transverse quadrupole waves propagating at the speed of light, carrying energy away from accelerating masses. These waves arise as perturbations in the metric tensor, representing ripples in spacetime itself.^[12] Subsequent theoretical advancements clarified the implications of these waves for energy conservation. In 1918, Emmy Noether's theorems linked continuous symmetries in the action of general relativity to conserved quantities, revealing that global conservation laws hold only through nonlocal surface integrals at spatial infinity; this framework implied that isolated systems emitting gravitational waves would lose energy carried to infinity. Noether's work, developed in collaboration with David Hilbert and Felix Klein, addressed foundational issues in variational principles for curved spacetimes. By 1957, Andrzej Trautman extended these ideas by applying Sommerfeld-like radiation conditions to general relativity, demonstrating that gravitational waves could be extracted from isolated systems, allowing explicit calculation of radiated energy and confirming the physical reality of wave emission in nonlinear regimes.^[13] Early predictions faced significant conceptual challenges, including doubts about whether gravitational waves could exist in the full nonlinear theory. Einstein himself, collaborating with Nathan Rosen in the 1930s, explored exact solutions and initially concluded that waves might be mere coordinate artifacts, as attempts to construct propagating wave metrics led to singularities rather than physical radiation. This skepticism stemmed from the absence of radiating solutions in closed spacetimes and difficulties in defining energy localization. The issue was resolved in 1962 by Hermann Bondi, M. G. J. van der Burg, A. W. K. Metzner, and Rainer K. Sachs, who analyzed gravitational fields in asymptotically flat spacetimes using null coordinates; their formalism showed that radiation reduces the Bondi mass at null infinity, providing a conserved flux for gravitational energy carried by waves.^[12]^[14] In the pre-detection era of the 1960s, theoretical maturity spurred experimental efforts despite the waves' expected weakness. Physicist Joseph Weber claimed in 1969 to have detected gravitational wave signals using resonant aluminum bars at sites 1,000 km apart, reporting coincident signals between the detectors attributed to galactic sources. These announcements, published in Physical Review Letters, generated intense interest but were later debunked by replicate experiments from groups at Bell Labs, the University of Rochester, and others, which found no correlated signals above noise levels. Nonetheless, Weber's pioneering resonant mass detectors validated the feasibility of direct searches and catalyzed international collaboration in gravitational wave research.^[15]

Discovery and confirmation

The first indirect evidence for gravitational waves came from observations of the binary pulsar PSR B1913+16, discovered in 1974 by Russell A. Hulse and Joseph H. Taylor, Jr., using the Arecibo radio telescope.^[3] This system consists of a pulsar orbiting a compact companion, with an orbital period of about 7.75 hours.^[16] By 1978, detailed timing measurements revealed that the orbit was decaying due to energy loss, precisely matching the predictions of general relativity for emission of gravitational waves at a level of 0.2% agreement. This orbital shrinkage, observed at a rate of approximately 2.4 × 10^{-12} s/s, provided strong confirmation of the quadrupole formula for gravitational radiation derived from Einstein's theory. For their discovery and the subsequent analysis demonstrating gravitational wave energy loss, Hulse and Taylor were awarded the 1993 Nobel Prize in Physics, recognizing the binary pulsar's role as a laboratory for testing strong-field general relativity.^[3] The first direct detection of gravitational waves occurred on September 14, 2015, when the Advanced Laser Interferometer Gravitational-Wave Observatory (LIGO) observed the signal GW150914. This event originated from the merger of two black holes with masses of approximately 36 and 29 solar masses, located about 1.3 billion light-years away in the southern sky. The detected strain amplitude was on the order of 10^{-21}, representing a spacetime distortion a thousand times smaller than the diameter of a proton, yet discernible due to LIGO's sensitivity. The signal's characteristic "chirp" waveform, from inspiral through merger and ringdown, matched numerical relativity simulations to high precision, confirming the event as a binary black hole coalescence. This achievement earned Rainer Weiss, Barry C. Barish, and Kip S. Thorne the 2017 Nobel Prize in Physics for decisive contributions to the LIGO detector and the observation of gravitational waves.^[17] A landmark subsequent confirmation arrived with GW170817, detected jointly by LIGO and Virgo on August 17, 2017.^[18] This signal arose from the merger of two neutron stars at a distance of about 140 million light-years, producing a gravitational wave strain of roughly 10^{-21} and enabling the first multimessenger astronomical event when followed 1.7 seconds later by a short gamma-ray burst (GRB 170817A) observed by Fermi and INTEGRAL.^[18] The coincidence of gravitational waves with electromagnetic counterparts, including optical kilonova emission, verified neutron star mergers as sources of such bursts and provided independent measurements of the wave speed matching light.^[18]

Theoretical Properties

Propagation and speed

Gravitational waves propagate through spacetime as ripples in the metric tensor, satisfying the linearized wave equation in vacuum under general relativity. In the transverse-traceless gauge, the perturbation h_{\mu\nu} obeys

\frac{\partial^2 h_{\mu\nu}}{\partial t^2} - c^2 \nabla^2 h_{\mu\nu} = 0,

where c is the speed of light in vacuum. This equation describes non-dispersive propagation at speed c, with waves exhibiting transverse quadrupolar polarization modes: the plus (h_+) mode, which stretches and compresses spacetime along perpendicular axes in the wave's propagation plane, and the cross (h_\times) mode, which does so at 45 degrees to those axes.^[19] The speed of gravitational waves is equal to the speed of light c in vacuum, as predicted by general relativity. This was directly confirmed by the multimessenger observation of GW170817, a binary neutron star merger, where the gravitational wave signal arrived 1.7 seconds before the associated gamma-ray burst GRB 170817A, consistent with both propagating at c over a luminosity distance of approximately 40 Mpc; the arrival time difference constrains the speed of gravitational waves to c (1 - 3 \times 10^{-15}) < v_g < c (1 + 7 \times 10^{-16}).^[20] Lunar laser ranging experiments further support this by testing general relativity's predictions in the Earth-Moon system, yielding no deviations that would indicate a differing propagation speed for gravitational effects, with constraints on violations of the equivalence principle at the level of $4 \times 10^{-14} (as of 2023).^[21] In general relativity, gravitational waves experience minimal dispersion during propagation, even through media, as the theory predicts a frequency-independent speed c to leading order. However, certain modified gravity theories, such as those with massive gravitons or Lorentz-violating terms, introduce dispersion relations where the phase velocity deviates from c at high frequencies, potentially leading to observable delays or amplitude modifications over cosmological distances.

Physical effects on spacetime

Gravitational waves produce tidal distortions in spacetime, manifesting as a fractional change in the proper distance between two points, given by \Delta L / L = h / 2, where h is the dimensionless strain amplitude of the wave.^[22] This effect causes stretching in one direction and compression in the perpendicular direction, with the distortions alternating as the wave passes, reflecting the transverse-traceless nature of gravitational waves in general relativity.^[23] In the presence of a gravitational wave, free test masses separated by a distance L experience relative displacements of order \delta L \sim h L, leading to oscillatory motion without net acceleration in the wave's transverse plane.^[24] This tidal forcing on test masses forms the foundational principle for detecting gravitational waves using laser interferometers, where minute changes in arm lengths are measured to infer the passing wave.^[25] Gravitational waves in general relativity exhibit two independent polarization states: the plus polarization, which stretches spacetime along the x-direction while compressing it along the y-direction (and vice versa), and the cross polarization, which produces distortions along axes rotated by 45 degrees relative to the plus mode.^[26] Unlike electromagnetic waves, general relativity predicts no scalar or longitudinal polarization modes for gravitational waves, ensuring their purely tensorial character. For a typical astrophysical gravitational wave with strain h \approx 10^{-21}—comparable to those detectable by Earth-based observatories—these effects over kilometer-scale separations result in displacements on the order of atomic scales, approximately $10^{-18} meters, illustrating the exquisite sensitivity required for observation.^[27]

Energy transport and redshift

Gravitational waves transport energy from their sources through the effective stress-energy pseudotensor derived in the high-frequency approximation. The averaged energy density and flux for a plane wave in the transverse-traceless gauge are given by \rho = \frac{c^3}{32\pi G} \langle \dot{h}_{ij}^{\rm TT} \dot{h}^{ij \rm TT} \rangle, with the power per unit area (energy flux) F = \frac{c^3}{16\pi G} \langle \dot{h}_{+}^2 + \dot{h}_{\times}^2 \rangle, where \dot{h} denotes the time derivative of the strain amplitude and the angle brackets indicate time averaging over several cycles. This flux quantifies the rate at which energy propagates outward at the speed of light, analogous to electromagnetic waves but rooted in spacetime curvature. The total gravitational wave luminosity from an isolated source is then L_{\rm GW} = \frac{c^5}{G} \int \langle \dot{h}^2 \rangle \, dA, integrated over a closed surface enclosing the source at large distance, where dA is the area element.^[28] In addition to energy, gravitational waves carry linear momentum and angular momentum, leading to observable dynamical effects on the emitting systems. The linear momentum flux equals the energy flux divided by c, so the total momentum radiated is p = E/c, where E is the total energy carried away. Angular momentum is radiated through higher multipoles, with the flux determined by the wave's polarization and directionality. In binary black hole mergers, this anisotropic emission imparts a recoil velocity (kick) to the final remnant, with magnitudes up to approximately 5000 km/s for highly spinning, unequal-mass systems in specific configurations. Such recoils arise from the net linear momentum loss during the inspiral, merger, and ringdown phases, as confirmed by numerical relativity simulations.^[29] For binary systems in circular orbits, the energy loss due to gravitational wave emission drives the inspiral, with the average rate given by the post-Newtonian quadrupole formula \frac{dE}{dt} = -\frac{32}{5} \frac{G^4 \mu^2 M^3}{c^5 r^5}, where \mu is the reduced mass, M is the total mass, G is the gravitational constant, c is the speed of light, and r is the orbital separation. This expression, derived to leading quadrupole order, captures the dominant dissipative effect in the weak-field, slow-motion limit and scales as the fifth power of the orbital velocity, leading to a characteristic inspiral timescale proportional to r^4. Higher-order post-Newtonian corrections refine this for more compact binaries, but the leading term establishes the essential scaling for energy dissipation. When gravitational waves propagate over cosmological distances, they experience redshift due to the expansion of the universe, analogous to electromagnetic radiation. The observed frequency scales as f_{\rm obs} = f_{\rm source} / (1 + z), where z is the cosmological redshift of the source, reflecting the stretching of spacetime during propagation.^[30] The strain amplitude similarly diminishes as h \propto 1/(1 + z), beyond the $1/d_L falloff with luminosity distance d_L, because the wave's wavelength elongates with the scale factor. In observations of binary inspirals, this manifests in the redshifted chirp mass \mathcal{M}_z = \mathcal{M} (1 + z), where the intrinsic chirp mass \mathcal{M} is inferred from the observed signal after accounting for z, enabling joint constraints on source properties and cosmology via the relation d_L(z) = (1 + z) \int_0^z \frac{c \, dz'}{H(z')} in a flat universe.^[30] These scalings allow gravitational wave events to serve as standard sirens for measuring the Hubble parameter and dark energy equation of state.

Quantum and Advanced Aspects

Wave-particle duality and gravitons

In quantum field theory formulated on curved spacetime, gravitational waves exhibit wave-particle duality, manifesting as classical wave phenomena that arise from coherent superpositions of gravitons, the hypothetical quanta of the gravitational field. These gravitons are massless bosons with spin 2 and two possible helicity states, +2 or -2, propagating at the speed of light. Each graviton carries an energy given by E = h f, where h is Planck's constant and f is the frequency, analogous to photons in electromagnetism. The classical wave description emerges in the limit of high occupation numbers, where the coherent state behaves collectively like a propagating ripple in spacetime. The graviton serves as the predicted mediator of the gravitational interaction in attempts to quantize general relativity, coupling to the stress-energy tensor of matter and fields with an extremely weak strength determined by the gravitational constant G. This coupling is characterized by a vertex factor proportional to \sqrt{G \hbar / c^3}, yielding an effective scale of approximately $10^{-19} GeV^{-1} in natural units, reflecting the immense energy scale of the Planck mass M_{\rm Pl} \approx 1.22 \times 10^{19} GeV. Due to this feeble interaction—five orders of magnitude weaker than the weak nuclear force—individual gravitons evade direct detection, as the cross-section for graviton-matter scattering is suppressed by factors of $1/M_{\rm Pl}^2.^[31]^[32]^[33] A full quantum theory of gravity incorporating gravitons remains elusive, as perturbative quantization of general relativity encounters non-renormalizable divergences beyond one-loop order, precluding a consistent ultraviolet completion within standard quantum field theory frameworks. Semiclassical approximations address this by treating gravitational waves as quantum fields propagating on a classical spacetime background, enabling calculations of emission processes from macroscopic sources; this approach parallels the computation of Hawking radiation, where quantum effects near a black hole horizon lead to particle creation in the semiclassical limit. Such methods successfully describe gravitational wave production from astrophysical events but break down at the Planck scale, where quantum fluctuations of spacetime become significant.^[34] No direct observation of gravitons has occurred, consistent with their predicted elusiveness. However, recent proposals (2024–2025) suggest that single gravitons from passing gravitational waves could be detected using advanced quantum sensing technologies, such as optomechanical resonators, though no such observations have been reported as of 2025.^[35]^[36] Indirect constraints arise from high-energy cosmic ray collisions, which reach center-of-mass energies up to $10^{17} eV and could probe quantum gravity effects like graviton pair production or emission; the lack of anomalous signatures in these events imposes bounds on deviations from general relativity, such as extra-dimensional models enhancing graviton couplings, imposing bounds on extra-dimensional models and quantum gravity effects up to scales of ∼10^9 GeV in some scenarios, without evidence for non-standard graviton interactions.

Relation to quantum gravity

General relativity breaks down when quantized, as the theory is non-renormalizable, leading to ultraviolet divergences that become severe at the Planck scale, where quantum gravitational effects are expected to dominate. The Planck length, defined as \ell_p = \sqrt{\frac{\hbar G}{c^3}} \approx 1.6 \times 10^{-35} m, marks this fundamental scale, below which classical notions of spacetime geometry fail.^[37] Gravitational waves offer a unique probe into these high-energy regimes, as their high-frequency components could reveal deviations from general relativity if quantum gravity modifies propagation or generation mechanisms. In string theory, one leading candidate for quantum gravity, the graviton emerges as the massless spin-2 mode of closed strings, providing a natural quantum description of gravitational interactions. Extra dimensions, a hallmark of string theory, can alter gravitational wave propagation by allowing energy to "leak" into compactified dimensions, potentially causing damping or modified dispersion at short wavelengths corresponding to high frequencies.^[38] Such effects would manifest as amplitude suppression or phase shifts in detected signals, offering testable predictions for deviations from four-dimensional general relativity. Loop quantum gravity, another approach to quantum gravity, posits that spacetime is fundamentally discrete at the Planck scale, composed of quantized loops rather than a smooth manifold.^[39] This discreteness implies modified dispersion relations for gravitational waves, where the propagation speed depends on frequency due to the underlying granular structure, potentially leading to delays in high-frequency wave arrival times compared to low-frequency components.^[40] Observations of gravitational waves from distant sources could thus constrain these quantum corrections by measuring any frequency-dependent travel times. To systematically test such quantum gravity effects, the parameterized post-Einsteinian (ppE) framework introduces generic deviations into the gravitational waveform phase and amplitude, allowing searches for modified dispersion relations beyond general relativity. The binary neutron star merger GW170817 provided stringent constraints within this framework, revealing no evidence for frequency-dependent dispersion in the observed signal (∼20–2000 Hz), providing stringent constraints on quantum gravity models predicting such modifications at accessible energies.

Implications for early universe cosmology

Primordial gravitational waves are tensor perturbations generated during the inflationary epoch of the early universe, arising from quantum fluctuations of the spacetime metric. These waves span an extraordinarily broad frequency range, approximately from $10^{-18} Hz to $10^{11} Hz, reflecting the vast scales probed by inflation. The strength of these primordial signals is characterized by the tensor-to-scalar ratio r, which quantifies the amplitude of tensor modes relative to scalar perturbations; observations from the BICEP/Keck collaboration have constrained this parameter to r < 0.036 at 95% confidence level, providing a key test of inflationary models. A stochastic gravitational wave background can also originate from early universe processes such as first-order phase transitions or the dynamics of cosmic strings, contributing to the relic radiation density. These backgrounds are described by the dimensionless energy density parameter \Omega_\mathrm{GW}(f) = \frac{1}{\rho_c} \frac{d\rho_\mathrm{GW}}{d \ln f}, where \rho_c is the critical density of the universe and \rho_\mathrm{GW} is the gravitational wave energy density; such signals would encode information about symmetry breaking in the very early cosmos. For cosmic strings, the background arises from the continuous emission and reconnection events in the string network, with the spectrum peaking at frequencies tied to the string tension. Phase transitions, potentially linked to electroweak or QCD scales, produce bursts of gravitational waves through bubble collisions and turbulence, yielding a characteristic peaked spectrum in \Omega_\mathrm{GW}(f). Gravitational waves from the reheating phase following inflation, as well as pre-Big Bang scenarios, stem from the coherent oscillations of scalar fields that populate the early universe. During reheating, the inflaton field's oscillations source gravitational waves through parametric amplification, potentially altering the primordial tensor spectrum at frequencies accessible to future detectors. In pre-Big Bang models inspired by string theory, dilaton and other scalar field dynamics prior to the hot Big Bang generate similar wave signals. These contributions could be detectable through their imprint on cosmic microwave background (CMB) B-mode polarization patterns or via low-frequency searches with pulsar timing arrays, offering probes of physics beyond standard inflation. Constraints from Big Bang nucleosynthesis (BBN) impose strict limits on high-frequency gravitational waves, as their energy density would otherwise heat baryons and disrupt the light element abundances predicted by standard cosmology. Specifically, waves at frequencies around 10 kHz are bounded by \Omega_\mathrm{GW} < 10^{-5}, ensuring that the extra energy injection does not exceed the tolerated deviations in deuterium or helium yields. These bounds highlight how gravitational waves serve as a complementary tool to electromagnetic observations in constraining early universe energetics.

Astrophysical Sources

Binary mergers and compact objects

Binary mergers involving compact objects, such as black holes and neutron stars, are among the most prominent astrophysical sources of gravitational waves. These events occur when two compact objects in a binary system spiral toward each other due to energy loss via gravitational radiation, eventually coalescing. The resulting gravitational waveform is characterized by three distinct phases: the inspiral, where the objects orbit each other at increasing speeds; the merger, where they collide; and the ringdown, where the final distorted black hole settles into a stable configuration.^[41] During the inspiral phase, the orbital frequency f evolves according to the post-Newtonian (PN) approximation derived from general relativity, following the relation f \propto (t_c - t)^{-3/8}, where t_c is the time of coalescence and t is the time before merger. This "chirp" signal arises from the leading-order quadrupole formula for gravitational wave emission, causing the frequency and amplitude to increase rapidly as the binary tightens. The merger phase peaks near the innermost stable circular orbit (ISCO), where strong-field effects dominate, transitioning from perturbative PN descriptions to full numerical relativity simulations. The ringdown phase then emits quasi-normal modes, resembling a damped sinusoid as the final black hole relaxes.^[42] Compact binary systems include black hole-black hole (BBH) mergers, neutron star-neutron star (NS-NS) mergers, and hybrid neutron star-black hole (NS-BH) systems. A canonical BBH example is GW150914, involving component masses of approximately 36 M_\odot and 29 M_\odot, which merged to form a 62 M_\odot black hole. For NS-NS binaries, GW170817 featured two neutron stars each with masses around 1.4 M_\odot, providing insights into nuclear physics through tidal deformation effects. NS-BH hybrids, such as those in GW200105 and GW200115, involve a neutron star of approximately 1.4 M_\odot orbiting a black hole of approximately 8.9 M_\odot for GW200105 and 5.7 M_\odot for GW200115, and are expected to produce distinct waveforms influenced by the neutron star's tidal disruption.^[43]^[44]^[45] The efficiency of gravitational wave emission in these mergers is quantified by the fraction \eta of the total rest mass converted to wave energy. For non-spinning, equal-mass BBH systems, \eta \approx 0.05, as exemplified by GW150914 where about 3 M_\odot (5% of the initial mass) was radiated away. Spin alignment can enhance this efficiency, with aligned spins allowing \eta up to approximately 0.1 in optimized configurations, as predicted by numerical relativity simulations. These emissions carry away not only energy but also angular momentum, shaping the final black hole's properties.^[43] Population studies from gravitational wave detections provide merger rate estimates for these systems. For BBH mergers, analyses of the LIGO-Virgo observing runs O3 and early O4 yield rates in the range of 10-100 Gpc^{-3} yr^{-1}, reflecting the abundance of stellar-mass black hole binaries in the local universe. These rates evolve with redshift and inform models of binary formation channels, such as isolated field evolution or dynamical interactions in dense environments. NS-NS and NS-BH rates are lower, constrained to below a few Gpc^{-3} yr^{-1} from current observations.^[46]^[47]

Core-collapse supernovae

Core-collapse supernovae (CCSNe) arise from the gravitational collapse of the iron cores in massive stars with initial masses exceeding 8 solar masses (M > 8 M_⊙), leading to a rapid core bounce that can produce detectable gravitational waves if asymmetries are present. During the collapse phase, the core implodes under its own gravity until nuclear densities are reached, at which point a shock wave forms and rebounds, creating a proto-neutron star. Non-axisymmetric features, such as those induced by rapid rotation or turbulent convection, perturb the collapsing material and generate time-varying mass quadrupole moments, the primary source of gravitational wave emission in general relativity. These asymmetries arise because stellar evolution often imparts differential rotation, and convective instabilities in the post-bounce phase further amplify deviations from spherical symmetry, enabling the emission of gravitational waves from deep within the stellar interior.^[48]^[49] The gravitational wave signals from CCSNe are characterized as short-duration bursts, typically lasting a few milliseconds, with dominant frequencies in the range of 100–1000 Hz, arising primarily from the excitation of fundamental (f-) and gravity (g-) modes in the proto-neutron star following the bounce. The peak strain amplitude for a Galactic event at a distance of 10 kpc is estimated to be around 10^{-21}, though this depends strongly on the degree of asymmetry and progenitor properties; such signals are generally too weak for detection beyond the Milky Way with current instruments, limiting prospects to our Galaxy or nearby satellites like the Large Magellanic Cloud. Simulations indicate that the waveform features a prominent initial spike from the core bounce, followed by a ring-down phase modulated by oscillations, with the signal's detectability enhanced in the most rapidly rotating progenitors.^[50]^[51] Numerical models of rotating core-collapse, incorporating general relativistic magnetohydrodynamics, predict that the gravitational wave strain scales proportionally with the angular momentum asymmetry δJ of the core, specifically h ∝ δJ / (G M / c²), where M is the core mass, reflecting the efficiency of quadrupole radiation from differential rotation during bounce and early post-bounce evolution. These simulations, often using progenitors with initial rotation periods of hours to days, demonstrate that even modest asymmetries (δJ ~ 10^{47}–10^{49} erg s) can yield strains approaching the sensitivity thresholds of advanced detectors, while highly symmetric collapses produce negligible emission. Such models underscore the role of rotation in driving explosion asymmetries and provide templates for waveform reconstruction.^[52]^[48] Historical searches for gravitational waves from CCSNe have focused on SN 1987A, the nearest such event in the Large Magellanic Cloud, yielding stringent upper limits on the emitted strain of h < 10^{-21}–10^{-20} in the 100–1000 Hz band from bar detectors like those operational at the time, with no detection reported despite the event's proximity. Modern analyses using archival data from Advanced LIGO further tighten these limits, constraining possible neutron star properties and explosion asymmetries. Looking ahead, next-generation observatories such as the Einstein Telescope (ETA) are projected to detect gravitational waves from virtually all Galactic CCSNe, enabling detailed inference of core dynamics and explosion mechanisms with signal-to-noise ratios exceeding 100 for typical events.^[53]^[54]^[55]

Continuous emission from rotating stars

Continuous gravitational waves arise from rapidly rotating neutron stars that exhibit non-axisymmetric deformations, primarily due to triaxial ellipticity in their mass distribution. The equatorial ellipticity is defined as \epsilon = \frac{I_{xx} - I_{yy}}{I_{zz}}, where I_{xx}, I_{yy}, and I_{zz} are the principal moments of inertia, with typical values around $10^{-6} arising from mechanisms such as magnetic field distortions or thermal asymmetries during formation.^[56] These deformations cause the star to emit nearly monochromatic gravitational waves at twice the rotation frequency, f \approx 2 f_{\rm rot}, spanning 20–2000 Hz for rotation rates of 10–1000 Hz, making them persistent sources over years or decades.^[57] The characteristic strain amplitude of these waves is given by

h_0 = \frac{4\pi^2 G}{c^4} \frac{\epsilon I_{zz} f^2}{d},

where I_{zz} \approx 10^{38} \, \rm kg \, m^2 is the axial moment of inertia for a typical neutron star of mass M \approx 1.4 M_\odot and radius R \approx 12 \, \rm km, f is the gravitational wave frequency, and d is the source distance.^[56] This amplitude scales quadratically with frequency and ellipticity, so faster-spinning stars are brighter emitters, but detectability diminishes with distance; for instance, a source at 1 kpc with \epsilon = 10^{-6} and f = 200 \, \rm Hz yields h_0 \approx 10^{-25}.^[58] Prominent candidate sources include precessing pulsars, where wobbling misalignments induce time-varying quadrupolar moments; magnetars, whose strong internal fields (B \gtrsim 10^{14} \, \rm G) can sustain deformations up to \epsilon \approx 10^{-5}; and recycled millisecond pulsars, spun up by accretion in binary systems, potentially harboring residual asymmetries from their evolutionary history.^[57] Ongoing all-sky searches by LIGO and Virgo have set stringent upper limits on ellipticity, with \epsilon < 10^{-8} for frequencies above 100 Hz in targeted surveys of known pulsars and even tighter bounds (\epsilon \lesssim 10^{-9}) for nearby isolated sources in broad-frequency scans.^[59]^[60] Pulsar glitches—sudden spin-ups attributed to superfluid vortex avalanches in the stellar interior—can transiently alter the crust's deformation, exciting torsional oscillations or f-modes that produce short-duration gravitational wave bursts lasting seconds to minutes, with peak strains potentially exceeding continuous levels by orders of magnitude if \Delta \epsilon \sim 10^{-4}.^[61] These events offer probes of neutron star interior physics, though no detections have occurred, with upper limits from glitch epochs constraining excited mode amplitudes to below $10^{-6} times the star's canonical energy.^[62]

Stochastic and primordial backgrounds

The stochastic gravitational wave background arises from the incoherent superposition of signals from numerous unresolved astrophysical sources, primarily compact binary systems such as white dwarf binaries, neutron star binaries, and black hole binaries spanning a wide frequency range from approximately $10^{-4} Hz to $10^3 Hz.^[63] This background forms a confusion noise that dominates the low-frequency spectrum in detectors like LISA, with predicted energy density parameter \Omega_\mathrm{GW} \approx 10^{-9} around the millihertz band due to the cumulative emission from galactic and extragalactic populations. Models based on population synthesis, such as those using the StarTrack code, indicate that Population I/II binaries contribute the dominant signal in the LISA band, with \Omega_\mathrm{GW} \sim 3 \times 10^{-12} at 4 mHz, while Population III binaries add a smaller but distinct component.^[63] Primordial gravitational waves, generated during cosmic inflation as tensor perturbations, form a nearly scale-invariant background that probes the earliest universe. The amplitude of these modes is characterized by the tensor-to-scalar ratio r, with current upper limits placing r < 0.036 at 95% confidence from BICEP/Keck CMB observations (as of 2021); a value of r \approx 0.01 would imply \Omega_\mathrm{GW} h^2 \approx 10^{-15} at frequencies around $10^{-16} Hz.^[64] These waves are redshifted to extremely low frequencies today, making them accessible via CMB B-mode polarization rather than direct interferometric detection, and they provide constraints on inflationary energy scales through the relation \Omega_\mathrm{GW} h^2 \propto r integrated over the tensor power spectrum.^[65] Exotic cosmological sources can also contribute to the stochastic background, including topological defects like cosmic strings and first-order phase transitions in the early universe. Cosmic strings, formed during symmetry-breaking phase transitions, develop cusps—points where the string moves near light speed—and kinks—sharp discontinuities—that emit short, high-frequency gravitational wave bursts, leading to a non-Gaussian background with power-law spectrum \propto f^{-1/3} for cusps and \propto f^{-2/3} for kinks.^[66] Similarly, first-order phase transitions at the electroweak scale (\sim 100 GeV) involve bubble nucleation and collisions, producing gravitational waves through scalar field gradients, acoustic waves in the plasma, and turbulence, with peak frequencies around 10 \muHz and \Omega_\mathrm{GW} potentially detectable by LISA for strong transitions (\alpha \gtrsim 0.01).^[67] Recent pulsar timing array observations have provided evidence for a nanohertz-frequency stochastic background. The NANOGrav 15-year dataset, analyzing 67 pulsars, revealed correlations in timing residuals matching the Hellings-Downs curve predicted by general relativity for a gravitational wave background, with a signal-to-noise ratio of approximately 3.4 and a strain amplitude h_c \approx 2.4 \times 10^{-15} at 1 yr^{-1}, likely sourced by a population of supermassive black hole binaries.^[68] Complementing this, the MeerKAT Pulsar Timing Array (MPTA) reported evidence in its 4.5-year dataset from 83 pulsars, detecting Hellings-Downs correlations at 3.2–3.4\sigma significance with strain amplitude h_c \approx 7.5 \times 10^{-15} at 1 yr^{-1} for a shallow spectral index, further supporting the astrophysical origin from supermassive black hole binaries while highlighting the need for refined noise modeling.^[69]

Detection Techniques

Indirect methods and challenges

Prior to direct detection, indirect evidence for gravitational waves was obtained through precise astronomical observations that confirmed general relativity's predictions of energy loss via wave emission. The binary pulsar PSR B1913+16, discovered in 1974, provided the first such evidence when timing measurements revealed an orbital decay rate matching the general relativistic prediction at the level of 0.9983 ± 0.0016 (or within 0.16%) as of 2016.^[70] This decay is attributed to quadrupole gravitational wave emission, far exceeding the expected non-relativistic effects. Solar system tests further supported the propagation properties of gravitational disturbances. The Cassini spacecraft's radio tracking experiment in 2002 measured the Shapiro time delay during a solar conjunction, yielding the post-Newtonian parameter \gamma = 1 + (2.1 \pm 2.3) \times 10^{-5}, consistent with general relativity's prediction of \gamma = 1.^[71] This bound implies that the speed of gravitational wave propagation deviates from the speed of light by less than about $10^{-5} c, ruling out slower or superluminal speeds that could alter wave signatures.^[71] Detecting gravitational waves directly faces profound challenges due to their extreme weakness. Expected strains from astrophysical sources, such as binary mergers in our galaxy, reach only h \sim 10^{-21}, corresponding to spacetime distortions of roughly one part in $10^{21}.^[72] Achieving sensitivity to such levels necessitates kilometer-scale interferometer arms, like the 4 km baselines in initial designs, to produce measurable path-length changes on the order of $10^{-18} m.^[72] Environmental noise sources impose additional limits, particularly at low frequencies. Seismic vibrations generate gravity-gradient noise through Earth density fluctuations, coupling to test masses and mimicking wave signals with a transfer function scaling as (2\pi f)^{-2}, dominant below 30 Hz.^[73] Thermal noise in suspensions and mirror coatings, arising from internal friction and thermoelastic losses, sets a floor around 50–100 Hz, requiring low-loss materials like fused silica with dissipation rates below $10^{-7}.^[74] Directionality exacerbates these issues, as interferometer antenna patterns reduce sensitivity by up to 50% for waves aligned with arm bisectors, necessitating networks for sky localization.^[73] Frequency-dependent obstacles further constrain ground-based efforts. Terrestrial detectors are limited to roughly 10–$10^4 Hz due to seismic interference below 10 Hz and reduced arm-length response above kilohertz. Lower bands, from nanohertz to millihertz, require space-based interferometers or pulsar timing arrays to avoid planetary noise. Technological hurdles include quantum limits from shot and radiation-pressure noise, which balance at the standard quantum limit. The displacement sensitivity is bounded by S_x \geq 2 \hbar / (m \Omega^2), translating to strain h \sim \sqrt{\hbar / (P \tau)} for laser power P and integration time \tau, where \Omega = 2\pi f.^[75] Overcoming this demands advanced squeezing techniques to redistribute vacuum fluctuations, though current implementations approach but do not yet surpass it broadly.^[75]

Ground-based interferometers

Ground-based interferometers detect gravitational waves in the audio-frequency band, typically from 10 Hz to several kHz, by measuring minute changes in the distance between test masses using laser interferometry. These detectors operate on the principle of a Michelson interferometer, where a laser beam is split by a beam splitter into two perpendicular arms of equal length, reflected back by mirrors at the ends, and recombined to produce an interference pattern sensitive to differential arm length changes induced by gravitational waves.^[76] For the Laser Interferometer Gravitational-Wave Observatory (LIGO), each arm measures 4 km in length, providing the baseline scale necessary for detecting strains on the order of $10^{-21}.^[76] To boost sensitivity, the arm cavities are configured as high-finesse Fabry-Pérot resonators, which trap and recycle the light for multiple round trips—up to hundreds—effectively multiplying the optical path length without increasing physical size.^[77] This design, combined with power recycling that reflects unused light back into the interferometer to increase circulating power, achieves a strain sensitivity of approximately \sqrt{S_h(f)} \sim 10^{-23} / \sqrt{\mathrm{Hz}} around 100 Hz in Advanced LIGO.^[78] The primary facilities include LIGO's two widely separated sites in the United States: one at Hanford, Washington, and the other at Livingston, Louisiana, both featuring 4 km arms to enable coincidence detection that rejects local noise. The Virgo interferometer, located near Pisa, Italy, employs a similar Michelson topology but with 3 km arms, complementing LIGO by improving sky localization of sources.^[79] KAGRA, situated underground in the Kamioka mine in Japan, also uses 3 km arms and incorporates cryogenic cooling for mirrors to reduce thermal noise, with the subterranean location minimizing seismic interference.^[80] During the fourth observing run (O4), which began in May 2023, these detectors have incorporated frequency-dependent quantum squeezing to suppress shot noise, enhancing overall sensitivity by up to 20% in key frequency bands compared to prior runs.^[81] In operation, these power-recycled interferometers continuously monitor phase differences between the arms, where a gravitational wave induces a differential phase shift given by

\Delta \phi = \frac{2\pi L}{\lambda} h,

with L the arm length, \lambda the laser wavelength (typically 1064 nm), and h the dimensionless strain. Various noise sources, including seismic vibrations from earthquakes or human activity, are mitigated through multi-stage isolation systems: passive elements like stacked pendulums and rubber absorbers handle low frequencies below 1 Hz, while active feedback using hydraulic actuators and inertial sensors suppresses disturbances up to tens of Hz.^[82] For searches targeting continuous gravitational waves from rapidly rotating neutron stars, the Einstein@Home project leverages volunteer distributed computing to process terabytes of LIGO data, performing all-sky hierarchical searches for nearly monochromatic signals with sensitivities reaching spin-down limits for nearby sources.^[83]

Space-based detectors and pulsar timing arrays

Space-based gravitational wave detectors aim to observe low-frequency signals in the millihertz regime, inaccessible to ground-based instruments due to seismic and gravity-gradient noise. The Laser Interferometer Space Antenna (LISA), a joint ESA-NASA mission planned for launch in the 2030s, will consist of three drag-free spacecraft forming a triangular constellation with arm lengths of 2.5 million km, trailing Earth in a heliocentric orbit.^[7]^[84] LISA's sensitivity spans frequencies from $10^{-4} to $10^{-1} Hz, enabling detection of inspiraling supermassive black hole binaries with masses up to $10^9 solar masses at redshifts z \lesssim 20.^[84] The drag-free technology isolates test masses from spacecraft disturbances using micro-Newton thrusters, achieving optical path-length noise below 10 pm/\sqrt{\mathrm{Hz}} at 1 mHz, which corresponds to a gravitational wave energy density sensitivity of \Omega_{\mathrm{GW}} \sim 10^{-11} for binary sources.^[84]^[85] Conceptual missions like TianQin and DECIGO extend this approach with alternative configurations. TianQin, a proposed Chinese space-based interferometer, features three satellites in geocentric orbits with arm lengths of approximately 170,000 km, targeting millihertz frequencies ($10^{-4} to 1 Hz) for sources such as galactic white dwarf binaries and extreme mass-ratio inspirals.^[86]^[87] DECIGO, a Japanese pathfinder concept, employs 1,000 km arms in a heliocentric orbit to probe the decihertz band (0.1 to 10 Hz), bridging LISA and ground-based detectors while focusing on intermediate-mass black hole mergers and primordial gravitational waves.^[88] Pulsar timing arrays (PTAs) provide an alternative method for detecting nanohertz-frequency gravitational waves ($10^{-9} to $10^{-7} Hz) through precise monitoring of millisecond pulsar pulse arrival times. Collaborations such as NANOGrav, the European Pulsar Timing Array (EPTA), and the Parkes Pulsar Timing Array (PPTA) observe 50 to 100 stably rotating millisecond pulsars, typically with spin periods of 1 to 10 ms, using radio telescopes like the Green Bank Telescope, Effelsberg, and Parkes.^[89]^[90] Gravitational waves induce timing residuals \delta t \sim h T, where h is the wave strain amplitude and T is the observation duration, accumulating over years-long baselines to reveal correlated signals across the array.^[91] For an isotropic stochastic background, these correlations follow the Hellings-Downs curve, a quadrupolar spatial pattern predicted by general relativity for waves propagating through the pulsar-Earth baselines.^[92] In 2023, EPTA's analysis of 24.7 years of data from 25 pulsars yielded marginal evidence for such a background (Bayes factor of 4), strengthening to evidence at approximately 3\sigma significance (Bayes factor of 60, false alarm probability \sim 0.1\%) using a 10.3-year subset with modern instrumentation.^[91] Similar results from NANOGrav's 15-year dataset and combined International PTA efforts confirmed the signal's consistency with the Hellings-Downs prediction, attributing it primarily to a supermassive black hole binary background.^[91]

Observations and Astronomy

Major detections to date

The LIGO-Virgo-KAGRA (LVK) collaboration has detected over 300 gravitational wave events since the first observation in 2015, with the majority classified as binary black hole (BBH) mergers and a smaller number involving neutron stars or mixed systems.^[5] By the conclusion of the fourth observing run (O4) in November 2025, the catalog includes approximately 310 confirmed detections, predominantly BBH events with component masses ranging from about 5 to 150 solar masses (M_⊙) and redshifts typically below z < 1, corresponding to distances within roughly 5 billion light-years.^[93] These events provide a census of compact binary coalescences in the local universe, revealing a population dominated by stellar-mass black holes formed through various astrophysical channels.^[94] Key highlights among early detections include GW150914, the inaugural observation on September 14, 2015, from a BBH merger at a luminosity distance of approximately 410 Mpc (about 3.6 billion light-years, or redshift z ≈ 0.09), involving black holes of roughly 36 M_⊙ and 29 M_⊙ that formed a 62 M_⊙ remnant. Another landmark was GW170817, detected on August 17, 2017, marking the first binary neutron star (BNS) merger observed at a distance of about 40 Mpc (redshift z ≈ 0.01), which produced a kilonova and enabled multimessenger confirmation through electromagnetic counterparts. In the realm of intermediate-mass black holes, GW190521, observed on May 21, 2019, represented a BBH merger with component masses around 85 M_⊙ and 66 M_⊙, yielding a remnant of approximately 142 M_⊙ and challenging models of black hole formation in dense star clusters. During O3 (2019–2020), the first confident neutron star-black hole (NSBH) mergers were identified, such as GW200105 and GW200115, both detected in January 2020, involving a neutron star of about 1.9 M_⊙ merging with black holes of roughly 8.9 M_⊙ and 6.0 M_⊙, respectively, at distances around 900 Mpc and 320 Mpc. The O4 run, spanning May 2023 to November 2025, has significantly expanded the catalog, adding over 200 candidates by mid-2025, including diverse BBH systems that probe the upper end of the mass spectrum.^[5] Notable recent events include GW231123, detected on November 23, 2023, the most massive BBH merger to date, where progenitors of approximately 106 M_⊙ and 132 M_⊙ coalesced into a 225 M_⊙ black hole at a distance of about 2 Gpc.^[95] Additionally, detections in late 2024 and 2025, such as the pair GW241011 and GW241110 observed about one month apart on October 11 and November 10, 2024, respectively, highlighted "second-generation" black holes formed from prior mergers, producing remnants of approximately 23 M_⊙ each, at distances of about 0.7 billion and 2.4 billion light-years.^[96] Searches for the stochastic gravitational wave background during O4 have refined upper limits on the energy density parameter Ω_GW, particularly from unresolved BBH populations, with constraints tightened to levels below 10^{-8} at frequencies around 100 Hz, though no definitive detection has been confirmed. Parameter estimation for these events relies on Bayesian inference to extract source properties, such as the effective inspiral spin parameter χ_eff, which quantifies the aligned component of the black holes' spins and typically ranges from -1 to 1, with most detections showing low values (|χ_eff| < 0.3) indicative of formation via isolated binary evolution. Distances are inferred from the luminosity distance d_L, related to redshift via the cosmological integral:

d_L = (1 + z) \int_0^z \frac{c \, dz'}{H(z')},

where c is the speed of light and H(z') is the Hubble parameter, allowing reconstruction of event locations within standard ΛCDM cosmology. These analyses, performed using tools like Bilby or LALInference, achieve uncertainties of order 10–20% on masses and distances for high-signal-to-noise events.

Multimessenger astronomy

Multimessenger astronomy involves the coordinated observation of gravitational waves (GWs) alongside electromagnetic (EM) signals and neutrinos, enabling a more complete understanding of astrophysical events. The landmark event GW170817, detected on August 17, 2017, by the LIGO and Virgo observatories, exemplified this approach when the GW signal from a binary neutron star (BNS) merger triggered rapid follow-up observations. Approximately 1.7 seconds after the GW detection, gamma-ray bursts were observed by the Fermi Gamma-ray Burst Monitor and the Integral SPI-ACS instrument, confirming the association with a short gamma-ray burst (sGRB 170817A).^[97] Subsequent optical and infrared observations identified the kilonova AT2017gfo in the galaxy NGC 4993, characterized by r-process nucleosynthesis signatures, along with a synchrotron afterglow detected across radio to X-ray wavelengths.^[97] The source was localized to a distance of approximately 40 Mpc.^[97] Subsequent events have highlighted both successes and limitations in multimessenger pursuits. For instance, GW190814, detected on August 14, 2019, by LIGO and Virgo, involved a merger between a 23 M⊙ black hole and a 2.6 M⊙ compact object, potentially a neutron star, but extensive EM follow-up searches across optical, radio, and gamma-ray bands yielded no counterpart, possibly due to the merger's geometry or environment suppressing emission. Neutrino associations remain elusive, though coordinated searches by IceCube have targeted high-energy neutrino emission from GW events during the third LIGO-Virgo observing run (O3), placing upper limits on neutrino fluence and constraining models of emission from BNS or neutron star-black hole (NSBH) mergers without confirmed detections.^[98] These joint observations provide unique benefits beyond GW-only analyses. The EM counterpart to GW170817 allowed an independent distance measurement via redshift, independent of standard candles like supernovae, yielding a Hubble constant estimate of H₀ ≈ 70 km s⁻¹ Mpc⁻¹ and helping resolve tensions in cosmological parameters.^[99] For neutron stars, the GW signal's tidal deformability parameter Λ ≈ 300 constrained the equation of state, implying a neutron star radius of R ≈ 11 km for a 1.4 M⊙ star and ruling out overly stiff or soft models. Looking ahead, the LIGO-Virgo-KAGRA (LVK) collaboration's low-latency alert system enables real-time notifications to EM and neutrino observatories, facilitating rapid follow-ups for transient events. The Astrophysical Multimessenger Observatory Network (AMON) coordinates cross-messenger searches for transients, integrating data from diverse instruments to identify coincidences. Future space-based detectors like LISA, operating in the millihertz band, will probe supermassive black hole binaries and other sources, with expectations for EM counterparts such as accretion flares to enhance multimessenger insights.

Tests of general relativity and future prospects

Observations of gravitational waves have provided stringent tests of general relativity in the strong-field regime. In the parametrized post-Einstein framework, which allows for deviations from general relativity through additional parameters, the binary neutron star merger GW170817 showed no evidence of dipole radiation during the inspiral phase, consistent with the parameter \phi = 0 predicted by general relativity.^[100] This absence constrains alternative theories that predict enhanced emission at lower multipole orders.^[101] The ringdown phase of binary black hole mergers further validates general relativity's no-hair theorem, which states that the final black hole is fully described by its mass, spin, and charge (with charge negligible). Quasi-normal modes, the damped oscillations emitted during ringdown, have frequencies and damping times matching those expected for Kerr black holes. For instance, in GW150914, the dominant mode's frequency scales approximately as f \sim \frac{c^3}{G M}, where M is the final black hole mass, with no significant deviations from general relativity.^[102] Similar consistency holds in subsequent detections, such as GW190521, where multiple modes align with the theorem's predictions.^[103] No non-general-relativistic signatures, such as extra polarizations or dispersion, have been observed in gravitational wave detections to date. The propagation speed of gravitational waves from GW170817 matching that of light to within $10^{-15} imposes tight bounds on massive gravity theories, limiting the graviton mass to m_g < 1.2 \times 10^{-22} eV/c^2. Future upgrades and detectors promise enhanced tests of general relativity. The LIGO A+ upgrade, targeting the 2030s observing run O5, will incorporate advanced optics and squeezing to increase the detectable volume by a factor of about 10, enabling more precise waveform comparisons.^[6] Space-based missions like LISA will observe supermassive black hole binaries, probing general relativity in the low-frequency regime with millihertz sensitivity.^[84] Third-generation ground-based detectors, such as the underground Einstein Telescope, aim for a tenfold sensitivity improvement over current instruments, facilitating multimode ringdown tests and searches for deviations in extreme-mass-ratio inspirals.^[104] Pulsar timing arrays, sensitive to nanohertz waves, will complement these by constraining cosmological parameters through stochastic backgrounds.^[105] Gravitational waves also offer applications beyond direct tests, including probes of dark matter and the universe's expansion. As standard sirens, they provide luminosity distances independent of cosmic distance ladders; with over 10 events, they could measure the Hubble constant H_0 to percent-level precision, resolving tensions in expansion rate estimates. Additionally, gravitational waves can detect dark matter effects, such as through modifications to binary waveforms from dark matter accretion or superradiance around spinning black holes.^[106]

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[PDF] Gravitational Waves from Spinning Neutron Stars
This paper reviews the search for continuous gravitational waves from spinning neutron stars, their emission mechanisms, and the expected signal strength. ...
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Inferring neutron star properties with continuous gravitational waves
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Deep Einstein@Home All-sky Search for Continuous Gravitational ...
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### Summary of PPN Parameter Gamma and Implications
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NANOGrav
### NANOGrav PTA Details
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The European Pulsar Timing Array - EPTA
### EPTA Details Summary
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The second data release from the European Pulsar Timing Array III ...
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LIGO, Virgo and KAGRA observed “second generation” black holes ...
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Multimessenger Parameter Estimation of GW170817
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Jul 1, 2019 · First, we study the general-relativistic dynamics of the source, in particular constraining dipole radiation in the strong-field and radiative ...
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[2505.00797] Pulsar Timing Arrays - arXiv
May 1, 2025 · Pulsar Timing Arrays (PTAs) have recently found strong evidence for low-frequency gravitational waves (GWs) in the nanohertz frequency regime.
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[1907.10610] Gravitational wave probes of dark matter - arXiv
Jul 24, 2019 · In this white paper, we discuss the prospects for characterizing and identifying dark matter using gravitational waves, covering a wide range of dark matter ...