Gravitational Waves: Science with Compact Binary Coalescences

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Frédérique Marion and Stephen Fairhurst (2024), Scholarpedia, 19(5):53863. doi:10.4249/scholarpedia.53863 revision #201127 [link to/cite this article]
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Curator: Frédérique Marion



Gravitational waves (GW) are ripples in space-time propagating at the speed of light, a prediction of general relativity (GR) confirmed by observations. GW emission is caused by accelerating masses. Strong emission requires astrophysical sources with large masses, significant asymmetry, high compactness (characterized by the ratio of mass to radius) and relativistic motion; see Figure 1. The most powerful and emblematic GW sources are binary systems of compact objects (black holes, neutron stars, white dwarfs) orbiting each other due to gravitational attraction. The dynamics of such a system are driven by the laws of motion, where GR must be taken into account. The emission of energy in GWs drives the system towards coalescence, the orbit shrinking with time due to energy loss, until the compact objects eventually merge.

Figure 1: Comparison of the space-time curvature and timescale involved in the dynamics of GW sources such as GW150914 and GW151226 to other systems or experiments, illustrating how they differ by many orders of magnitude. From (Yunes et al., 2016).

Neutron star binaries historically provided the first evidence for the existence of GWs, the Hulse-Taylor binary pulsar exhibiting a decline of its orbital period matching the prediction of GR (M. Weisberg et al., 2010). More recently, compact binary mergers have provided the first GW signals detected with ground-based interferometric GW detectors (the Laser Interferometer Gravitational-Wave Observatory or LIGO in the United-States, Virgo in Europe, recently joined by KAGRA in Japan). The first observation of GWs was from the merger of two black holes; the event (named after the year, month and day it was observed) is denoted GW150914 (LIGO and Virgo, 2016a). The bandwidth of these detectors (in the range of 10 Hz to 10 kHz) makes them sensitive to the final stage of the binary coalescence for systems involving neutron stars or stellar-mass black holes. Observing GW signals from binary systems at an earlier stage of their evolution or/and with higher masses requires detectors operating at lower frequencies. This article focuses on the compact binary mergers observed in the high-frequency end of the GW spectrum.

Compact binary mergers are rare events which, though powerful, generate only a tiny tidal deformation of space-time requiring extremely sensitive detectors able to probe a large volume of the Universe. Since they were built, LIGO and Virgo have been alternating observing runs and upgrades to the instruments to improve their sensitivity and stability. In recent years, the O1 run (2015-2016) provided order 1 detections, the O2 run (2016-2017) provided order 10, and the O3 run (2019-2020) provided order 100, the cumulative number of event candidates from these runs reaching 91 events identified by the collaborations (LIGO, Virgo and KAGRA, 2023a), with several additional events identified by independent teams analyzing the data. The O4 run commenced in May 2023 and is expected to provide several hundred additional events. Binary black hole (BBH) mergers dominate the observed events, which also includes a few binary neutron star (BNS) and neutron star - black hole (NSBH) mergers. Events are named after the date they occurred (and time of the day if needed). These spectacular detections have opened a broad field of physical and astrophysical interpretation.

The Signal: basic features

Figure 2: The various stages of a compact binary coalescence and the GW signal reconstructed from the GW150914 event observed in the LIGO detectors. From (LIGO and Virgo, 2016a).

Inspiral-merger-ringdown: The evolution of a binary system, and the associated GW signal, during and after coalescence can be divided into three parts: the long inspiral stage, when the orbit shrinks adiabatically (slowly and smoothly), is followed by a stage where the two objects plunge toward each other and merge, then a stage where the subsequent final compact object (typically a black hole) relaxes to its quiescent state. The three stages are usually referred to as the inspiral, merger and ringdown; see Figure 2. Let us first describe the basic features of the signal, with a focus on the inspiral part.

Polarization: GWs are transverse waves and have two tensor, independent modes of polarization in GR, referred to as \(+\) (plus) and \(\times\) (cross). For a given polarization mode, space contracts or expands along orthogonal axes, with the axes of the \(+\) mode being rotated by 45 degrees with respect to the axes of the \(\times\) mode. The GW signal emitted by the binary is not the same in all directions. It is circularly polarized (and strongest) for directions perpendicular to the orbital plane (i.e. for a system viewed face-on, from the point of view of the observer), linearly polarized (and weakest) for directions in the orbital plane (i.e. for a system viewed edge-on), and elliptically polarized in other directions.

Figure 3: The event GW150914 as it appeared in the LIGO Hanford and LIGO Livingston detectors, in the time domain and in the time-frequency domain. The chirp pattern is visible. From (LIGO and Virgo, 2016a).

Chirp pattern and signal-to-noise ratio: The dominant frequency of the emitted GWs corresponds to twice the orbital frequency of the binary (but see Higher-order modes). As the orbit gradually shrinks, the frequency, frequency derivative and amplitude of the GW signal increase, shaping it into the famous chirp pattern; see Figure 3. Although the radiated GW strain is largest at the end of the inspiral process, the signal spends more cycles at lower frequencies than at higher frequencies, so that its spectral amplitude decreases with frequency, following a \(f^{-7/6}\) dependency; see Figure 4. The ratio of the signal power to the detector noise power spectrum determines the density of the squared signal-to-noise ratio (SNR), whose integral over the detector bandwidth gives the squared SNR.

Figure 4: Left: Amplitude spectral density of the strain noise for the LIGO Hanford and Livingston detectors and the recovered signals of GW150914 (blue), GW151226 (orange), and GW151012 (green) plotted so that the relative amplitudes can be related to the SNR of the signal. Right: Time evolution of the signals in the sensitive band of the detectors. Adapted from (LIGO and Virgo, 2016b).

Frequency content of signal: The inspiral typically starts at a frequency well below the sensitivity of the ground-based detectors, and slowly increases. The inspiral phase ends when the system reaches the last stable orbit, at a frequency that at leading order scales with the inverse of the system total mass. The maximum GW frequency is about 1500 Hz for a BNS with a total mass of 3 solar masses (M). Given typical detector sensitivity curves, this means that for such light systems, most of the SNR comes from the inspiral part, with negligible SNR carried by the merger and ringdown parts. On the other hand, for heavier systems, with masses on the order of tens of solar masses, the inspiral ends in a frequency band where detectors are most sensitive, leading to the merger and ringdown contributing a significant or dominant fraction of the SNR. This can been seen in Figure 4, which shows the frequency evolution of the first three detections. GW190514 was the most massive and merges at the lowest frequency, while GW151226 was less massive and merges at a higher frequency.

Doppler shift: During their propagation from source to receiver, GWs experience a Doppler shift due to the expansion of the Universe, in the same way as electromagnetic waves are redshifted. This brings the detected signal to lower frequencies compared to the emitted signal, which makes it look like the source is heavier than it is. The apparent mass is larger than the true source mass by a factor \((1+z)\), with \(z\) the redshift of the source. The degeneracy between mass and redshift is irreducible from the GW signal itself; a cosmological model has to be assumed in order to translate the measured source distance into a redshift to infer source masses from apparent masses.

Transient signals: Because ground-based detectors have sensitivity curves that deteriorate sharply below 20 Hz, they register only the late stages of the binary evolution, which means the GW is observed as a transient signal. The time spent within the detector bandwidth by the chirp depends on the (apparent) source mass. BNS signals typically spend a few minutes and thousands of cycles, whereas BBH signals spend at most a few seconds and a few hundred cycles, or as little as a few cycles for the heaviest sources; see Figure 4.

Source parameters: The detected waveform depends on the parameters of the source, which can therefore be inferred from the signal measured in one or several detectors. Parameter estimation is performed via Bayesian inference, exploring a multi-dimensional space to identify the set of parameters that best matches the data, and plausible ranges for the various parameters. Degeneracies among parameters cause correlations in measurement uncertainties. In the most general case, a binary source can be described with 19 parameters, of which 12 are intrinsic and 7 extrinsic.

Figure 5: Estimated component masses for the sources observed during the LIGO-Virgo O1 and O2 observing runs, where contours show the 90% confidence region for the masses. Correlations in the measurement of \(m_1\) and \(m_2\) are visible, with contours following lines of constant chirp mass for low mass events. Lines of constant mass ratio \(q = m_2/m_1\) are shown. Figure from (LIGO and Virgo, 2019).

Intrinsic parameters: Intrinsic parameters affect the dynamics of the system and drive the amplitude and phase evolution of the signal. They include the masses of the two orbiting objects, \(m_1\) (heavier object) and \(m_2\) (lighter object), their spins (each characterized by its direction and dimensionless magnitude, relative to the maximum theoretically allowed spin of a same-mass black hole), the eccentricity of the binary orbit (two parameters), and the tidal deformability of each object. The eccentricity is expected to be small in most formation scenarios, as the orbit of the binary is expected to have circularized (through GW emission) long before the signal reaches detectable frequencies (see Orbital eccentricity). Unlike black holes, neutron stars are expected to have non-zero tidal deformability (see Matter effects).

Masses and spins: For most of the inspiral phase, the system can be described as two point particles and its evolution as a perturbative process, leading to the so-called post-Newtonian expansion (the terms of the expansion being powers of the orbital velocity compared to the speed of light) (Blanchet, 2014). At lowest order, the waveform phase evolution is driven by a combination of the masses called the chirp mass, \({\cal M}_c=(m_1 m_2)^{3/5}/(m_1 + m_2)^{1/5}\). This parameter is therefore well measured, while the mass ratio \(q=m_2/m_1\) and the spins, which enter at higher orders, are more difficult to measure. This means that measurements of the individual masses, \(m_1\) and \(m_2\), are correlated, with a primary degeneracy along a line of constant \({\cal M}_c\); see Figure 5. This degeneracy is partially broken with the measurement of \(q\) through its subtle effects on the inspiral, and/or by the observation of the ringdown, which is driven by the total mass of the system. There is also a degeneracy in the impact of the mass ratio and the spins on the inspiral, which is a further source of correlated uncertainty.

Figure 6: Sky locations of the GW sources observed in the O1 and O2 LIGO-Virgo observing runs. The qualitative difference in localization accuracy with a 3-detector network (denoted with -HLV), compared to a 2-detector network, is visible. Figure from (LIGO, Virgo and KAGRA, 2020a).

Extrinsic parameters: The source’s extrinsic parameters include the time and phase at coalescence. The others are related to the location of the source with respect to the detector (luminosity distance and two angles for direction) and the orientation of the binary with respect to the detector (inclination angle and polarization angle). The luminosity distance and the inclination affect the strength of the GW signal received. In particular, it should be noted that the measured GW strain \(h\) corresponds to the wave amplitude (not the energy), which therefore decreases linearly with the luminosity distance. The degeneracy between inclination and distance impacts the distance measurement, as it is difficult to distinguish e.g. a distant source viewed face-on from a closer source viewed edge-on. The other angles (direction and polarization angle) affect how the wave couples to the detector and determine the detector response. Interferometric detectors are not very directional, but their response is nonetheless not isotropic. GWs arriving from directions perpendicular to the detector plane couple optimally, while a detector has blind spots in the direction of the interferometer bisectors. Source localization relies primarily on triangulation, which uses the observed time delays of the signal at different detectors (caused by the finite propagation speed of GWs) to reconstruct the direction of the source in the sky. This requires at least two detectors and is much more accurate with three or more (LIGO, Virgo and KAGRA, 2020a); see Figure 6 for an illustration of how events observed with the LIGO Hanford, LIGO Livingston and Virgo detectors (denoted HLV) are better localized than those observed with the LIGO detectors only.

A wealth of additional features

Several subtle effects can modify the simple picture outlined above. When observed, they are associated with a better understanding of the source.

Precession: One such effect is the precession of the binary orbital axis, which changes the inclination of the orbit with respect to the line of sight over time, resulting in amplitude and phase modulation of the chirp waveform. Precession occurs when one or both objects have a significant spin component in the orbital plane (i.e. the spin is not aligned with the orbital momentum). Precession effects are most easily observable when the binary is viewed close to edge-on. To date, GW200129 is the event showing the strongest evidence for precession (Hannam, 2022, LIGO, Virgo and KAGRA, 2023a).

Figure 7: A representation of the quadrupolar \(_{-2}Y_{2 \pm 2}\), octupolar \(_{-2}Y_{3 \pm 3}\), hexadecupolar \(_{-2}Y_{4 \pm 4}\) and 32-polar \(_{-2}Y_{5 \pm 5}\) modes present in the GW190814 signal. Figure courtesy of N. Fischer, S. Ossokine, H. Pfeiffer, A. Buonanno (Max Planck Institute for Gravitational Physics), Simulating eXtreme Spacetimes (SXS) Collaboration. A visualisation of the higher-order modes in the GW190814 signal is available in this movie.
Figure 8: Estimated distance and inclination of the binary system observed as GW190814. The measurements of these two parameters are correlated, due to the difficulty of distinguishing a well-oriented distant source from an ill-oriented nearby source. The improvement brought by taking into account higher-order modes and precession is also visible. Figure from (LIGO and Virgo, 2020c).

Higher-order modes: We have already mentioned that a binary emits GWs primarily at twice the orbital frequency. However, in general, the dominant GW emission is accompanied by harmonics at higher frequencies. Much like a periodic signal can be decomposed into a sum of sine and cosine functions, a GW signal can be described as a superposition of spin-weighted spherical harmonics, \(_{-2}Y_{lm}\). The quadrupolar \(_{-2}Y_{2 \pm 2}\) term is the dominant mode, and other modes have different angular emission patterns and different frequency content than the quadrupole. A number of the modes vanish when the system is equal mass or when viewed face-on. For two equal-mass black holes, all of the modes with odd \(m\) must vanish, to ensure the system is unchanged under interchange of the black holes, or equivalently a rotation through half a cycle. Clear evidence for higher-order modes has been observed for the asymmetric BBH systems GW190412 and GW190814; see Figure 7. When present, higher-order modes are most easily observable when the binary is viewed close to edge-on. When observed, they help break the degeneracy between distance and inclination, leading to a more precise measurement of the distance, see Figure 8, while observation of specific modes, such as the \(_{-2}Y_{3 \pm 3}\), provide clear evidence of mass asymmetry.

Orbital eccentricity: Binaries on an elliptical orbit will radiate a different GW signal than those on a circular orbit, with the amplitude and frequency of the signal modulated due to the eccentricity of the orbit. In addition to causing the binary's orbit to shrink, the GW emission also causes an elliptic orbit to circularize, so that the eccentricity decreases over time. Thus, even if a binary forms on an eccentric orbit, in many cases the orbit will be essentially circular before the GW signal is observed by ground-based observatories. However, for binaries which form dynamically (see Formation Scenarios), it is possible that the orbital eccentricity is observable. To date, there have been several claims of orbital eccentricity in the observed GW population, with GW190521 (LIGO and Virgo, 2020b) a potential candidate, but no definitive observation.

Matter effects: Whereas black holes can be treated like point particles throughout the inspiral, it is not the case for neutron stars, whose internal structure becomes important as the orbital separation approaches the size of the objects. The tidal field of the companion induces a mass-quadrupole moment and accelerates the inspiral. The induced quadrupole moment depends on the (to date unknown) neutron star tidal deformability. Tidal effects affect the phase of the waveform and increase with frequency, becoming significant and potentially observable above a few hundred hertz. To date, matter effects have not been unambiguously observed in any of the BNS or NSBH events, which only allows setting upper limits on the tidal deformability of neutron stars (corresponding to lower limits on their compactness). This is however significant progress towards elucidating the structure of matter under the extreme densities (several times above nuclear density) found in neutron stars.

Post-merger signal – black hole ringdown: The mass of compact object binaries is large enough for the merger remnant to likely be a black hole in most, if not all, cases. The final black hole is characterized by its mass (which is smaller than the total mass of the binary, due to the energy lost to GWs) and spin. It is created in a perturbed state and reaches equilibrium by radiating GWs in the form of a ringdown, i.e. a linear superposition of exponentially damped sinusoidal oscillations. These 'quasinormal modes' occur at a specific set of frequencies over characteristic timescales (damping times) that are completely determined by the mass and spin of the remnant black hole (see Remnant properties). The amplitude and phase of each quasinormal mode, however, depend on the specifics of the binary. The fraction of the final black hole mass radiated away via the ringdown is small and is largest for an equal-mass binary (~1%). Ringdown frequencies decrease with the final black hole mass, and lie in the best part of typical ground-based detector sensitivity curves only for larger stellar-mass BBH binaries.

Post-merger signal – BNS case: In the case of BNS coalescences, the nature of the compact remnant resulting from the merger depends primarily on the masses of the inspiralling objects and on the properties of nuclear matter. The outcome of the merger can be the prompt formation of a black hole, the formation of a stable neutron star, or the formation of a short-lived neutron star that is too massive for stability, therefore eventually collapsing to a black hole. The ringdown GW signal associated with a remnant black hole was discussed above, but is extremely difficult to observe for a BNS, as it is a very weak signal occurring at high frequency where current detectors have poor sensitivity. If the remnant is a long-lived or short-lived neutron star, some oscillation modes of the neutron star may be excited in the aftermath of the merger and produce potentially detectable GW radiation. To date, such a post-merger signal has not been observed, but it is a target that would help understand neutron star properties.

Lensing: Like electromagnetic waves, GW propagation is affected by the space-time curvature induced by massive astronomical objects (stars, black holes, galaxies, galaxy clusters) lying along the line of sight between source and observer. Gravitational lensing of GWs from compact binary coalescences has specificities, however, because of the transient nature of the signal and because GW wavelengths are in some cases comparable to the size of the lens. Lensing can manifest through several (spectacular or subtle) signatures. One is magnification of the signal, which would lead to underestimating the distance of the source and overestimating its mass. Another one is the analog of multiple images for light, in the form of repeated events, separated by time delays of minutes to months (for lensing by a galaxy) or years (for galaxy clusters). If the lens is a smaller object, however, like a star or a compact object, the time delay would be so small that the repeated events overlap, leading to beating patterns in a single, distorted signal. No conclusive evidence of lensing has been observed so far in the available sample of binary mergers. This is not surprising, as potential lenses are rare in the nearby Universe accessible to GW detectors, given their current sensitivities. The estimation is that only one in 103 to 104 events would be affected by gravitational lensing.

Figure 9: An artist's illustration of a BNS merger such as GW170817. Figure courtesy of National Science Foundation/LIGO/Sonoma State University/A. Simonnet. An animation representing phenomena observed around the time of the GW170817 merger is available in this video.

Multi-messenger counterparts: Beyond the features of the GW signal itself, there are other ways by which some compact binary coalescences may be observed, involving electromagnetic waves and possibly neutrinos. No counterpart is expected to accompany stellar-mass BBH mergers, and indeed none has been confirmed to date (although speculative scenarios predict there might be electromagnetic transients from matter in the close environment of BBH binaries). On the other hand, counterparts are expected in the case of BNS coalescences, and possibly in the NSBH case if the neutron star is not swallowed whole by the black hole, i.e. some matter is tidally disrupted before merger. To date, there is one famous event with a wealth of confirmed counterparts across the electromagnetic spectrum, namely GW170817, the first BNS event (LIGO and Virgo, 2017; LIGO, Virgo and Astronomers, 2017). Seconds after the coalescence witnessed with GWs, a short gamma-ray burst (GRB) was detected, associated with a jet of highly-relativistic matter. Following the detection of the GW signal and the GRB, an extensive observing campaign led to the discovery of a bright optical transient in the hours that followed, then of X-ray and radio emission days to weeks later. The latter are interpreted as an afterglow of the GRB, due to the late interaction of the slowing jet with the interstellar medium, while the former is associated with the so-called kilonova phenomenon, i.e. thermal radiation from the mildly relativistic, expanding merger ejecta, powered by the radioactive decay of unstable nuclei rapidly synthesized in the ejecta neutron-rich matter (this nucleosynthesis process is responsible for producing the heaviest elements in the Universe); see Figure 9. GW170817 therefore confirmed that BNS mergers are progenitors of short GRBs and kilonovae and lead to the production of heavy-elements. The possibility of multi-messenger signatures is the reason why the LIGO and Virgo data are analyzed immediately as they are collected, so that low-latency alerts can be issued to trigger follow-up observations of GW candidate events.

Characterizing the astrophysical source population

The initial detections confirmed the existence of black holes and the possibility for BBH systems to form and merge within the age of the Universe. Subsequent observations of BNS and NSBH mergers confirmed their existence as well as providing direct evidence that short GRBs are associated with binary mergers. Moreover, the sample of compact binary coalescence signals observed to date, see Figure 10, allows a first characterization of the statistical properties of the source population. The results discussed below are presented in the paper "Population of Merging Compact Binaries Inferred Using Gravitational Waves through GWTC-3" (LIGO, Virgo and KAGRA, 2023b).

Merger rates: The simplest assessment that can be made consists in counting the number of events and inferring the rate of mergers per unit volume and unit time, for each type of binary (BNS, NSBH, BBH). Going from the observed sample size to the merger rate in the Universe requires unfolding the probability for a merger of a given type to be observed (the detection efficiency). This depends primarily on the detector sensitivity and on the mass distribution of the source population, as well as the (unknown) boundary between NS and BH masses. The mass distribution is only partially known (see Masses and spins) and is therefore a source of systematic uncertainty, which adds onto the statistical uncertainty arising from the finite sample size (in particular, only a few BNS and NSBH events have been detected to date). Drawing the line between neutron stars and stellar-mass black holes at 2.5 M, the rate of each type of merger is inferred to be in the range 13 – 1900, 7.4 – 320 and 16 – 130 per cubic Gigaparsec per year for BNS, NSBH, and BBH, respectively. It is worth noting that BBH mergers are intrinsically rarer than BNS or NSBH mergers. They dominate the observed sample, however, because massive BBH mergers are stronger sources of GWs that are detectable out to greater distances, meaning that a larger volume is probed for those binaries.

Evolution of BBH merger rates with redshift: The sample of observed BBH mergers is now large enough to allow investigating whether the rate evolves with redshift, i.e. changes over cosmic time. There is now indeed strong evidence that the BBH merger rate increases with redshift over the limited range probed with current detectors (up to redshift 1.5). This is unsurprising as the star-formation rate is known to evolve similarly with redshift, meaning that the density of stars likely to become black holes in binaries that eventually merge changes over cosmic time. To date, the evolution of merger rate is consistent with the cosmic star-formation rate.

Informing binary formation scenarios: A major reason for characterizing the distributions of the intrinsic parameters of merging compact binaries is to gain a better understanding of their formation and of the evolution of the progenitor stars that predated the compact objects (Mandel, 2022). There are two main classes of scenarios, or channels, for the formation of merging compact binaries. One is referred to as the isolated binary evolution channel, in which the binary originates from a pair of heavy stars that evolve until both stars collapse to compact objects. The other one is referred to as the dynamical formation channel, in which the compact objects predate the binary and pair up via gravitational capture in dense environments like globular clusters. A third, speculative, formation channel for BBH binaries assumes that primordial black holes were formed from density fluctuations in the early Universe, then paired to form binaries. The various channels are expected to lead to different parameter distributions. In particular, the isolated binary evolution channel favors symmetric systems (where both compact objects have similar masses), whereas more asymmetric systems are possible in the dynamical formation channel. Primordial black hole binaries could have masses below the minimum mass expected for compact objects of stellar origin (of approximately 1 M), and could dominate at high redshifts where there is insufficient time for astrophysical black hole binaries to form and merge. Compact objects in binaries formed in isolation are expected to have their spin axes aligned with the axis of the orbit, while those formed dynamically can have their spins pointing in random directions. Dynamically formed binaries could also have non-negligible eccentricity, if the final interaction occurs close to merger. Several channels may contribute to the observed sample of binary mergers, and characterizing the statistical properties of the source population will help determine their relative contribution.

Figure 10: The sample of binary systems observed in the LIGO-Virgo O1, O2 and O3 observing runs and used to characterize the population of GW sources, shown in the \(m_1\) and \(m_2\) space LIGO, Virgo and KAGRA (2023b). Figure courtesy of LIGO-Virgo-KAGRA Collaboration / IGFAE / Thomas Dent.

Informing stellar evolution: Properties of the population of merging binaries, masses in particular, can also shed light on the history of the compact object progenitors. The black hole mass distribution, for instance, is related to the chemical environment in which their massive star progenitors were formed, as stars with low metallicity (the abundance of elements heavier than hydrogen and helium) are expected to lose less matter due to winds and therefore to form heavier black holes. The high end of the black hole mass distribution is also interesting to understand the role of a particular class of supernovae called pair-instability supernovae. Stars heavier than 140 M are expected to undergo these catastrophic explosions that blow the star completely apart and leave no remnant, inducing a gap in the black hole mass distribution in the range of about 50 to 120 M. At the low end of the compact object mass distribution, prominent questions focus on the maximum mass of neutron stars and on whether there is a continuum or a mass gap between the heaviest neutron stars and the lightest black holes.

Black hole masses: The masses of black holes in merging binaries observed with GWs extend to higher values (up to ~80 M) than those of the black holes in X-ray binaries observed in our Galaxy (which only reach ~20 M), indicating that some of them were likely formed in environments with a lower metallicity than the Milky Way. Considering the distribution of masses for the heavier black hole in the binary, it is not simply described by a power law but shows additional structure, with two significant peaks at ~10 M and ~35 M and another tentative peak at ~18 M. The distribution does not exhibit a sharp cutoff at ~50 M as expected from pair-instability supernovae, raising the question of how the black holes heavier than 50 M were formed. They could be formed hierarchically as the products of previous mergers that have subsequently merged again, or could have evolved from lighter black holes that have accreted mass in gaseous environments, or the pair-instability model may need to be revisited. At the low mass end, the distribution also does not exhibit a sharp cutoff, rather showing that binaries with components in the 3 to 5 M range are uncommon but not ruled out.

Neutron star masses: Regarding neutron stars in merging binaries, the currently available sample shows a mass distribution that is relatively uniform between ~1.2 M and ~2 M. This is in contrast to the Galactic population of neutron stars (observed as pulsars) whose mass distribution is bimodal, with peaks around ~1.3 M and around 1.5 – 1.7 M (Özel, 2016). GW observations indicate that the maximum neutron star mass is in the range of 1.8 to 2.3 M, consistent with pulsar observations. However, with only a handful of neutron stars observed through GWs, the mass distribution remains rather unconstrained.

Spins: Although spins are difficult to measure, as their effect on the waveform is most of the time subtle, the spin distribution of BBH sources is starting to be informative. It shows that BH spins tend to have small but non-zero magnitudes. In terms of directions, they tend to have a non-zero component in the orbital plane, meaning that they are not completely aligned with the orbital axis, and sometimes even point in the opposite direction from the orbital axis. This suggests that at least some of the observed BBH systems formed dynamically, as this is unlikely in the isolated binary evolution scenario. To date, NS spin measurements are less informative. The handful of NS observed in merging binaries are consistent with NS having low, or zero, spin.

Fundamental physics

In previous sections, we have briefly mentioned how compact binary coalescences can be used to probe the properties of matter under extreme conditions in neutron stars. This is one aspect of how they can be exploited as a tool for fundamental physics. In this section, we turn to another aspect, which is testing general relativity (LIGO, Virgo and KAGRA, 2023c). GR predicts basic properties of GWs such as their polarization and propagation speed, which can be compared to observations. GR also describes the dynamics of space-time that lead to the generation of GWs. The dynamics is highly nonlinear and complex in the case of a compact binary approaching coalescence, allowing tests of GR in a deeply relativistic regime. Loud events, i.e. events with a high SNR, are particularly useful for such tests, as the low measurement noise allows for precision testing.

GW propagation: The multi-messenger event GW170817 offered an extraordinary opportunity to check the speed of GWs. The short time delay (1.7 s) between the end of the GW signal and the GRB (see Multi-messenger counterparts) can be explained by the time taken to launch the jet powering the GRB but can also be used to bound the speed difference between light and GWs. Compared to the >85 million years travel time for both signals, it means that light and GWs propagate at the same speed to very high precision, the fractional difference being at the level of 10-15. The same delay allowed checking to high precision that light and GWs are affected in the same way by background gravitational potentials along their propagation (by comparing their so-called Shapiro delays). These simple but powerful results have ruled out many alternative theories of gravity whose predictions differed from GR on that point. Another prediction of GR regarding GW propagation is that it does not involve dispersion, i.e. all frequencies are expected to travel at the same speed. If they didn’t, this would distort the measured waveforms of merging binaries. No evidence of this happening has been seen in the current sample of events. This allows us to constrain the mass of the graviton (the hypothetical quantum of gravity, mediating the gravitational interaction in a quantum theory of gravity), as a massive graviton would induce dispersion in GW propagation. The graviton has to be lighter than the electron by more than 28 orders of magnitude.

GW polarization: While GWs can have only two polarization modes in GR, they could have up to six in a more general theory of gravity. Recording signals in at least two non-co-aligned detectors allows probing for additional polarization modes beyond the two GR tensor modes. Results to date have not shown evidence for the presence of the latter.

GW emission: The measured signal waveform can be scrutinized from several angles in search for possible deviation from GR expectations. The simplest test that can be done, called a residual test, is to check whether the data in a network of detectors, after subtracting the best-fitting GR waveform, is consistent with measurement noise (uncorrelated across detectors) or shows evidence for an extra signal component (coherent across detectors). Another approach is to check whether the inspiral and post-inspiral parts of the waveform are consistent with each other (for events with sufficient SNR in both), by comparing the mass and spin of the final black hole inferred from each. More detailed tests look at specific parameterizations of the waveform and check for parameters deviating from their GR values. This can be done for instance by looking at the parameters of the post-Newtonian expansion describing the inspiral, or – for spinning BBH systems – by checking if the subtle imprint of spins on the inspiral waveform is consistent with GR’s prediction for Kerr black holes.

Remnant properties: Scrutinizing the late part of the measured signal allows investigation of the properties of the remnant and checks on whether they are consistent with GR predictions for Kerr black holes. In particular, spectroscopy of the ringdown signal will reveal if the frequencies and damping times of the quasinormal modes have the predicted (exclusive) dependence on the remnant mass and spin. Current sensitivities do not allow for precision spectroscopy yet, but are paving the way for this fundamental test. Another approach is to search explicitly for features in the signal that could be present if the remnant is not a GR black hole but an exotic compact object. For instance, the absence of a horizon around the remnant could lead to multiple reverberations of the GW signal, manifesting as repeating echoes in the measured ringdown signal. Dedicated searches have not shown any evidence of echoes to date.


Intermediate-mass black holes: At the crossroads of astrophysics and cosmology lies an open question about the origin of supermassive black holes. These are located at the center of most, if not all, large galaxies and – with masses from hundreds of thousands to billions of solar masses – lie at the high end of the black hole mass spectrum, compared to the stellar-mass black holes we have been discussing so far. Since black holes can grow by accreting matter or merging with other black holes, it is hypothesized that supermassive black holes may have grown over time from lighter seeds, which suggests a continuum of masses involving intermediate-mass black holes. Observational evidence for the latter was however missing, until the GW190521 merger was observed (LIGO and Virgo, 2020b), a most interesting event from many angles, not least because the remnant is estimated to be a ~140 M intermediate-mass black hole, demonstrating a way to populate this class of objects.

Primordial black holes: Another, more speculative, class of black holes of great interest for cosmology consists in primordial black holes that could have formed through the gravitational collapse of overdensities in the early Universe, possibly with masses well below 1 M (as they would not be the remnant of stars). If part of merging compact binaries, these subsolar-mass black holes could be detected with GW detectors (provided their masses are not much below 1 M), providing evidence for black hole formation in the early Universe. Searches for coalescences involving subsolar-mass objects have not revealed any candidate so far, therefore constraining the abundance of such objects.

Measuring the expansion of the Universe with standard sirens: The local expansion rate of the Universe is characterized by the Hubble constant, \(H_0\), whose value has not yet been determined accurately due to inconsistencies between results produced by different methods, a persistent tension of major significance for the standard model of cosmology. Interestingly, GW observations of compact binary coalescences provide a new and independent way of measuring \(H_0\) (Schutz, 1986). Measuring \(H_0\) requires relating the recession velocity of a source to its distance. The key feature is that for compact binary coalescences the distance can be inferred directly from the GW signal (however not necessarily with great accuracy, due to degeneracies with other extrinsic parameters; see Extrinsic parameters), making them “standard sirens”. On the other hand, the recession velocity is related to the redshift of the source and cannot be extracted from the GW signal, being fully degenerate with the system mass (see Doppler shift). A measure of the redshift is most straightforwardly provided if an electromagnetic counterpart allows identification of the galaxy hosting the source and its redshift, as in the case of GW170817. This event provided the first proof of concept for measuring \(H_0\) with GWs, with a ~15% accuracy. Reaching significant accuracy will require many more events.

Dark sirens: Since events with counterparts are rare, while BBH mergers dominate the event sample, it is useful to also include events without counterparts, the so-called “dark sirens”, in the measurement. This can be done statistically, in two different ways. One way is to average the redshift of a given GW source over all the possible host galaxies consistent with its localization, using a galaxy catalog. The other way is to exploit the fact that the distribution of the observed (redshifted) black hole masses depends both on the distribution of the true masses and the distribution of source redshifts, therefore sharp features (such as a peak) in the mass distribution can be used to estimate the redshift distribution. While the \(H_0\) measurement is expected to remain dominated by events with counterparts, including dark sirens improves the accuracy.


By way of conclusion, we highlight the connection between the beautiful science prospects offered by GW observations of compact binary coalescences and the instrumental challenges of the detectors making the observations. Further developing these pioneering instruments requires overcoming numerous difficulties, albeit with great reward.

An overall improvement in sensitivity will both increase the number of detections – further revealing the source population – and the SNR for nearby sources – allowing for precision measurements. Improving the sensitivity at low frequencies – a notoriously difficult task – has several benefits: for a given source, it allows measuring more cycles of the chirp, therefore achieving better accuracy in estimating the source parameters; it also gives access to sources with higher (apparent) masses. On the other hand, improving the high-frequency sensitivity will be key to revealing features in the late inspiral signal of binary neutron stars and detect the post-merger signal. In between, reaching excellent sensitivity in the medium-frequency range will allow for precision spectroscopy of remnant black holes.

The size of the detector network is also an important aspect, as it impacts both the duty cycle with which observations can be made and the ability to localize sources on the sky. The latter is crucial for multi-messenger observations, as the search for counterparts relies on GW detectors providing an accurate estimate of the source position. The prospect of KAGRA joining observations in the near future and a third LIGO detector being built in India in the coming years is therefore very encouraging. Moreover, improved low-frequency sensitivity will increase the chances of detecting BNS signals ahead of merger and issuing early-warning alerts that would boost the opportunity of observing associated prompt electromagnetic emission.

Beyond the planned improvements of the current ground-based detectors, efforts are ongoing to materialize a new generation of instruments – Einstein Telescope and Cosmic Explorer (Kalogera et al., 2021) – capable of realizing the full science potential outlined in this article. In particular, Einstein Telescope and Cosmic Explorer will be sensitive to the whole population of BBHs in the Universe and will probe physics connected to major open questions such as quantum gravity, dark matter and dark energy. Additionally, new types of instruments offer the fascinating prospect of exploring more of the GW spectrum covered by compact binaries. The Laser Interferometer Space Antenna (LISA) will open the mHz frequency range, while pulsar timing arrays have revealed evidence for GWs in the nHz range, giving access to compact binaries with much heavier objects or/and at a much earlier stage of their evolution. Interestingly, some of the binaries merging in the ground-based detectors frequency band will be observable in LISA months or years before merger, opening the opportunity to optimize their multi-wavelength observation.


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See also

Gravitational Waves: Ground-Based Interferometric Detectors

Gamma ray bursts theory

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