The era of gravitational wave astronomy began in 2015 with the observation of the signal from the merger of two black holes by the LIGO detectors by 2021 almost 100 more such transient signals from coalescences of compact binaries of black holes and neutron stars were catalogued. With improvements to the groundbased interferometer network consisting of LIGO Virgo and KAGRA now promising to bring the total number of detections into the hundreds we review the observational signatures and analysis methods for the most prolific gravitationalwave source the coalescence of compact binaries.
['General Relativity and Quantum Cosmology', 'Astrophysics - Cosmology and Nongalactic Astrophysics', 'Astrophysics - High Energy Astrophysical Phenomena']
Compact binary coalescences gravitationalwave astronomy with groundbased detectors
{'Compact binary coalescences: gravitational-wave astronomy with ground-based detectors': 'K. Chatziioannou · T. Dent ·  \nM. Fishbach · F. Ohme · M. P¨urrer ·  \n- V. Raymond · J. Veitch  \nReceived: date / Accepted: date  \nAbstract The era of gravitational wave astronomy began in 2015 with the observation of the signal from the merger of two black holes by the LIGO detectors; by 2021, almost 100 more such transient signals from coalescences of compact binaries of black holes and neutron stars were catalogued. With improvements to the ground-based interferometer network consisting of LIGO, Virgo, and KAGRA now promising to bring the total number of detections into the hundreds, we review the observational signatures and analysis methods for the most prolific gravitational-wave source: the coalescence of compact binaries.  \nKeywords Gravitational Waves · Compact Binaries · Data Analysis', 'K. Chatziioannou': 'Department of Physics, California Institute of Technology, Pasadena, California 91125, USA', 'T. Dent': 'IGFAE, University of Santiago de Compostela, E-15782, Spain', 'M. Fishbach': 'Canadian Institute for Theoretical Astrophysics, 60 St George St, University of Toronto, Toronto, ON M5S 3H8, Canada', 'F. Ohme': 'Max-Planck-Institut fur Gravitationsphysik, Albert-Einstein-Institut, Callinstr. 38, D-30167 Hannover, Germany  \nLeibniz Universitat Hannover, D-30167 Hannover, Germany', 'M. Purrer': 'Department of Physics and Center for Computational Research, University of Rhode Island, Kingston, RI 02881, USA', 'V. Raymond': 'Cardiff University, Cardiff CF24 3AA, United Kingdom', 'J. Veitch': 'Institute for Gravitational Research, School of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, United Kingdom', 'Definitions and conventions': "Except where other units are specified, we use 'geometric units' where c = G = 1. The following symbols and definitions are used throughout:  \n- -d , d ( t ) : detector strain data stream\n- -n , n ( t  \n) : noise component of data stream  \n- -h , h ( t ) : gravitational-wave induced strain on the detector\n- -˜ d , ˜ d ( f ) : Fourier transform of d , etc.\n- -h + , h × : gravitational-wave plus and cross polarizations\n- -˜ x ( f ) ≡ ∫ x ( t ) exp( -i 2 πft ) dt : forward Fourier transform\n- -H ≡ h + -ih × : complex combination of gravitational-wave polarizations\n- -x ( t ) ≡ ∫ ˜ x ( f ) exp( i 2 πft ) df : inverse Fourier transform\n- -x j ≡ 1 N ∑ N -1 k =0 ˜ x k exp( i 2 πjk/N ) : inverse discrete Fourier transform\n- -˜ x k ≡ ∑ N -1 j =0 x j exp( -i 2 πjk/N ) : forward discrete Fourier transform\n- -S n : power spectral density (units Hz -1 ) of noise; we use the one-sided density unless otherwise noted\n- -P ( A | B ) : probability of a logical proposition A given information B\n- -⟨ a | b ⟩ ≡ 4 Re ∫ ∞ 0 ˜ a ∗ ( f ) ˜ b ( f ) S n ( f ) -1 df : noise-weighted inner product\n- -p ( a | B ) : probability distribution over a parameter a given information B\n- -H N , H S : noise hypothesis and signal-plus-noise hypothesis\n- -θ : vector of source parameters\n- - C : a matrix named C\n- -m 1 , m 2 : primary and secondary component masses, m 1 ≥ m 2 by convention\n- -q ≡ m 2 /m 1 ≤\n- -M ≡ m 1 + m 2 : total mass -η ≡ m 1 m 2 /M 2 : symmetric mass ratio  \n1 : mass ratio  \n- -M≡ Mη 3 / 5 : chirp mass\n- -S 1 , S 2 : dimensionful component spin vectors\n- -M ⊙ : solar mass\n- -χ 1 ≡ S 1 /m 2 1 : dimensionless spin vector\n- -L : Newtonian orbital angular momentum vector\n- -χ ≡ | χ | : dimensionless spin magnitude\n- -cos θ 1 ≡ ( S 1 · L ) / ( | S 1 || L | ) : spin tilt\n- -ϕ 1 : spin azimuthal angle in the L frame\n- -χ eff ≡ ( m 1 χ 1 cos θ 1 + m 2 χ 2 cos θ 2 ) /M : effective aligned spin\n- -χ p ≡ max[ χ 1 sin θ 1 , q (4 q +3) / (4 + 3 q ) χ 2 sin θ 2 ] : effective precessing spin", '1.1 (Pre-)history of compact binary GW sources': "Coalescences of compact binaries involving neutron stars and black holes their inspiraling evolution due to GW emission, and ensuing merger to form a remnant object - are some of the most extreme astronomical phenomena known. Each event can convert a few percent of the total binary mass to gravitational radiation, typically a few times the solar mass for coalescences involving ∼ 30 M ⊙ black holes [31], and the peak GW luminosity at merger can reach ≳ 10 56 erg/s [35]. Coalescences involving neutron stars, in addition to their GW signature, can eject debris from the merger site that emits electromagnetic radiation across the spectrum, enabling spectacular multi-messenger observations [28,30]. However, while the calculation of GW emission from gravitationally bound systems was promptly performed within Einstein's General Relativity [467,256,252,257,556], for decades it was thought that the effects would be too weak to ever be of observational interest. This pessimistic estimate was based partly on considering astrophysical sources known at the time, i.e. planets and stars, and partly on the extremely weak expected effects, of order 10 -20 or less even for optimistic assumptions.  \nThe order of magnitude of the GW strain h from a gravitationally bound binary of masses m 1 , m 2 with orbital radius r , at distance D from Earth, is  \n| h | ∼ 1 r 1 D 2 Gm 1 c 2 2 Gm 2 c 2 ≡ R S, 1 R S, 2 rD , (1)  \nwhere R S, ∗ denotes the component Schwarzschild radius. For planetary or binary star systems at realistic distances, the strain is thus vanishingly small. Furthermore, a bound system with total mass M and size R has a maximum dynamical frequency ω d ∼ √ ( GM/R 3 ) ∼ ( Gρ ) 1 / 2 , where ρ is the system's average density, and we expect the frequencies of GW emission to be comparable. Considering 'non-compact' objects, this dynamical frequency may attain ∼ 10 -3 Hz for rocky planets, a range impractical for terrestrial detector operation due to overwhelming seismic disturbances. The discovery of white dwarfs - coincidentally close in time to the formulation of General Relativity [65] - provided a source whose dynamical frequency could reach the decihertz to Hz range; indeed, white dwarf binaries are a major source for the future space-based detector LISA [86]. Only with the discovery of Galactic neutron stars (NSs) [101,385], as both isolated rotating (non-axisymmetric) sources and as binaries with emission frequencies up to ∼ 10 3 Hz, was a credible GW source for Earth-based detectors known to exist [250,312,635].  \nA major impetus was given to the investigation of compact binaries by the discovery of a double NS binary [394] and subsequent confirmation that its orbit decays as expected due to GW emission, including post-Newtonian (PN) corrections to the lowest order of GW emission [669]. While known Galactic binary neutron stars (BNSs) are very far from the point of merger, their existence puts a lower limit on the rate of detectable coalescences in the local uni-  \nverse [557,416]. Furthermore, the mass- and spin-dependence of PN emission contributions carry nontrivial information about the properties of the source binary [216]. This combination of detectability and science potential confirmed BNS as a prime source for the first generation of interferometric detectors. Binaries of black holes (BHs) or mixed neutron-star-black-hole (NSBH) binaries were also considered as possible sources, but with much more uncertain merger rates [7].", '1.2 The Era of Interferometric Detectors': 'While resonant bar GW detectors are sensitive at specific frequencies (see [69] for a review), a broad-band antenna is necessary both to detect and characterize chirping compact binary coalescence (CBC) signals [54]. The initial detector network of LIGO, GEO 600 and Virgo, as essentially Michelson interferometers with many enhancements that are beyond the scope of this review, realized a first technical sensitivity goal in their observations up to 2007 [13, 356,56]. CBC searches covering Initial data up to 2010 obtained null results [11, 5], as expected under all but the most optimistic rate predictions. At the same time, a revolution in numerical methods for solving the Einstein equations for black-hole spacetimes enabled the full merger-ringdown GW emission from binary black hole (BBH) systems to be investigated [570,169,106] and combined with previous detailed knowledge of the PN inspiral [155,156,73]; thus, in addition to searches for CBC inspirals, higher-mass BBH mergers could also be targeted [9].  \nWith the upgraded Advanced detectors coming online from 2015 onward [6, 57], improvements both in the response to GW, and in suppression of noise over the full ∼ 10-1000 Hz band [21], as shown in Fig. 1, were immediately rewarded with the discovery of a merging stellar-mass BBH via its GW signal, GW150914 [19]. Not only was this system the first of its kind observed, it also contained stellar-mass BHs more massive than any confirmed via electromagnetic emission [14]. Within two years, the first joint detection of a BNS coalescence through both GW [29] and electromagnetic emission confirmed the exceptionally broad science potential of multimessenger astronomy with GWs: yielding, among other results, confirmation that BNS mergers source short gamma-ray bursts (GRBs) and kilonovae, and that GW propagate at the speed of light [28,30]. These overnight astrophysical revolutions were enabled by advances both in our understanding of GW emission and in analysis methods, as we will review in detail.  \nAfter the gamut of stellar-mass compact binary types was completed with the discovery of mixed NS-BH binaries [46] thanks again to increased detector sensitivities [151], with the Advanced era O4 run now underway, and as the community is designing future detectors to reach compact binary sources at the edge of the observable universe, this review revisits the basics of CBCs and of current data analysis techniques. We discuss the theory behind the signals we observe in Sec. 2; then in Sec. 3 we describe how we can use GW signals  \nFig. 1: The strain sensitivity for the LIGO Livingston detector (L1) and the LIGO Hanford detector (H1) during the first observing run, O1. Also shown is the noise level for the Advanced LIGO design (gray curve) and the sensitivity during the final data collection run (S6) of the initial detectors. Figure reproduced from [21].  \n<!-- image -->  \nrecorded in detector data to make statistical statements on the presence and properties of possible sources. Based on this framework, we consider methods for CBC signal detection in Sec. 4, and parameter estimation in Sec. 5. In Sec. 6 we generalize from one signal to the catalog of all detections, and describe how large sets of events can be analyzed while taking into account their uncertainties and the instrument selection effects. We conclude in Sec. 7 with an outlook over upcoming CBC observations in the near and medium term and discussion of the associated challenges for data analysis.', '2 Compact binary signals': "Data analysis techniques for compact binaries consisting of BHs and NSs require detailed knowledge of the GW signal. In principle this can be achieved by calculating the orbital dynamics and waveform by solving Einstein's equations for a source binary. This section starts out by providing an overview of the generation and emission of GWs under Newtonian and post-Newtonian assumptions in Secs. 2.1 and 2.2, including tidal effects and precession, as well as relevant cosmological effects on GW propagation; we then discuss salient characteristics of GW signals in terms of the parameters describing compact binary sources and the effect of these parameters on the GW strain in Sec. 2.3.  \nThe coalescence of compact binaries and the emitted GWs consist of several stages: (i) inspiral, where the orbital velocity is much smaller than the speed of light, and which is well described by post-Newtonian theory, (ii) the highly non-linear merger which requires full numerical solutions of Einstein's equations (see Sec. 2.4), and, (iii), for BBHs, the ringdown stage which is governed by quasi-normal modes emitted from a perturbed final BH (see Sec. 2.5). Modern waveform models may rely on semi-analytical techniques to model the complete GW signal from these stages, as exemplified by the effective-one-body and phenomenological modeling frameworks discussed in Sec. 2.6, or follow a full data-driven approach as used by surrogate models summarized in Sec. 2.7. We close out with a comparison of the accuracy and efficiency of waveform models in Sec. 2.8.", '2.1 Emission of compact binaries: the quadrupole formula': "We introduce the basic structure of the gravitational wave (GW) signal from compact binaries by considering a weak perturbation of Minkowski space:  \ng αβ = η αβ + h αβ , (2)  \nwhere η αβ = diag( -1 , 1 , 1 , 1) is the Minkowski metric in Cartesian coordinates, and | h αβ | ≪ 1 is a small perturbation. 1 This perturbation tensor will represent GWs on a Minkowski background. The trace-reversed perturbation tensor  \n¯ h αβ ≡ h αβ -1 2 η αβ h σ σ , (3)  \nsatisfies a simple wave equation if an appropriate coordinate system (gauge) is chosen in which ∂ β ¯ h β α = 0. This choice is commonly referred to as Einstein-de Donder gauge, or Lorenz gauge. 2 With this choice of coordinates, the Einstein  \nequation expanded to first order in ¯ h αβ becomes a flat-space wave equation,  \n□ ¯ h αβ ≡ ( -∂ 2 ∂t 2 + ∂ 2 ∂x 2 + ∂ 2 ∂y 2 + ∂ 2 ∂z 2 ) ¯ h αβ = -16 πT αβ , (4)  \nwith the stress-energy tensor T αβ as source term; see, e.g., [376,633] for a more detailed derivation and discussion. The solution to Eq. (4) for the spatial components of the metric perturbation can be approximated far away from the source as  \n¯ h ij ( t, d ) = 2 d d 2 dt 2 ∫ S x i x j ρ ( t -d, x ) d 3 x . (5)  \nHere x are the spatial coordinates, d is the distance between the observer and the source, ρ is the mass density, and the integral covers a three-dimensional space S that fully contains the source. The time components of ¯ h αβ can be set to zero in an appropriate gauge, as we discuss next.  \nThe Einstein-de Donder gauge does not define a unique coordinate system. The additional gauge freedom can be used to impose further constraints on ¯ h αβ . A common choice is the transverse-traceless gauge, which can only be imposed in vacuum away from the source. In this gauge, all time components ¯ h 0 β and the trace ¯ h α α vanish. Therefore, if one has transformed to this gauge (a discussion on how to do that follows below), Eq. (5) specifies the complete metric perturbation and we can ignore the time components. In addition, we can drop the distinction between h αβ and its trace-reversed counterpart ¯ h αβ .  \nEquation (5), often referred to as the quadrupole formula of GW emission, states that the GW tensor h ij originates from the second time derivative of the mass quadrupole  \nI ij ( t ) = ∫ S x i x j ρ ( t, x ) d 3 x . (6)  \nWe now consider a binary system in a circular orbit in the x -y plane, see Fig. 2. Due to energy being carried away by the emitted gravitational radiation, the binary orbit cannot remain perfectly closed and circular; but considered as an idealized example, the quasicircular case serves as a good starting point. The system is characterized by the two masses m 1 ≥ m 2 , which correspond to the gravitational (rest) masses of the individual binary components in isolation. As we consider widely separated orbits, the size of the objects is negligible. For binary components orbiting with a separation r and an orbital frequency ω , the location of each object in the center-of-mass coordinate system is  \nx 1 = m 2 r M (cos( ωt ) , sin( ωt ) , 0) , x 2 = -m 1 m 2 x 1 , (7)  \nwhere M = m 1 + m 2 is the total mass. With the simplifying assumption of point masses, the mass density is  \nρ ( t, x ) = m 1 δ [ x -x 1 ( t )] + m 2 δ [ x -x 2 ( t )] . (8)  \nFig. 2: Illustration of a binary in an idealized circular orbit of radius r . The two orbiting (point) masses are defined by their mass m i and coordinates x i .  \n<!-- image -->  \nThe mass quadrupole moment and its second derivative are then  \nI ij ( t ) = m 1 x i 1 ( t ) x j 1 ( t ) + m 2 x i 2 ( t ) x j 2 ( t ) , (9)  \nd 2 I ij dt 2 ≡ I ij = m 1 ( x i 1 x j 1 + x i 1 x j 1 +2˙ x i 1 ˙ x j 1 ) +(1 ↔ 2) . (10)  \nUsing the explicit coordinates of the binary given in Eq. (7), one can calculate the GW tensor from Eq. (5) as  \nh ij ( t, d ) = -4 Mω 2 r 2 η d   cos(2 ωt ' ) sin(2 ωt ' ) 0 sin(2 ωt ' ) -cos(2 ωt ' ) 0 0 0 0   , (11)  \nwhere we use t ' to denote the retarded time,  \nt ' = t -d, (12)  \nand the symmetric mass ratio, η , is defined by  \nη = m 1 m 2 M 2 . (13)  \nThe GW tensor in the transverse-traceless gauge only has two independent components. In our example, we oriented the coordinate system such that the binary orbits in the x -y plane. This leads to the structure of h ij that can be observed in Eq. (11). The tensor is traceless, because h xx = -h yy . For waves traveling along the z -direction, the 'transverse' condition of the transversetraceless gauge is also satisfied as only spatial components perpendicular to the direction of propagation contain non-zero values.  \nBecause h ij is a metric perturbation, it has to be symmetric, i.e., h xy = h xy . The two independent components are identified as the two polarizations of the GW, h xx = h + , h xy = h × . In this idealised case of a circular binary,  \nthe polarizations are harmonic oscillation whose frequency is twice that of the orbital motion; h + and h × differ only by a phase offset of π/ 2.  \nAn observer (e.g. a GW detector such as a LIGO or Virgo) will not necessarily be located along the z -axis (i.e., exactly above or below the orbital plane). For more general scenarios, one has to project h ij to the two-dimensional subspace perpendicular to the direction of propagation to remain in the appropriate transverse-traceless gauge. If ˆ N is the unit vector from the source to the detector, then we can find the projection operator P kl and the appropriately transformed GW perturbation in the transverse-traceless gauge as  \nP kl = I kl -ˆ N k ˆ N l , (14)  \nh ' kl = P k i P l j h ij -1 2 P kl P ij h ij . (15)  \nHere, I denotes the three-dimensional identity matrix. The first term in Eq. (15) projects the components of h ij to the subspace perpendicular to ˆ N . The second term removes the trace.  \nThe two polarizations can be read off the transformed GW perturbation tensor. An L-shaped interferometer will only be sensitive to a linear combination of the two polarisations, depending on its orientation. We denote this observable signal by h and provide more details in Sec. 3.4. For the case of a binary signal in the form of Eq. (11), any linear combination of both polarizations can be expressed by introducing an amplitude factor A and a phase offset ϕ 0 that each depend on the geometry of the source and the detector. Using the relative velocity v = ωr we may then express the GW strain due to stationary circular binary motion as  \nh ( t, θ ) = -A 4 Mv 2 η d cos (2 ωt + ϕ 0 ) . (16)  \nHere, θ = { M,v,η,d,ω,ϕ 0 , A} is the vector of parameters that characterizes the orbital motion and orientation of the source.  \nNot all of those parameters are independent. According to Kepler's third law, the total mass, velocity and orbital frequency of a system in a stable, circular orbit are related by  \nv 2 = M r ⇒ v 3 = Mω. (17)  \nThis relation will enable us to heuristically estimate the rate of change of the binary velocity and separation. Although we started by assuming a stable, perfectly circular orbit, the GW signal will carry energy away from the system. The GW flux is proportional to the square of the time derivative of GW tensor in the TT gauge, ˙ h ij ˙ h ij . The luminosity, L , is the flux averaged over an orbit and integrated over a sphere of radius d . For our binary system, it is proportional to  \nL ∝ | d ˙ h | 2 ∝ M 2 v 4 η 2 ω 2 = η 2 v 10 . (18)  \nDeriving a general expression for the energy carried by a GW is beyond the scope of this introduction. It can be found in many excellent textbooks, e.g., [183,376]: the result reads  \nL = 32 5 η 2 v 10 . (19)  \nThe orbital energy of the binary system can be written using Kepler's third law as  \nE = -Mη 2 v 2 . (20)  \nGiven the energy loss of the system due to GW luminosity, Eq. (19), and taking the mass parameters M and η to be constant, one can use energy balance to calculate how the velocity is changing over time:  \ndE ( t ) dt = -L ( t ) , (21)  \n⇒ dv ( t ) dt = -L ( t ) dE ( v ) /dv = 32 5 M ηv 9 ( t ) (22)  \n⇒ v ( t ) = 1 2 8 √ 5 M η ( t c -t ) . (23)  \nThus, the relative velocity of the binary increases over time as ( t c -t ) -1 / 8 , apparently diverging at the so-called coalescence time t c , although the simplifying assumptions made here are not valid all the way up to the time of the binary's merger. Following Eq. (17), the separation is  \nr ( t ) = 4 M 4 √ η ( t c -t ) 5 M . (24)  \nThus, the approximate time to coalescence for a binary with an initial separation of r 0 is  \nT = 5 r 4 0 256 M 3 η . (25)  \nFinally, the orbital frequency, using Eq. (17) again, behaves as  \nω ( t ) = v 3 ( t ) M = 1 8 ( 5 M 5 / 3 η ( t c -t ) ) 3 / 8 = 1 8 M 5 / 8 ( 5 t c -t ) 3 / 8 , (26)  \nwhere we have introduced the chirp mass  \nM = Mη 3 / 5 = ( m 1 m 2 ) 3 / 5 ( m 1 + m 2 ) 1 / 5 . (27)  \nFig. 3: The evolution of the relative velocity v ( t ), the (re-scaled) GW signal h ( t ) and the inspiral track of a Newtonian binary in quasi-circular orbits that evolve only due to the energy carried away by the GW.  \n<!-- image -->  \n-  \nWith an explicit expression for the orbital frequency, we can also calculate the orbital phase  \nϕ ( t ) = ∫ ω ( t ) dt = ϕ c -( t c -t 5 M ) 5 / 8 . (28)  \nHere, we label the integration constant ϕ c as it turns out to be the orbital phase at the divergence time t c .  \nThis is a remarkable result: the inspiral of a binary under the emission of GWs is predominantly governed by one combination of the component masses, the chirp mass M . The functions v ( t ), h ( t ), as well as the binary separation vector ( r cos ϕ, r sin ϕ ) in the plane of the orbit are shown in Fig. 3.  \nThe total mass M acts as a trivial scale factor in the binary's dynamics and GW signal. The signal amplitude scales with M/d , and the time, frequency, and separation (at fixed velocity) are proportional to M . This will have important consequences when considering binary sources at cosmological distances, as we will discuss in Sec. 2.2.3. The values in units of M shown in Fig. 3 may be converted to physical (SI) units via  \nt -t c = ( t -t c M ) × M G c 3 ≈ ( t -t c M ) × 4 . 9 ms M M ⊙ , (29)  \nx i ( t ) = ( x i ( t ) M ) × M G c 2 ≈ ( x i ( t ) M ) × 1 . 5 km M M ⊙ , (30)  \nexplicitly reintroducing the gravitational constant G and speed of light c .  \nTable 1: Overview of PN orders and effects that enter the gravitational waveform at this order. Here, k is the exponent of the orbital velocity above leading order, as used in Eqs. (31) and (32). PN terms marked with an 'x' are only partially unknown. Solid gray boxes indicate which binary parameters or physical effects contribute at the corresponding PN order.  \n<!-- image -->", '2.2 Post-Newtonian signal calculation': "The calculation presented above illustrates the basic behaviour of a compact binary inspiral under the emission of GWs. Although General Relativity is a complex, non-linear theory, for which even approximate solutions require much careful work, the strategy used in Sec. 2.1 to calculate the GW emission of a binary can be generalized to include non-linear effects by means of a series expansion.  \nConsidering energy and luminosity as functions of the binary's source parameters θ = { M,η,... } and of the orbital velocity v , the post-Newtonian (PN) framework may be applicable, 3 allowing us to express these quantities as a series in v (assumed much smaller than the speed of light), thus:  \nE ( v, θ ) ≈ -Mη 2 v 2 ( 1 + n ∑ k =2 E k ( θ ) v k +ln v n ∑ k =8 E ( l ) k ( θ ) v k ) , (31)  \nL ( v, θ ) ≈ 32 5 η 2 v 10 ( 1 + n ∑ k =2 L k ( θ ) v k +ln v n ∑ k =6 L ( l ) k ( θ ) v k ) . (32)  \nThe leading-order term in those expressions is often referred to as the Newtonian order term, while each additional contribution is known as the k/ 2 PN order. In General Relativity, the 0.5PN order term (i.e., k = 1) vanishes, and higher-order corrections include specific physical effects that may not be present at lower orders. The energy also contains logarithmic terms ∝ ln v at 4PN order and beyond, and starting from 3PN order in the flux.  \nExamples of higher-order effects include the coupling of the individual objects' spins with the orbital motion, which enters at 1.5PN order ( k = 3);  \nnon-linear tail terms, also entering at 1.5PN; the self-spin and spin-spin coupling entering at 2.5PN; horizon absorption effects starting at 2.5PN; and tidal effects for compact objects that are not black holes, entering with a v -dependence corresponding to 5PN order (see Sec. 2.2.1). A list of effects and references is given in Table 1.  \nWe conclude our brief introduction to the PN framework by focussing on how the expressions of energy in Eq. (31) and flux in Eq. (32) can be used directly to model the GW signal of an inspiralling compact binary. Various strategies have been developed to calculate the GW strain h , written via a generalisation of Eq. (16) as  \nh ( t, θ ) = A 4 Mv ( t ) 2 η d cos [ ϕ GW ( t ) + ϕ 0 ] . (33)  \nHere, we restrict the amplitude to the leading order term that is proportional to v 2 . However, higher PN corrections are included in the GW phase and velocity as we discuss next.  \nThe GW phase, ϕ GW ( t ) = 2 ϕ ( t ), is twice the orbital phase, which in turn can be calculated from the integrated orbital frequency ω ,  \ndϕ dt ( t ) = ω ( t ) = v 3 ( t ) M , (34)  \ncf., Eq. (26). Finally, as in the Newtonian case, the relative velocity is given by an ordinary differential equation,  \ndv dt ( t ) = -L ( t ) dE ( v ) /dv . (35)  \nUnlike the Newtonian case, we now think of the energy E and flux L as series expansions up to a given PN order.  \nDifferent strategies to solve this system of equations are commonly referred to as PN approximants : we name and summarize each such strategy below, see Ref. [157] for a more detailed discussion and comparison.  \nTaylorT1 Integrate Eqs. (35) and (34) numerically.  \nTaylorT2 Solve the inverse of Eq. (35),  \ndt dv ( v ) = -dE ( v ) /dv L ( v ) , (36)  \nby re-expanding the expression into a series truncated at the same order as E and L and integrate analytically to obtain t ( v ). Equation (34) is solved in a similar way, by re-expanding  \ndϕ dv ( v ) = v 3 ( t ) M dt dv ( v ) , (37)  \ninto a series in powers of v and integrating analytically. One then has analytical descriptions of time and phase, both parameterized by the relative velocity. Finding ϕ ( ˆ t ) for a particular time ˆ t requires a root-finding algorithm to solve t ( v ) = ˆ t .  \nTaylorT3 This approach starts with the same steps as TaylorT2, but inverts t ( v ) analytically (as a series expansion to consistent order) to obtain a series expression for v ( t ). This is then used in ϕ ( v ) to obtain an analytical, closed-form series expansion for ϕ ( t ).  \nTaylorT4 This approach is similar in spirit to TaylorT1 in that it integrates both differential equations numerically. However, Eq. (35) is first re-expanded and truncated, and this form is then integrated.  \nTaylorF2 Under the [248] the GW phase in the Fourier domain can be derived from the time domain phase. A simplified expression for the dominant mode is given by [563,  \nstationary phase approximation 96,223,224,152,157,70]  \n˜ h ( f ) = A f -7 / 6 e -iψ ( f ) , (38)  \nwhere the amplitude  \nA ∝ M 5 / 6 Q (angles) /d, (39)  \nincludes the response of the detector which depends on the location of the source with respect to the reference frame of the GW detector and the inclination angle under which the source is observed. The TaylorF2 phasing is fully known up to 3.5PN for the sake of illustration we only highlight the most important terms 4  \nψ ( f ) = 2 πft c -2 ϕ c (angles) -π 4 + 3 128 ηv 5 { 1 + v 2 [ 3715 756 + 55 9 η ] -v 3 [ 16 π -( 113 3 -76 3 η ) χ s -113 3 δ PN χ a ] + v 4 [ g ( η ) + σ ] + · · · -v 10 39 2 ˜ Λ + · · · } , (40)  \nwhere v = ( πMf ) 1 / 3 is the expansion parameter (assuming c = 1, we expand in powers of v/c ). The Newtonian term (0PN) only depends on binary parameters through the chirp mass M as 1 / ( ηv 5 ) = ( π M f ) -5 / 3 . At 1.5PN order ( v 3 term within the braces) spinorbit contributions enter the phase which we write as symmetric and anti-symmetric dimensionless spins aligned with the orbital angular momentum 5 , χ s = ( χ 1 + χ 2 ) / 2 and χ a = ( χ 1 -χ 2 ) / 2, as well as δ PN = ( m 1 -m 2 ) /M and g ( η ) is an unspecified function. Similarly, σ represents the spin-spin contribution at 2PN order. The v 10 term describes tidal effects which we discuss below.  \nAll these approximants correspond to different ways of obtaining expressions for the GW given truncated series expressions for the binding energy and luminosity. In essence, they differ by the order in which they (numerically or analytically) solve the equations vs. re-expand the series. Given the truncated nature of the PN series, no method is a priori preferred and their relative strengths and weaknesses are determined by how well they approximate the full non-linear GW calculation.", '2.2.1 Tidal effects': 'An extended body in a spatially inhomogeneous external field will experience varying forces throughout its extent. This is an example of a tidal interaction which is well understood in Newtonian gravity. In the following we define the tidal deformability of a NS which quantifies how easily the star is deformed in an external field. A static, spherically symmetric star of mass m and radius R in an external quadrupolar field E ij will develop an induced quadrupole moment Q ij . The (dimensionful) tidal deformability of the star is given by the ratio [307,188]  \nλ = -Q ij E ij , (41)  \nwhich can be shown to be proportional to R 5 . Hence, the dimensionless tidal deformability is defined as  \nΛ = λ m 5 = 2 3 k 2 R 5 m 5 , (42)  \nwhere k 2 is the dimensionless Love number ; typically k 2 ∼ 0 . 2 -0 . 3. For given m , both k 2 and R depend on the equation of state which relates the interior pressure to the energy density and temperature of the NS; the tidal deformability of a BH is zero [118].  \nIn the TaylorF2 phase of Eq. (40) we included the leading-order tidal contribution, which, though a Newtonian physical effect, has a velocity dependence equivalent to 5PN expansion order, with a prefactor given by the effective tidal parameter [307,291]  \n˜ Λ = 16 3 ( m 1 +12 m 2 ) m 4 1 Λ 1 +( m 2 +12 m 1 ) m 4 2 Λ 2 ( m 1 + m 2 ) 5 , (43)  \nwhere Λ i are defined for each binary component via Eq. (42). Although this tidal term enters at very high PN order we include it here, since it enables the measurement of the neutron star equation of state from binary inspiral signals; this is possible for moderate signal-to-noise ratios due to the large, O (100), value of ˜ Λ . A detailed discussion of measuring tidal effects and the neutron star equation of state is given in Sec. 6.4.', '2.2.2 Precessing binaries': 'If the compact objects have spin vectors S i that are not perfectly aligned with (equivalently, they have components orthogonal to) the Newtonian orbital angular momentum of the binary, then the spin vectors and the orbital angular momentum will change direction in space. The resulting motion resembles precession around the total angular momentum, whose direction is approximately fixed when averaged over the timescale of precession. This spin-precession effect enters through a combination of the 1.5PN spin-orbit and 2PN spin-spin (and higher) interaction terms [476]. The equation of motion for the spin of BH a is  \nd S a dt = Ω a × S a , (44)  \nwith a ∈ 1 , 2 denoting each binary component. The angular velocity of the precessional motion for the two BHs labelled a and b is  \nΩ a = 1 r 3 [( 2 + 3 m b 2 m a ) L N -S b +3(ˆ x · S b )ˆ x ] , (45)  \nwhere L N is the Newtonian orbital angular momentum, x the relative separation vector of the two BHs, and r = | x | , and ˆ x = x /r .  \nThe typical timescale of precession  \nt pr ≡ | S i | | ˙ S i | ∼ r 5 / 2 ∼ v -5 , (46)  \nis shorter than the radiation reaction timescale  \nt r ≡ E L ∼ v -8 , (47)  \nby 1.5PN orders [191,192]. In the adiabatic limit we can approximately ignore radiation reaction over one precession cycle. Then the total angular momentum J ≃ L N + S 1 + S 2 is conserved, d J /dt = 0, and therefore d L N /dt = -d ( S 1 + S 2 ) /dt . This implies that the orbital plane precesses with the same angular velocity as the total spin. In practice the total angular momentum direction is conserved when averaged over spin-precession, while its magnitude shrinks due to radiation reaction [94].', '2.2.3 Cosmological effects': "So far, we have assumed that our signal is propagating on Minkowski spacetime. However, CBC sources can be observed at large enough distances (on the order of Gpc) that the effect of the cosmological expansion of the Universe becomes important. Just as for electromagnetic radiation, the GW signal is redshifted, reducing the instantaneous frequency observed at the detector by a factor 1 / (1 + z ) relative to frequencies in the source reference frame, where z is the redshift. As we have seen in Sec. 2.1, the phase evolution of a signal from a BBH system (or any compact binary, in the point mass approximation)  \nis determined by the combination t/M , thus a rescaling of the time coordinate is equivalent to rescaling the binary mass: for an observer with time coordinate defined by dt obs = dt (1 + z ), the phase is a function of t obs / ( M (1 + z )). This scaling property also holds for numerical solutions of the full Einstein equations in vacuum (Sec. 2.4), although it is broken by finite-size effects of neutron-star matter, which are strongly dependent on the NS mass (Sec. 2.2.1, see also [496]). If such effects can be neglected, there is a complete degeneracy in phase between a source at nonzero redshift and one with a mass (1 + z ) times greater [437].  \nThe amplitude or power of radiation propagating in cosmological models is described by the luminosity distance: a source's measured bolometric flux F is to be related to an intrinsic bolometric luminosity L , which for an isotropically emitting source gives d L = √ L/ (4 πF ). The luminosity distance may be derived as a function of redshift for any given model; for cosmologies with a spatially flat metric, it is related to the comoving distance d c by d L ( z ) = (1+ z ) d c ( z ) [388]. It can then be shown that the GW signal received at Earth from a binary of masses m 1 , 2 at redshift z and luminosity distance d L ( z ) is identical (again neglecting NS matter effects) to a signal propagating in flat spacetime from a binary with masses m 1 , 2 (1 + z ) at distance d L , e.g. [308]. A physical interpretation of luminosity distance is given in Sec. 2.3.2.  \nFor convenience, it is common to define 'detector frame', i.e. redshifted mass M z = M (1 + z ) as the parameter determining the observed phase of a signal. This parameter and d L then suffice to determine the amplitude. Unless otherwise noted (e.g. in Sec 6.3), we will take masses, considered as waveform parameters in the context of data analysis, to be redshifted masses, but drop the z subscript for simplicity.  \nGravitational lensing The previous discussion assumed that GW propagate through a homogeneous and isotropic Universe (for some choice of spatial slicing): in reality, this is not a perfect approximation. Inhomogeneities in the distribution of matter along the line of sight to a source, and the associated perturbations in the cosmological metric, will affect the propagation of GW. The resulting effects go under the name of 'gravitational lensing', by analogy with the distortion or magnification of optical images. Such effects may be broadly classified as 'weak' lensing, relatively small (perturbative) changes to the GW signal caused by the inhomogeneous large-scale structure of matter; and 'strong' lensing, large but rare effects caused by the presence of a massive body - most often considered as a galaxy or cluster of galaxies - aligned with the source binary [711,524], which can significantly alter the signal morphology as well as the amplitude, and even give rise to multiple copies of a given transient, in analogy to multiple lensed optical images (e.g. [536]).  \nThe effects of weak lensing, as a perturbation to the GW signal amplitude which averages to zero over many events, have generally been neglected for the ground-based network, though becoming significant for future observations [386,201]. The presence of strong lensing, while a priori rare (e.g. [527]), will imply specialized strategies, models and methods, both for signal detection  \nand parameter estimation as well for astrophysical and cosmological interpretation (see [367,47,53] and references therein). Details of such analyses are beyond the scope of this review, thus unless otherwise stated we will consider only cases where strong lensing is absent.", '2.3.1 Description of binary parameters': "A BBH undergoing quasi-circular inspiral can be described by the two component masses m i and the spin vectors S i (or angular momenta) of its component Kerr black holes. We refer to these eight parameters as intrinsic parameters. The mass space can alternatively be parametrized by the total mass M or chirp mass M = Mη 3 / 5 and the (asymmetric) q = m 2 /m 1 ≤ 1 or symmetric mass-ratio η = m 1 m 2 /M 2 . Since the maximum spin a Kerr black hole of mass m can reach is | S max | = ( Gm 2 ) /c it is customary to work with dimensionless spin vectors χ i = c S i / ( Gm 2 i ). These dimensionless spin vectors are usually projected onto the (Newtonian) orbital angular momentum unit vector ˆ L N of the binary in order to separate the spin vectors into parallel and orthogonal components. As described in Sec. 2.2.2, non-zero components in the orbital plane give rise to spin-precession of the orbital plane and the spin vectors around the total angular momentum J = L + S 1 + S 2 [94,424]. For precessing binaries the spin vectors and derived quantities evolve with time and therefore they are defined at a reference frequency (e.g. the time at which the dominant mode GW frequency reaches the reference frequency).  \nOften, only the dominant dependence of the spins is of interest, motivating the use of the effective aligned spin χ eff = ( m 1 χ 1 + m 2 χ 2 ) · ˆ L N /M [74, 624] which is approximately constant during the inspiral to at least the 2PN order [581]. In a similar vein, the effective precession parameter [631]  \nχ p = max { χ ⊥ 1 , q (4 q +3) 4 + 3 q χ ⊥ 2 } , (48)  \naverages the in-plane spins (with χ ⊥ i being their magnitudes) over a number of precession cycles and assigns the effective precession spin to the primary BH; this is done because spins on the primary have a larger impact on the waveform than spins on the secondary when the binary has unequal masses. Generalizations of the χ p definition have also been proposed [336,676]. In parameter estimation, the spin vectors are often expressed in spherical polar coordinates with respect to to ˆ L N by spin magnitudes χ i and polar ('tilt') angles θ i along with an appropriate choice of the inclination angle for precessing binaries (see below) [283]. Instead of the azimuthal angles ϕ i , the angle ϕ 12 between ϕ 2 and ϕ 1 and the angle ϕ JL (the difference between the total and orbital angular momentum azimuthal angles) are used.  \nIn addition to the above set of eight intrinsic parameters seven more extrinsic parameters are needed to describe a quasi-circular BBH in relation  \nto a GW detector. These are the sky location (right ascension α and declination δ defined in the equatorial celestial coordinate system), polarization angle ψ , luminosity distance d L , an orbital inclination angle, and the time t c and phase ϕ c at coalescence. The inclination of the binary's orbit as seen from the observer is traditionally parametrized by an angle ι , the angle between L and the direction toward the observer. Because L is not a stable direction for precessing binaries and the direction of the total angular momentum ˆ J is approximately fixed throughout the inspiral (except in the unusual case of binaries undergoing transitional rather than simple precession [94]) we prefer to instead use θ JN , the angle between the total angular momentum and the unit vector ˆ N directed toward the observer as a measure of inclination. Transitional precession [94,740] happens when the total angular momentum J goes through zero during inspiral, because S 1 + S 2 ≃ -L , and thus flips its direction. This can happen for sufficiently asymmetric precessing binaries with mass-ratios of q ≲ 0 . 3.  \nBBHs do not necessarily undergo quasi-circular inspiral where the BHs trace out a sequence of shrinking circular orbits. Instead, the orbit is in general tangent to an ellipse at any one point. Of course, the orbit is not a closed elliptical orbit because of gravitational radiation. There is no unique definition of eccentricity in General Relativity. Locally we can choose a Keplerian parametrization of the binary's orbit r = p/ (1+ e cos ν ), where r is the radius, p the semi-latus rectum, and we have introduced an eccentricity parameter e and a radial phase parameter ν which describes the position of a point on an elliptical orbit. Eccentricity is radiated away during the inspiral and merger and most binaries are expected to circularize by the time they enter the frequency band of a GW detector [555]. Therefore, the eccentricity parameters must be defined at a reference frequency in a similar way as for spin parameters.  \nFor a generic, possibly eccentric and precessing, compact binary the relationship between the binary's source frame ( x, y, z ) (spanned by the instantaneous orbital angular momentum vector and the orbital plane at a reference frequency) and a wave frame ( X,Y,Z ) in which the emitted GW propagates toward a GW detector is illustrated in Fig. 4 and the documentation of the LALSimulation package [459]. The Z -axis of the wave frame points toward Earth. Therefore, the X -Y plane is the plane of the sky. The angle between the Z -axis of the wave frame and the z -axis of the source frame is the inclination ι . To describe generic binaries we need to introduce three more (positional) orbital elements [564] in addition to the inclination and reference phase. The longitude of the ascending node is the angle on the plane of the sky from the X-axis of the reference direction in the wave frame to the ascending node , which is the point at which the binary's orbit cuts the plane of the sky from below. The true anomaly is the angle ν (introduced above) along the orbital plane from the periapsis or pericenter (the point of closest approach in the orbit) to the present position of the primary orbiting body. In addition, the argument of the periapsis needs to be specified to define the orientation of an elliptical orbit in the orbital plane. For non-eccentric orbits it is degenerate with the reference phase. It is also worth mentioning that the longitude of the  \nascending node parameter is degenerate with the polarization angle (as both effect rotations on the plane of the sky).  \nFor binaries containing NSs we need to take into account additional degrees of freedom related to their response to a tidal field as discussed in Sec. 2.2. The dominant quadrupolar ( ℓ = 2) tidal deformation is described by the dimensionless tidal deformability of each neutron star Λ i = (2 / 3) k 2 [( c 2 /G )( R i /m i )] 5 and the effective parameter ˜ Λ is a mass-weighted linear combination of tidal deformabilities.  \nFor reference we have collected all of the above intrinsic and extrinsic binary parameters discussed in Table 2. For an extended table of parameters used by the bilby inference code see App. E of Ref. [603].", '2.3.2 The effect of binary parameters on the GW strain': "Masses and mass-ratio A change in the total mass M of a binary while holding all other parameters fixed results in a reciprocal change in the binary's frequency evolution, so that the dimensionless product Mω ( t ) or Mf is unchanged. Therefore, an increase of M is equivalent to shifting the GW strain in the Fourier domain to lower frequencies by the same factor, which makes the observable signal in the sensitive frequency band of a GW detector shorter.  \nFor fixed total mass the symmetric mass-ratio enters the phasing at 0PN order (see Eq. (40)). A binary becomes more and more asymmetric as the mass-ratio q decreases from its maximum q = 1 (equal mass) and this also decreases the symmetric mass-ratio η from its maximum 1 / 4 to lower values. The main effects of decreasing η for a GW signal with constant total mass starting from a fixed initial frequency are: (i) a steep increase in the time it takes for the binary to merge; as in Eq. (25) the so-called chirp time [627] is inversely proportional to η , and (ii) an increase in the number of GW orbits or cycles until merger, as the 0PN phase term is inversely proportional to η .  \nThe chirp mass depends both on total mass and symmetric mass-ratio. An increase of the chirp mass has two main effects: (i) the GW signal from a binary becomes brighter as its Fourier domain amplitude in Eq. (39) scales as M 5 / 6 , and (ii) the number of GW orbits or cycles from a fixed initial frequency until merger are reduced because the 0PN phase term in Eq. (40) goes as M -5 / 3 .  \nWe have seen in Sec. 2.2 that the chirp mass M enters the phase evolution of Eq. (40) at leading order, while the mass-ratio q and effective aligned spin χ eff appear in the phasing at higher orders. It would not be unreasonable to expect that terms at lower PN order which make larger contributions to the phasing will be easier to measure than terms entering at higher PN order, although this is not always true. See Sec. 5.5.4 for a discussion. Similarly, we expect that the two component masses m 1 , m 2 will be harder to constrain than the chirp mass and symmetric mass-ratio because they will involve correlations between chirp mass and mass-ratio.  \nAligned spins The dominant effect for spins aligned with the orbital angular momentum of the binary is captured by an effective aligned spin parame-  \nTable 2: Commonly used parameters to describe compact binary coalescences, as found in LIGO-Virgo-KAGRA parameter estimation releases. Mass parameters can be defined either as redshifted ('detector frame') or source frame masses, related by m zi = m src i (1 + z ). Spins are labeled with an index i = 1 , 2 indicating the primary or secondary object.Most spin quantities evolve during the coalescence and are therefore defined at a reference frequency f ref .  \nter. The parameter χ eff is a simplified version of the combination which appears in the spin-orbit term of the PN phasing in Eq. (40). The full term is χ eff -76 226 η ( χ 1 + χ 2 ) [563,70]. The fact that the phasing depends on such a combination implies that it is in general difficult to measure the aligned spins on the individual BHs; for very unequal mass binaries only the spin on the primary will be measurable [575,193]. In a similar way as for the symmetric mass-ratio, increasing the aligned spin relative to a non-spinning compact binary (i.e. positive χ i ) starting from a fixed initial frequency, and holding other parameters constant, increases the time and the number of GW cycles until merger. If the spins are anti-aligned with the orbital angular momentum (i.e. negative χ i ) the time to merger is decreased relative to the spinless case. The  \nFig. 4: The orientation of the binary system, defined by the parameters L , the orbital angular momentum; J the total angular momentum; S 1 , 2 the spin angular momentum vectors for the two bodies with masses m 1 , 2 ; ψ the polarisation angle; θ JN the angle between the direction to the observer ˆ N and the total angular momentum. Also shown are the binary's source frame ( x, y, z ) with L pointing in the z -direction and wave frame ( X,Y,Z ) where ˆ N coincides with the Z -axis. Figure generated using [719].  \n<!-- image -->  \nslower merger process of binaries with positive aligned spins is due to additional angular momentum that needs to be shed before the binary can settle down into a stationary Kerr black hole. This effect has been called 'orbital hang up' [170].  \nLuminosity distance We saw in Eq. (33) that the GW strain in Minkowski space depends inversely on the distance: GW signals from closer binaries are  \nstronger, as well as 'louder' in a given detector. We also saw in Sec. 2.2.3 that cosmological effects can be expressed by using the Minkowski space signal model but defining 'redshifted' masses larger by a factor (1 + z ), and with d L taking the place of d .  \nThe phase evolution of a coalescing compact binary which radiates GWs determines the redshifted masses, and, together with the measurement of the GWsignal amplitude, the source luminosity distance can thus be deduced [634, 627,642]; however the effects of angular parameters, particularly inclination as discussed below, on signal amplitude can induce large uncertainties in this measurement.  \nSky location and polarization angle As will be discussed in Sec. 3.4, the GW strain recorded by a GW detector is a weighted linear combination of the GW polarizations h = F + h + + F × h × , where the (real) detector pattern functions F + , F × depend on the sky position ( α, δ ) and the polarization angle ψ . The GW polarizations are defined with respect to a system of axes in the plane perpendicular to the propagation direction of the wave. The polarization angle effectively rotates this system of axes (see Fig. 4). The triple ( α, δ, ψ ) defines the location of the source with respect to the frame of a GW detector on Earth.  \nSpherical harmonic modes and inclination In the wave-zone, far away from the coalescing binary, it is customary to expand the complex combination of gravitational-wave polarizations H = h + -ih × on a space-like 2-sphere centered on the source in terms of appropriate basis functions for metric perturbations around Minkowski space. These basis functions are spin-weighted spherical harmonics [344] -s Y ℓm (of spin weight s = -2) which depend on the usual spherical polar angles ( θ, φ )  \nh + -ih × = H ( t ; θ, φ ) = ∞ ∑ ℓ =2 ℓ ∑ m = -ℓ h ℓm ( t ) -2 Y ℓm ( θ, φ ) , (49)  \nwhere h ℓm ( t ) are the GW modes or harmonics . At leading order, for the dominant ( ℓ, m ) = (2 , ± 2) modes in the GW signal, we have  \nH ( t ; θ, φ ) = h 22 ( t ) -2 Y 22 ( θ, φ ) + h 2 , -2 ( t ) -2 Y 2 , -2 ( θ, φ ) , (50)  \nwhere  \n-2 Y 2 , ± 2 ( θ, φ ) = √ 5 64 π (1 ± cos θ ) 2 e ± 2 iφ  \nwith h 22 ( t ) = A 22 ( t ) e -iϕ GW ( t ) , and, assuming equatorial symmetry of the binary, h 2 , -2 = ( h 22 ) ∗ . This can be simplified to  \nH ( t ; θ, φ ) = √ 5 4 π A 22 ( t ) [ 1 + cos 2 θ 2 cos ( ϕ GW ( t ) -2 φ )  \n-i cos θ sin ( ϕ GW ( t ) -2 φ ) ] . (51)  \nThe angles θ and φ are related to the direction under which the observer sees the orbital plane defined in the source frame of the binary. In general the orbital plane of the binary will be inclined with respect to the plane perpendicular to the direction of wave propagation ˆ N , which is described by an inclination angle ι ( θ JN for precessing binaries). This inclination angle takes the place of the angle θ in the spherical harmonics. The angle φ can be substituted by the phase of the binary at coalescence ϕ c at time t c . The effect of changing these parameters ϕ c , t c is a simple offset in phase and shift in the time of the GW strain, say the time at which the signal reaches its maximum amplitude. An expansion into spin-weighted spherical harmonic modes allows us to separate the dependence of the waveform on inclination and a phase angle so that the modes h ℓm ( t ) only depend on the remaining binary parameters.  \nAs can be seen from the above expression for the GW strain (substituting ι for θ ), if the source is seen 'face-on' ( ι = 0) or 'face-off' ( ι = π ) the GW signal leads to a much stronger response than if the inclination tends towards 'edge-on' ( ι = π/ 2) where the response is weakest. Moreover, for ι = 0 the strain is a linear combination h + -ih × ∝ A 22 exp[ -i ( ϕ GW -2 φ )] and the GW is circularly polarized, whereas for ι = π/ 2 only h + ∝ ( A 22 / 2) cos( ϕ GW -2 φ ) is non-zero and the GW is linearly polarized.  \nAdapted reference frames for precessing binaries Figure 5 illustrates how waveform modes of a precessing binary exhibit modulations when analyzed in the inertial reference frame of the binary. The modes simplify greatly when one switches to a non-inertial, rotating reference frame which tracks the timedependent motion of the precession: the co-precessing frame. There, the (2 , 2) mode is clearly dominant over the weaker higher order modes shown. Mathematically, the transformation law for spin-weighted spherical harmonics -s Y ℓm under arbitrary rotations can be used to find that the inertial GW modes h I ℓm transform to the co-precessing modes h copr ℓm as follows [630,142,366]  \nh copr ℓm ( t ) = e imα ( t ) ℓ max ∑ ℓ =2 ℓ ∑ m ' = -ℓ e im ' γ ( t ) d ℓ m ' ,m ( -β ( t )) h I ℓ,m ' ( t ) , (52)  \nwhere d ℓ m ' ,m denote Wigner d-matrices, and the time dependent Euler angles α ( t ) , β ( t ) , γ ( t ) parametrize the evolution of ˆ L N in a frame with ˆ z = ˆ J . An arguably superior representation of time-dependent frame rotations is given by quaternions [142].  \n̸  \nIn the co-precessing frame the precessing waveform is fairly well approximated by the waveform which has spin components in the orbital plane set to zero, a binary with spins aligned with the orbital angular momentum. However, in general h ℓ,m ( t ) = ( -1) ℓ h ∗ ℓ, -m ( t ) and this approximation therefore discards asymmetry in the waveform modes [143,695,678,432]. A further simplification can be achieved by taking out the orbital phase times the m number of each mode, h coorb ℓm ( t ) = h copr ℓm ( t ) e imϕ orb , where a quantity that approximates the orbital phase is defined as the average of the dominant mode phases in the co-precessing frame ϕ orb = (arg[ h copr 2 , -2 ( t )] -arg[ h copr 2 , 2 ( t )]) / 4.  \nAn example of the waveform emitted by a strongly precessing binary decomposed into modes is shown in Fig. 5 and the precession cones traced by the orbital angular momentum and spin vectors for the same binary are displayed in Fig. 6. In this configuration both spin vectors have non-zero projections in the orbital plane, and the binary has unequal masses which amplifies the effect of precession on the waveform.  \nThe impact of sizable precessing spins as subsumed in the effective precession parameter χ p is mainly visible in strong modulations in the inertial frame waveform as seen in the modes shown in the top left panel of Fig. 5. The strength of these modulations increases as the binary becomes more unequal in mass-ratio and becomes more prominent for an inclination angle tending towards edge-on rather than face-on inclination. Precession also has more subtle effects on the phasing which are not visible in Fig. 5.  \nEccentricity The waveform emitted from a highly eccentric binary is highly deformed and resembles bursts of radiation compared to a quasi-circular inspiral. The binary alternates orbital passages by the periastron (where the two bodies have their closest approach) and the apastron (where they are the furthest apart within one orbit). Contrary to quasi-circular binaries, waveforms emitted from eccentric binaries (even if they have aligned spins) have oscillatory amplitudes and frequencies. The periastron of the orbit is also no longer fixed and instead precesses or advances during the inspiral. The emitted gravitational radiation is enhanced for highly eccentric binaries and radiates peak power in the higher harmonics in the inspiral regime. The increased GW emission also radiates away eccentricity during the inspiral of an eccentric binary so that the orbit eventually circularizes [555]. Therefore, as the binary approaches the merger, the higher harmonics become subdominant. The time to coalescence for eccentric binaries is shortened compared to quasi-circular binaries. A radial phase parameter l fixes where in the orbit a binary finds itself at a reference frequency and so this parameter will shift the oscillation pattern in the waveform in time.", '2.4 Numerical relativity simulations': "As a coalescing binary approaches merger, the orbital velocity reaches a sizable fraction of the speed of light. In this regime the PN expansion breaks down and the solution of Einstein's equations can only be approximated by full numerical simulations. Simulations of the merger of BBHs have become routine since the breakthroughs by several independent groups in 2005 [570,169,106]. The Einstein equations formally depend on the ten degrees of freedom in the metric tensor g αβ 6 . The introduction of a coordinate system (usually one time-like and three space-like coordinates) allows us to separate Einstein's equations  \nFig. 5: Selected spherical harmonic modes ((2 , 2) , (2 , 1) , (3 , 3) , (4 , 4)) of the GW signal emitted by a precessing black hole binary. Shown is the real part of the modes in three reference frames. Top left: inertial frame, top right: coprecessing frame, bottom left: co-orbital frame, bottom right: one half the averaged phase of the co-precessing frame (2 , ± 2) modes. The binary has mass-ratio q = 1 / 4, and spin vectors χ 1 = (0 . 2 , 0 . 3 , 0 . 35), χ 2 = (0 . 5 , 0 . 05 , 0 . 42) defined at a reference frequency of 20 Hz for a total mass of 50 M ⊙ . The waveform model used is SEOBNRv5PHM [583].  \n<!-- image -->  \ninto six dynamical and four constraint equations of second order. The dynamical equations describe the time evolution of the spatial metric tensor and its time derivative (or, equivalently, the extrinsic curvature variable) given some initial data on an initial hypersurface which in turn needs to satisfy the elliptic constraint equations. To stably evolve the Einstein equations numerically it is important to choose a formulation of Einstein's equations which makes the dynamical equations hyperbolic of a strong enough degree so that the system is well-posed 7 . This is satisfied by the BSSN and the generalized harmonic formulations [108,78]. To fix the coordinate or gauge freedom four variables need to be specified. For instance, they can describe the proper time of an observer's normal to the current spatial hypersurface (lapse function) and how the timeline of an observer deviates from the normal to the hypersurface (shift vector).  \nStarting from a numerical solution of the constraint equations for a BBH system, one can evolve the system in time until the merger with the help of special techniques to deal with the curvature singularity inside each BH: either the BH interior inside the apparent horizon (a quasi-local definition of the BH horizon which is based on the spatial locations where outgoing light rays  \n<!-- image -->  \nFig. 6: Evolution of Newtonian orbital angular momentum ˆ L N and spin vectors χ i over time for the precessing binary with waveform shown in Fig. 5. The initial point is indicated by a filled circle and the end point (merger) by a diamond. The evolution of each vector traces out a separate precession cone about the direction of the total angular momentum ˆ J 0 at the initial point (slightly scaled).  \n<!-- image -->  \ncannot escape the BH 8 ) is excised from the computational domain and the excision surface carefully follows the BH motion, or one can use the so-called 'moving puncture' method which places the singularity in-between points of the numerical grid. Special prescriptions for the gauge conditions are also used to avoid singularities. The spatial derivative operators in the dynamical equations are usually discretized with high order finite difference stencils or  \nspectral methods. The spatial resolution is set by the size of the apparent horizons of the BHs and therefore scales with the size of the smaller BH if the masses are unequal, while the total mass can be scaled out. Time evolution is performed by methods for numerically solving ordinary differential equations. The spatial discretization scale then determines the size of the largest time step for which the system can be stably evolved. Furthermore, BBH simulations need to resolve both the small scales of the BH horizons, and extract the GWs in the wave zone, sufficiently far away from the binary. Covering these disparate scales with a single uniform grid is infeasible; instead, practical simulations require the use of fixed or adaptive mesh refinement methods. Errors due to extraction of the waves at finite distances from the source can be mitigated by extrapolation methods or, ideally, by transporting the radiation to future null infinity.  \nA non-eccentric binary black hole coalescence can be divided into three regimes: (i) inspiral where the orbital velocity is low, v ≪ c , and the system undergoes quasi-circular inspiral, (ii) merger where the two separate BH horizons combine to form a single distorted potato-like horizon and orbital velocities reach v ≃ 0 . 5 c or more in this highly nonlinear regime, and (iii) ringdown , where the final black hole radiates away excess energy in terms of quasinormal modes (QNMs) and settles down to a stationary Kerr black hole solution. This last stage can be well described by BH perturbation theory. The emitted waveform is a chirp signal that peaks in amplitude and frequency at the merger, before decaying exponentially in the ringdown. The mass and spin (angular momentum) of the BHs can be determined numerically from their apparent horizon properties.  \nFor an equal-mass non-spinning BH binary, about 5% of the total mass of the system is radiated away in GWs and therefore the mass of the final Kerr black hole is about M f /M ≃ 95% of the initial mass of the binary. The final Kerr black hole of an equal-mass non-spinning binary turns out to have a spin of about χ f ≃ 0 . 7 acquired from remaining angular momentum of the binary that has not been radiated away during the coalescence. For unequal mass binaries the radiated energy is lower than for the equal mass case. In general, the dependence of the final mass and the final spin is a function of the initial binary masses and spins which is needed for waveform models that include a description of the merger and ringdown parts of the signal. This information can be fitted from numerical relativity (NR) simulation data available over the binary parameter space.  \nNR simulations can produce highly accurate waveforms (with phase errors a fraction of a radian required by data analysis applications), but the high computational cost limits the number of simulations and their length in time which can be carried out in practice. Moreover, the cost of simulations also varies strongly over the binary parameter space and, in particular, increases steeply the more unequal the mass-ratio and the higher the aligned spin are, based on the induced size of the apparent horizons to be resolved and the allowed time step. A single simulation can take several months on a supercomputer while running in parallel on hundreds of CPU cores. Over 4000 publicly available  \nBBH NR simulations have been performed so far and have been collected in several catalogs for use in modeling and GW data analysis [1,518,145,2,381, 380,378,379,3,405,4,361].", '2.5 Quasi-normal modes': 'After the apparent horizons of the two individual BHs in a binary have merged, the BH can be treated as a deformed Kerr BH in linear perturbation theory. It is therefore natural to seek to describe the gravitational radiation emitted by this object by metric perturbations of an analytical BH solution. The linear perturbations around a Schwarzschild black hole are described by the ReggeWheeler (RW) and Zerilli equations and perturbations of a Kerr black hole by the Teukolsky equation [476]. Perturbations are in general decomposed into spheroidal harmonics of spin-weight -2 which depend on quantum numbers ℓ, m and the (complex) mode frequency ω . In the Schwarzschild case, for a fixed ℓ mode, these equations are formally of the form  \nd 2 ϕ dx 2 ( ω, x ) + [ ω 2 -V ( x ) ] ϕ ( ω, x ) = 0 , (53)  \nwith x ∈ R a rescaled version of the Schwarzschild tortoise coordinate, ϕ ( ω, x ) the Fourier transform with respect to time of the RW or Zerilli function, and V ( x ) the RW or Zerilli potential. These potentials have a peak in the vicinity of small and positive x and fall off smoothly to zero as x →±∞ .  \nConsider for a moment perturbations of a one-dimensional string with fixed endpoints. In this case, a general perturbation can be expressed as a sum over normal modes ϕ n ( t, x ) = e iω n t ψ n ( x ), which form a complete set. In contrast to the fixed (Dirichlet) boundary conditions used in the string example, the metric perturbations of a BH are not describing a stationary system, but will either escape into the BH horizon or towards infinity with radiation boundary conditions ϕ ( ω, x ) ∝ e iω | x | as x → ±∞ . We will obtain a discrete set of complex frequencies ω QNM = ω R + iω I belonging to the so-called quasi-normal mode (QNM) solutions of the system. The QNMs vanish in time as damped oscillations of the form  \ne ω I t [ a sin( ω R t ) + b cos( ω R t )] , (54)  \nwhere ω I < 0. For a Schwarzschild black hole, QNM solutions depend on the azimuthal quantum number ℓ which scales the frequency ω R , and the overtone index n which increases ω I and therefore determines how quickly the mode is damped in time. QNMs for Kerr also depend on the (magnetic) quantum number m . The Kerr QNM modes can co-rotate ( m> 0) or counter-rotate ( m< 0) with the Kerr BH. We are most interested in the least-damped QNM modes which will dominate the emitted GW signal. The least-damped mode for a nonspinning black hole has ω n =1 ,ℓ =2 ≈ (0 . 75 -i 0 . 18) c/R S , with the Schwarzschild radius R S = 2 GM/c 2 . This yields a frequency f ≃ 12 kHz( M ⊙ /M ) and a damping time τ = 1 / | ω I | ≃ 55 µ s( M/M ⊙ ).  \nPerturbations of the Kerr metric have been analyzed in terms of curvature perturbations in the Newman-Penrose null tetrad formalism which leads to the Teukolsky equation. The QNMs of Kerr BHs are denoted as ω nℓm , with ω 122 being the least-damped mode, now depend on the Kerr parameter of the background BH around which we are perturbing.', '2.6 Modeling the signal from inspiral to ringdown': "We next discuss two approaches that are used to construct accurate waveform models covering all regimes of GWs emitted in a black hole binary coalescence: the phenomenological (Phenom) and effective-one-body (EOB) frameworks. Both combine theoretical understanding from PN and perturbation theory with NR simulations, and both have also been modified to model waveforms from binaries which include NSs. Another approach that builds a direct surrogate to numerical simulations is discussed in Sec. 2.7.  \nIn the data-driven Phenom and surrogate approaches, and to a lesser degree for EOB, the waveform is usually decomposed to simplify modeling. This is informed by physical intuition about the class of binaries to be modeled, in particular taking into account the effect of certain parameters on the signal as described in Sec. 2.3. For non-precessing binaries the dominant ( ℓ, m ) = (2 , ± 2) spherical harmonic mode of the GW signal can be well represented by its amplitude and phase h ( t ) = A ( t ) e iϕ ( t ) . If higher harmonics are included in the model the amplitude - phase decomposition can lead to zero-crossings which are hard to model and it is advantageous to extract the orbital motion of the two bodies by multiplying the complex modes with an exponential of half the phase of the (2 , 2) mode. In the resulting 'co-orbital frame' [695] the real parts of the modes become non-oscillatory and amplitude-like, while the imaginary parts add some fine structure. If the signals include precession effects, one can transform into a time-dependent reference frame in which the orbital plane of the binary is fixed. In this 'co-precessing frame' [630,540,144], precessioninduced modulation effects are significantly reduced. Both the motion of the reference frame and the signal in the precession-adapted frame need to be modeled.  \nEffective-one-body framework In the late inspiral the PN approximation deteriorates and eventually breaks down as the orbital velocity becomes a sizable fraction of the speed of light. This difficulty can be addressed by the effectiveone-body (EOB) framework [155,156]. In the following we qualitatively describe the main idea of EOB following Ref. [476]. We start by considering the equations of motion of a two-body system. Up to 2PN order and assuming zero spins, no radiation reaction force is present and the system is conservative. After introducing a particular set of coordinates (ADM coordinates) we can write the equations of motion in terms of a Hamiltonian in the center-of-mass frame of the binary, using relative position and conjugate momentum coordinates ( q , p ). Due to the invariance of the Hamiltonian under time translations and  \nrotations, energy and angular momentum are conserved. We write energy as a function of the angular momentum and a radial action variable which is an adiabatic invariant.  \nThe basic idea of EOB is to find a one-body problem in an external spacetime that reproduces the solution of the given two-body problem (written in a gauge invariant form). We can consider a test particle of reduced mass µ = m 1 m 2 / ( m 1 + m 2 ) = ηM which moves along the geodesic of a static spherically symmetric metric g µν eff written in terms of some effective coordinates. At the Newtonian level this is the Schwarzschild metric which describes the gravitational potential of a mass M = m 1 + m 2 . The EOB line element is  \nds 2 eff = g µν eff dx µ eff dx ν eff = -A ( R ) dt 2 0 + B ( R ) dR 2 + R 2 ( dθ 2 +sin 2 θdϕ 2 ) , (55)  \nwhere the functions A ( R ) and B ( R ) need to be determined. The EOB approach then defines a natural mapping between the real two-body problem and the effective one-body problem. This can be motivated by a quantum mechanical analogy (the Bohr-Sommerfeld quantization condition of the hydrogen atom). Adiabatic invariants corresponding to the principal and angular momentum quantum numbers can be directly related between the two descriptions, whereas the mapping of the energy is nontrivial. One arrives at the following result at 2PN order:  \nA ( R ) = 1 -2 M R +2 η ( M R ) 3 , (56)  \nB ( R ) = 1 + 2 M R +(4 -6 η ) ( M R ) 2 , (57)  \nalong with an energy mapping. Therefore, within this approximation we can substitute the original two-body problem by an easier to solve effective onebody problem and solve it for η > 0.  \nConsidering the conservative dynamics of the system one finds that, as is the case for the Schwarzschild metric, the EOB metric has a horizon (because A ( R ) has a simple zero at R > 0 for all η ) and an innermost stable circular orbit (ISCO). Similarly, one can define the light ring : the smallest possible (unstable) circular orbit for a massless particle. For very unequal masses a small mass inspirals through a sequence of quasi-circular orbits until it reaches the ISCO and suddenly starts to plunge toward the central BH. In contrast, for comparable masses, there is no sharp transition between inspiral and plunge.  \nThe effect of the emission of GWs (first entering at 2.5PN order) can be expressed as a radiation reaction force ˆ F φ which can be added into the Hamiltonian equations of motion for the conservative dynamics. Written in  \npolar coordinates in the orbital plane these equations read  \ndr d ˆ t = ∂ ˆ H ∂p r , (58)  \ndφ d ˆ t = ∂ ˆ H ∂p φ , (59)  \ndp r d ˆ t = -∂ ˆ H ∂r , (60)  \ndp φ d ˆ t = ˆ F φ ( r, p r , p φ ) , (61)  \nwhere  \nˆ H ( r, p r , p φ ) = 1 η √ √ √ √ 1 + 2 η [ √ A ( R ) ( 1 + p 2 r B ( r ) + p 2 φ r 2 ) -1 ] , (62)  \nand r = R/M , ˆ t = t/M .  \nAfter solving these equations of motions numerically to obtain the inspiral and plunge dynamics, we can compute the emitted GWs via the quadrupole formula. To complete the waveform, a merger part needs to be smoothly attached after the plunge. This ringdown waveform consists of a superposition of damped sinusoids which results from the oscillation of the final Kerr BH in its quasi-normal mode oscillations. This yields a qualitatively correct waveform of the inspiral, merger and ringdown. The above EOB construction can be extended up to higher PN orders (4PN is used is practical models) and can be generalized to include spin effects. In practice, EOB models also add corrections for neglected non quasi-circular effects and introduce unknown higher order PN terms in the Hamiltonian which are then tuned or calibrated to NR simulations to increase the accuracy of the model.  \nEOB models from two different groups have been used for LVK analyses. This includes the 'SEOBNR'-type models for aligned spin and the dominant mode [667,668,135], models including higher harmonics [207,565] and precession [548,103,541,583], as well as prescriptions for tidal models [654,490], and models incorporating further optimizations [233,430]. Of these, the most complete recent model for BBHs is 'SEOBNRv5PHM' [583] for generically precessing binaries which includes higher harmonics. 'TEOBResum'-type models have been constructed for non-precessing binaries including tidal effects [522, 520,75], and for precessing binaries [76,327].  \nPhenomenological waveform model framework The class of phenomenological models takes a very practical approach at constructing models of the GW signal emitted from compact binaries. This approach matches a post-Newtonian inspiral to a phenomenological description of the merger and ringdown which aims at encapsulating the waveform morphology from numerical-relativity simulations. The method posits ansatz functions in the Fourier domain for  \ndifferent regimes of the coalescence along with a polynomial dependence on mass-ratio and spins. Hybrid inspiral - NR waveforms are created by stitching together NR simulations with inspiral waveforms produced by an uncalibrated EOB model (evaluated at the same binary parameters as the respective NR simulation). These 'hybrids' are created for all mass-ratios and aligned spins where accurate NR simulations are available and are used to calibrate the model. The resulting phenomenological models focus on computational efficiency by avoiding the numerical solution of systems of complicated ODEs (as used in EOB) and instead provide analytical expressions that can be directly evaluated.  \nTraditionally, phenomenological models have been constructed in the Fourier domain [74], so that inner products between waveforms used in data analysis can be directly computed. Therefore, modeling starts by considering the morphology of waveforms in the Fourier domain in contrast to the EOB time domain method. Ansatz functions are picked in three frequency regimes: inspiral, an intermediate late-inspiral regime, and merger-ringdown. In the following we describe some elements of the 'IMRPhenomXAS' waveform model [568].  \nIn the inspiral regime the phase model is written as the 3.5PN TaylorF2 approximant plus higher order 'pseudo-PN' terms  \nφ Ins ( f ) -φ TF2 ( f ) = 1 η [ σ 0 + 4 ∑ i =0 σ i +1 f ( i +3) / 3 ] , (63)  \nwhere the σ i are unknown PN coefficients that have not yet been calculated. The σ i are determined by fitting a rescaled difference between frequency derivatives of hybrid waveform and TaylorF2 phases f -8 / 3 ( φ ' Hybrid ( f ) -φ ' TF2 ( f )).  \nA similar PN-based ansatz is made for the inspiral amplitude  \nA Ins ( f ) = A PN + A 0 3 ∑ i =1 ρ i ( πf ) (6+ i ) / 3 , (64)  \nwith 'pseudo-PN' coefficients ρ i , again determined from the hybrids.  \nAs discussed in Sec. 2.5 the ringdown can be described by BH perturbations in the time domain. It is therefore instructive to consider the Fourier transform of a damped oscillation h ( t ) = Θ ( t ) e 2 πt ( if RD -f damp ) , where Θ ( t ) is the Heaviside step function. This leads to a Lorentzian function (Cauchy distribution) for the phase derivative and for the square of the Fourier domain amplitude | ˜ h ( f ) | 2 ∝ 1 / [( f -f RD ) 2 + f 2 damp ]. Since the physical waveform should fall off faster than 1 /f due to its smoothness, the ansatz for the merger-ringdown amplitude multiplies the Lorentzian by a decaying exponential  \nA MR = [ a R ( f damp σ ) ( f -f RD ) 2 +( f damp σ ) 2 ] e -λ ( f -f RD ) / ( f damp σ ) , (65)  \nwith fitting parameters a R , σ, λ . The merger-ringdown model for the phase derivative consists of a Lorentzian plus terms with negative powers in fre-  \nency to capture to capture steep gradients  \nηφ ' RD = c RD + n ∑ i c -p i i + c 0 a φ ( f -f RD ) 2 + f 2 damp , (66)  \nagain introducing a number of new parameters c RD , c i , a φ . Moreover, the ringdown model depends on a fitting formula that predicts the final BH's mass and spin from the progenitor BHs. From these one can determine the ringdown and damping frequencies f RD and f damp .  \nLastly, an intermediate regime bridges the gap between the inspiral and merger-ringdown parts of the model and uses ansatz functions inspired by the neighboring regimes. For a description of the model in the intermediate regime and details about where in frequency the models in the three frequency regions are situated and how they are matched together, see [568].  \nWith the amplitude and phase ansatz functions defined, one can fix the model coefficients for any given point in parameter space where an accurate NR simulation is available and a hybrid waveform has been computed. To obtain a practical model one posits a polynomial or rational dependence of the model coefficients over parameter space, motivated by the mass-ratio and spin dependence of PN approximants. The model uses hierarchical fits in symmetric mass-ratio, an effective aligned spin parameter, and the difference between the BH's aligned spins.  \nThis approach leads to fast, closed form models that are convenient to use for data analysis. Examples include the aligned-spin BBH models up to 'IMRPhenomD' [74,624,423] and the recent 'IMRPhenomXAS' [568]. Newer models are available in versions that include higher-order modes [465,328]. Models for precessing binaries require additional modeling of the time-dependent rotation of the reference frame, in a similar way as for EOB models. To use this prescription in the Fourier domain, additional approximations are introduced via the stationary phase approximation [366]. An extension to two-spin dynamics in given in [421]. A more accurate solution of the frame dynamics for generically precessing binaries is used in 'IMRPhenomXPHM' [569]. A recent model includes tuning against numerical waveforms in the precessing sector [360]. Further models incorporate tidal effects for BNSs [240,243] or NS - BH binaries [677]. Recently, a family of phenomenological models has been created in the time domain [270,269,268] in an effort to avoid some of the approximations used by Fourier domain phenomenological models.", '2.7 Reduced-order and surrogate models': "Over the past decade, data driven models of GWs have come to prominence in GW data analysis with the promise of delivering high accuracy and fast evaluation speeds. Reduced-order and surrogate modeling aims at creating an efficient and accurate surrogate 9 or substitute model for a given set of gravi-  \ntational waveform data, with the help of numerical methods for decomposing, compressing, and fitting the original data.  \nWaveform data set and data pieces. A data set { h ( t ; λ i ) } N i =1 can be generated from a GW model h ( t ; θ ) 10 (or a set of NR waveforms) at parameter values T = { λ i } N i =1 on a time grid t = { t j } m j =1 . The input data is then a set of waveforms given over the space of relevant intrinsic binary parameters λ (a subset of the set of all physical parameters θ ). The values T can form a grid or, more generally, can be scattered data points in the domain of the original model. Each waveform in this input data set is decomposed into a number of data pieces f k ( t ; λ ) chosen in such a way as to obtain as simple a function of time and binary parameters as possible as discussed in Sec. 2.6.  \nReduced basis. Each waveform data piece f ( t ; λ ) in a surrogate model aims to approximate its discrete training data set H = { f ( t ; λ i ) } N i =1 as closely as possible. Usually, one first computes a discrete orthonormal basis { B i } N i =1 , ⟨ B i , B j ⟩ = δ ij with elements B i = B i ( t ) = ( B i ( t 1 ) , . . . , B i ( t m )) T from the training data, with ⟨· , ·⟩ the Euclidean inner product on R m . Often a reduced basis with dim B = n < N is computed which amounts to a truncated basis expansion  \nf ( λ ) = f ( t ; λ ) ≈ n ∑ i =1 c i ( λ ) B i ( t ) , (67)  \nwith n chosen according to a desired accuracy threshold. The basis expansion coefficients are given by  \nc i ( λ ) = ⟨ f ( λ ) , B i ⟩ . (68)  \nTwo methods have been used in the literature to build a reduced basis for a data piece f : (i) a greedy algorithm which at step k adds waveform data at the point in parameter space where the data piece vector has the highest projection error ∥ f ( · ; λ ) -P k f ( · ; λ ) ∥ ; here, P k f ( · ; λ ) is the orthogonal projection of the data piece onto the span of the basis { B i } k i =1 constructed in the first k steps of the algorithm (see Appendix A of [293] for details), or (ii) a method where the waveform data is suitably downsampled in the independent variable, arranged in a matrix with elements F ij = f ( t j ; λ i ) and the (truncated) singular value decomposition [349] F = U T ΣV is used to compute the basis [573,574]. If the training set is sufficiently dense and the data are smooth, waveforms inside the domain covered by the training set, but not part of the training set itself will be well approximated by the reduced basis.  \nSurrogate construction. Given a reduced orthonormal basis B i ( t ) for a waveform data piece f ( t ; λ ) two contrasting approaches have been used to build  \na surrogate for this piece by fitting or interpolating coefficients over parameter space: (i) one may directly fit or interpolate the coefficients c i ( λ ) 11 as in Ref. [573,574], or (ii) the problem can be recast as interpolation in time by selecting a 'good' set of sparse empirical time nodes { T 1 , . . . , T n } from the full time grid such that the resulting interpolant recovers the waveform data piece exactly at the T i while minimizing the interpolation error. As discussed in Ref. [293] this is achieved by the so-called empirical interpolant [471,187] (EI)  \nI n [ f ]( t ; λ ) = n ∑ j =1 B j ( t ) f ( T j ; λ ) , (69)  \nwhere B j ( t ) = ∑ n i =1 B i ( t ) ( V -1 ) ij and V ij = B j ( T i ), V ∈ R n × n . The EI time nodes are chosen such that V has a small condition number. This can be accomplished with an algorithmic complexity of O ( n 3 ) as shown in App. A of Ref. [89]. With this transformation in hand, the data piece f ( · ; λ ) then needs to be fitted at each of the EI time nodes T j .  \nFitting and interpolation methods. A variety of methods have been used to predict expansion coefficients c i ( λ ) or the waveform data piece at a fixed EI node f ( T j ; λ ) at unknown parameter space values λ given training data. See Ref. [639] for an introductory review. Interpolation methods enforce that the interpolant coincides with the given data at the training set points, while a fitting method provides no such guarantee and minimizes the residual between the given data and e.g. a polynomial ansatz made by the method. In low dimensional parameter spaces interpolation by tensor-product splines [136] is straightforward and reliable [573,574] whenever a large enough training set can be generated on a regular grid. A combination of radial basis functions with monomials has been used in Ref. [552] which does not require a grid. For moderately large parameter spaces with dimensions ≳ 5 methods for fitting scattered data must be used. This includes polynomial regression with feature selection such as the greedy forward stepwise regression method with cross-validation error estimates described in Appendix A of Ref. [128], Gaussian process regression (GPR) [585] which can also provide a measure of uncertainty for each fit [245,442,696,695], but is costly for large training sets ( O ( N 3 )), and, more recently, neural networks [351] which have been shown to handle problems of moderate dimensionality with good accuracy and are fast to evaluate [422,632,675].  \nAvailable surrogate models. Surrogate models have been built for dominant mode aligned-spin binaries [293,573,574,442], models including higher harmonics [696,208] and precession [128,129,695,323]. Many surrogate models (or ROMs) have been built for EOB models, see Sec. 2.6, which are in general significantly slower to evaluate than phenomenological models, see Sec. 2.6,  \nlimiting their applicability for data analysis purposes. Similar to phenomenological models EOB surrogates tend to be built directly in the frequency domain [573,574,442,208]. Over the past several years surrogate models of NR simulations have been successfully constructed including effects of precession, higher harmonics, memory, and eccentricity [128,129,695,696,730,404]. As NR simulations are extremely costly and sparse, these studies have worked with carefully designed training sets to keep the number of simulations small without sacrificing high accuracy. Naturally, their parameter space coverage is more restrictive than for semi-analytic models, but where they are available, NR surrogates are arguably more accurate than models based on semi-analytical techniques. Surrogates have also been constructed for remnant properties of BH mergers (final mass and spin, as well as recoil velocity) [697].  \nThe evaluation speed of surrogate models is typically several orders of magnitude faster than it took to generate one of the training space waveforms; the speedup is especially high for NR surrogates. Surrogate models usually achieve accuracies of 10 -3 (or better) in the mismatch , a normalized and maximized inner product between waveforms, (see Eq. (102) and the following discussion in Sec. 3.7) against original data which is in general sufficient for most GW data analysis tasks for current ground based interferometers, but see [576] for estimated accuracy requirements for future ground-based GW detectors.", '2.8 Comparison of waveform accuracy and efficiency': "Waveform models aim to provide accurate approximations of the true underlying waveform (as a solution to Einstein's equations in GR). Given that the most accurate merger waveforms come from a sparse set of expensive NR simulations, but are too short to cover the frequency sensitivity of ground based GW detectors, practical models invariably need to combine analytical and numerical waveforms in addition to some interpolation or fitting. Therefore, waveform models compound various sources of errors from their constituent parts and model assumptions. Given these imperfections and the undesirable consequences of waveform systematics in data analysis applications (missed signals, bias in binary parameters) it is crucial to study how these models compare among themselves and against NR simulations in an effort to further improve their fidelity.  \nA detailed comparison of the accuracy of current waveform models is beyond the scope of this review. The following articles provide comparisons between a number of models in terms of mismatch or in terms of a shift ( bias ) in recovered parameters compared to the parameters of a synthetic signal in parameter estimation. For comparisons of aligned spin models see Refs. [423, 135,568,696], for models including precessing see Refs. [421,695,541,569,360]. Tidal models are being compared in Refs. [243,677].  \nFor data analysis applications the efficiency of waveform models is very important. A comparison of the computational cost of BBH models implemented in the LSC's Algorithm Library Suite [460] is shown in Fig. 7. The evaluation  \n<!-- image -->  \n<!-- image -->  \nFig. 7: Evaluation times for Fourier domain polarizations of a number of BBH waveform models as a function of the total mass of the binary. Top Left: aligned-spin models for the dominant (2 , 2) mode, Top Right: aligned-spin models including higher harmonics, Bottom: models for precessing binaries. The total mass was varied from 10 M ⊙ to 100 M ⊙ , while all other parameters were kept fixed (mass-ratio 1:3, all spin components set to zero except χ 1 z = χ 2 z = 0 . 5 and, for models supporting precession, χ 1 x = 0 . 5; the starting and maximum frequencies were set to 20 and 16384 Hz, with the frequency spacing automatically determined by LALSuite's [460] SimInspiralFD function.) PA refers to the post-adiabatic approximation and MB to multi-banding techniques. The NR surrogate model in the precession panel is limited by the number of cycles available, which translates to a lower total mass for which it can be evaluated. In contrast, the NR surrogate in the middle panel has been hybridized with PN waveforms in the inspiral, allowing it to produce longer waveforms.  \n<!-- image -->  \ntime increases steeply for low-mass binaries driven by the waveform length. Models including higher harmonics and/or precession are in general slower that simpler models. Phenomenological and surrogate / ROM models are several orders of magnitude faster than time domain EOB models. Further acceleration for frequency domain models is possible with multi-banding techniques as used by IMRPhenomXPHM [703,329], while time domain EOB models can trade efficiency for accuracy through the post-adiabatic approximation [521, 502].", '3 Statistical framework': 'In this section we develop a statistical model of the GW signal and detector noise, showing how the noise curves from Sec. 1 and the parameter-dependence of the waveforms from Sec. 2 combine to produce a natural geometric structure for the parameter space. This structure gives rise to the typical correlations in the parameter estimation results discussed in Sec. 5, and gives us a method for covering the parameter space with templates for a matched-filter search, discussed in Sec. 4.  \nDetector noise is generally treated as a stationary Gaussian process with a correlation structure described by a given power spectral density (PSD), resulting in the Whittle likelihood function [717]. The noise PSD used may be estimated from data or predicted via the physics of the detector(s) considered. This noise model was initially applied to GW data analysis studying both detectability and parameter estimation for inspiral signals in [295]. The likelihood ratio of a signal under the Gaussian noise model also gives rise to the classical matched filter for signal detection [689].  \nMore recent works have examined relaxing some of the assumptions behind the Whittle likelihood: for example, the PSD may not be known exactly, so its uncertainty may be marginalised over [609,463,190,663,121], or the PSD may be jointly estimated along with the signal parameters [189,561].', '3.1 Likelihood function for detector noise': "The calibrated output of a GW detector d ( t ) = h ( t )+ n ( t ) contains the GW signal h ( t ) and additive noise n ( t ). The noise is a combination of fundamental, environmental and technical noise sources for the particular detector; references in Sec. 1.2 provide more details. Mathematically, we can say that the noise is drawn from a stochastic process, n ∼ N described by a statistical model H N that gives us the probability of observing a particular noise realisation p ( n ( t ) | H N ).  \nThe noise model is chosen in the light of available information, which we can quantify in terms of the moments of the process, written as expectation values integrating over the space of noise realisations. 12 The first moment, the mean, may for simplicity be set to zero by redefining the zero of detector strain output:  \nE N [ n ( t )] = ∫ n ( t ) p ( n ( t ) | H N ) dn ( t ) = 0 . (70)  \nThe second moment of the stochastic process  \nE N [ n ( t ) n ( t ' )] = ∫ n ( t ) n ( t ' ) p ( n ( t ) | H N ) dn ( t ) ≡ C N ( t, t ' ) , (71)  \nspecifies a two-point kernel C N ( t, t ' ) known as the auto-covariance function.  \nWe may continue in this fashion to higher-order moments to completely specify the noise stochastic process. However, the common case where we only have constraints for the first and second moments, i.e., the mean and variance, the probability distribution that maximises the entropy, therefore representing the least informed distribution, is the multivariate Gaussian distribution. This motivates the use of the Gaussian likelihood function in our analyses, although it is possible to derive a likelihood with higher order information.  \nStationarity Astochastic process is said to be stationary if it is invariant under a translation in time,  \np ( n ( t ) | H N ) = p ( n ( t + τ ) | H N ) , (72)  \nwhich is satisfied if all its moments are time-invariant. A less restrictive requirement is that of weak-sense stationarity , where only the first two moments are required to be invariant. The mean may be set to zero at all times, but for the second moment stationarity implies that  \nC N ( t, t ' ) = C N (0 , t ' -t ) ≡ C N ( τ ) , (73)  \ni.e., the auto-covariance is a function only of the time lag τ ≡ t ' -t .  \nDiscretization The noise considered above as a continuous process n ( t ) is a mathematical idealization of actual detector output, which is a set of discrete samples from the generating process. For data sampled at a finite set of J times, t j , separated by uniform intervals ∆t , the sampled values can be thought of as a vector n in a J -dimensional space, and the covariance function becomes a symmetric matrix  \nC jk ≡ E [ n j n k ] = ∫ n j n k p ( n | H N ) d J n . (74)  \nNeglecting information from higher moments, the probability of a particular noise realisation n is given by the multivariate Gaussian distribution  \np ( n | C N , H N ) = 1 √ det 2 π C N exp   -1 2 ∑ j ∑ k n j ( C -1 N ) jk n k   . (75)  \nThis time-domain likelihood function requires a double sum over the data, thus its computational cost scales as J 2 . The likelihood may be much more efficiently evaluated in the frequency domain using the discrete Fourier transform (DFT): the frequency-domain noise realization is expressed via a matrix operation as ˜ n = F n ∆t , where F jk = J -1 / 2 exp( -2 πijk/J ) is a (unitary)  \ndiscrete Fourier transformation matrix. Since FF ∗ = I , the time-domain likelihood can be rewritten as  \np ( n | C N , H N ) = 1 √ det 2 π C N exp [ -1 2 ( n T F )( F -1 C -1 N F )( F -1 n ) ] = 1 √ det 2 π C N exp [ -1 2 ∆t 2 ˜ n T ˜ C -1 N ˜ n ∗ ] , (76)", 'where ˜ C N = F -1 C N F .': 'If the noise is stationary, then C N is a circulant matrix, and ˜ C N is diagonal, allowing us to replace the double sum with a single sum over the diagonal elements. We define the one-sided noise PSD S N ( f ) via the Wiener-Khinchin theorem as the Fourier transform of the noise auto-covariance function,  \n1 2 S N ( f ) = ∫ ∞ -∞ C N ( τ ) exp( -2 πifτ ) dτ , (77)  \nallowing the diagonal of ˜ C N to be written as ˜ C ii = J 2 ∆t S N ( f i ). Therefore, the frequency domain likelihood can be written as  \np ( ˜ n | S N , H N ) = det (2 π C N ) -1 / 2 exp [ -1 2 J∆t ∑ i ˜ n i ˜ n ∗ i 1 2 S N ( f i ) ] = det (2 π C N ) -1 / 2 exp [ -1 T Re ∑ i> 0 ˜ n i ˜ n ∗ i 1 2 S N ( f i ) ] , (78)  \nusing the fact that the original time-series n is real, implying ˜ n -i = ˜ n ∗ i , and we have introduced T = J∆t . If we further define the noise-weighted inner product  \n⟨ a | b ⟩ ≡ 2 Re ∫ ∞ 0 ˜ a ∗ ( f ) ˜ b ( f ) 1 2 S N ( f ) df , (79)  \nthe noise likelihood can be written in terms of the frequency-domain noise realisation ˜ n ( f ) as  \np (˜ n ( f ) | S N , H N ) = det (2 π C N ) -1 / 2 exp ( -1 2 ⟨ n | n ⟩ ) . (80)  \nWith this definition the space of data is formulated as an inner product vector space , which we will later use to access various (differential)-geometric methods, for example considering distances between neighbouring signals. See [601, 38] for further discussion of the derivation of the Gaussian likelihood and its assumptions.', '3.2 Likelihood for a signal model': 'We now return to the case where the data contains both noise and a signal, which we denote as H S for signal hypothesis. This hypothesis includes our previous assumptions about the noise n given by the noise model H N with PSD S N , but adds the assumption that the mean of the data is non-zero due to the presence of a signal h , such that d = h + n . Calculating the precise form of the signal requires assuming a particular waveform model; for a given set of source parameters θ , the waveform model yields the signal time series h , or signal model , and, from 80, the likelihood of the residuals h -d is  \np ( d | h ( θ ) , H S ) = det(2 π C N ) -1 / 2 exp ( -1 2 ⟨ d -h | d -h ⟩ ) . (81)  \nThe shape of this function across the parameter space θ will determine the  \nprecision with which we can estimate the physical parameters of the source, see Sec. 5. The likelihood function over θ is also of importance when searching for a signal, as values of the likelihood close to those at any global or local maximum are generally only obtained over a tiny fraction of the possible parameter space. We therefore need to know how finely to sample the space to avoid missing possible signals, i.e. how to lay out a template bank , as discussed in Sec. 4.3. The inner product in Eq. (81) can be expanded as ⟨ d -h | d -h ⟩ = ⟨ d | d ⟩ + ⟨ h | h ⟩-2 ⟨ d | h ⟩ . The quantity ρ 2 opt = ⟨ h | h ⟩ is the square of the optimal signalto-noise ratio (SNR), a useful measure of the detectability of a signal in a detector with given noise PSD. In the picture where data of fixed length is described as a J -dimensional vector space with the inner product Eq. (79), the optimal SNR is the length of the vector representing a particular signal. We may consider the space of data streams d to have an associated metric Σ : for data in the frequency domain, the metric is a diagonal matrix with nonzero elements given by Σ ii = ( T/ 2) S N ( f i ) -1 , indicating that the Fourier frequency bins are orthogonal dimensions of this space. The operation of whitening the data - achieved by multiplying a data stream ˜ d by S N ( f ) -1 / 2 - can then be seen as the normalisation of these frequency bins, such that the metric becomes the identity matrix.', '3.3 The Fisher information metric': "The space of possible signals h can be considered as a sub-manifold of the full vector space of possible data streams d . This sub-manifold of signals, with dimensionality equal to the number of independent parameters used to describe the signal, is not itself a vector space, since a linear combination of signals in general is not another valid signal. Still, provided the waveform is a smooth function of these parameters we may consider it as a continuous manifold.  \nWe can then use the signal parameters as a coordinate system to identify points on the signal manifold; the flat-space metric restricted to the signal  \nmanifold will induce a metric Γ in the parameter space, determined by the 'closeness' of neighbouring signals. Formally, for an infinitesimal change in parameters d θ we have a change in the waveform d θ ( ∂ h /∂ θ ) and a squared interval  \nd s 2 = ∥ d θ ∥ 2 = ( d θ ∂ ˜ h ∂ θ ) T ˜ C -1 N ∂ ˜ h ∂ θ d θ = d θ T Γ d θ , (82)  \nwhere the metric in parameter space is given by  \nΓ ij = 〈 ∂ h ∂ θ i ∣ ∣ ∣ ∣ ∂ h ∂ θ j 〉 . (83)  \nThe matrix Γ is known as the Fisher information matrix, or metric in this context [84].  \nThe Fisher matrix formalism is widely used to determine parameter estimation precision under a Gaussian approximation of the likelihood function [295, 214,692]. If we have identified the parameter values θ 0 which maximise the log-likelihood at l 0 = log L ( θ 0 ), the expansion of the log-likelihood to second order around θ 0 is given by  \nlog L ( θ 0 + δ θ ) ≈ l 0 -1 2 δ θ i Γ ij δ θ j ≡ l 0 -1 2 ∥ δ θ i ∥ 2 . (84)  \nNeglecting higher order terms results in a Gaussian function on parameter space with covariance matrix Γ -1 . Assuming a uniform prior distribution, the resulting posterior probability distribution for the parameters is also a multivariate Gaussian. The diagonal elements of Γ -1 will then give the variance of the marginal posterior, i.e. the precision at which a given parameter may be measured. In particular, the variance is proportional to ⟨ h | h ⟩ -1 , i.e. to ρ -2 opt .  \nIn practice, this predicted precision may be inaccurate if the prior distribution is not uniform over the scale of the likelihood peak, or if the Gaussian approximation of the likelihood fails, e.g. if there are multiple maxima of comparable height due to symmetries of the likelihood. These limitations restrict the applicability of the Fisher matrix formalism for parameter estimation precision to cases with high SNRs and a single, well-defined peak [599] - though even in the low SNR case where we cannot neglect higher-order terms, it provides the Cramer-Rao bound on the covariance of an unbiased point estimator [692]. One can alleviate these restrictions using a higher order expansion of the likelihood to predict parameter estimation precision [706], but in general the parameters of a signal must be estimated by sampling the posterior probability distribution, as described in Sec. 5.", '3.4 Detector response': 'Having described the generic likelihood function for a transient GW signal, we now introduce some specifics for the CBC case, with the ultimate aim of understanding the design of searches and parameter estimation methods.  \nThe strain from a generic frequency-domain compact binary signal can be written as  \n˜ h + ( f ) = A + ( f ) exp( -iϕ + ( f ) + ϕ 0 ) , (85)  \n˜ h × ( f ) = A × ( f ) exp( -iϕ × ( f ) + ϕ 0 ) , (86)  \nfor the + and × polarisations respectively, see Sec. 2.3. A detector, labelled by I , responds linearly to a weighted combination of these,  \n˜ h I ( f ) = F + ˜ h + ( f ) + F × ˜ h × ( f ) , (87)  \nwhere F + and F × are the detector response functions (or pattern functions ). These are independent of signal frequency in the long-wavelength approximation, but depend on the direction of propagation of the waves with respect to the detector, as parameterised by the right ascension α and declination δ of the source and the GW polarisation angle ψ , as in Sec. 2.3.1 and Table 2. For an interferometric detector with arms along the x - and y -axes, the response functions are given by, e.g. [475],  \nF + ( θ, ϕ ; ψ = 0) = 1 2 (1 + cos 2 θ ) cos 2 ϕ, F × ( θ, ϕ ; ψ = 0) = cos θ sin 2 ϕ,  \n(88)  \nwhere θ and ϕ are the polar and azimuthal angles in the detector frame; for nonzero polarization angle ψ the responses follow from considering a rotation about the direction of propagation.  \nIn the simplified case of emission in the l = m = 2 mode of a binary whose component spins are parallel to the orbital angular momentum vector, the orbital plane will not precess over time and the two polarization phases are related by ϕ + = ϕ × + π/ 2, thus we have  \nA + ( f ) = 1 2 (1 + cos 2 θ JN ) A 0 ( f ) d L , (89)  \nA × ( f ) = cos θ JN A 0 ( f ) d L , (90)  \nwhere A 0 ( f ) is a common frequency-dependent amplitude, and other parameters are defined in Table 2.  \nThe response in detector I for a signal whose arrival time and coalescence phase are t c,I and ϕ c,I respectively may then be written  \n˜ h I ( f ) = A I ( f ) exp ( i ( Φ ( f ) -2 πft c,I -ϕ c,I )) , (91)  \nwhere the amplitude A I ( f ) = A 0 ( f ) /D eff ,I ( α, δ, ψ, θ JN , t c ), with D eff ,I being the effective distance to the source for detector I , which folds together the dependence on position and orientation angles [82]. Equation (91) implicitly defines Φ ( f ); see Eq. (40) as an example of the phase of the TaylorF2 approximant.', '3.5 Measurement precision of signal parameters': "Now that we have a definition of the notion of 'closeness' of two signals or templates via the Fisher matrix, we consider the case of actual compact binary signals. This will let us understand which quantities make the largest contributions to measurable waveform differences, and therefore, about which ones observed signals are expected to yield the most information.  \nIn Sec. 3.3 we saw that the definition of the Fisher matrix, Eq. (83), requires waveform derivatives ∂ ˜ h /∂ θ . In the frequency domain, considering a single detector, we can write the waveform as Eq. (91), i.e., a product of amplitude and phase factors, to obtain the derivatives with respect to θ :  \n˜ h ( f ) = A ( f ) exp iΦ ( f ) ,  \n(92) ∂ ˜ h ( f ) ∂ θ = ∂A ( f ) ∂ θ ˜ h ( f ) A ( f ) + i ∂Φ ( f ) ∂ θ ˜ h ( f ) = ( ∂ log A ( f ) ∂ θ + i ∂Φ ( f ) ∂ θ ) ˜ h ( f ) . (93)  \nFrom this we can see the relative importance of the signal amplitude and phase to the measurement of parameters: roughly speaking, a change in the phase by the addition of one radian (counted over the entire observed bandwidth of the signal) is equivalent to a change in the amplitude by a factor of e . Since the waveform may spend many thousands of cycles in the detector's sensitive frequency band, the signal phase evolution is of primary importance: parameters which affect the phase are typically much more precisely measured than those that vary only the amplitude. The derivative is also proportional to the signal itself: as expected, louder signals provide more information about their parameters.  \nWe can take the post-Newtonian waveform expansion as a starting point to investigate the influence of some key parameters. As detailed in Sec. 2.2 and Eq. (40), the phase is written as a sum over powers of frequency f ( k -5) / 3 :  \nΦ ( f ) = 2 πft c -ϕ c -π 4 + ∑ k ψ k ( M , q, a 1 , a 2 ) f ( k -5) / 3 + · · · , (94)  \nomitting terms in log f for simplicity. 13 The derivative with respect to the reference phase ϕ c is trivial; for the other parameters we have  \n∂Φ ( f ) ∂t c = 2 πf , (95)  \n∂Φ ( f ) ∂ θ = ∑ k f ( k -5) / 3 ∂ψ k ∂ θ . (96)  \nIn Table 1 we show the order at which different parameters appear. At lowest order only M contributes: we have ψ 0 ( M ) = (3 / 4)(8 π M ) -5 / 3 , such that ∂ψ 0 ( f ) /∂ M∝M -8 / 3 . We therefore expect that only the (redshifted) chirp mass, not the individual masses m 1 , m 2 , will be well-measured for inspiraldominated waveforms. Only by continuing the expansion to higher orders in k will a second combination of masses, the symmetric mass ratio η , enter the expansion; contributing less to the integral of Eq. (83), it will be less well measured. Therefore we should expect the measurement uncertainties in the two component masses to show a high degree of correlation, as for instance in [36].  \nAt higher order still, the spin vectors a 1 , a 2 enter the expansion, and are accordingly less well determined. As with the individual masses, certain combinations of spins appear earlier than others, the most important being the effective aligned spin χ eff , and then the effective precessing spin χ p . Since the spin orientations are defined on the sphere, the corresponding parameters have nontrivial prior boundaries and/or distributions; for relatively low SNRs, the Fisher matrix expansion is thus less useful in predicting their measurability, and the full apparatus of Bayesian inference should be used, as in Sec. 5. Tidal deformability, finally, enters for systems containing one or more NSs only from 5PN order, as in Sec. 2.2.1, thus requires much higher SNRs for a nontrivial measurement, although the high order is partly counteracted by a large numerical coefficient, compare [36,707].  \nFurther insights arise by considering the integrand of Fisher matrix elements: for a parameter θ , the diagonal matrix element is the frequency integral of | ∂ ˜ h /∂θ | 2 /S n ( f ), This may be thought of as a spectral density of information or parameter precision; Fig. 8 illustrates the density over log frequency for various parameters of BNS systems. Moreover, for parameters having similar frequency dependence of ∂ ˜ h /∂θ , the off-diagonal matrix elements are expected to be relatively large, implying strongly correlated measurement errors, for instance between orbit-aligned spins and mass ratio [214,105].  \nPost-Newtonian coefficient space As the coefficients of many PN expansion terms ψ k are complicated non-linear functions of physical binary parameters, the Fisher matrix varies strongly over parameter space, leading to additional pitfalls for low- or moderate-SNR signals. While the nonlinearity can be partly addressed by careful choice of parameter basis, e.g. [700], an alternative strategy considers the space of PN coefficient values , over which the Fisher matrix is  \nFig. 8: Frequency dependence of Fisher matrix element integrands, reproduced from [370] (see also [225]). The quantity plotted versus frequency is the density of information about each binary parameter over log(frequency), | ∂ ˜ h /∂θ | 2 / ( fS n ( f )) where S n is taken to be the zero-detuned high power Advanced LIGO model PSD; each curve is normalized to peak at unity, except the tidal parameter, normalized to unity at 1 kHz. Although the tidal parameter information appears to increases monotonically towards high frequency, this calculation will become inapplicable by the BNS merger phase, at frequencies beyond the scale of this plot.  \n<!-- image -->  \n(near)-constant [666,545,149,537] for low-mass binaries. Linear combinations of ψ k may then be found for which the matrix is diagonal, making tasks such as template placement in a multi-dimensional space (Sec. 4.3) significantly simpler.", '3.6 Multiple detectors': 'We can extend the single-detector analysis to the case of multiple detectors by applying the chain rule of probability. For example, if we are analysing data from Hanford (H), Livingston (L) and Virgo (V), the joint likelihood is  \np ( d H , d L , d V | H S ) = p ( d H | d L , d V , H S ) p ( d L | d V , H S ) p ( d V | H S ) = ∏ I ∈{ H,L,V } p ( d I | H S ) , (97)  \nwhere the expression reduces to a product over the detectors if the noise in each detector is assumed to be statistically independent. 14 The multi-detector  \nlikelihood may then be written as  \np net ( { d I }| H S ) ∝ exp ( -1 2 ∑ I ⟨ d I -h I | d I -h I ⟩ I ) , (98)  \nwhere ⟨ a | b ⟩ I denotes an inner product using the PSD of detector I and h I is the signal as measured at detector I , including the detector responses and time shifts due to propagation time between different sites. This coherent likelihood function gives a complete model of the detector network response as a whole: the signals at each detector h I ≡ h I ( θ ) are dependent on a single parameter vector θ . This form is used in coherent analyses, including for parameter estimation of the signal in Sec. 5.  \nIf we allowed the signal parameters to (unphysically) vary independently in each detector I , which we may write as θ I , we could use this additional freedom to find a higher maximum of the product of likelihoods: ∏ I max θ I p ( d I | θ I , H S ) ≥ max θ p net ( { d I }| θ , H S ). Thus we can simplify the problem of maximising the likelihood, for the purposes of detection, by maximising the separate perdetector likelihoods, at the cost of some inaccuracy if unphysical combinations of parameters are allowed. In practice, constraints are imposed upon maxima to ensure compatible signals in all detectors, for example by enforcing consistent relative arrival times. This approach is referred to as a coincident likelihood, as opposed to the fully coherent likelihood of Eq. (98), and is commonly applied in matched filter searches, as discussed further in Sec. 4.7.', '3.7 Normalized signals and similarity measures': "Since CBC sources occur at a wide range of a priori unknown distances, the absolute amplitudes of GWs at a detector also have a wide range of uncertainty. We may discuss the properties of signals without specifying an amplitude or distance via normalized signals with an optimal SNR (for some notional or real detector sensitivity) of unity,  \nˆ h = h ⟨ h | h ⟩ 1 / 2 = ⇒ h = ρ opt ˆ h . (99)  \nThe Fisher matrix is then  \nΓ ij = ρ 2 opt 〈 ∂ ˆ h ∂ θ i ∣ ∣ ∣ ∣ ∣ ∂ ˆ h ∂ θ j 〉 , (100)  \ngiving an expected inverse relation of parameter estimation accuracy with optimal SNR. For data containing a signal ρ opt ˆ h plus noise n , if we consider a model that is identical to the signal except for an unknown amplitude A ˆ h , the likelihood is maximized for A ML = ρ opt + ⟨ ˆ h | n ⟩ , where the noise term averages to zero over data realizations.  \nVarious measures of similarity between different signal models h ( θ ) are linked to aspects of parameter estimation and detection. While the amplitude of a predicted signal has an influence on astrophysical interpretation, similarity of signals is generally assessed independently of amplitude: the most basic measure is the overlap of two signals h 1 , h 2  \nO ( h 1 , h 2 ) = ⟨ h 1 | h 2 ⟩ ( ⟨ h 1 | h 1 ⟩ ⟨ h 2 | h 2 ⟩ ) 1 / 2 ≡ ⟨ ˆ h 1 | ˆ h 2 ⟩ . (101)  \nThe overlap can be shown to be maximized at unity, when h 1 = h 2 .  \nFurthermore, the absolute values of the orbital phase of a binary or its time of merger are rarely of astrophysical interest (in the absence of non-GW observations that may depend on these quantities), and detection algorithms do not generally make use of them. Allowing arbitrary differences in phase and coalescence/merger time between signal models, we arrive at a less restrictive measure called the match  \nM ( h 1 ( t c 1 , ϕ c 1 , θ 1 ) , h 2 ( t c 2 , ϕ c 2 , θ 2 )) = max t c 1 -t c 2 ,ϕ c 1 ,ϕ c 2 O ( h 1 , h 2 ) , (102)  \nwhere θ represents source parameters excluding time and phase. 15 The mismatch is simply 1 -M , which like M ranges between 0 and 1.  \nIf, conversely, we are comparing different signal models, for example two approximants discussed in Sec. 2.6, our goal is to assess systematic differences. The comparison between signals for binaries with identical parameter values θ ' (including intrinsic parameters), M ( h 1 ( t c 1 , ϕ c 1 , θ ' ) , h 2 ( t c 2 , ϕ c 2 , θ ' )), is then called the faithfulness . This quantity informs how far we expect the models to give the same inferences on such parameters; differences between models will contribute to systematic uncertainties and/or biases in measured source properties (see Sec. 2.8).", '4 Detection': "In this chapter we present the problem of detection for compact binary merger signals in the data of a GW detector network, and describe solutions deployed or proposed for real data as well as the principles underlying them.  \nIn the ground-based detector network, sensitive at frequencies of a few Hz and above, CBC signals are comparatively rare and weak relative to the noise. By contrast, future space-based detectors are expected to face the opposite case where signals (not necessarily all from compact binaries) dominate the output stream, see e.g. [85]. Even for 3rd generation detectors with a sensitivity to BBH mergers covering nearly the whole of the observable Universe, the rate of signals is expected to be at most one per tens or hundreds of seconds; while these 'chirp' signals will overlap in time within the detector's frequency band, they will not overlap significantly in the time-frequency plane [591,493,409].  \nDetection thus implies selecting the small fraction of data containing distinguishable signals, as opposed to noise or indistinguishably weak signals. This problem is close, but not identical, to classic detection theory with a well-defined 'null hypothesis' denoting the lack of signal. In realistic cases, there are important differences from idealized theory.  \nFor one, possible binary signals cover a large parameter space: we do not a priori know the true distribution of signals over that space. A theoretically optimal search maximizes the number of detected signals for a given rate of false alarms, but we are unable to evaluate this for an unknown population. We also might choose a figure of merit that differs from simply the expected number of detections.  \nSecond, even for a known signal distribution, the optimal search method depends on the noise content of the data. Since real GW detector data contains transient excess noise events whose rate and morphology cannot be predicted, search optimization requires empirical information on such noise properties, and thus is necessarily limited by the finite statistics of real noise.  \nFurthermore, searches precede a much more computationally intensive exploration of the properties of candidate signals, see Sec. 5. Thus, beyond the mere presence of a signal, sufficiently accurate point estimates of arrival time and of binary component masses provided by the search algorithm will significantly reduce the cost of subsequent analysis. Searches also provide detailed timing information used to estimate the source's location for low-latency followup [641,642].  \nCBC detection methods are also subject to technical and computational limits. The technical challenges of setting up a (near-)optimal search are currently solved only for signals in the four-dimensional parameter space { m 1 , m 2 , χ 1 z , χ 2 z } of quasi-circular, non-precessing coalescences neglecting subdominant GW multipoles, see Sec 2. Methods for detection of more complex (and thus more realistic) signals are under active investigation, as discussed below.  \nIn the rest of this chapter we revise standard detection theory, before discussing alterations or extensions for real GW data and reviewing the challenges  \nof more complex CBC signals, computational issues and different astrophysical search applications.", '4.1 Theory of transient signal detection': "Given a stretch of data d ( t ), we want to distinguish between the hypothesis H S that it contains a signal of known form, and the hypothesis H N that it contains no signal. 16 For GW data spanning several months, more than one signal may arrive; to begin we consider a short period containing no more than one signal 17 , see Sec. 6 for joint analysis of multiple signals. The data are supposed to contain noise with known statistical properties, usually described via the correlation function, which in the simplest case is assumed stationary, see Sec. 3.1. The detection algorithm calculates a scalar 'test statistic' ϱ ( d ) which will as far as possible take a higher value under the H S hypothesis than under H N .  \nEfficiency and false alarm rate The performance of a test statistic may be evaluated by imposing a threshold value ϱ ∗ : the efficiency ϵ ( ϱ ∗ ) is the expected fraction of cases actually containing a signal for which ϱ > ϱ ∗ , i.e. P ( ϱ > ϱ ∗ | H S ). The false alarm rate 18 F ( ϱ ∗ ) is the expected fraction of cases containing no signal for which ϱ > ϱ ∗ , i.e. P ( ϱ > ϱ ∗ | H N ). In terms of probability density over the data realization d ,  \nϵ ( ϱ ∗ ) = ∫ Θ( ϱ ( d ) -ϱ ∗ ) p ( d | H S ) d d, (103)  \nF ( ϱ ∗ ) = ∫ Θ( ϱ ( d ) -ϱ ∗ ) p ( d | H N ) d d. (104)  \nReceiver operating curve and likelihood ratio We may calculate ϵ ( ϱ ∗ ) and F ( ϱ ∗ ) over a range of ϱ ∗ values, parameterizing a curve in the ( F , ϵ ) plane called the 'receiver operating curve' (ROC). Any monotonic increasing function of ϱ ( d ) will yield the same ROC, giving the same ranking of data from noise-like to signal-like; hence ϱ ( d ) is called a 'ranking statistic'.  \nA 'Neyman-Pearson' optimal statistic is one that yields the largest possible ϵ for a given value of F . It may be shown, e.g. [126] that the likelihood ratio Λ SN is an optimal statistic, where  \nΛ SN ( d ) ≡ p ( d | H S ) p ( d | H N ) . (105)  \nIf this likelihood ratio can be efficiently and accurately computed for the true GW signal population under all possible data realizations, the problem is solved; however it will not be surprising that in real applications the classic solution, the matched filter, is only the first of many tools required.  \nThe idealized matched filter For a signal of known form h ( t ), the likelihood of a data set d ( t ) in a single detector as defined in Sec. 3.2 is  \np ( d | H S , h ( t )) = p ( n ( t ) ≡ d ( t ) -h ( t ) | H N ) = const · e -⟨ d -h | d -h ⟩ / 2 , (106)  \nwhere the inner product ⟨ a | b ⟩ is given by Eq. (79) for Gaussian noise with a known spectrum S N . The likelihood in the absence of signal is just  \np ( d | H N ) = p ( n ( t ) ≡ d ( t ) | H N ) = const · e -⟨ d | d ⟩ / 2 , (107)  \nthus the likelihood ratio is Λ SN ( d ) = e ⟨ d | h ⟩-⟨ h | h ⟩ / 2 . For a fixed signal, ⟨ h | h ⟩ is constant, thus ⟨ d | h ⟩ is an equivalent ranking statistic to Λ SN . As a product of the (whitened) data with a whitened template in the frequency domain, ⟨ d | h ⟩ is called a 'matched filter'.", '4.1.1 Signal with unknown parameters: composite hypotheses': "In astrophysical applications, some properties of the signal are unknown; we parameterise them by a set of values θ , for instance the amplitude or time of arrival at the detector. Hence, we have a 'composite signal hypothesis'. We write the signal hypothesis for specific parameter values as h ( θ ) , H S or just θ , H S . In order to determine an optimal detection method, we require the distribution over θ , notated as p ( θ ) , H S ; this distribution is analogous to a prior in parameter estimation, see Sec. 5.2. Considering transient signals as an inhomogeneous Poisson process with rate λ ( θ ), the PDF of a single merger's parameters is proportional to λ ( θ ).  \nFor some parameters, p ( θ ) is trivial: the distributions over arrival time t c and coalescence phase ϕ c are uniform. Since we believe the Earth is not in a special location in the Universe, the distribution of source locations is uniform over spatial volume, up to cosmological effects at high redshift: hence we expect a distribution of luminosity distances p ( d L ) ∝ d 2 L , and of signal amplitudes p ( A ) ∝ A -4 [636]. Similarly the orbital axis direction is isotropic, thus for the inclination p ( ι ) ∝ sin ι .  \nFor other parameters, in particular binary masses and spins, there is no clear expectation. Published LIGO-Virgo searches generally specify only the boundaries of the region to be searched: for instance in [15] a range of (redshifted) component masses [1 , 99] M ⊙ and binary total masses [2 , 100] M ⊙ was considered, with a maximum component spin magnitude χ < 0 . 9895. However, search methods were not designed for optimal sensitivity to a specific distribution over this space.  \nFor a signal distribution p ( θ ), the optimal likelihood ratio statistic is  \nΛ SN ( d ; p ( θ )) = ∫ p ( θ ) p ( d | θ , H S ) d n θ p ( d | H N ) . (108)  \nThe numerator here is mathematically identical to the evidence for a signal model specified by parameters θ with prior p ( θ ), see Sec. 5.  \nCurrent search methods treat the signal likelihood p ( d | h ( θ ) , H S ) as a strongly peaked function of θ , and approximate the integral over a finite parameter space Θ by a small region around the global maximum. For a signal of optimal SNR ρ opt as in Sec. 3.2, the peak likelihood is higher by a factor ∼ exp( ρ 2 opt / 2) than at parameters far from the peak; thus, such an approximation may be useful for ρ opt ∼ (5 -10) and above. Approximate evaluation of Λ SN is also possible if one records only the maxima of p ( d | θ , H S ), or better of p ( θ ) p ( d | θ , H S ), that exceed a predetermined threshold: these maxima are called triggers , a term borrowed from high-energy physics. The threshold value for real (non-ideal) data results from a tradeoff between search sensitivity vs. computational and data storage limits.", '4.2 Matched filtering for single-mode CBC signals': "For a general merging binary system, the signal has a complicated structure if including effects of precessing component spins and GW emission in nondominant modes; yet more so if the orbit has non-negligible eccentricity, i.e. is not quasi-circular. We begin, though, with the simple case of dominant mode GW emission from quasicircular binaries with orbit-aligned component spins, where the signal at a given detector has the form of a sinusoid with frequency-dependent amplitude and phase factors, Eq. (91). The likelihood may easily be maximized over the signal amplitude parameter and coalescence phase [626]. Marginalizing over these parameters yields corrections, which at least for high-SNR signals are small [148,232].  \nThe maximized likelihood ratio is exp( ρ ( t c , θ -) 2 / 2), where ρ is the phaseand amplitude-maximized matched filter SNR for a signal with coalescence time t c and template parameters θ -(excluding amplitude, coalescence time and phase). We find [82]  \nρ ( t c , θ -) 2 = ⟨ d | ˆ h ( t ; t c , ϕ c = 0 , θ -) ⟩ 2 + ⟨ d | ˆ h ( t ; t c , ϕ c = π 2 , θ -) ⟩ 2 (109)  \n= ∣ ∣ ∣ ∣ ∣ 2 ∫ ∞ 0 ˜ d ( f ) ˆ ˜ h ∗ ( f ; θ -) 1 2 S N ( f ) e 2 πift c df ∣ ∣ ∣ ∣ ∣ 2 , (110)  \nwhere ˆ h ( θ -) is a normalized template waveform with ϕ c , t c set to 0. This 'complex matched filter' uses the fact that ˜ h ( t c ; ϕ c = π 2 ) = i ˜ h ( t c ; ϕ c = 0). The DFT [82] allows us to straightforwardly evaluate ρ ( t c , θ -) as a series over t c , although time-domain implementations have also been considered [176,468]. It is appropriate to call ρ a SNR, as each scalar product of the form ⟨ d | ˆ h ⟩ has  \nzero mean and unit variance in stationary Gaussian noise; thus in the absence of signal we have the distribution p ( ρ | H N ) = ρ exp( -ρ 2 / 2), peaking at unity.  \nFor data containing a signal A ˆ h ( t ; t c , ϕ c , θ -) that exactly matches a template, the most probable value of ρ is the optimal SNR ρ opt = A , see Sec. 3.7. Heuristically, with some fairly large number of independent templates (see discussion below in Sec. 4.3), the rate of noise triggers will fall to a sufficiently low level at high SNRs such that signals with ρ opt > 8 have a high probability of being distinguishable from noise. The horizon distance D h of a detector for a binary with given intrinsic parameters is the largest distance for which ρ opt > 8: this is attained for a system located directly overhead the detector and with zero orbital inclination to the line of sight. Often, D h for a BNS system with non-spinning components of mass 1 . 4 M ⊙ is considered as a 'standard candle' to quantify detector sensitivity.  \nBy contrast, the sensitive range of a detector is an averaged distance, considering the distribution of angular parameters specifying the binary position and orientation. It can be defined by considering a distribution of mergers with a homogeneous rate density R ∗ . The expected rate of mergers having ρ opt > 8, R det , may then be calculated [296]; the range D det is the radius of a sphere such that (4 π/ 3) D 3 det R ∗ = R det [82,658]. Equivalently, drawing a merger with random position and orientation within a distance D h , the probability that it has ρ opt > 8 is ( D det /D h ) 3 ≃ 2 . 26 -3 .  \nThese definitions neglect cosmological expansion of space, which affects both the signal seen at the detectors [437], see Sec. 2.2.3, and the geometric relation between luminosity distance and volume of space. While for Advanced detectors the redshift z of detectable BNS is small compared to unity, for more massive and distant source redshift effects must be included [195]. The actual sensitivity of a search over real data will be determined by the detector network , and as we discuss in Secs. 4.6 and 4.8, does not correspond to a fixed SNR threshold. The rate of mergers is also likely not constant even in comoving cosmological units [303,52], implying further corrections to the D det calculation. Sensitive range is, though, still a useful standard measure of broadband detector performance.", '4.3 Signal manifolds and template banks': "In searching for signals from binaries with unknown parameters (component masses, spins, etc.), in addition to the scientific problem of determining the correct range and distribution of signal parameters, we face a technical challenge of computational cost to evaluate the likelihood at a 'densely spaced' set of points in the parameter space. This challenge has a well-defined solution via the construction of template banks, the topic of this section.  \nGiven a desired parameter search region, for data with a given PSD there will be some rate of noise events (maxima of likelihood or matched filter SNR) inside this region. As the rate density of noise events is a rapidly falling function  \nof SNR, for a statistically confident detection a signal must have SNR above a specific threshold.  \nThe matched filter SNR obtained for a template exactly matching a signal with parameters θ has a non-central χ 2 distribution with mode equal to ρ opt , as above. For a template ˆ h ( θ ' ) differing from the signal, the 'match' of Eq. (102), i.e. the normalized overlap maximized over arrival time and orbital phase, notated here as  \nM ( θ , θ ' ) ≡ M ( h ( t 1 , ϕ 1 , θ ' ) , h ( t 2 , ϕ 2 , θ )) , (111)  \nis bounded to be ≤ 1. The most likely value of SNR recovered by ˆ h ( θ ' ) is then M ρ opt ; thus for a given signal the probability of detection drops progressively with decreasing M . For a bank, we consider only the maximum of M ( θ , θ ' p ) over templates ˆ h ( θ ' p ). For a given probability of detection, the optimal SNR required scales as 1 / M , which for fixed intrinsic parameters θ implies a maximum source distance scaling as M . The expected number of detectable signals then scales as the volume of space ∝ M 3 (neglecting cosmological effects). If all signals had the same maximum match M we would lose a fraction 1 -M 3 of signals relative to the ideal case of an infinitely dense bank.  \nIn general the match ranges between 1 and the smallest value in the target parameter region, the 'minimal match' (MM), i.e. the largest 'hole' in the bank. Thus a minimax strategy places templates to obtain a sufficiently high MM value, losing a fraction 1 -MM 3 of signals in the worst case. Typical target values are MM = 0 . 965 (0 . 97, 0 . 98), losing at worst ∼ 10% ( ∼ 9%, ∼ 6%). For a broad and smooth distribution of signal parameter values, the average match will be significantly higher than MM [81].  \nFor a template ˆ h ( θ ' p ), the region of parameter space 'covered' with match ≥ MMis that within a given distance of θ ' p using the information (Fisher) metric of Sec. 3.3, after projecting out the coalescence time and phase dimensions; the covered region is an approximate ellipsoid in suitable coordinates [542, 543]. To completely cover an extended region, we require overlapping ellipsoids; the larger such overlaps are the more templates are required in total, increasing computational cost. If coordinates can be found with an approximately constant metric, geometrical grid methods will minimize the overlaps [543,200,149,375,605]: this is the case where the PN parts of the inspiral waveform dominate the SNR. In general for inspiral-merger-ringdown (IMR) signals there is no convenient coordinate basis with a near-constant metric, and stochastic or random methods must be used [102,373,477], with a modest increase in computational cost relative to theoretically optimal placement.  \nRandom placement using a uniform distribution over specific coordinates is likely to be inefficient, requiring a large number of templates to cover a given fraction of the space, as first, the ideal density of templates indicated by the metric may vary significantly over the space, and second, due to random fluctuations many templates may be much closer than required for coverage. Stochastic methods address both issues by sequentially proposing new random templates, where a template is added to the bank only if its highest match  \nover the current template set is below some threshold. The match may either be evaluated via an approximate metric (e.g. [71,413]) or by directly computing the maximized overlap with existing templates. For both methods the computational cost of building stochastic banks can become large: there exist several strategies to limit this cost while maintaining efficiency of coverage, e.g. [178], including combinations of geometric and stochastic placement [611, 218,610]. High-frequency effects arising from tidal deformation of NS components may invalidate the geometric placement framework, as a quadratic approximation of the overlap is insufficient, however stochastic methods are still applicable [371].  \nFor target signals considered as the dominant multipole emission from quasi-circular BBHs with non-precessing spins, which may be written as a single sinusoidal term, the template bank ranges over two masses and two aligned spin components, though in practice only the combination χ eff has a large enough effect on the signal to affect placement [149,71,571,220]. This parameter space is essentially completely covered by banks in use for LVK searches, up to limits imposed by the waveform models. The case of searches for low-mass eccentric binaries may also be treated via this standard route, considering the effect of eccentricity as a correction to the dominant mode PN evolution [532,238], though the larger parameter space leads to severe computational limitations.  \nHierarchical search and alternative templated methods The above discussion assumes the use of a single, fixed bank for a given data set; however, alternative schemes may yield computational savings. Hierarchical search first applies a 'coarse' bank with lower minimal match, then selectively filters an additional higher-density bank around times and parameter points with high SNR, e.g. [322,649,237]. Fixed banks may also be dispensed with altogether, in favour of time-dependent stochastic selection of template parameters, given a suitable optimization method [365]; thus, particle swarm optimization (PSO) promises to reduce computational cost in cases where a very large fixed bank is required [715,653], as for instance eccentric inspirals [546].  \nEffectualness and bank robustness We do not expect any real CBC signal to be exactly represented by the template waveforms - real signals will be 'outside' the submanifold in signal space over which a bank is placed, even if numerical relativity (NR) waveforms are used directly as templates [440]. The bank's usefulness is measured by its effectualness or fitting factor [93] for a given signal h test ( θ ), defined as max p M ( h test ( θ ) , h ( θ ' p )), where the phase and coalescence time of the template h ( θ ' p ) are maximized over when calculating M ; for multimodal signals, this is not equivalent to maximizing over the signal phase. In practice, various strategies exist to evaluate the robustness of banks: using h test from different waveform models or approximants, or from NR; taking the test signal's parameters θ to lie outside the region covered by the bank; and including the effects of non-dominant multipoles, spin-induced orbital preces-  \nrbital eccentricity (beyond the leading PN effect). We discuss methods for improving coverage of such general signals below in Sec. 4.5.", '4.3.1 Optimizing searches for a target signal population': 'We now consider implementation of the theoretically optimal search method for a signal of unknown parameters, i.e. the marginalized likelihood ratio of Eq. (108). If sufficient computational resources are available, the likelihood integral may in principle be evaluated by directly sampling the entire search parameter space, using methods adapted to source parameter estimation in place of template filtering, Sec. 5.4; see also [647]. In practice, the integral over parameter space can be approximated by a sum over templates (marginalizing over phase and coalescence time for each template) [232]. In such a sum, the density or distribution of signals over the search parameter space p ( θ ) is replaced by a signal probability per template (at given SNR); this probability will then scale inversely with the density of templates.  \nThe expected behaviour of the likelihood in the neighbourhood of a local maximum is given by the Fisher metric, see Sec. 3.3, thus the marginalized likelihood over a region around the template with maximum SNR may also be approximated analytically. The resulting ranking statistic contains a term for the ratio of the astrophysical signal distribution to the template bank density induced by the Fisher metric [232] (assuming a near constant correlation/match between neighbouring templates). The bank or metric density may be thought of as the density of independent noise events over parameter space, yielding an interesting connection between parameter measurability and detection: the more precisely the parameters of a signal can be measured at a given SNR, the higher is the rate of noise events in a matched filter search. Template or noise density is very highly concentrated in the low component mass regions of CBC parameter space, in contrast to signal density which is concentrated mainly over component masses 10-50 M ⊙ [52]. For a signal density that varies rapidly over the search space, it may also be important to account for the effect of noise, and/or of biases in the search templates relative to expected signal waveforms, on the position of likelihood maxima [309].  \nWhile the detection rate is maximized by a ranking where the signal model corresponds to the best current estimate of the astrophysical merger distribution [439], other figures of merit may be optimized by suitable model choices. Specific subpopulations of astrophysical interest may be targeted via a strongly peaked signal density [473,454]; conversely, search sensitivity may be maintained over a broad parameter space, accounting for template density effects, via a simple monotonic model, for instance a merger rate uniform over log m 1 , log m 2 [309,44,558].', '4.4 Computational optimization and low latency search': "The evaluation of matched filter time series via DFT [82] is realized by instruction sets that optimize throughput per CPU or GPU core when processing large batches of data. However, DFT is not immediately adapted to all search applications: as one requires a segment's worth of samples to be convolved with the template, waiting for a complete input data segment may introduce tens or hundreds of seconds additional latency. In practice, DFT optimization allows for the use of redundant, overlapping data segments to reduce latency to ≤ O (10 s) [529,219].  \nSignificant computational savings may also be realized by exploiting the time-dependence of the inspiral-merger signal: the cost of filtering scales with the sample rate, but the highest rate is only required for the last few template cycles (considering only the dominant sinusoidal mode), the previous cycles being well represented at a lower rate. Thus, separate filter banks operating with different bandwidths may be combined to reconstruct the SNR of the full template at much lower total cost [158,64,100].  \nAn alternative strategy for streaming low latency search is the time domain filter [176,468,389] which introduces negligible latency, at the cost of significantly higher basic computational cost. (Time-domain whitenening or overwhitening filters still require a nontrivial data buffer, unless carefully designed [687].) In practice, the number of template filters must be minimized. Given a bank with high minimal match, the outputs of neighbouring templates are highly correlated: one template may be written as a sum of neighbouring templates plus a small (few %) additional term. The number of matched filtering operations can be reduced by decomposition of a bank into a smaller number of (near) orthogonal filters having a negligible match with each other, for instance singular value decomposition [173]. However, reconstructing the SNR time series of the original bank from a smaller basis requires a large number of matrix multiplications: careful tuning is required to realize a computationally workable search [498,199].  \nIf our figure of merit for the detection algorithm gives enough weight to how early a signal may be identified, as against sensitivity (probability of detection) at fixed F ( ϱ ∗ ), then a low bandwidth filter which can produce an output before the coalescence time , even at the cost of underestimating the likelihood ratio, may form part of an optimized strategy [614,533,434]. While such 'early warning' filters also imply reduced parameter precision, even a rough indication of sky position and distance may be valuable for guiding initial followup observations.", '4.5 Searches beyond a single GW emission mode': "The machinery of complex matched filters and matches (overlaps) maximized over phase, which underlies current CBC searches, requires templates with a single sinusoidal component whose phase and amplitude are functions of  \neither time or frequency. This 'dominant mode' assumption is a good approximation over a significant range of parameter space, as the effects of orbital precession or non-dominant emission multipoles on signals seen at the detectors are often small. Furthermore, the actual detected population to date lies predominantly in this range, with mostly near-equal binary masses and small in-plane spins. 19 Nevertheless, the single-component banks have low effectualness for signals whose (optimal) SNR receives large contributions from non-dominant modes: specifically, unequal-mass ( q ≲ 1 / 4) binaries, especially when viewed edge-on [179,162]; signals with strong amplitude modulation during the inspiral phase due to orbital precession, which likewise implies total angular momentum nearly perpendicular to the line of sight; signals for binaries undergoing transitional precession; and very high mass binaries, for which dominant mode emission is at frequencies below the detector sensitive band. Non-negligible eccentricity within the detectors' sensitive frequency range will also decrease effectualness and sensitivity, given that the template bank assumes quasi-circular orbits [150,244,324].  \nLow effectualness does not imply such signals cannot be detected, but significant losses in sensitivity are expected. Methods to suppress instrumental artefacts in non-ideal data by penalizing signals that deviate from singlecomponent templates, see Sec. 4.6.2, will in general further reduce sensitivity to such signals. As their sources, though rare, are of high interest for both astrophysics and fundamental gravitational physics, there is a strong motivation to improve detection prospects.  \nMethods to do so fall broadly into two categories. One is to supplant the matched filter by a 'weakly modelled' algorithm that places only generic constraints on transient signal morphology - for instance on GW signal duration, frequency range and evolution, and polarization - rather than matching to exact templates. The details of weakly modelled methods are beyond the scope of this review: see for instance [429,20]. Such methods are expected to approach the detection efficiency of an optimal matched filter for high-SNR signals, regardless of their match to any given template model; their sensitivity to quieter signals will depend on implementation and on the parameter range targeted.  \nThe second basic approach to detection of multi-component signals is to extend the standard matched filtering machinery in order to account for the higher signal complexity, by either maximizing or marginalizing over the extrinsic parameters which control the relative amplitudes and phases of the component modes. These parameters comprise the sky location of the source and its spatial orientation (i.e. the inclination, orbital phase and polarization angle); Ref. [236] gives a general discussion of such marginalization.  \nPrecessing binaries For the inspiral of single-spin precessing binaries, Ref. [547] gives a semi-analytic maximization over these angular parameters, considering the l = 2 GW emission formulated as a sum of 5 'harmonic' components.  \nA search for precessing binaries along these lines requires a template bank covering spin magnitude and orientation parameters, which so far has only been achieved by imposing strong restrictions on other source parameters [396] in order to reduce the dimensionality of the problem: even then, the size of the bank is significantly larger than the non-precessing case. To circumvent the complexity of signals from generic spinning binaries, approximate search strategies have considered a simpler space of templates that also contains non-physical signals [154], or have maximized over extrinsic parameters without applying all physical constraints [547,153]. However, the resulting gain in SNR for precessing signals is largely offset by a higher noise level relative to a standard single-component search [693]. Restricting to the physical signal space, [372] demonstrates a method for maximizing likelihood over extrinsic parameters, with the exception of inclination, which significantly affects how precession appears in the detected signal. The resulting search is in principle more sensitive for some precessing systems, particularly unequal-mass BBH and NSBH, despite the higher noise level; however a prohibitively large number of templates is still required.  \nMore recently, the multi-component model of [547] was reformulated as a power law in tan( β/ 2), where β is the precession cone opening angle (Eq. (52)), equivalent to θ JL in the notation of Sec. 2.3.1. Over most of parameter space where tan( β/ 2) ≪ 1, the signal is well approximated by a subset of harmonics, often as few as two [276]. Using this hierarchy, [492] constructed a bank with effectualness over 0 . 9 for a full range of NSBH spins, without excessive increases in computational cost and noise background, to yield substantially increased sensitivity to highly precessing signals.  \nNon-dominant multipoles A multi-component search involving multipoles of different l (emission 'modes') is conceptually similar to the precessing inspiral l = 2 case, but the component GW frequencies are widely separated and cannot be approximated as a single sinusoid with 'sidebands'. The additional complexity of multi-mode templates is expected to increase the noise level in searches [179], requiring targeted strategies to achieve improved sensitivity. In [369] an approach similar to [372], but without analytic maximization over the source orbital rotation angle (corresponding to the coalescence phase) was given, treating both this angle and the inclination as parameters to be covered by a bank of multi-mode templates, in addition to binary masses and spins. The two additional angular parameters control the relative amplitudes and phases of different multipoles. Despite the larger bank, this strategy is more sensitive than the standard dominant mode search for binaries with significantly unequal masses and/or high (redshifted) total mass, and with nearly edge-on inclination [185]. Recently, a general numerical method of likelihood marginalization over extrinsic parameters [606] was used to effectively reduce the multi-mode search to a combination of separate matched filters for each mode, potentially yielding significant efficiency gains [709]. For searches using 'higher mode' templates, as for standard single-mode templates, in addition to false alarms caused by Gaussian noise, sensitivity is also impacted by tran-  \nsient noise artefacts [185,708]; the next section presents the general problem of non-ideal detector data, before discussing relevant strategies and methods to maintain search sensitivity.", '4.6.1 Noise spectrum drift and calibration': 'The theoretically ideal matched filter assumes stationary noise with a known PSD, but the mere fact that real GW detectors do not observe for indefinitely long times, and are subject to variations in sensitivity, breaks this assumption [38]. Even within segment lengths analyzed in practice, i.e. 10 2 -10 3 s, PSDs can disagree significantly between shorter stretches of data, typically by a few percent, e.g. [702]. Such drift is not necessarily surprising, as GW interferometers have myriad moving parts, subject to complex and sometimes nonlinear passive isolation and active control systems [21,151] to suppress lowfrequency ground motion; alignment degrees of freedom affecting the strain noise spectrum may then vary on few-second timescales. Mild short-term PSD variations may be accounted for in detection algorithms by suitable moving averages or ad-hoc correction factors [498,735,517].  \nSearch sensitivity may also be affected by inaccuracies in detector calibration: specifically, large calibration errors may increase the mismatch of templates with GW signals in the calibrated strain. A constant amplitude or phase calibration error has no effect on search sensitivity in a single detector, though if errors are different between detectors a multi-detector search may be affected, see Sec. 4.7.3. The effect of frequency-dependent errors on singledetector sensitivity scales quadratically with the size of the miscalibration considered as a fractional amplitude error, or phase error in radians [79,159] - implying a negligible loss given the scale of current errors [26,657,58,59].', '4.6.2 Transient excess noise: glitch suppression': "While the great majority of existing data can be well described as quasistationary or locally stationary Gaussian noise [38], it has been evident since the first LIGO observing run [12] that multiple, short ( ≲ 1 s) stretches of data exist that are highly inconsistent with any Gaussian noise model. 20 Significant excess power is observed for short times over some (narrow or wide) frequency range: such noise transients are termed glitches , e.g. [16]. These times are in general visible as large deviations in the whitened strain time series. Without a method for subtracting them from the data, glitches invalidate the assumptions of the optimal matched filter and can generate arbitrarily high spurious matched filter SNR and Λ SN values.  \nImplementing an optimal search in the presence of glitches would require an adequate model of their morphology and amplitude and time distributions. For some types of glitch, these parameters may in part be predicted or inferred from detector auxiliary data or environmental monitor data monitoring the physical effects, internal or external to the instrument, that cause excess measured strain power [228,61]. However, many glitch types remain partly or completely undiagnosed, e.g. 'blips' [161]. Hence, glitch suppression strategies have so far been based on more heuristic or empirical methods. These essentially assume that glitch morphology is significantly different from CBC signal morphology, and thus aim to find discriminator statistics which optimally separate the two populations of transient signals. In principle, glitches in a given detector might replicate the form of CBC signals, though this is highly implausible, except for very high-mass systems with only a few signal cycles at detectable frequencies. In this 'worst case', no suppression is possible except within a multi-detector analysis, where a single-detector glitch will be inconsistent with a signal present in the network, see Sec. 4.7. Discriminators based on multi-detector signal properties have also been proposed using methods from parameter estimation [701,403].  \nLoud glitch removal: gating and inpainting Very loud glitches that are not suppressed or removed may cause severe artefacts in any frequency domain (i.e. Fourier-transform based) analysis, and will also mask real signals if the glitch has even a small overlap with the relevant CBC template. Very high amplitude outliers in whitened strain (10 2 -10 3 standard deviations) are sufficiently unlikely to be caused by GW signals that the affected data can safely be ignored. Gating achieves this by zeroing (windowing) out short data segments ( ≲ 1 s) around these outliers, at the possible cost of introducing edge artifacts [691,498,613]. Searches using gating successfully detected and localized the GW170817 signal, a BNS where one detector's data contained a loud glitch ∼ 1 s before merger [29], see Fig. 9. Gating is restricted to short segments, as zeroing significant fractions of the data would introduce significant biases in PSD estimation which may invalidate the matched filter output (on top of reducing expected signal SNR).  \nA theoretically better justified but computationally more expensive technique is inpainting : rather than zeroing a short segment, it is replaced by a time sequence determined by the condition that the resulting likelihood or matched filter is completely insensitive to template values inside the 'hole' [735]; i.e. the inpainted data contributes identically zero to the likelihood. While inpainting removes the issue of gating edge artefacts, it still introduces some SNR bias which requires correction. In practice, the matched filter is not valid or useful if a significant fraction of the template duration is affected by gating or inpainting, thus short duration templates are typically vetoed in times surrounding gates/'holes' [735,186]. While these techniques prevent very loud glitches from raising the overall search noise distribution, they cannot avoid a loss of sensitivity around the glitch time, as they necessarily remove a portion of any GW signal.  \nSignal consistency: chi-squared and friends Glitches that are not a priori too loud to be confused with CBC signals can still produce significant ( ρ ≳ 10) matched filter SNRs. If the morphology of a glitch differs significantly from the template that best matches it, the data will remain inconsistent with Gaussian noise even after subtracting the corresponding signal: i.e. there is excess power orthogonal to the template. In addition, properties of the matched filter apart from the SNR itself are unlikely to be typical of an astrophysical signal. These considerations may be exploited to exclude or suppress triggers caused by glitches.  \nA classic example is the time-frequency chi-squared test [80]: the template is divided into disjoint frequency bins and the distribution of SNR over the bins (at the supposed merger time of the trigger) is employed to calculate a measure of fit with the expected signal behaviour. Typically, while the matched filter  \nFig. 9: Mitigation of the glitch in LIGO-Livingston data during the GW170817 signal. Top left: A time-frequency representation of the raw LIGO-Livingston data; the time-frequency track of GW170817 is clearly visible despite the presence of the glitch. Bottom left: Band-passed LLO strain data (orange curve), showing the glitch in the time domain. To mitigate the glitch in rapid reanalysis to produce a 3-detector sky map, the raw detector data were multiplied by an inverse Tukey window (gray curve, right axis) that zeroed out ('gated') the data around the glitch. The blue curve shows a wavelet-based reconstruction of the glitch [206] which was subtracted from the data for subsequent analysis of the source properties. Figure reproduced from [29]. Right: SNR time series from the PyCBC search, without gating (top) and with gating applied to LLO data (bottom).  \n<!-- image -->  \nintegrand of a true signal is supported over the entire template bandwidth, that of glitches is concentrated in a narrow frequency band; this chi-squared test is thus typically most effective for long duration or high bandwidth templates. Comparable signal consistency tests use the expected time dependence of SNR based on the filter template autocorrelation [374,498,363], or the distribution of trigger SNRs over different templates [374], the so-called 'bank chi-squared' test [374,688,363].  \nThe above tests consider only the glitch power that matches (is contained within) the search templates. The ability to discriminate glitches can be improved by also measuring power orthogonal to the templates, see [235] for a general discussion. For instance, the 'sine-gaussian' veto of [528] deploys additional filters at frequencies higher than the end (ringdown) frequency of the binary template, in order to exclude or suppress very short duration transients with excess power over a wide frequency range [161]. If some assumption is made on the morphology of glitches, or if their morphology can be inferred or 'learned' from bulk data, e.g. [651,342], then near-optimal strategies may be designed to distinguish them from binary signals, e.g. [411,495]: the idea is to calculate or approximate the likelihood ratio between CBC signal and glitch hypotheses. The feasibility of such methods depends on the number of parameters or principal components required to model glitches accurately.  \nCare must be taken when applying signal-based vetoes and discriminators, as true CBC signals do not necessarily match the search templates closely. For current searches that only model the dominant emission mode from nonprecessing binaries, GW power orthogonal to the best-matched template may arise from precession and/or 'higher modes', posing a risk of false positives in identifying glitches (see e.g. [528]). Any reduction in sensitivity to signals with such physical effects can be checked via analysis of simulations (as in [178]) and may be mitigated by the design of signal-based tests.  \nExtended and/or broad-band excess noise Some types of non-ideal detector behaviour violate the assumption that departures from Gaussian-distributed noise take the form of isolated, short glitches. Longer periods of excess noise, for instance 'clusters' of glitches correlated with a temporary instrumental problem, may be diagnosed and suppressed by monitoring the time variation of an averaged search trigger rate [100]. More serious noise contamination over extended periods can arise from light scattering off moving optical components into the main beam, lasting several seconds to tens of seconds over specific frequency bands [55,650,714], see Fig. 10. The matched filters for templates intersecting the affected time-frequency region may be invalidated: short of ignoring such data altogether and effectively zeroing search sensitivity, there is at present no clear strategy for tackling these artefacts. They are an increasing concern as detector low-frequency sensitivity improves, and hence the duration of signals in the sensitive frequency band increases: the longer the signal duration, the higher the probability it will intersect an excess noise region. Gating/inpainting may still be applied over small fractions of the template duration, but in general more complex and potentially costly methods are re-  \nFig. 10: Spectrogram of the Virgo interferometer output signal during intense microseism activity (logarithmic color scale), showing extended broad-band artefacts due to light scattering. Overlaid are curves predicting the frequency of excess power from the measured velocity of the source of unwanted beam reflection, an optical bench. Figure reproduced from [55].  \n<!-- image -->  \nquired, for instance explicitly modelling and subtracting the excess noise time series [227,690,685], currently not adapted for automatic use in searches.", '4.7 Multi-detector searches: coherent, coincident & externally triggered': 'The discussion has, so far, been restricted to data from a single detector, but we now consider strategies for searching global network data. While noise time series in different observatories can be considered as statistically independent (with minor possible exceptions [679]), binary signals will appear consistently in each detector with predictable variations in amplitude, time of arrival and other relevant parameters over the network. Network search methods account for these facts, while respecting constraints from computational cost.', '4.7.1 Co-located detectors and composite data streams': "The most trivial network consists of detectors with the same location and orientation: in the past, this described the two LIGO Hanford interferometers [449,313] and it may be relevant for a future Einstein Telescope configuration[316,572]. 21 The joint likelihood is then a product over detectors, taking the same signal in each. For two co-located detectors with independent noise  \nprocesses we have a log likelihood ratio  \nlog Λ SN = ⟨ d 1 | h ⟩ 1 + ⟨ d 2 | h ⟩ 2 -1 2 ( ⟨ h | h ⟩ 1 + ⟨ h | h ⟩ 2 ) , (112)  \nwhere ⟨ a | b ⟩ A denotes an inner product using the PSD of detector A . The first two terms may be rewritten as ⟨ d co | h ⟩ co via a composite or coherent data stream ˜ d co = S co ( f )( ˜ d 1 S -1 n, 1 ( f ) + ˜ d 2 S -1 n, 2 ( f )), where S co ( f ) is an initially arbitrary function defining the inner product ⟨|⟩ co . Requiring that S co ( f ) in fact be the PSD of the data stream ˜ d co , we obtain S co ( f ) -1 = ∑ A S n,A ( f ) -1 , and the terms in brackets may be rewritten as -(1 / 2) ⟨ h | h ⟩ co . Thus, co-located and -aligned detectors behave as a single composite detector with a lower noise floor than any individual one. This remains the case if there are significant correlations between the detectors' noise streams, although the resulting expressions are more complicated [449]. In addition to the composite detector, a 'null' stream may be derived for which the GW signal vanishes: while under ideal conditions the null stream is not useful for detection, it may be used to diagnose non-ideal behaviour such as glitches and/or calibration systematic error [72,637,350].", '4.7.2 General networks': "For non-co-aligned detectors, we consider two a priori independent signals h + , × ( t ) with different amplitudes and phases. The signals seen at different detectors are then controlled by the extrinsic parameters of source direction, inclination, and polarization angle, which for the single-detector dominant mode case discussed earlier collapse into a single amplitude factor. 22 In general, an optimal search is obtained by marginalizing the joint multi-detector likelihood over these extrinsic parameters given their known priors, as well as over the intrinsic mass and spin parameters. For short transient signals the dependences on extrinsic ('sky') and intrinsic parameters can be approximately separated [419] such that component masses and (orbit-aligned) spins may be covered by a fixed template bank, as in the single-detector case. The intrinsic space metric is determined by a composite PSD which, as in the co-located case, is the harmonic sum of individual detector PSDs [419].  \nCurrently implemented approaches treat the extrinsic parameter dependence approximately via maximization. For a binary of given masses and spins, the two signals in the radiation frame can be written as, e.g. [374],  \nh + , × = A 1 , 2 h 0 ( t ) + A 3 , 4 h π/ 2 ( t ) , (113)  \nwhere h 0 ,π/ 2 ( t ) are the two waveform phases for a source at a fixed fiducial distance such that ⟨ h 0 | h 0 ⟩ A = ⟨ h π/ 2 | h π/ 2 ⟩ A ≡ σ 2 A , and the A µ ( µ = 1 , . . . , 4)  \nare amplitudes dependent on the source's distance and orientation relative to the given frame. 23 The signal seen by detector A is decomposed as  \nh A ( t ) = ∑ 1 , 2 A 1 , 2 F A + , × h 0 ( t A ) + ∑ 3 , 4 A 3 , 4 F A + , × h π/ 2 ( t A ) ≡ 4 ∑ µ =1 A µ h A µ ( t A ) , (114)  \ndefining the four components h A µ ( t A ), where t A is the signal arrival time at the detector. The multi-detector likelihood ratio generalises from Eq. (112) as  \nlog Λ SN = ∑ A ( ⟨ s A | h A ⟩ A -1 2 ⟨ h A | h A ⟩ A ) ≡ ∑ A ( A µ ⟨ s A | h A µ ⟩ A -1 2 A µ M µν A ν ) , (115)  \nwhere M µν = ∑ A ⟨ h A µ | h A ν ⟩ A [374]. Since ⟨ h 0 | h π/ 2 ⟩ ≃ 0, we take the matrix M µν to be block-diagonal. Maximizing Λ SN freely over A µ we find a coherent SNR  \nρ 2 coh ≡ 2 log max A Λ SN = ∑ A ⟨ s A | h A µ ⟩ A ( M µν ) -1 ∑ B ⟨ s B | h B ν ⟩ B . (116)  \nThis coherent SNR 2 , comparable to the F -statistic for continuous wave detection [406] is chi-squared distributed with 4 degrees of freedom [215]. Further redefining the radiation frame via a rotation, 24 the cross-terms between + and × elements in M µν can be eliminated, rendering the matrix diagonal. The resulting 'dominant polarization' SNR 2 , e.g. Eq. (2.34) of [374], is a sum of independent power terms over the two waveform phases and two polarizations.  \nFor a template of given intrinsic parameters, the signal arrival times in each detector remain to be searched over. As these times are determined by the direction to the source, i.e. sky position, this aspect can be handled by setting up a discrete grid or bank of sky points with spacing determined by a condition on minimal match, based on expected loss of SNR from slightly inaccurate relative arrival times [419,217,470]; this sky mismatch contributes in addition to intrinsic parameter mismatch. The computational cost of constructing the coherent SNR from individual matched filters scales directly with the number of sky points, which grows rapidly for larger networks of widely separated detectors (see Chapter 4 of [217]), as expected given the localization accuracy of such networks [274,716]. It is then the dominant cost except for networks with relatively poor localization, e.g. the initial LIGO configuration with two co-located Hanford interferometers H1 and H2 [470], or the LHO-LLO-Virgo network before reaching Advanced design sensitivity.  \nCoherent search computational cost is thus strongly dependent on the maximal mismatch (i.e. expected loss in SNR) used to construct both the intrinsic  \nand sky banks. At realistic fixed computational cost, one is forced to mismatches higher than the typically considered ∼ 3%, implying a nontrivial loss of sensitivity [217]. (This limitation only becomes stronger considering additional searches of time-shifted data for estimating the noise background, see Sec. 4.8.1.) Therefore for covering a complete range of sky directions, rather than coherent search, coincident search has been preferred, where triggers are first obtained as peaks in SNR for each separate detector's data, then combined over detectors in a subsequent stage, see Sec. 4.7.3.  \nExternally triggered searches If, however, the signal arrival time and source direction are more or less exactly known, such as for sources detected via electromagnetic (EM) transient emission, a coherent search is viable at realistic computational cost with negligible loss of sensitivity [374,721]. Such externally triggered searches have been employed for short GRB counterparts [8,10,28, 51], and have also been considered for fast radio bursts (FRBs) [50]. As these searches cover small volumes of parameter space, particularly coalescence time, significantly lower SNRs are required to obtain a high evidence in favour of H S than in 'all-sky' searches, where all source parameters are unknown. Beyond optimizing sensitivity in the ideal Gaussian noise case, the coherent search offers specialized methods for suppressing glitches, for example the presence of excess power or SNR in a null stream; these tests become more powerful if the sky direction, and thus the detector antenna factors, are known. It is believed that GRB are emitted within a narrow cone around the binary orbital axis (so-called 'beaming'), thus the range of orientations to be maximized over is significantly reduced, essentially assuming circular GW polarization, which further increases sensitivity relative to the all-sky case [721].", '4.7.3 Coincident search and multi-detector consistency': "Published searches of LIGO-Virgo data using CBC templates have employed the coincident method, which conducts a single-detector search for each detector and then combines candidate signals over the detector network, e.g. [104]. While coincident search is in general less sensitive than coherent, it is computationally feasible and simpler to implement. The product of the single-detector stage is a set of triggers above a low SNR threshold, typically 4-5; for large template banks and long stretches of data, typically only a subset are stored. For instance, 'clustering' may be applied to discard triggers with significantly lower SNR than others in nearby templates within a small time window.  \nThen, candidate events for the observing network are derived from combinations of single-detector triggers which are sufficiently likely to be generated by the same GW signal [597]. (Depending on detector sensitivities and pattern functions, a signal may also give rise to only one single-detector trigger [167, 613,531].) Typically, 'coincidence' between triggers in different detectors is determined in two stages: initially, triggers are enforced to be generated in the same template and to have consistent merger times, allowing for GW propagation time between observatories plus a small buffer. For multi-detector candi-  \ndates surviving these cuts, the likelihood of the remaining trigger parameters - SNRs, relative merger times and relative GW phases - is estimated based on the expected signal distribution [175,530,364]. As these trigger properties are determined by the putative source's extrinsic parameters, this step is analogous to maximization over source direction and orientation in the coherent search. However, as the coincident search only considers maxima of singledetector SNR above threshold, it will not in general be able to reproduce the maximum signal likelihood of the coherent search. Instead, the coincident search estimates the likelihood of the set of single-detector triggers .  \nThe single-detector SNR threshold, arising from computational limitations, is the main limitation on the sensitivity of coincident search, as compared to an ideal (and computationally infeasible) coherent search. For a 2-detector network the two approaches are expected to be nearly equivalent, but the loss relative to optimal sensitivity increases rapidly with the number of observatories of similar sensitivity in the network from 3 upwards (see e.g. [217]). For networks with many detectors, a signal may have moderate or low SNR in several detectors, thus thresholding may cause an inaccurate estimate of its likelihood ratio.  \nHierarchical coherent search A more complicated scheme to obtain some benefits of coherent search with moderate computational cost involves making an initial selection of trigger times using a coincident search, then running the full coherent search only around these trigger times. With careful selection of thresholds, such a hierarchical approach may approach the coherent search sensitivity for large networks at manageable cost [138,217] although the need for accurate background estimation, to be discussed in the next section, places computational limitations. The complexity of coherent hierarchical search, as well as the status of the Advanced detector network to date, have restricted its implementation in practice, though see [199] for an exception with large scale GPU use to cover a 3-detector sky grid, and [538] for a recent application of hierarchical coherent search to the LHO-LLO network.", '4.8 Statistical methods for significance and search sensitivity': 'Having searched a stretch of data, the remaining task is to interpret the output: have signals actually been detected, and which possible signals could have been detected? Given the unpredictable real detector noise and unknown signal distribution, regardless which function of trigger parameters is chosen as detection statistic, the statistic value cannot be directly interpreted in terms of odds, i.e. probability of signal vs. noise origin; instead the relation must be estimated empirically. This section will describe statistical methods for interpreting search results, both concerning individual candidate detection and the overall rate of detections, and ultimately the rate of mergers for an astrophysical population of sources. Inference on population properties beyond the rate requires more sophisticated inference, which is treated in Sec. 6.', '4.8.1 Detection significance in non-ideal data': 'The initial objective is to measure the (frequentist) significance of a candidate event, in terms of the probability that noise would generate a comparable event. The false alarm probability of an event with ranking statistic ϱ ∗ , in a search of data with duration T , is the probability at least one event with ϱ ≥ ϱ ∗ would occur if searching this duration of instrumental noise, i.e. in the absence of GW signals. Considering noise events as an inhomogeneous Poisson process with differential rate µ N ( ϱ ), we have  \np ( ≥ 1 noise events with ϱ ≥ ϱ ∗ | T ) = 1 -p (0 noise events with ϱ ≥ ϱ ∗ | T )  \n= 1 -exp ( -T ∫ ∞ ϱ ∗ µ N ( ϱ ) d ϱ ) . (117)  \nThis may be written in terms of false alarm rate (FAR) at a threshold ϱ ∗ , i.e. ∫ ∞ ϱ ∗ µ N ( ϱ ) d ϱ .  \nConsidering a duration ∼ T of noise data, the FAR may only be directly measured with precision ∼ 1 /T . However, the noise rate density µ N ( ϱ ) is estimated in practice via bootstrap resampling of the actual data (although this data may contain signals [174,177]): in order to reach very small false alarm probabilities, resampling schemes must generate much larger durations of notional noise data [175,691]. The most conceptually straightforward method is to apply a large range of relative time-shifts to data or triggers from different detectors, obtaining samples comparable with the actual search but without coincident signals [713,104]; the time-shifts must thus be larger than signal arrival time differences over the network. Additional steps may be taken to remove potential signals for a mean-unbiased estimate of µ N ( ϱ ), e.g. [688, 412]; however, to ensure the correct rate of false positive outcomes, i.e. candidates with a very small estimated false alarm probability in the absence of signals, the time-shifted analysis must use the same data selection as the actual search [177]. For events observed only in a single detector, the precision of background estimates is severely limited unless extrapolation beyond the limit ∼ 1 /T is considered [167,531,226].  \nTypically, estimates of µ N ( ϱ ) and candidate FAR are produced by resampling hours up to a few weeks of data; on such timescales, detector behaviour (e.g. average rate of glitches) is assumed stable. FAR may be interpreted by comparison with an expected rate of astrophysical signals, as discussed further in Sec. 4.8.2, or by finding the p -value, i.e. the false alarm probability over a total duration of data T , via p = 1 -exp( T · FAR) [691,18].', '4.8.2 Search sensitivity, merger rate and probability of astrophysical origin': "To connect the results of a search with astrophysical processes, its response to GW signals must be estimated: this is achieved by analyzing large sets of simulated signals ('injections'), added to the data searched, or at least data with the same noise properties. The search output for any individual injection  \nshould be the same as for an equivalent astrophysical signal, up to possible waveform systematic errors. Injected signal parameters are drawn randomly from a population distribution which may be intended as astrophysically realistic, or which may be reweighted via importance sampling [681] to represent other distributions of interest.  \nFor relatively low-mass systems in the Initial detector network, the potential sources within a detectable range could not be approximated as homogeneous, thus search results were interpreted using an estimate of the distribution of blue light luminosity over the local Universe, an observable which is assumed to trace compact binary formation [275]. For more recent observations, sources are assumed uniformly distributed in (comoving) volume and time, with random orientations [9]: thus, search sensitivity is described by the fraction of signals ε det up to a maximum distance d max that are detected with inverse false alarm rate (IFAR) 25 or ranking statistic above a threshold value. If a total time T is searched, the search sensitive volume-time is given by [24, 22]  \nV T = ∫ d max 0 T 1 + z ( d L ) ϵ det ( d L ) dV c dd L dd L ≡ V max ε det , (118)  \nwhere ϵ det ( d L ) is the average probability of detection at a given distance, V c ( d L ) is the comoving volume within a sphere out to distance d L , and the factor 1 / (1 + z ) accounts for time dilation; the total comoving volume-time V max is defined by evaluating the integral with ϵ det set to 1. If a total of N inj injections are performed with the desired distribution and N det are recovered above threshold, we estimate a detection fraction ε det ≃ N det /N inj . The sensitivity V T may be evaluated for an arbitrary distribution of masses and spins: in particular, for a delta function in (source frame) masses and spins, V T is a function of these intrinsic parameters, proportional to the detection probability for a merger with randomly drawn extrinsic parameters within d max .  \nFor a constant (in comoving units) astrophysical merger rate R , the expected number of detections is then R · V T . The result of the search is then considered as a counting experiment: how many signals above threshold were actually seen? The likelihood of this outcome given a value of R allows us to infer statistical rate limits or estimates. For this procedure to be consistent, the detection threshold should be set such that, with high probability, any event above it is a true signal; in practice, the threshold choice is somewhat arbitrary, and may bias rate estimates [125]. In addition, the value of V is strongly dependent on the assumed binary mass and spin distribution, causing, in general, large unknown systematic errors in astrophysical rate estimates.  \nWhile this dependence on the target distribution will be addressed by astrophysical population inference in Sec. 6.2, the number of detections may be inferred more precisely, and avoiding arbitrary bias, by including potential noise events in the inference [286] (cf. [497]). We consider both the rate density of noise events µ N ( ϱ ) ≡ r N b ( ϱ ) and that of signals µ S ( ϱ ) ≡ r S f ( ϱ ),  \nwith probability density functions (PDFs) over detection statistic b ( ϱ ) , f ( ϱ ) respectively. We assume that these rates and PDFs are measured with negligible uncertainty, except for r S ; f ( ϱ ) may be estimated via injections. In the resulting Poisson mixture model, the likelihood that a single search trigger has statistic ϱ is p ( ϱ | r S ) ∝ r N b ( ϱ ) + r S f ( ϱ ), and the likelihood of the entire set of events obtained in the search { ϱ p } given a value of r S is [286,22]  \nP ( { ϱ p }| r S ) ∝ exp[ -T ( r N + r S )] ∏ p ( r S f ( ϱ p ) + r N b ( ϱ p )) . (119)  \nChoosing a prior over r S (e.g. of 'Poisson-Jeffreys' form p ( r S ) ∝ r -1 / 2 S ) we obtain a posterior distribution p ( r S |{ ϱ p } ). If r S were known exactly, the probability that a given event is of astrophysical rather than terrestrial (noise) origin would be p astro ( ϱ ) = r S f ( ϱ ) / ( r S f ( ϱ ) + r N b ( ϱ )); accounting for rate uncertainty, we find [24]  \np astro ( ϱ ) ≡ 1 -p terr ( ϱ ) = ∫ p ( r S |{ ϱ p } ) r S f ( ϱ ) r S f ( ϱ ) + r N b ( ϱ ) d r S . (120)  \nThe signal and noise PDFs may be extended to a multi-dimensional space of search outputs beyond the ranking statistic ϱ , notably template masses and spins; this extension allows for signal model features (for instance a specific mass distribution) which are not included in the ranking ϱ [87]. The method has also been extended to include multiple distinct signal components [418].  \nThe values of p astro obtained in catalogues of CBC events, e.g. [45], are mainly determined by the strongly falling dependence of the background distribution b ( ϱ ): significant variations appear between different search pipelines for some events with odds p astro /p terr ∼ O (1), reflecting both fluctuations in event parameters, e.g. in SNR, and differences in methods used to suppress noise events. The p astro values of past events may also be revisited when new search results enable a more precise determination of the rate and distribution of true signals.", '4.9 Machine learning applications': "Since CBC searches still face complex issues of implementation, machine learning methods have seen significant interest in recent years, ranging from random forest and similar binary classifiers to convolutional deep neural networks. Among the motivations for such methods are the unpredictable nature of glitches, motivating algorithms which can 'learn' glitch properties empirically from data rather than using hand-tuned vetoes. A related problem is finding a near-optimal detection statistic over the multi-dimensional parameter space of multi-detector events, where the 'curse of dimensionality' may impede accurate empirical estimation of the noise and signal distributions.  \nA second broad motivation for applying deep learning techniques is computational speed. Search sensitivity or latency are typically still limited by the  \ncomputational cost of matched filtering, or of reconstructing template outputs from a smaller set of filters; in principle, classifiers such as neural networks may be evaluated with very little latency, depending on the compute architecture and constraints such as memory.  \nClassical machine-learning methods, notably maximum likelihood fitting [530] and kernel density estimation [174], are already employed to characterize noise distributions in LVK searches. Multivariate classifiers such as random forest and simple neural networks have been considered since the Initial detector era [107,426,417,427], though finding limited application due to the complexity of incorporating them into existing analysis pipeline architecture.  \nMore recently, image recognition methods have been applied to time-frequency representations of short data segments, in order to identify and suppress glitches [739] or to directly classify signal vs. noise segments, bypassing matched filtering, e.g. [83]. While such 'visual' methods can easily identify contaminated data, they discard the signal phase information and so cannot approach the sensitivity of matched filtering for longer duration signals.  \nA more fundamental approach that has dominated recent research applies the convolutional neural network (CNN) directly to (whitened) strain data streams, thus preserving their information content [334]. With suitable training, a CNN may reproduce the matched filter's detection efficiency [321] and could address further search issues, for instance optimizing sensitivity for a given signal population, or in non-ideal data, e.g. [330]. Deep learning training typically optimizes sensitivity at operating FAR thresholds of 1 per ∼ few hours, leaving a challenge of performance at lower (false) event rates, which requires more complicated schemes [393,629].  \nWhen comparing CNN performance to 'classic' templated searches, the large-scale compute resources required for training, as well as hardware for evaluation (typically high-grade GPUs) should be borne in mind - see e.g. [488] for a recent practical application. There is an intriguing similarity between the compute operations in templated (time domain) low latency searches, i.e. fast convolution of filters with the data and reconstruction of physical template outputs via large-scale matrix multiplication, and the steps required to train and evaluate CNNs [729]. Application to longer duration (low mass) signals also appears to be impeded by memory constraints given the large number of waveform cycles, although see [436]. A hybrid approach using aspects of existing templated searches within a deep learning compute architecture may yield further advances.", '5.1 Introduction to Bayesian Inference': "As an observational science, GW astronomy is well suited to the Bayesian probability interpretation, whereby uncertainties resulting from noisy observations are quantified through a probability distribution. This statistical approach stands in contrast to the frequentist methods usually employed in the detection problem, where the significance of a potential signal is estimated through comparison to a large number of noise-generated events. Bayesian inference has been widely employed in astronomy and cosmology where one is often dealing with a small amount of data, or a single realisation of a phenomenon. Historically, its use for anything but simple problems was hindered by computational difficulty, but with the advent of modern computers and the development of algorithms such as Markov Chain Monte Carlo the method has seen a resurgence. We will introduce the building blocks necessary to understand Bayesian analysis of GW signals; many textbooks are available for a more in-depth understanding of relevant techniques [643,332] and their historical and theoretical underpinnings [408,407].  \nSince the binary mergers giving rise to GW signals are unique events rather than repeated trials of the same experiment, statements about their parameters have evidential probability, i.e. corresponding to degrees of belief based on the available evidence. In the usual case where the parameters are continuous (masses, angles, etc.), rather than finite, discrete probabilities, we must use a probability density function (PDF) defined on the parameter space: integrating the PDF over (part of) its domain then gives a finite probability. As a density, the PDF depends on the specific parameterisation used, thus a Jacobian must be applied to transform between different parameterisations (see Sec. 5.5).  \nIn Bayesian inference, we update our state of knowledge about a model by calculating the posterior PDF over the model parameters θ using Bayes' theorem:  \np ( θ | d, H S ) = p ( θ | H S ) p ( d | θ , H S ) p ( d | H S ) . (121)  \nThe posterior PDF is computed from the prior PDF p ( θ | H S ), the likelihood function p ( d | θ , H S ), and the normalisation constant p ( d | H S ). The prior PDF describes our prior knowledge about the parameters, before we consider the observations. For example, without fore-knowledge of the binary's location we use a distribution uniform on the celestial sphere (see Sec. 5.2). The likelihood function is the probability of observing the data d if the parameters are θ , see Eq. (81). H S denotes the 'signal hypothesis', a model of the data that defines what parameters are contained in θ , along with models of the waveform and the residual noise, and any additional information required to completely specify the likelihood function. The constant p ( d | H S ) is the 'marginal likelihood' of the data given H S , it is computed by marginalising the likelihood over all  \nparameters, i.e. integration weighted by the prior density:  \np ( d | H S ) = ∫ p ( θ | H S ) p ( d | θ , H S ) d θ ≡ Z S . (122)  \nThis term is also often called the 'Bayesian evidence' for a model, and is commonly denoted by the symbol Z in analogy to the partition function of thermodynamics. It serves as a normalisation constant for the posterior probability distribution, being independent of the parameters θ , though varying between one model and another. For a given model, the posterior distribution is often defined only up to a constant of proportionality:  \np ( θ | d, H S ) ∝ p ( θ | H S ) p ( d | θ , H S ) . (123)  \nThe marginal likelihood is also the expectation value of the likelihood of the data, taken over the prior for a given model: thus, as it quantifies the mean likelihood for a particular model, it can also be used to compare the posterior probability of two models. The 'posterior odds' O 1 , 2 between two models H 1 and H 2 is given by the ratio of their posterior probabilities  \nP ( H 1 | d ) P ( H 2 | d ) = P ( H 1 ) P ( H 2 ) p ( d | H 1 ) p ( d | H 2 ) ≡ P ( H 1 ) P ( H 2 ) B 1 , 2 , (124)  \nwhere the Bayes factor B 1 , 2 is the ratio of the marginal likelihoods for hypotheses 1 and 2. The prior odds P ( H 1 ) /P ( H 2 ) is the ratio of model probabilities prior to considering an observation, for example the two models might be 'the source is a binary black hole' versus 'the source is a binary neutron star'. Since in this example we might have different prior PDFs for the two models, based on assuming the presence or absence of tidal deformability, we see that the models do not need to have the same number of parameters; in general they do not even need to have any common parameters. Nevertheless, the Bayesian framework allows one to compare different assumptions quantitatively via the posterior odds ratio. To go even further, we can allow the prior PDF to be dependent on model 'hyper-parameters', which then may themselves be estimated in hierarchical inference, for which see Sec. 6.  \nIn practical cases, while one seeks to compute a ratio of marginal likelihoods for generic models (e.g. General Relativity versus a particular modified gravity theory), typically the actual computation requires much more precisely specified models, which may define a particular PSD, waveform model, and additional signal processing parameters as required to unambiguously evaluate the likelihood. Since the marginal likelihood is integrated over the model's priors, it also requires that we fully specify both the parameters of the model, and the form of the prior distribution over these parameters. This makes a Bayesian hypothesis test sensitive to very specific model assumptions, which is desirable when relevant information is available a-priori; such assumptions should be made explicit when reporting such a result.", '5.2 Priors': 'Bayesian inference makes explicit the need for a prior probability distribution over the model parameters, which can be chosen to incorporate any prior information or constraints. For some parameters the prior is determined unambiguously by symmetry arguments: this is the case for the direction of the source on the sky, its spatial orientation relative to Earth, and the polarization angle, assuming that no other information is available. However a model assuming an electromagnetic counterpart to a binary coalescence, such as a gamma-ray burst, may use a prior encoding its (known) sky direction rather than the default prior PDF uniform on the celestial sphere. All parameters present in the model (see Table 2) must have a prior specified before inference can be done; in practice, priors are set using a combination of such symmetry arguments, astrophysical foreknowledge or modelling, and convenience in interpreting the resulting posteriors.', '5.2.1 Prior from astrophysics': 'The prior may be chosen as a functional form derived from studies of independent observations or other physics considerations. A standard example is the a-priori spatial distribution of GW sources. Following the cosmological principle and the Copernican principle, given the very large distances of typical GW sources, a prior uniform in co-moving volume is most commonly used (Eq.(156) in Sec. 6.3). While this forces the analysis to assume a cosmology, the choice can be amended in post-processing (see Sec. 5.2.3 below). A prior uniform in co-moving volume also has the convenient advantage of being normalisable, as the integrated volume is finite.  \nIn some cases parameter estimation (PE) results are obtained for different choices of prior representing different astrophysical assumptions, for example allowing wider or smaller ranges of NS spin magnitude [36]. Another example would involve deriving a prior over the masses of BHs using previous electromagnetic or GW observations (in conjunction with other model assumptions), e.g. [36].', '5.2.2 Computational considerations': 'The computational cost of most parameter-estimation analyses is dominated by calculating the likelihood, not the prior. However, it should still be possible to evaluate the prior quickly, as some sampling strategies (in particular nested sampling based on MCMC, see Sec. 5.4.2) require many evaluations. Furthermore as noted above, the posterior and the prior being probability densities, any change of parametrisation x i → y i (using generic notation) requires computing a Jacobian,  \np ( y | H ) = p ( x ( y ) | H ) ∣ ∣ ∣ ∣ ∂x i ∂y j ∣ ∣ ∣ ∣ . (125)  \nThis computation is much more convenient for priors of closed (analytic) form.', '5.2.3 Change of priors via reweighting': 'From Eq. (121), it is clear that having calculated the posterior PDF p 1 ( θ | d, H S ) for a prior p 1 ( θ | H S ) with given data, it is possible to calculate the posterior p 2 ( θ | d, H S ) for a different prior p 2 ( θ | H S ) over the same parameter space:  \np 2 ( θ | d, H S ) = p 2 ( θ | H S ) p 1 ( θ | H S ) p 1 ( θ | d, H S ) . (126)  \nThus, provided that the weight p 2 ( θ | H S ) p 1 ( θ | H S ) -1 can be computed, it is possible to reweight the posterior to accommodate a different prior assumption (even a parametrised one - see Sec. 6 for details). To obtain an accurate result, the estimation of the original posterior p 1 ( θ | d, H S ) must have delivered enough information (usually in the form of posterior samples, see Sec.5.4, or an accurate interpolant) where the weight is large, so where p 2 ( θ | H S ) is large and p 1 ( θ | H S ) small. This is made usually more difficult as most PE schemes are designed to spend most computing resources in regions where p 1 ( θ | d, H S ) (and p 1 ( θ | H S )) is large.  \nTo enable flexible use of posterior PDF computations, it is thus usually better to avoid priors that exclude part of the parameter-space, and instead be very conservative for the computationally expensive first estimation of the posterior, relying on reweighting for subsequent analyses with different priors. This often leads in practice to adopting priors that are uniform over parameters which are of particular interest for further studies: this choice allows a wide range of reweighting options, and also makes summary statistics easier to interpret.', '5.2.4 Other considerations on priors': 'Another approach involves uninformative priors, or the so-called Jeffreys prior, which is a density proportional to the square root of the determinant of the Fisher information matrix. It is invariant under a change of parametrisation, but can require extra computation to determine its density at points of interest, since the Fisher information matrix depends on the signal and noise models for a particular observation.  \nAs mentioned above, a prior PDF represents a distribution of possible signals over parameter space, thus prior choices are necessarily connected with issues of astrophysical or cosmological population modelling. A systematic treatment of such population models will require Bayesian hierarchical inference, where the goal is to determine properties of the global population, rather than of individual sources; this will be presented in Section 6.', '5.3 Likelihood function': "In the context of Bayesian PE, the statistical model describes the entire process by which the data is created, including both signal and noise models and all relevant parameters required to predict the signal. In Sec. 3 we modeled the noise as a stationary Gaussian process, giving us a noise likelihood function. We have seen in Sec. 2 how the signal model is defined using a parameter vector θ ; together with a stationary Gaussian noise model, this yields the likelihood function in Eq. (81), defined in terms of the data d , signal model h ( θ ), and the signal hypothesis H S , which includes the statistical description of the noise via its power spectral density S N ( f ):  \np ( d | h ( θ ) , H S ) ∝ exp ( -1 2 ⟨ d -h | d -h ⟩ ) . (127)  \nThe inner product inside the exponential is in practice evaluated with a finite amount of data sampled over time T , with a sampling rate f s , yielding a discrete sum over N = T/f s points in the time domain instead of an integral. In this case the maximum frequency that can be represented in this data is the Nyquist frequency f s / 2, and since the data time series is real, we have in the frequency domain ˜ d ( -f ) = ˜ d ( f ) ∗ . The (log) likelihood for a GW signal template ˜ h is then a sum over a finite range of positive frequencies,  \np ( d | h ( θ ) , H S ) ∝ exp   -2 T N/ 2 ∑ i =0 | ˜ d i -˜ h i | 2 S N ( f i )   . (128)  \nMultiple detectors can be incorporated by simply multiplying their individual likelihood functions, assuming their noise is statistically independent,  \np ( d | h ( θ ) , H S ) ∝ ∏ A p ( d A | h A ( θ ) , H S , A ) , (129)  \nwhere the index A runs over available detectors and its use as conditioning information denotes that the corresponding noise PSD is to be used.  \nIn cases where the waveform is calculated in the time domain, care needs to be taken to transform into the frequency domain. To avoid aliasing, f s must be chosen so that the maximum frequency of the signal is no greater than the Nyquist frequency f s / 2, and a window function is applied before applying a discrete Fourier transform. The window function is used to taper the start and end of the waveform, reducing the spurious high frequency content introduced by the discontinuity between the start and end time if no window is applied. This windowing may lead to an overall loss of the signal's power: the Fourier transform of the data ˜ d , after applying a window whose Fourier transform is ˜ w , is ˜ d w = ˜ d ∗ ˜ w , and the likelihood needs to be modified to take this into account [665].  \nFor signal models generated natively in the frequency domain, particular care needs to be taken of the waveform's abrupt start and end: such unphysical  \nfeatures may match against data and lead to artificially precise and biased results [479]. In general with CBC templates including merger and ringdown, the signal model naturally tapers at its end, while the start is in general at low enough frequencies so as not to contribute to the likelihood integral.  \nThe likelihood of Eq. (128) is the most commonly used form, but relies on assumptions of a stationary Gaussian noise process with exactly known PSD and no windowing, damping or correlation effects. Various works have looked at relaxing these assumptions, e.g. [609,463,189,190,663,390,97].", '5.4 Exploring parameter space': "The posterior PDF contains all the information about the model parameters that can be extracted from the data. However, its domain is the entire prior, which for compact binaries is a high-dimensional space containing in practice 15 parameters or more for a generic quasi-circular coalescing black hole binary. While we can evaluate the density at any point in parameter space using Eq. (121), it is not practical to evaluate the posterior entirely on a grid due to the size of the parameter space. One is often concerned with the marginal distributions, given by integrating the posterior over 'nuisance' parameters, for example we may only be interested in the chirp mass M , in which case all the other parameters are nuisance parameters denoted θ ' . The marginal posterior is  \np ( M| d, H S ) = ∫ p ( M , θ ' | d, H S ) d θ ' = ∫ p ( d |M , θ ' , H S ) p ( M , θ ' | H S ) p ( d | H S ) d θ ' .  \nEvaluating this integral using standard quadrature would be very difficult in high dimensional spaces. Instead, in practice, Monte Carlo methods are often used to compute quantities of interest. These rely on a sampling of the posterior distribution at various points, chosen with a particular application in mind.  \nFinal PE results are typically provided as samples from the posterior distribution itself, i.e. a set of points in parameter space, where the density of the samples is proportional to the posterior PDF. With such a sampling, integrals weighted by the posterior probability are easily evaluated by Monte Carlo methods, with each sample given an equal weight in the integrand. For example the marginal distribution above can be approximated by a histogram, where the number of samples in each M bin would be proportional to the probability density in that bin. Using posterior samples also makes it trivial to produce credible intervals for individual parameters, defined to be an interval of one-dimensional parameter space ( θ min , θ max ) that contains a chosen fraction X of the probability, and therefore posterior samples, i.e.  \n∫ θ max θ min p ( θ | d ) dθ = X. (130)  \nThere are however many choices of θ min and θ max that satisfy this equation. A common choice of interval to quote is the minimum credible interval , which spans the region of highest probability density that contains X .  \nA number of methods have been developed to draw the posterior samples efficiently, based on a variety of statistical methods, with different degrees of specialisation. We give an overview of the main methods and their historical development below.", '5.4.1 Markov Chain Monte Carlo': "Markov-Chain Monte-Carlo (MCMC) methods are a class of algorithms that enable efficient exploration of high-dimensional probability distributions. First introduced in 1953 in the context of simulating the behavior of particles in a gas [499], they are often a building block of other sampling algorithms.  \nMCMC methods work by constructing a Markov chain whose stationary distribution is the desired target distribution: the chain is thus designed to converge to the target distribution after a sufficient number of steps. At each step of the chain, the algorithm proposes a new state, which is accepted or rejected based on a probabilistic criterion that preserves the desired stationary distribution.  \nOne of the most commonly used such methods is the Metropolis-Hastings algorithm introduced in 1953 [500] and extended in 1970 [377]. In order to sample from a target probability distribution π ( θ ), the algorithm consists of the following steps:  \n- -Start with an initial parameter vector θ 0 , often drawn from the prior.\n- -At each iteration t , propose a new parameter vector θ ' from a proposal distribution q ( θ ' | θ t -1 ), where q is a proposal distribution that depends on the current parameter value θ t -1 (also knowns as a 'jump proposal distribution'). A common choice for q is a Gaussian distribution with mean θ t -1 and some fixed covariance matrix.\n- -Calculate the acceptance ratio  \nA = π ( θ ' ) π ( θ t -1 ) q ( θ t -1 | θ ' ) q ( θ ' | θ t -1 ) , (131)  \nwhere π ( θ ) is the target distribution and q ( θ ' | θ t -1 ) is the probability of proposing θ ' given the current parameter value θ t -1 . The ratio of the proposal distributions q cancels out if q is symmetric, but if not it is required to maintain detailed balance.  \n- -Repeat steps 2-4 for a sufficient number of iterations to obtain a stationary distribution.\n- -Set θ t = { θ ' with probability min(1 , A ) θ t -1 with probability 1 -min(1 , A ) .  \nIn step 3, the acceptance ratio A determines whether the proposed parameter vector θ ' is accepted or rejected. If A > 1, then θ ' is accepted with  \nprobability 1, since the new value has a higher probability density than the current value. If A < 1, then θ ' is accepted with probability A , which is proportional to the likelihood of the proposed value relative to the current value. This ensures that the chain eventually converges to the target distribution π ( θ ).  \nPTMCMC(Parallel Tempering Markov Chain Monte Carlo) is an advanced variant of MCMC that incorporates parallel tempering to improve the exploration of complex probability distributions ([659,420,251], and see [645,700, 282] for examples of GW-specific applications). In PTMCMC, multiple chains, each associated with a specific temperature, are run simultaneously. These different temperatures act as scaling factors, controlling the distribution's 'flatness'. By having chains at higher temperatures explore the distribution more broadly and chains at lower temperatures focus on regions of higher density, PTMCMC enables a more efficient exploration of the entire parameter space. To achieve this, the PTMCMC chains periodically swap parameters between neighboring temperatures, so that information about the tails of the distribution from the higher temperature chains can percolate to the lower temperature ones. This involves a special jump proposal that exchanges states between adjacent chains, which is accepted or rejected based on the temperatures and the likelihoods of the current states in the chains. This process promotes better chain mixing and facilitates the movement of states between different regions of the distribution. Since different temperature chains can run in parallel when a swap is not taking place, this aspect of PTMCMC can be efficiently parallelised.  \nThere are several other MCMC methods that are commonly used in addition to the Metropolis-Hastings algorithm. One popular method is the Gibbs sampler ([333,331], and see [700] for examples of GW-specific applications), which is efficient when the joint distribution of the parameters can be factorized into conditional distributions. The Gibbs sampler updates one parameter at a time, by drawing from its conditional distribution given the current values of the other parameters. This can be more efficient than the MetropolisHastings algorithm when the conditional distributions are easier to sample from than the full joint distribution.  \nAnother commonly used MCMC method is the Hamiltonian Monte Carlo (HMC) algorithm ([249,359], and see [567,315] for examples of GW-specific applications), which is designed to move more efficiently through high-dimensional parameter spaces. HMC generates proposals by simulating the Hamiltonian dynamics of the system, which involves using the gradient of the log-posterior distribution to simulate the motion of a particle in a higher-dimensional space. HMC can be particularly useful for sampling from target distributions with complex geometries or correlations between parameters.  \nThere are also variations of the Metropolis-Hastings algorithm, such as the adaptive Metropolis-Hastings algorithm [358,88], which adjusts the proposal distribution during the sampling process to improve efficiency. The slice sampling algorithm [221,526] is another variation, which generates proposals by slicing the target distribution at a random point and using a bounded interval  \nto select a new value. Slice sampling can be particularly useful when the target distribution is difficult to sample from using other methods.  \nIn general, MCMC methods tend not to suffer as much from the curse of dimensionality as other sampling algorithms, but they can be computationally expensive, and require careful tuning of the proposal distribution to ensure efficient exploration of the parameter space. Problem-specific proposal distributions are almost always required, and in the context of GW astronomy depend on the detector network configuration and the signal properties.", '5.4.2 Nested Sampling': "Nested sampling is a more recently developed Bayesian inference algorithm, proposed initially in [644]. Its aim is to evaluate the marginal likelihood integral of Eq. (122) through a stochastic sampling of the parameter space. As a byproduct, it can also produce samples from the posterior PDF. The method relies on a mapping of the D -dimensional integral to a one-dimensional integral  \nZ = ∫ p ( d | θ , H ) p ( θ | H ) d θ = ∫ 1 0 L ( X ) dX , (132)  \nwhere X is defined as the fraction of the prior mass enclosed by a ( D -1)dimensional surface at constant likelihood L ( X ):  \nX ( L ' ) = ∫ L ( θ ) >L ' p ( θ | H ) d θ . (133)  \nAs X decreases from 1 to 0, L ( X ) becomes progressively higher, so that L (0) is the maximum of the likelihood function. Finding the actual surface that encloses any particular likelihood (i.e. finding X ( L )) is prohibitively difficult in general, so the method instead uses a statistical estimate of X obtained through nested sampling of the prior distribution. This relies on the fact that the prior is by definition normalised over the entire parameter space, so we can initiate the algorithm with X 0 = 1. The algorithm then draws N live samples θ i from the prior, and evaluates their likelihoods to get L i . The samples are then ordered by likelihood value, such that L i < L i +1 . Since the samples are drawn from the prior, the fraction of the prior mass t contained within a likelihood contour that passes through sample j is given by the beta distribution p ( t j ) = β ( t j ; N live , 1).  \nAt each iteration of the algorithm the lowest-likelihood sample is removed, and replaced by another drawn from the prior but with a higher likelihood. The number of samples above the current likelihood contour, the live points , is then kept constant, and at each iteration the prior mass enclosed by the contour shrinks by a factor t , so X i +1 = t i X i . By repeatedly drawing nested samples, X shrinks geometrically toward 0, and we force the samples into a smaller volume of higher likelihood. Working with logarithms we have the expected shrinkage ⟨ log t ⟩ = -N -1 live , with variance σ 2 log t = N -2 live , so after i iterations log X ≈ -i/N live .  \nThe marginal likelihood integral can be approximated using the trapezoid rule,  \nZ ≈ 1 2 ∑ i ( L i + L i +1 ) w i , (134)  \nwhere w i = X i +1 -X i . The algorithm is typically run until a termination condition is met, which is often chosen via an upper bound on the remaining evidence L max X i /Z i < exp(0 . 1). The set of samples visited during the run is referred to as the nested sampling chain, and provides a scan of how the parameter values are constrained as the likelihood increases. Once the algorithm is complete, samples from the posterior can be produced by re-sampling the nested sampling chain with weight L i w i /Z , in a procedure similar to that described in Sec. 5.2.3.  \nThe challenge in nested sampling lies in being able to draw samples efficiently from the likelihood-limited prior distribution. In this regard it shares many common features with the challenge of efficiently sampling the posterior distribution that is encountered in Markov-chain Monte Carlo (MCMC) methods, and can benefit from many of the same techniques to improve efficiency. In initial applications of nested sampling to CBC data analysis, the samples were drawn using a short MCMC chain [701] with customised jump proposals to explore the symmetries of the likelihood function. This was developed further with LALInferenceNest [700] which combined the custom jump proposal with adaptive MCMC steps based on the affine-invariant sampling method of [352] (also used in the popular Emcee sampler [311]). An alternative method requires modelling the constrained region in such a way that it can be sampled exactly. This has the advantage that the samples can be independently drawn, eliminating the need for a mini-MCMC chain at each iteration of nested sampling. The popular MultiNest sampler thus fits the target regions with a set of ellipsoids that can be sampled exactly [292]. However, if the likelihood contour does not match the ellipsoids exactly, rejection sampling will be needed, introducing an inefficiency that can be as bad as running an MCMC chain.  \nAs nested samplers have become more sophisticated, the need for custom CBC-specific jump proposals has reduced somewhat, and many generalpurpose sampling algorithms have been applied. The PyCBC [127] and Bilby [99] libraries provide user-friendly python interfaces for Bayesian inference of CBC signals, and can inter-operate with many different sampling algorithms, including MultiNest and Dynesty [652].", '5.4.3 Hybrid grid-sampler exploration': 'An alternative to the purely stochastic sampling methods described above is a mixed approach, where different parameters are tackled with different methods. This is particularly relevant for the division between intrinsic parameters,  \nwhich affect the phase or amplitude evolution of the signal, and extrinsic parameters which only result in an overall amplitude or phase shift. Since exploration of the intrinsic parameter space requires recomputing a potentially costly waveform, there is a strong motivation to use a directed search method for that subspace, while the less costly extrinsic space can still be sampled in a pseudo-random fashion. The RIFT pipeline has adopted such a hybrid grid plus stochastic sampling scheme [551,445], which has enabled the use of otherwise prohibitively expensive waveform models for PE, and even the direct use of NR waveforms [446,17].', '5.4.4 Machine Learning methods': "Recently, the rapid development of machine learning (ML) techniques based on large neural networks has enabled novel approaches to gravitational wave analysis (see e.g. [213] for a review). There are a wide array of methods that have been proposed, which can be broadly categorised as enhancements to existing sampling algorithms, or being complete replacements of more traditional methods. Generally these have the goal of decreasing the time required to obtain a PE result.  \nEarly applications of ML were based around improving a particular aspect of the sampling algorithm. For instance, the difficulty in drawing samples from the likelihood-constrained prior distribution in the nested sampling algorithm. One approach that has been explored by the BAMBI sampler was to enhance the efficiency of these draws by training a neural network to approximate the likelihood surface [354], which was available within the LALInference software as far back as the first observing run. More recently, the technique of normalizing flows has been used to accelerate both nested sampling [720] and MCMC [98] within the Bilby framework, and can offer at least an order of magnitude speedup over previous samplers.  \nNormalizing flows are a technique that is particularly well-suited to Bayesian inference problems, see [431] for a recent review. Their goal is to learn an invertible transformation that maps a complicated distribution in physical parameter space p Θ ( θ ) to a simple distribution p Z ( z ) (often a multivariate normal - hence the name normalizing flow) in a latent space . The transformation function is defined by z = f ( θ ; Φ ) where Φ is the output of a neural network, and f is typically built up from multiple layers of simple transformations (for example an affine transformation). Using ML optimization methods such as stochastic gradient descent it is possible to find the optimum Φ that results in the output of the flow p Θ ( θ ) = p Z ( z ) | J ( θ , z ) | approximating a desired target distribution, where J ij ( θ , z ) = ∂z i ∂θ j is the Jacobian matrix of the transformation. The output of the flow can then be used to produce a trainable proposal distribution for use in MCMC and nested sampling algorithms.  \nAnother hybrid application of normalizing flows was explored with a combination of normalising flow jump proposals and Hamiltonian Monte Carlo, using the auto-differentiation tools of modern machine learning packages to accelerate the computation of gradients [723].  \nA particularly appealing application of machine learning methods is the possibility of completely replacing the stochastic sampling algorithms which require repeated draws, and explicit likelihood calculations which are computationally costly. Likelihood-free inference techniques have been applied which completely replace the likelihood and prior in Bayesian inference with a joint distribution of the data and parameters, which is then later conditioned on a specific piece of observed data. Two techniques which have been used for this are conditional variational auto-encoders [320], and normalising flows [229] (in a different configuration from the 'enhancements' mentioned above). Both these methods have the advantage of being very low-latency compared to stochastic sampling, as they rely on a neural network that is trained in advance on a large number of simulated signals and noise realisations. This moves the dominant cost of inference up-front, so that when an actual observation is made the network simply conditions its output on observed data. Such methods can be slow to train, especially on large datasets, and so their application to longer BNS signals is still to be fully demonstrated. However there have been recent works showing promising combinations of amortised inference with classical principal component analysis, a technique used elsewhere [173] for reducing data volumes [447].  \nAnother simulation-based inference technique that has been developed for computationally expensive problems is neural ratio estimation [503], which has been shown to reduce the number of likelihood evaluations required to perform full parameter estimation by a factor of ∼ 50[117].", '5.5 Degeneracies of the likelihood': "Parameter estimation results for CBC signals, and of gravitational waves in general, often exhibit certain characteristic features in the posterior PDF, a classic example being the 'V'-shaped degeneracy between distance and inclination angle. These degeneracies may result in correlations between parameters, or even in multi-modal posteriors, corresponding to similar waveforms that cannot be distinguished from each other due either to an exact symmetry in the waveform model, or because the noise level is too high to resolve their differences. Understanding these degeneracies is essential to the interpretation of PE results, but it can also help us in the design of PE methods, since these structures can be difficult to sample with a naive approach (for example MCMC with fixed jump scales leads to poor mixing of chains). Several works have proposed specific MCMC jump proposals or transformations of the parameter space to reduce or remove these degeneracies [588,700,283,607].", '5.5.1 Distance - inclination': "Expanding the logarithm of the likelihood function defined in Eq. (81) results in a quadratic in the waveform h ,  \nlog L = -1 2 ( ⟨ d | d ⟩ + ⟨ h | h ⟩ -2 ⟨ d | h ⟩ ) + k (135)  \nwhere k = -1 2 log det 2 π C N is a normalisation constant that depends only on the (usually fixed) PSD. (If the noise model is fixed then ⟨ d | d ⟩ is also a constant.)  \nFrom Eq. (91) we see that the response of a single detector labelled I to the dominant l = m = 2 mode of a non-precessing signal can be separated into frequency-dependent and non-frequency dependent parts as  \nh I = A 0 ( f ) d L ( F I + 2 (1 + cos 2 θ JN ) + F I × cos θ JN ) exp i [ Φ ( f ) -2 πft c,I -ϕ c,I ] . (136)  \nSince the inner product is an integral over frequency, this allows us to move the common non-frequency-dependent factors α I out of the inner product, to obtain  \nlog L = -1 2 ( α I ) 2 | A 0 ( f ) | 2 + α I ⟨ A 0 ( f ) exp i [ Φ ( f ) -2 πft c,I ] | d ⟩ cos ϕ c,I +const . ,  \n(137)  \nα I = 1 2 d L ( F I + (1 + cos 2 θ JN ) + 2 F I × cos θ JN ) , (138)  \nwhere | A 0 | 2 = ⟨ A 0 ( f ) | A 0 ( f ) ⟩ . Since the single parameter α I incorporates the effect of both d L and θ JN , as well as the antenna response for a particular detector I , there exists a degeneracy among these parameters where α is held constant. The transformation θ JN → π -θ JN , F I × → -F I × , results in the same signal appearing in the detector, and corresponds to a flip between lefthanded and right-handed elliptical polarisations of the gravitational wave, with a change in the polarisation angle. Intuitively we can understand this perfect symmetry as a single detector only sees one of the two polarisation modes of a general waveform. The detector response functions themselves depend on three parameters α , δ and ψ , and it is always possible for a single detector to find a new sky position and polarisation angle that maps F × → -F × while keeping F + . This can be achieved by changing the sky location, or by mapping ψ → π -ψ . For a two-detector network, there are two α I 's, and the sky location can be inferred from the time delay between detectors (see below). However the degeneracy in ψ still exists. A third detector in the network is required to fully break this degeneracy, and therefore infer the polarisation state of the GW, although in the case where detectors are of unequal sensitivity, or where a signal originates near the null of one detector's response function, it may be impossible in practice to do so due to the limited signal-to-noise ratio.  \nEven in a multiple-detector case, the measurement of the amplitudes α I is still subject to a statistical uncertainty, therefore there is a anti-correlation between d -1 L and cos θ JN when 0 < θ JN < π/ 2, which reduces the accuracy to which distance can be determined, unless the inclination angle can be measured in another way (for example by multi-messenger observation). This correlation limits the possible precision of inference on the Hubble constant using GW distance estimates [25].", '5.5.2 Sky localization': 'The localisation of compact binary sources is a highly important output of parameter estimation used to enable multi-messenger followup observations and correlations with other astrophysical transients, e.g. GRBs. The location of the source on the celestial sphere is typically described by the right ascension α and declination δ coordinates. As a single gravitational wave detector is approximately omnidirectional, a network of two or more detectors is required to achieve any reasonable precision. As we have seen, the antenna response functions determine the amplitude of the signal at each detector, and so provide weak information about the probable location; far more information, though, is obtained from differences in signal arrival times between observatories.  \nFig. 11 shows the localisation of a M = 10 M ⊙ source by different twodetector networks, illustrating the difference between localisation with timing alone versus that including the true antenna response function. From timing alone, each pair of detectors located at x is able to localise a source with propagation direction k to a ring on the sky consistent with a constant time offset between them ∆t IJ = ( x J -x I ) · k /c . The width of the ring is produced by the uncertainty in the difference in the time of arrivals at the two sites, which is inversely proportional to the effective bandwidth of the signal, weighted by the noise PSDs [274].  \nIf we consider the three pairs of detectors possible from a three detector network, the three resulting rings intersect at the true location of the source, but also at the point which is the reflection of the source in the plane containing the detectors, which generates the same times of arrival (see left panel of Fig. 11). There are therefore always two locations on the sky consistent with any observed time of arrival in a three detector network. With the inclusion of the antenna response function, the symmetry of the two modes is broken, since in general only one has a consistent set of antenna response functions that lead to the observed signal in all three detectors. However if one detector has significantly poorer sensitivity than the others, then the signal to noise ratio may be low in one detector, and the degeneracy only weakly broken.  \nSampling the sky location posterior can be challenging for algorithms because of the spherical topology of the parameter space and the ring-like structures which even when broken must be explored at lower likelihood levels for nested sampling algorithms, or at higher temperatures for annealed MCMC methods. Local Gaussian jump proposals that are not adaptive to the structure will have difficulty achieving good acceptance rates at all points in the  \nFig. 11: Localisation of a gravitational wave source by a detector network using (top) only timing information and (bottom) timing and antenna response information. Colours indicate which network is used to perform localisation, and the true location is marked with a +.  \n<!-- image -->  \nspace. One solution is to perform a reparameterisation that rotates the sky so the north pole is aligned with the vector separating the detectors. If the dominant pair of detectors is chosen then the posterior ring will align with a constant latitude in the new coordinate system, decoupling the two angular parameters so that they can be sampled more effectively. The time of arrival of the signal can also be remapped as part of this transformation: the time of arrival is typically well measured in each detector individually, but the time of arrival at the geocentre, relative to the detectors, is dependent on the direction of propagation of the signal. Therefore using the common reference position of the geocenter leads to degeneracy between the sky location and the reference time. When performing the sky rotation it is also useful to remap the reference time to that of arrival in the most sensitive detector.', '5.5.3 Phase and polarisation angle': 'The orientation of a quasi-circular binary is given by the three angular parameters, θ JN , ψ and ϕ c , which completely determine the polarisation content of  \nthe signal through the decomposition into spin-weighted spherical harmonics, as in Eq. (49). For the dominant ℓ = m = 2 mode, the signal can be written as  \nh + -ih × = h 22 -2 Y 22 ( ι, ϕ c ) + h 2 -2 -2 Y 2 -2 ( θ JN , ϕ c )  \nwhere -2 Y ℓm ( θ JN , ϕ c ) = f lm ( θ JN ) e imϕ c , and f lm ( θ JN ) is the inclinationdependent emission pattern determined by the spin-weighted spherical harmonics [491]. The effect of the phase parameter on the m = 2 mode is therefore to rotate between the + and × parts of the waveform when written in this complex format. Recall that the effect of the ψ parameter is also to rotate between + and × polarisations with angle 2 ψ . When the source is viewed directly face-on or face-off ( θ JN = 0 , π ), varying the ϕ c and ψ parameters produces equivalent effects on the m = 2 mode, causing a positive or negative degeneracy, ϕ + ψ = const or ϕ -ψ = const. Each condition describes a characteristic stripe in the ϕ, ψ parameter space that can be difficult to sample, being periodic in both parameters and potentially quite narrow. For a generic orientation θ JN , the posterior exhibits a combination of both positive and negative degeneracies.', '5.5.4 Masses and spins': "As discussed in Sec. 3.5, those parameters which most influence the phase evolution of the signal tend to be the best measured. These are referred to as intrinsic parameters, including both the masses and spins of the binary components, combinations of which enter into the phasing formulae at different PN orders for the inspiral (see Tab. 1). Generally, the lower the order, the more effect the parameter has on the phasing, and the easier it will be to measure it for lower-mass signals. Unfortunately the parameter combinations that enter into the PN expansion are not necessarily those of the greatest astrophysical interest. The chirp mass, entering the PN expansion at lowest order, is by far the best determined intrinsic parameter for inspiral-dominated signals. As a consequence, the posterior on component masses is constrained to lie on a line of approximately constant chirp mass. The length of the line in the direction perpendicular to chirp mass is determined by the uncertainty in the second mass combination, which can be taken to be the symmetric mass ratio η which enters at 1PN order.  \nHigh-mass signals, though can have a significant or even dominant contribution from the merger and ringdown parts. Since the quasi-normal mode frequencies of the ringdown signal scale inversely with the final mass, in these cases the final mass measurement may be as effective as the chirp mass measurement [699].  \nSampling the highly correlated ( m 1 , m 2 ) joint distribution can be challenging, especially for binary neutron star signals where the chirp mass is extremely well measured. It is therefore common to use the ( M , q ) parameterisation of the mass space, where q = m 2 /m 1 < 1 provides a naturally bounded domain that is still easily interpretable.  \nTurning to the spin parameters, the first combination that enters the PN expansion is the χ eff parameter, a mass-weighted combination of the aligned spin components that determines the spin-orbit coupling and has a dominant effect on the rate of inspiral in the near-equal-mass regime [74]. It is conventionally defined as  \nχ eff ≡ ( m 1 χ 1 + m 2 χ 2 ) · ˆ L m 1 + m 2 . (139)  \nAs χ eff enters at 1.5PN order, it is typically poorly measured for near-threshold signals. Furthermore, the prior on this parameter is not trivial: if one assumes that the spin vectors χ 1 , χ 2 have a uniform prior within the 3-ball | χ | < 1, the marginal prior on the z -components of the spin vectors contains an integrable singularity at χ z = 0, p ( χ z ) = 1 2 χ max ( -ln | χ z /χ max | ) [445]. This in turn creates a singularity in the marginal prior at χ eff = 0 since it is the weighted sum of the χ z 's. This makes interpreting the results of parameter estimation difficult: typically one is looking primarily for evidence that the posterior is significantly shifted compared to the prior (see for example the χ eff analysis in [484]).  \nFurther spin measurements enter at even higher PN order in the phasing, although the non-aligned parts of the spin vectors can be measured by another means: through the precession of the orbital plane that occurs due to spinorbit coupling, introduced in Sec. 2.2.2. This effect produces an amplitude and a phase modulation of the signal, as the system's orbital angular momentum vector tips toward and away from the observer. The measurability of this effect is dependent on the rate of change of the emission pattern -2 Y lm ( ι, ϕ c ) with respect to the instantaneous inclination angle ι = tan -1 ˆ L · ˆ N , which determines the size of the amplitude modulation seen by an observer viewing the system along the N vector. This time-dependence of ι for a precessing binary makes the choice of angular parameter θ JN a more useful choice in parameter estimation, since the total angular momentum of the source, J is more closely conserved during the merger process. To encode the dominant information about the precession, the χ p parameter has been introduced, which measures the relevant in-plane spin combination [366,631]. This too has a non-uniform prior induced by the uniform component spins, which complicates the interpretation of its posterior distribution. See [163] for a detailed comparison of spin priors under different constraints.  \nThe use of these two dominant parameters is also complicated in practice by the need to specify 4 additional spin degrees of freedom to fully determine the source properties. It is therefore more common to use an angular coordinate system determined by the spin magnitudes a 1 , 2 , tilts θ 1 , 2 , and the azimuthal angles ϕ JL and ϕ 12 which measure the difference between J and L an between the component spin vectors respectively [283].  \nResults on the spin angular can be visualised by the marginal posteriors on the various parameters of interest, particularly χ eff , χ p , θ 1 , 2 , a 1 , a 2 . To examine the 4D parameter space of component spin magnitudes and tilts a 'spin disk' plot, (e.g. Fig. 12), can be useful.  \nFig. 12: An example 'spin disk' plot for GW170729, reproduced from [35]. The plot shows the posterior on the spin vectors' magnitudes and tilts, marginalised over the spin azimuthal angles. Bin sizes are adjusted to contain equal prior probability, and the colour density represents the posterior probability in each bin. Left and right panels show the primary and secondary spin, respectively.  \n<!-- image -->", '5.6 Optimization': "Parameter estimation usually involves millions of likelihood evaluations, dominating the total computational cost. Numerous schemes have therefore been developed to reduce the cost of a likelihood evaluation. One can reduce the number of summation elements in the likelihood sum from Eq. (128) with various techiques. Multi-banding [176,703,329,511] involves downsampling the data at different sampling rates depending on the signal's frequency content. As CBCs increase in frequency as a function of time, one can compute the overlap for earlier parts of the signal with a much lower sampling rate than later parts of the signal, without loss of accuracy.  \nAnother technique that can reduce the signal bandwidth to even greater extremes involves computing the waveform relative to a reference signal. The resulting likelihood can be written in terms of the difference between the signals, which occupies a much smaller bandwidth than the full signal, provided they are sufficiently similar. This technique is employed in various forms as the heterodyned likelihood [204], or relative binning [734] methods. It can result in a speed-up of up to four orders of magnitude in the likelihood computation [205]. This technique has also been shown to apply to signals containing multiple modes, which can otherwise prove a challenge to acceleration schemes [452] since the modes interfere with each other, and the signal cannot be described as having an instantaneous frequency.  \nFurthermore, using the reduced order basis approach described in section 2.7, it is possible to precompute a reduced set of quadrature weights that  \naccurately approximate the waveform likelihood integrals, called a Reduced Order Quadrature (ROQ) rule [90,172,171,646,512,577,513,514]. Instead of evaluating the likelihood integral at each frequency sample, ROQ exploits the precomputed quadrature weights to efficiently compute the likelihood by taking a linear combination of the reduced basis waveforms. As show in section 5.5, one can expand the log-likelihood of Eq. (81) into a linear ⟨ d | h ⟩ and a quadratic ⟨ h | h ⟩ term, and a constant. The quadratic term is the same for all waveforms, and can be precomputed. The linear term can be written as a sum over the quadrature weights w i and the inner product of the reduced basis waveforms h i with the data d , ⟨ d | h ⟩ = ∑ i w i ⟨ d | h i ⟩ . The inner product of the data with the reduced basis waveforms can be precomputed and stored in a matrix R , so that the likelihood can be written as log L = -1 2 ⟨ d | d ⟩ + w T R d +const . , where w is the vector of quadrature weights. The ROQ rule is then the set of quadrature weights w that minimise the error in the likelihood integral. This can be computed by solving the linear system R w = 1 , where 1 is a vector of ones. The ROQ rule can be used to compute the likelihood for any waveform in the reduced basis, and so can be used to compute the likelihood for any signal in the parameter space. This approach significantly reduces the computational burden, as the number of reduced order bases is typically much smaller than the number of frequency points in the likelihood integral. Integrals over more points, typically those for longer, lower-mass signals, will see a greater speed-up.  \nAn effective optimization strategy is to marginalise (either analytically or numerically) over as many parameters as possible when evaluating the likelihood. While this usually makes each likelihood evaluation more computationally expensive, the reduced dimensionality almost always yields an overall speed-up. Marginalization has been applied to the orbital phase, coalescence time and distance parameters [700,236,99]. It is possible to recover the probability density function of the marginalised parameters in post-processing, at the cost of evaluating more likelihood values; however, as this post-processing step is done only on samples after thinning, the extra cost is small.", '6.1 From one to many: Hierarchical inference': "Section 5 discussed the inference of the parameters θ describing an individual source. The posterior PDF p ( θ | d ) with given data d depends on a choice of prior p ( θ | H S ). Section 5 also explained how to select between two models H 1 and H 2 , each with its corresponding prior distribution, using Bayes factors and posterior odds. The goal of hierarchical inference, discussed in this section, is to find the 'best' prior, or equivalently population model that is common to all events. This can be thought of as an extension of the model selection presented in Sec. 5.  \nIn general we consider a family of models for the prior PDF parametrized by Λ , a vector of population hyper-parameters, writing the source prior as p ( θ | Λ ). A hierarchical inference aims to use the data from all events { d i } N i =1 to infer the hyper-parameters Λ , in addition to the source parameters { θ i } N i =1 for every event i . Such an analysis stacks information across events by jointly inferring parameters specific to each event as well as (hyper-)parameters common to all events. Thus, when additional events are observed, we update not only the population from which they are drawn, but also the parameters of previously seen events.  \nWe can use Bayes' theorem to write the joint posterior PDF over the source parameters and population hyper-parameters as [466,478,485]:  \np ( { θ i } N i =1 , Λ | { d i } N i =1 , H S ) = p ( { θ i } N i =1 | Λ ) p ( { d i } N i =1 | { θ i } N i =1 , H S ) p ( Λ ) p ( { d i } N i =1 | H S ) , (140)  \nwhere p ( Λ ) is a prior distribution of hyper-parameters (or hyper-prior); note that this is a simplified expression that neglects observational selection effects, which are treated in the following Section 6.2. Assuming that each observation i is independent, we have  \np ( { θ i } N i =1 , Λ | { d i } N i =1 , H S ) = p ( Λ ) ∏ N i =1 p ( d i | θ i , H S ) p ( θ i | Λ ) p ( { d i } N i =1 | H S ) . (141)  \nIn many situations, we are only interested in inferring the population hyperparameters Λ , so we can marginalize over the single-event parameters { θ i } N i =1 :  \np ( Λ | { d i } N i =1 , H S ) = p ( Λ ) ∏ N i =1 ∫ p ( d i | θ i , H S ) p ( θ i | Λ ) d θ i p ( { d i } N i =1 | H S ) . (142)  \nThe numerator of Eq. (142) is the hyper-prior multiplied by the 'single-event evidences' ∫ p ( d i | θ i , H S ) p ( θ i | Λ ) d θ i . This makes intuitive sense because the 'best' (maximum-likelihood) Λ corresponds to the choice of prior on θ that maximizes the product of single-event evidences.  \nSingle-event evidences may be evaluated by various methods. Assuming that parameter estimation has already been performed on each event using  \nsome 'interim prior' π PE ( θ ), resulting in J PE samples θ j i for each event i , importance sampling is often convenient to apply. The PE samples are drawn from a posterior PDF given by the likelihood p ( d i | θ i ) with the 'PE' prior,  \nθ j i ∼ p ( d i | θ i ) π PE ( θ ) . (143)  \nThe single-event evidence can then be estimated as the following importance sampling, or Monte Carlo, approximation (in what follows we will drop the explicit dependence on the signal hypothesis H S ):  \n∫ p ( d i | θ i ) p ( θ i | Λ ) dθ ≈ 1 J J ∑ j =1 p ( θ j i | Λ ) π PE ( θ j i ) . (144)  \nThe error in this Monte Carlo approximation is small when the number of PE samples is large and the interim prior is not too different from the population model p ( θ | Λ ). For this error to be small compared to the uncertainty in the population posterior, the effective number of PE, which depends on the chosen population model(s) p ( θ | Λ ) for all choices of Λ in the hyper-prior, must be large enough [259]. Certain population models, especially those that allow for rapid variation of the population PDF p ( θ | Λ ) over parameter space, may require an especially large number of PE samples for each event, with a possibly prohibitive computational cost.  \nInstead of estimating the single-event evidences via importance sampling, one could first obtain a continuous functional approximation to the singleevent likelihood p ( d i | θ i ) using techniques such as kernel density estimates, Gaussian mixture models or Gaussian processes, based on the PE samples [443, 727,231,346,230]. Equipped with an estimate for p ( d i | θ i ), it is straightforward to evaluate the joint posterior over the single-event parameters and population hyper-parameters from Eq. (141). The density estimate still carries an uncertainty that scales inversely with the number of PE samples. Nevertheless, this approach can be better-behaved than importance sampling, for instance it correctly assigns regions of θ i parameter space without discrete samples a nonzero (but small) likelihood.  \nThe hierarchical Bayesian likelihood may also be evaluated without the use of PE samples by directly sampling source parameters ( θ i ) and population hyper-parameters ( Λ ) from the joint likelihood (Eq. (141)). Thus, rather than sampling from the PE posterior for each event separately with a fixed 'interim' prior, PE samples are drawn jointly with the population hyper-parameters across all events simultaneously. This may be desirable when we are interested in the population-informed parameters for individual events. However, directly evaluating the joint likelihood may require many more evaluations of the PE likelihood p ( d i | θ i ) in total, in part due to the much higher dimensionality of the joint parameter space, requiring in turn many more waveform evaluations and higher computational cost. Therefore, in order to update our inference of individual events' parameters with our knowledge of the population, we can reweight pre-existing PE samples (drawn with an interim prior) in post-processing, once we have sampled from the marginalized likelihood of  \nEq. (142) [299,326,506,261,510,516]. This remains true when including selection effects (discussed in the following section).", '6.1.1 Astrophysical events as a Poisson process': "So far we have written the source prior, or 'population model', as a probability density p ( θ | Λ ). For any fixed Λ , such a PDF, integrating to unity across the θ parameter space, describes the relative proportion of systems with different source properties. However, if considering absolute numbers of systems, for instance the astrophysical rate of binary mergers within some volume of space, we prefer to consider number densities dN/d θ instead of PDFs p ( θ ), the difference being that dN/d θ integrates to the total expected number of systems N within the considered range of θ ,  \ndN d θ ( θ | Λ ) = N p ( θ | Λ ) . (145)  \nWe often consider dN/d θ as the intensity function of an inhomogeneous Poisson process - 'inhomogeneous' because the intensity is not constant across the space of θ . The corresponding likelihood is given by  \np ( { θ i } , { d i } | Λ , N ) ∝ N N e -N N ∏ i =1 p ( d i | θ i ) p ( θ i | Λ ) (146)  \nwhere { . . . } denotes the set over N events. Using this likelihood, we can then write the posterior over the source parameters and population hyperparameters (which now include the total number of systems N ):  \np ( { θ i } , Λ , N|{ d i } ) ∝ p ( Λ , N ) N N e -N N ∏ i =1 p ( d i | θ i ) p ( θ i | Λ ) . (147)  \nWe have hidden the p ( { d i } ) factors in the proportionality because they are independent of Λ and N . We discuss the relation between Eq. (147) and Eq. (141) in the following subsection. Again, these expressions only hold in the absence of observational selection effects, where the expected number of observed events N is the same as the number of events occurring in the astrophysical population.", '6.2 Accounting for selection biases': "GWobservations, like most astrophysical observations, are subject to selection effects: most CBCs that occur in the Universe produce GWs that are too quiet to be confidently distinguished from noise in current detectors, thus only a small subset of mergers on our past light-cone make it into detection catalogs. Nearby, face-on binaries with high masses and large spins aligned with the orbital angular momentum are over-represented among GW detections,  \nbecause they emit louder signals. To learn about the underlying astrophysical population of CBCs, we must account for such observational selection effects in the hierarchical Bayesian inference.  \nEach CBC, with source properties θ , produces a signal lying within data d and is either detected, meaning that its estimated statistical significance, in terms of false alarm rate or probability of astrophysical origin, satisfies some numerical threshold, in which case we label it 'det' and record its corresponding data d i ; or non-detected, in which case we do not record it. Detection is therefore a binary property of the data resulting from a CBC signal within the detector noise at a particular time. We can write the probability that some hypothetical data is detected as P (det | d ). Typically, P (det | d ) is deterministically 0 or 1 for any piece of data d , based on whether the data passes some threshold in, for example, FAR or SNR [483], but this need not be the case [288].  \nIn order to correct for selection effects, we must account for the fact that our analysis only includes data from detected sources. Given N detections, we explicitly include the label { det i } N i =1 among { d i } N i =1 , { θ i } N i =1 and Λ when describing joint probabilities:  \np (det , d, θ , Λ ) = P (det | d ) p ( d | θ ) p ( θ | Λ ) p ( Λ ) . (148)  \nNote the conditional structure implied by the above equation: given data d , detectability det is conditionally independent of the source properties θ [466, 483,262]. Marginalizing the above expression over data realizations d and then over the source parameters θ , we obtain P (det | Λ ), which also appears in the GWliterature with the notation β ( Λ ) [15] or α ( Λ ) [483]. This is the fraction of sources that are detected (out of the total number N of astrophysical sources) for a hypothetical CBC population described by hyper-parameters Λ :  \nP (det | Λ ) ≡ ∫∫ P (det | d ) p ( d | θ ) p ( θ | Λ ) d d d θ . (149)  \nThe inhomogeneous Poisson process likelihood of Eq. (146) is then modified as follows. The expected number of observed events in the exponent term is now µ ( Λ ) ≡ N P (det | Λ ), while each factor i in the product over events is multiplied by P (det i | d i ) [466,483,680,705,262]:  \np ( { θ i } , { d i } , { det i } | Λ , N ) ∝ N N e -µ ( Λ ) N ∏ i =1 P (det i | d i ) p ( d i | θ i ) p ( θ i | Λ ) ∝ µ ( Λ ) N e -µ ( Λ ) P (det | Λ ) N N ∏ i =1 p ( d i | θ i ) p ( θ i | Λ ) . (150)  \nIn the last line, we substitute for N and hide the P (det i | d i ) terms in the proportionality because they are independent of the population hyperparameters Λ and N [262].  \nNow we can write the joint posterior over { θ i } , Λ , and µ (or alternatively N ) as  \np ( { θ i } , Λ , µ | { d i } , { det i } ) ∝ p ( Λ , µ ) µ N e -µ P (det | Λ ) N N ∏ i =1 p ( d i | θ i ) p ( θ i | Λ ) . (151)  \nIf we are only interested in inferring the shape of the population rather than the number of sources, we can marginalize Eq. (151) over µ . If we adopt a separable prior on µ and Λ , we recover the following posterior PDF for { θ i } and Λ :  \np ( { θ i } , Λ | { d i } { det i } ) ∝ p ( Λ ) P (det | Λ ) N N ∏ i =1 p ( d i | θ i ) p ( θ i | Λ ) , (152)  \nwhich reduces to Eq. (141) if P (det | Λ ) = 1, implying there are no selection effects.  \nIf we choose to work with the parameter N rather than µ , an arbitrary prior on N may not generally lead to a separable induced prior on µ and Λ . However, the choice of a uniform prior over ln N will always induce a uniform prior on ln µ , independently of Λ : with this choice, marginalization over N recovers Eq. (152). This allows us to sample from the joint posterior over the shape of the population Λ and its amplitude N or µ in two steps. First we can sample Λ from the marginalized posterior of Eq. (152). Then we can draw µ samples from the distribution p ( µ | N ) ∝ p ( N | µ ) p ( µ ), where the observed number of events follows a Poisson distribution, p ( N | µ ) ∝ µ N e -µ , and p ( µ ) is the prior on µ .  \nWenow discuss how to evaluate the detection fraction P (det | Λ ). In Eq. (149), it is defined as a double integral over data and single-event parameters: this may be estimated by Monte Carlo sampling. We draw θ j samples from p ( θ | Λ ), simulate the GW signal for each sample, add it to a realization of the noise distribution to get the corresponding mock detector data d j , and evaluate P (det | d j ) by processing the mock data with search pipelines. The integral P (det | Λ ) can then be approximated as the average over samples j :  \nP (det | Λ ) ≈ ⟨ P (det | d j ) ⟩ . (153)  \nIf P (det | d j ) is deterministic (either 0 or 1 for each simulated data d j ), this is simply the fraction of simulated signals that pass the detection threshold. As discussed in 4.8.2, we refer to these simulated signals as 'injections,' and the injections that pass the detection threshold are known as 'found injections.'  \nTo save computational cost, we often evaluate P (det | d j ) for just one set of θ j samples, drawn from some reference PDF p draw ( θ ) (see the discussion in 4.8.2). We then estimate P (det | Λ ) by reweighting these injections:  \nP (det | Λ ) = 〈 P (det | d j ) p ( θ j | Λ ) p draw ( θ j ) 〉 . (154)  \nThe use of importance sampling here is analogous to estimating single-event evidences using PE samples drawn under a reference prior distribution, Eq. (144); in both cases, the uncertainty in the Monte Carlo integral must be under control in order to obtain valid inferences [259]. 26  \nFor estimating β ( Λ ), the effective number of injections required to avoid biasing the likelihood increases in proportion to the number of detected events. In principle, the injected distribution p draw ( θ ) may be adjusted to better match the inferred astrophysical distribution p ( θ | Λ ), so that fewer injections are required; however this requires a priori knowledge of the true population, which is necessarily incomplete if derived from previous observations. As an alternative to importance sampling, we can compute the integral P (det | Λ ) by first estimating a smooth approximation to P (det | d ) [725,338,664]. This is analogous to calculating the single-event evidence terms with a smooth approximation to the PE likelihood instead of averaging over PE samples.", '6.3 Inference on astrophysical populations': "A major goal of GW astrophysics is to understand the origin of merging binary compact objects. The formation and evolutionary histories of neutron stars and black holes in merging binaries is an open problem involving many astrophysical processes that are currently poorly understood. Many such processes also play a fundamental role in a variety of astrophysical contexts outside the production of CBC sources, including stellar evolution, galaxy assembly, and the production of heavy elements. The population of CBC sources also has implications for cosmology, as GWs from cosmological distances encode information about the cosmic expansion history of the Universe.  \nBinary compact object mergers likely represent extremely rare outcomes of massive stellar evolution. Stars more massive than ∼ 8 M ⊙ end their lives as NSs or BHs [382]; in rare circumstances, such a NS or BH will be close enough to another compact object that they eventually merge within the age of the Universe. There are, broadly, two distinct channels to achieve this: either the two objects started off in a binary star system that remained gravitationally bound throughout its evolution [116,113,415], or they started off in a dense star cluster, interacting dynamically with other objects before assembling a tight binary [438,640]; see Refs. [486,481] for recent reviews. In either case, the two compact objects must reach a sufficiently tight orbit that will shrink through gravitational radiation to the point of merger. Alternatively, it has been proposed that primordial BHs produced shortly after the Big Bang could contribute to the CBC population [119]. The existence of a subpopulation of primordial origin, if discovered (for example, by finding BHs at subsolar masses or at redshifts higher than the first star formation), would have dramatic implications for early Universe cosmology and the composition of dark matter [182].  \nThe hierarchical Bayesian approach discussed in the previous subsections allows us to infer properties of the compact binary population, such as their merger rate and how BHs and NSs are distributed in mass, spin and merger redshifts. These population distributions depend on the evolutionary pathways of BHs and NSs in merging binaries. By measuring the population properties of GW sources, we ultimately hope to reverse engineer the histories of CBC progenitors. The population of CBCs also depends on the cosmological parameters, particularly the relation between luminosity distance, which controls the GW signal amplitude, and redshift, which affects the signal frequency in the same way as a change in component masses (see Sec. 2.2.3). Any inference on the (source-frame) mass distribution thus depends on the events' redshifts, derived from their luminosity distances via a cosmological model. The cosmological redshift-distance relation may also be simultaneously inferred together with the astrophysical population parameters [634,670,285,489,272].  \nIn general, a BBH population model describes the distribution over sourceframe primary and secondary masses (or alternatively primary mass and mass ratio), component spin magnitudes and orientations (or alternatively, 'effective' spins like χ eff and χ p ; see Section 2.3.1), merger redshift (plus background cosmology relating luminosity distance to redshift), and the binary position and orientation. Populations with NS components also have hyperparameters describing their tidal deformabilities, which are discussed in the following subsection.  \nSeveral different strategies for population modeling have then been followed. The first approach aims to infer physical parameters by directly comparing observations to simulations. The formation of CBCs is subject to several uncertainties, ranging from unknown initial conditions (for example, the initial mass distribution of stars) to uncertain physical quantities (for example, nuclear reaction rates in stellar cores), to evolutionary stages where direct modelling is too computationally difficult and we instead use approximate prescriptions (for example, the common envelope in binary stellar evolution). These uncertain elements may all leave some imprint on the CBC population, so one could in principle model the CBC population in terms of the corresponding physical parameters, and directly infer their values via hierarchical Bayesian analysis. We refer to such population models as physics-driven models .  \nAt the opposite pole of methodology, data-driven models , also known as non-parametric models, aim to be flexible enough to recover an arbitrary CBC population distribution. For example, the 'parameters' of these population models might simply be the merger rate density at each point in θ -space, and the number of parameters approaches infinity as one considers arbitrarily high resolution in θ space (e.g. a histogram in the limit of infinitely many, closelyspaced bins). In practice, the range of distributions covered is limited both by computational and statistical considerations, particularly in the limit of densities which vary rapidly over a small range of parameter space. Data-driven models give the most reliable description of the population in that they are unbiased by the systematic uncertainties that can plague (incomplete) physics-  \niven models. However, for the same number of observations, the statistical uncertainties in the inferred population tend to be larger because of the high model flexibility. Furthermore, unlike physics-driven models, the parameter inferences often cannot be immediately read off as physical or astrophysical implications, usually requiring further interpretation.  \nFalling in between physics-driven and data-driven models, parametric phenomenological models are designed to consist of relatively simple functional forms, such as Gaussians or (truncated) power laws, but contain features that are motivated by astrophysical theory: e.g. a gap in the BH mass spectrum or a local excess of density for BH spin magnitudes close to zero. The main advantage of parametric phenomenological models is simplicity in writing them down and drawing initial astrophysical conclusions. Oftentimes, these models are also cheaper to fit to data than data-driven or physics-driven models, although this stops being the case if the parametric model is over-prescriptive and struggles to fit the data.  \nIn the simplest parametric and data-driven population models, the probability distribution over source parameters is assumed to be separable, and thus can be written as a product,  \np ( θ | Λ ) = p m ( m 1 , m 2 | Λ m ) p χ ( χ 1 , χ 2 | Λ χ ) p z ( z | Λ z ) p Ω ( Ω | Λ Ω ) p i ( ι | Λ i ) . (155)  \nHere Ω and ι are the angular parameters describing the position on the sky and the binary orientation, respectively; the corresponding distributions are usually assumed to be isotropic, but see for instance [260,401,704] for counterexamples.  \nIn general, model parameters include cosmological parameters that set the redshift-distance relation and also the distribution p z , which may be written  \np z ( z ) ∝ 1 1 + z dV c dz f ( z ) , (156)  \nwith V c the comoving volume, such that the merger rate density per comoving volume per source-frame time evolves with redshift as  \ndN dV c dt ∝ f ( z ) , (157)  \nand dV c /dz also depends on the cosmological parameters. Common phenomenological models for f ( z ) include a power law in (1 + z ) or functional forms inspired by the star formation rate [303,164].  \nThe two-dimensional mass distribution is often factorized as [15,301,435]:  \np m ( m 1 , m 2 ) = p 1 ( m 1 ) p 2 ( m 2 | m 1 ) , (158)  \nor similarly using the mass ratio q = m 2 /m 1 in place of m 2 . Alternative factorizations may, though, be more natural in some contexts; for example, if we believe that both component BH masses are drawn from the same distribution, such that they both share features like local overdensities in the same locations, we may consider a 'pairing function' parametrization [302,280]:  \np m ( m 1 , m 2 ) ∝ p ( m 1 ) p ( m 2 ) f ( q, M tot ) , (159)  \nwhere p is the component mass distribution and f is a pairing function that depends on the mass ratio and/or total mass; two BH masses are randomly paired in a binary if f ≡ 1.  \nTypical phenomenological parametric models for the primary mass distribution p 1 ( m 1 ) or the component mass distribution p ( m ) include mixtures of (broken) power laws and Gaussians, with some smoothing applied at the minimum and maximum mass ends to avoid sharp cutoffs, and possible notch filters to represent dips or gaps [301,435,662,726,298,253,40]. The conditional secondary mass distribution, conditional mass ratio distribution, or pairing function is most commonly taken as a power law in mass ratio [301,435]. However, there are already indications of additional structure in the mass ratio distribution; for example there are hints that at higher masses, the mass ratio distribution more strongly prefers symmetric ( q ≈ 1) systems compared to low masses [458,277,615].  \nThe spin distribution can be parameterized in terms of 'effective' spin parameters χ eff and χ p . Past work has modeled these distributions as truncated Gaussians [608,506] or histograms with 3-5 bins [289,281]. Alternatively, one can consider all six spin degrees of freedom; in spherical coordinates, the dimensionless spin magnitude and spin tilt angle and azimuthal angle of each component. 27 A common parameterization for the spin magnitude distribution is a beta distribution, because it is naturally truncated over the physical range between 0 and 1 [726]. To avoid statistical sampling issues, the prior is typically restricted to non-singular beta distributions; however, this choice excludes spin magnitude distributions that peak sharply at zero, which can be undesirable if we wish to measure a possible excess of nonspinning BHs, as some theories suggest [318,604,325,168]. The cosine of the spin tilt distribution is often modeled as a truncated Gaussian; Ref. [661] introduced a mixture model between such a truncated Gaussian centered at zero (aligned spin) and a flat distribution in cosine tilt (isotropic spins); motivated by expectations from a mixture between isolated binary evolution formation channels, which are thought to produce binaries with nearly-aligned spins ([414] ( although the double pulsar system may provide a counterexample [287])) and dynamical assembly, which, at least in gas-free environments like globular clusters, is thought to produce binaries with isotropic spins [566,600]. The distributions of azimuthal spin angles are typically fixed as uniform, but theoretical predictions for spin-orbit resonances may also motivate explicit inference over these parameters [337, 698,694]. The simplest models assume that the two compact object spins in a binary are independently drawn from the same distribution, but this contradicts some predictions in which, for example, first-born and second-born BHs preferentially follow different spin distributions [736,110]. Recent work has thus relaxed this assumption, allowing primary and secondary BH spins to be drawn (independently or not) from different distributions [304,62].  \nEach phenomenological parametrization of the mass, spin or merger redshift distributions can be replaced by a data-driven or non-parametric model. Data-driven models commonly used in the GW literature include Gaussian mixture models [682,684], splines [254,255,348], Gaussian processes (e.g. [278], including Gaussian-process regularized histograms [482,456,587] and autoregressive processes [165]), Dirichlet processes [596], and optimized kernel density estimates [616,615]. Many such methods achieve flexibility by setting up an underlying parametric model, such as a power law, and allowing more or less general forms of deviation from it. Data-driven models typically impose some structure on the correlations between neighboring points in θ -space [384], in order to control the (lack of) smoothness of the population PDF, and thus limit the effective number of independent degrees of freedom to be fit. Such strategies to regularize data-driven models will become less necessary in the limit of large catalogs.  \nOnce data-driven models are fit to the data, there is usually a second 'feature extraction' step, where interesting features, such as local maxima, minima or trends, are identified. Although the model is not parameterized in terms of such features, posteriors on, e.g., the location of local maxima in the mass distribution may be calculated in post-processing [279,165]. The significance of any features can also be evaluated by comparing the posterior against the prior of the data-driven model, noting that although such models are designed to be flexible they can induce nontrivial priors on the population distribution.  \nIn both the parametric and data-driven models, the next level of complexity in modeling the CBC population is to allow parameter correlations rather than assuming that masses, redshifts and spins are independent. Twoparameter correlations investigated up to now include evolution of the mass or spin distribution with redshift [617,297,120,109,120], or the spin or mass ratio distributions varying over primary mass [618,660,314,305,712]. Empirically, a statistically significant correlation between the effective inspiral spin and the mass ratio has been identified [166], which may indicate that primary and secondary BH spins are drawn from different distributions, as discussed above.  \nThe simplest such model allows for only a linear correlation, although quadratic and higher order polynomial terms may straightforwardly be included. For instance one can allow the location of a Gaussian peak µ in the primary mass distribution to vary linearly with redshift, µ ( z ) = µ 0 + z · dµ/dz | z =0 , with parameters µ 0 and dµ/dz | z =0 . Another simple parametric model for twoparameter correlations is a switch point analysis: in such a model the parameters describing the distribution in one variable (e.g. mass) are allowed to take on different values depending if the second variable is above or below the switch point (e.g. at low versus high redshifts). Although a discontinuous jump in the distribution may not be physically well-motivated, switch point analysis can be a quick way to check if the data prefer correlations between two variables [297]. Correlations can also be encoded via mixture models, in which the population consists of different components (subpopulations) with  \ndifferent properties; for example, allowing distinct parameterized power law or Gaussian peak components in the mass distribution to take on different spin distributions [712,343,457]. Copulas also naturally allow the analyst to fix the form of the one-dimensional marginal distributions, while exploring different two-dimensional correlations specified by a copula density function [63]. Datadriven models can also be extended to include correlations between variables, either by directly modeling multiple dimensions non-parametrically (e.g., a multi-dimensional Gaussian mixture [682,683], a multi-dimensional Gaussianprocess regulated histogram [587,586] or a kernel density estimate [615]); or combining one-dimensional parametric models with flexible functions that describe correlations, for instance using a spline (rather than a linear function) to model evolution of a mass feature with redshift [383].  \nFitting physics-driven models to GW catalogs requires a different strategy, based on population synthesis simulations. The model may be described in terms of a mixture of formation channels, such as isolated binary evolution with common envelope, isolated binary evolution with stable mass transfer, chemically homogeneous evolution, triple star evolution, stellar evolution in young star clusters, dynamical assembly in globular clusters, dynamical assembly in the disks of active galactic nuclei, primordial black holes, and so on. The relative mixture weights (also known as branching ratios) of different channels may be considered as free parameters, together with some of the uncertain astrophysical parameters within each channel [656,738,737,140]. Such parameters may include the mass accretion efficiency, common envelope description, black hole natal spins and natal kicks, the metallicity-specific star formation rate, and the mass and radius distribution of globular clusters [539,141,655, 593,300,196]. For physics-driven models, it is computationally challenging to simulate enough populations that cover the resulting high-dimensional parameter space. Strategies to alleviate this issue such as interpolating between simulations and identifying where to run new simulations (e.g. using Gaussian processes) have been proposed [671,147]. Simulation-based inference may also be a promising technique for comparing physics-driven models to GW data [211,453]. An even bigger challenge, however, is that many of the uncertainties in physics-driven models are systematic, stemming from the lack of an adequate model, and therefore defy parameterization [146,114]. This limitation might be tackled by combining physics-driven models with phenomenological or data-driven models: incorporating specific predictions from population synthesis simulations, such as the remnant mass function or the distribution of delay times between star formation and merger, into a phenomenological population model [111,306,345]. This hybrid approach can be used to test one aspect of the population synthesis model at a time against the data; for example, checking whether the predicted delay time distribution is consistent with the inferred merger rate and the assumed star formation rate.  \nAs a technical aside, especially for data-driven or physics-driven CBC population models, it can be tempting to fit the 'detected distribution' (i.e., the distribution of GW sources conditioned on detection), rather than the underlying, astrophysical distribution. However, one has to be careful that doing so  \ndoes not violate the conditional structure of the data-generating process [262] to obtain a correct hierarchical Bayesian likelihood.", '6.3.1 Population model comparisons and checking': "With so many population modeling choices, the question naturally arises of which model best describes the data. One answer is to compare models using their Bayes factors (the ratio of their evidences) or their posterior odds (see Section 5.1). Often, the population models we consider are nested, which means that we can easily compute a Bayes factor between two models with a Savage-Dickey density ratio [239] rather than directly computing their evidences. A simple application is to mixture models: either a mixture between two formation channels in physics-driven models, or a mixture between (e.g.) power laws and Gaussians as in commonly-used phenomenological parametric models. Given the posterior on the branching ratio f between two components, we can compute the posterior probability density at f = 0 compared to the prior probability density: this density ratio gives us the Bayes factor between the simpler one-component model and the two-component mixture model.  \nHowever, in practice the Bayes factor or evidence ratio has various drawbacks. In the mixture model example, if the data poorly constrain f , the posterior will closely resemble the prior, and the Bayes factor between the two models will be unity: the naive expectation that Bayesian model selection will penalize the more complex model fails when the additional parameters are not well-constrained by the data. More generally, the choice of hyper-prior can significantly affect the Bayes factor: this is easy to see in the mixture model example. Such dependence on the hyper-priors can be problematic for interpreting population results: particularly for phenomenological models, there may not be a single well-motivated choice of hyper-prior. For instance, two possible prior choices for the power-law slope of the BBH primary mass distribution would be a uniform distribution between limits -10 and 10, or uniform between -5 and 5; no more fundamental principle enables us to prefer one of these priors to the other. Nevertheless, over a higher-dimensional space of hyper-parameters, different prior choices can yield evidences differing by orders of magnitude for otherwise identical models.  \nAn alternative to comparing two models is to merely check whether a model adequately fits the data, with a goodness-of-fit test. In the Bayesian context, such tests are often provided by posterior predictive checks [299,40, 602,507]. After fitting a given population model to the data, one can generate mock observations from that population, which represent draws from the posterior predictive distribution. Ideally, the mock observations include both measurement uncertainty and selection effects, such that the posterior predictive distribution is a distribution over data-space d ; a less powerful posterior predictive check can be carried out at the level of true source parameters θ if one includes selection effects but not measurement uncertainty in generating mock observations. The predicted mock observations can then be checked for consistency with the actual observations: in one dimension, this can be done  \nby comparing the cumulative distribution function (CDF) of the predicted versus observed source parameters, e.g. via a Kolmogorov-Smirnov (K-S) or Anderson-Darling test. In higher dimensions, one can compare the predicted versus observed number of detections inside one or more regions of interest within source parameter space, e.g. [297]. For a choice of test statistic - in the K-S test case the maximum distance between two CDFs, or the number of detections in a region of parameter space - the tension between the population model and the data can be quantified by first constructing a null distribution of the test statistic by drawing many mock catalog realizations from the posterior predictive distribution. Then the 'posterior predictive p-value' is calculated as the fraction of mock catalogs that are more extreme than the observed catalog with respect to the test statistic, quantifying where the observed value falls in the null distribution [299]. In addition to goodness-of-fit tests, posterior predictive checks have been used to classify population outliers in the GW literature [299,41,40,258].  \nIn general, posterior predictive p-values are also sensitive to the choice of model prior, insofar as this choice affects the posterior PDF, and consequently also the posterior predictive distribution. However, they are less sensitive than Bayes factors, which are affected by changes only in the prior (e.g. changes in prior boundaries) even in the case where the posterior is unaffected.", '6.4 Matter effects and equation of state of dense matter': "During the late stages of a binary coalescence that involves at least one neutron star, finite-size effects modify the expected binary evolution [188,242]. The two lowest-order effects in the PN phase expansion are that of the spin-induced quadrupole moment at 2PN [562] and tidal interactions at 5PN [307]. Despite the higher PN order, the tidal interactions are more prominent in the signal due to the larger numerical prefactor [387], as well as the fact that the former effect depends on the neutron star spin quadratically [370]. Tidal interactions originate from the fact that NSs have a finite size, and are therefore distorted by the gravitational field of their companion, with two effects on the binary orbit: (i) the tidal deformation drains orbital energy from the system, and (ii) the tidally-induced quadrupole moment acts as another source of gravitational radiation. Black holes do not exhibit such tidal interactions [118,197].  \nThe degree of tidal deformation depends on the internal composition of the NS and the (unknown) equation of state of dense nuclear matter in beta equilibrium. For stable, cold neutron stars the equation of state is a relation between the pressure p and the energy density ϵ which is expected to be common among all NSs [448,535,112]. In general, softer equations of state predict lower internal pressures and smaller NSs that are less deformable, while stiffer equations of state predict higher pressures, larger NSs, and stronger tidal interactions. Further nuclear phenomena such as the emergence of strange degrees of freedom or deconfined quarks and phase transitions [362] might  \ncomplicate this picture or even predict the existence of 'twin stars', i.e., NSs with the same mass but different radii.  \nInference for the common p ( ϵ ) equation of state is a sub-case of the hierarchical inference described above. Specifically, the generic hierarchical posterior from Eq. (142) now becomes [444,188]  \np ( {E , Λ }| { d i } N i =1 ) = p ( {E , Λ } ) ∏ N i =1 ∫ p ( d i | θ i ) p ( θ i |{E , Λ } ) d θ i p ( { d i } N i =1 ) , (160)  \nwhere we have separated the equation of state E ≡ p ( ϵ ) from the remaining population hyperparameters Λ . The latter could now represent the mass or spin distribution of NSs [544]; this distribution must be inferred simultaneously with E , as their uncertainties can be correlated [728,347]. Restricting Λ to the NS mass distribution, the relevant event-level parameters are the binary masses and tidal deformabilities θ i = { m i , Λ i } and the population model (or source prior) is now  \np ( m i , Λ i |{E , Λ } ) = p ( Λ i | m i , {E , Λ } ) p ( m i |{E , Λ } ) = δ ( Λ i -E ( m i )) p ( m i |{E , Λ } ) , (161)  \nwhere δ ( x ) is the Dirac delta function. The equation of state term δ ( Λ i -E ( m i )) represents the fact that the tidal deformability is unambiguously defined given an equation of state and a NS mass. The p ( m i |{E , Λ } ) term couples the mass distribution parametrized by Λ and the equation of state encoded in E , and in general reflects assumptions about NS astrophysics. For example, a simple uniform mass distribution between limits M l and M h ,  \np ( m i |{E , Λ } ) = Θ ( M l ≤ m i ) Θ ( m i ≤ M h ) M h -M l , (162)  \ncould contain an implicit dependence on the equation of state, M h = M max ( E ), if we assume that the maximum NS mass realized in the Universe is determined by nuclear physics, M max ( E ) being the maximum mass of nonrotating NSs predicted by E . If we instead assume that the astrophysical conditions dictating the formation of neutron stars limit neutron star masses to be below M max ( E ), M h would be independent of E ; the effects of such assumptions are explored further in [450].  \nEquation (161) requires some functional form for E . This may be achieved via parametric or nonparametric priors that prescribe candidate p ( ϵ ) curves (or equivalent functions). For example, the piecewise-polytropic parametrization expresses the pressure as a piecewise function of the baryon density ρ [589, 534], expressed in its simplest form, as  \np ( ρ ; p 1 = p ( ρ 1 ) , Γ 1 , Γ 2 , Γ 3 ) =      K 1 ρ Γ 1 : ρ < ρ 1 K 2 ρ Γ 2 : ρ 1 < ρ < ρ 2 K 3 ρ Γ 3 : ρ 2 < ρ, (163)  \nwhere ρ 1 and ρ 2 are fixed by optimization against nuclear models [589], while K 1 and K 2 are fixed by continuity. A closely related form is the spectral parametrization that instead expands the polytropic index in a polynomial of the pressure [461,462]  \np ( ρ ; γ i ) = ρ Γ ( x ) , Γ ( x ) = 3 ∑ i =0 γ i log( x ) i , (164)  \nwhere x ≡ p/p 0 and p 0 is the pressure where the equation of state transitions to a fixed crust model. An alternative approach instead parameterizes the dependence of the speed of sound [672,355]. Broad and uninformative priors are typically chosen for the parameters in each case, cf. [181,728].  \nA parallel method expresses the equation of state in a nonparametric way, a term used to refer to methods such as Gaussian processes [443,264,505,353] that parametrize the correlations between the values of a function at different points rather than the function itself. This approach offers the flexibility of directly selecting whether the equation of state prior includes strong or weak correlations [451] across different density scales while simultaneously imposing physical constraints such as thermodynamic stability and causality through the formulation of the Gaussian process.  \nBoth of the above phenomenological approaches can be augmented to include information from nuclear theory, and the associated uncertainties. Specific examples include the use of chiral effective field theory [464,673,672, 246,247] results at lower densities and perturbative quantum chromodynamics [353,648,433] results at high densities.  \nThe improving understanding about NSs properties in the recent years is the outcome of combining GW data [29,33,36,39] with other probes: radio [92, 212,310] and X-ray [504,594,505,595] observations of galactic neutron stars, observations of the electromagnetic counterpart of GW170817 [210,209,582, 487], nuclear experiments [267,598,66,590,265], and nuclear theory [469,672, 247,246]. Discussion of the intricacies involved in using such diverse data sets is beyond the scope of this review, however the emerging picture suggests that typical 1 . 4 M ⊙ NSs have radii around 12 -12 . 5 km (with uncertainties of a few kilometers) and maximum masses above 2 M ⊙ [504,505,579,578,580,444, 450,241,550,123,266]. Many questions remain about the maximum mass, the possibility of phase transitions, the effect of NSs rotation, as well as the nature of the 2 -3 M ⊙ objects observed with gravitational waves [42,263,290,674,234, 124] that upcoming GW observations have the potential to offer information about.", '6.5 Tests of General Relativity with multiple sources': "The binary dynamics described in Sec. 2.1 are derived assuming General Relativity (GR): modified gravity theories can introduce both qualitative and quantitative changes in observable binary coalescence signals. GR is commonly  \nexpected to be the low-energy limit of an as yet unknown full quantum theory of gravity, implying modifications at some, also unknown, energy or length scale [732]. An additional motivation for testing the theory is the fact that exploration of beyond-GR phenomena and dynamics can illuminate the properties of the theory itself. GW data offer the only probe of the theory in the strongly nonlinear and dynamical regime of compact binary mergers [23,733, 37,48,49].  \nEfforts to test GR are hampered by the fact that no self-consistent alternative theory has been studied in comparable depth or thoroughness. Therefore, most tests are characterized as 'phenomenological' or 'model-dependent', meaning that though they might be motivated by specific beyond-GR phenomena, they are not tuned to any specific high-energy completion of gravity. Tests of GR may be classified according to various criteria. The first criterion distinguishes between 'generation' vs. 'propagation' phenomena: those relating to how waves are generated at the source, or how they propagate to the detectors respectively. An alternative classification splits tests into those that target the fundamental or universal properties of the theory and of wave solutions (e.g., speed of propagation or existence of two transverse polarization modes), vs . 'quantitative' results (e.g., measuring the phase evolution of binary emission at some PN order). Beyond these model-independent tests, theory-specific tests have also been pursued in certain cases, e.g. [554,523].  \nCombining information about tests of GR with multiple sources proceeds again under the same hierarchical framework, with the likelihood given in Eq. (142) where Λ are now the hyperparameters that describe the population distribution of the beyond-GR effect(s). The population model, p ( θ i | Λ ) depends on the test under consideration, with two limiting cases commonly employed. The first corresponds to tests that introduce event-level parameters that are common to all events: it is equivalent to the traditional practice of 'multiplying the likelihoods of individual events' [741]. Examples here would be a massive graviton and other propagation effects, or a theory-specific test that is formulated directly in terms of the theory coupling constant(s). The population model now simplifies to a Dirac delta function  \np ( θ i | Λ 0 ) = δ ( Λ 0 -θ i ) , (165)  \nwhere Λ 0 is a common parameter that is shared by all events. The second approach instead corresponds to tests that introduce a new event-level parameter for each new event , and it is equivalent to the other traditional practice of 'multiplying the Bayes factors of individual events' [741]. The Bayes Factors here compare the GR and beyond-GR hypothesis and are obtained by marginalizing over each event's astrophysical parameters (such as masses and spins) and (in the case of parametrized GR tests) the beyond-GR parameters themselves. In this case the population model has no hyperparameters and corresponds to the fixed prior used when computing the Bayes Factor  \np ( θ i | Λ 0 ) = p ( θ i ) . (166)  \nThe assumption of completely independent event-level parameters corresponds to gravity theories that introduce a new coupling constant for each event, akin to a charge. This is a fairly strong assumption, as it prevents the analysis from 'inferring' the population prior and assumes it is fixed to what was selected during PE inference, as described in Sec. 5. If this assumption is violated, the analysis could result in incorrect physical conclusions [399].  \nBetween those two limiting cases lies the situation where the beyond-GR parameter values are not common among events, but come from a common, unknown underlying population distribution: a non-trivial population model p ( θ i | Λ ) is thus required. This is the most common situation, encompassing cases such as the parametrized post-Einsteinian inspiral test [731,194,455,68, 494,203,622,623], the inspiral-merger-ringdown consistency test [341,340,160, 400], parametrized ringdowns [184,402,398,294], and even the generic residual test [339]. Since these tests introduce phenomenological deviations to the GW signals predicted by GR [410], defining a first-principles population model is impossible. To first order, then, analyses attempt to extract the mean and the standard deviation of the population model, with GR predicting a zero value for both.  \nThen without loss of generality, the population model can be taken as a Gaussian distribution for the beyond-GR parameters [397], motivated from the fact that the Gaussian distribution maximizes the entropy (i.e., minimizes the information) for a fixed mean and standard deviation. Moreover, it is desirable to consider the population distribution of beyond-GR and 'within-GR' parameters (for example, component masses and spins) simultaneously as their measurement uncertainties are typically correlated [553], similar to the discussion of Sec. 6.3. Results from both beyond-GR-only and joint analyses find no evidence for a nonzero mean or standard deviation [397,48,49], though the latter are in general more constraining as they incorporate more information about the astrophysical properties of the detected binaries [553]. A generalization of the Gaussian population model also considers the possibility that the population consists of a mixture of two types of binary: those described within, and those beyond GR [619]. In this case the population model is a mixture of a delta function at zero, for binaries that obey GR, and a Gaussian, for binaries that violate it.  \nFinally, population constraints on beyond-GR effects are subject to selection effects, that can be quantified and included in the analysis following the steps discussed in Sec. 6.2 by simulating beyond-GR waveforms and analyzing them with existing detection pipelines. In the context of parametrized tests of binary GW phase evolution, Ref. [472] showed that current constraints after ∼ 100 detections are sufficiently constraining that the impact of selection effects is subdominant. However, this does not rule out the possibility of a 'hidden' subpopulation of binaries with large deviations from GR, as population constraints derived from the detected events would not be applicable if binaries originate from multiple populations. Unmodeled burst searches [43] that do not make use of GR compact binary models could alleviate the concern that binaries exhibiting strong deviations from GR would not be detectable in  \nthe first place. Efforts to quantify these effects for specific beyond-GR models, and to extend existing search methods, are underway, e.g., [198,202,525].", '7.1 Current and upcoming generations of compact binary observations': "The ∼ 100 transient GW signals detected during the first three LIGO-Virgo observing runs are all consistent with a compact binary origin: the coalescence of two black holes, two neutron stars, or a neutron star and a black hole. Figure 13 summarizes some key properties of these events: their source-frame chirp mass, motivated and defined in Sec. 2, and their distance from Earth. This catalog arose from an ever-expanding network of kilometre-scale advanced ground-based detectors operating in the ∼ 10 -1000 Hz regime which now includes, besides the twin LIGO [6] detectors, the European Virgo [57] and the Japanese cryogenic KAGRA [77] detector.  \nAs interferometric detectors sense GW amplitude, rather than energy, a two-fold increase in sensitivity leads approximately to an eight-fold (volumetric) increase in the detection rate. During the first and second observing runs (O1 and O2) approximately 1 compact binary coalescence was detected per month, while the third observing run (O3) brought approximately 1.5 events per week [44], see Fig. 13. Future estimates depend on the exact sensitivity  \nFig. 13: Source-frame chirp mass and luminosity distances of detected compact binaries during O3 (blue), O2 (orange), O1 (green). The selection effect due to the amplitude scaling of CBC signals for a given detector network is apparent in the 'exclusion region' at low masses and high distances. Increasing detector sensitivity allows for detection of more distant (and potentially rare) events.  \n<!-- image -->  \nM  \n/circledot  \nachieved, but the fourth observing run (O4) that started on May 24, 2023, scheduled to last for 20 months, is yielding 1 event per few days, which will eventually yield a catalog of hundreds. The next steps for the current detector network include a fifth observing run starting around 2027 [34,60], the addition of the LIGO-Aundha detector in India [620], and proposals for further LIGO improvements toward the A# and/or Voyager [67] configurations. The ultimate sensitivity of the detector network is subject to upgrade plans, but it is reasonably expected to raise the total number of detections to the thousands by the end of its lifetime.", '7.2 Next generation challenges': "The next generation of ground-based gravitational wave observatories, planned for operation in the 2030s and beyond, are expected to have a ten-fold improvement in sensitivity compared to the design sensitivity of the current generation [572,27]: this translates naively to a thousand-fold increase in the detection rate - with, though, significant corrections for cosmological evolution [195]. Such a network would be capable of detecting nearly all binary black hole mergers, and roughly half of binary neutron star mergers in the observable universe (see e.g. [612,137]). In addition to the sheer number of detections, the reduction in noise level, while undoubtedly a boon for GW astronomy, will present novel data analysis challenges for future compact binary work. The greatly increased SNR of the loudest signals means that improvements will be needed both in waveform modelling, and in noise modelling, to fully exploit future observatories.  \nThe Einstein Telescope [572] and Cosmic Explorer [271] detector designs anticipate a greatly reduced low frequency noise 'wall', allowing the early inspiral at frequencies of a few Hz to be observable in the data; thus, the lightest BNS signals will be visible for many hours and millions of signal cycles. With a possible detection rate of 10 3 events per day, signals will inevitably overlap in the sensitive band of the detectors, violating the usual model of a single signal buried in detector noise. Fortunately, unlike in the case of overlapping white dwarf binaries in LISA, the rapidly evolving CBC signal will not result in true confusion noise [409], but parameter inference can be affected when the coalescence times of two signals are within ∼ 0 . 1 s of each other [621,560,592, 409]. The residual power in overlapping signals may produce weak systematic biases in parameter estimation that could affect constraints on environmental and non-GR effects [91,391], and, unless overlapping signals are subtracted, can also impact detection efficiency [724].  \nThe greatly increased population of detections, and therefore the amount of available data, will inflate the computational cost of inference for hierarchical models which marginalise over individual event uncertainties, as explained in 6.1. More complicated population models will probably also be required to explain, or even just accurately represent, as-yet-unknown structures in the astrophysical distribution of sources. Non-parametric methods, discussed in  \nSec. 6.3, then offer a potentially useful means of modelling structure without specific assumptions for the functional form of distributions, as do classical statistical methods valid in the limit of high event count, e.g. density estimation [615].  \nFor high-SNR signals, systematic errors in the waveform models themselves will become even more important [576]: this motivates the further improvement of such models, informed both by NR simulations and advances in analytic methods, as well as methods to mitigate remaining waveform errors and uncertainties within the analysis stage [508,718]. The future development of modelling and analysis tools to address these challenges will uncover a range of possible new observables, including deviations from general relativity [549, 509], and effects of the source's environment on the signal - for instance, the presence of a third body in a coalescing system may produce an observable net acceleration along the line of sight (e.g. [395]). Louder signals will also require yet more accurate modeling of subdominant waveform effects, such as higher mode content, orbital precession, and orbital eccentricity, e.g. [584,519]. The greater distance reach of future networks will reveal sources at higher redshifts (up to z ∼ 10), which, along with improved low-frequency sensitivity, will result in detection of many more signals with chirp masses ≳ O (100) M ⊙ , whether from intermediate mass or redshifted stellar-mass black hole binaries. Such massive mergers will enable detailed investigation of the merger and ringdown portions of the signal [115,398].  \nThe greatly increased length of low-mass signals will expand the quantity of data covered by both search and parameter estimation analyses. Thanks to the chirping nature of the CBC signal, information does not arrive uniformly throughout the signal: to limit computing cost, one may thus use a reduced data rate in the early inspiral for a multi-band analysis [511], or use a reference template to perform heterodyning of the data, reducing its effective bandwidth [205,734]. Long signal durations also offer the ability to provide early warning detection and pre-merger localisation of the source, to enable electromagnetic observations from the moment of coalescence onward (see [474, 533] and discussion in Sec. 4.4); realization of this goal requires sufficiently rapid localization, and an associated low latency infrastructure to distribute pre-merger alerts [392].  \nMachine learning methods offer an avenue for performing extremely rapid inference which has been successfully demonstrated for signal durations of a few seconds, typical of BBH observations in the current detector network (see Sec. 5.4.4). These methods typically use the data stream as conditioning information for a neural network; however, as the data volume increases, the amount of information per datapoint will be reduced, motivating the development of compressed representations of the data that can efficiently encode the signal information. Large-scale use of advanced machine learning inference in production will also require more standardized and dedicated platforms to make efficient use of available processors, while retaining sufficient reliability and flexibility/scalability (e.g. Inference-as-a-Service [357]).  \nAs a fundamental consideration, any method of identifying a modelled source, whether based on ML or a classical template bank to cover the signal space, must contend with the fact that the number of independent templates will explode with longer signal durations, as seen from the fact that the fractional precision of chirp mass measurement scales inversely with the number of observed signal cycles. The presence of higher modes, precession and eccentricity also increase the number of independent templates that must be considered by an analysis This overall increase in our ability to resolve, and extract astrophysical information from, the complexity of CBC signals, implies that analysis methods are sure to continue development over the coming years.  \nIt is not only CBC signals that will pose increasing challenges in a nextgeneration network: the characterization of detector noise will likely require more complex and sophisticated methods. We cannot predict what types of non-ideal behaviour will occur in detectors whose locations are not yet decided, but with their broad-band sensitivity pushing the limits from many disparate types of noise in different frequency ranges, there are more possible sources of glitches or non-stationary disturbance. Although such effects, for instance Newtonian noise (reviewed in [368,686]), are not limiting on our current observations, with increased sensitivity weaker glitches may become visible above the stationary noise floor. Also in general, the longer signal durations implied by larger bandwidths imply a higher probability that a glitch will intersect with any given signal, placing an increasing emphasis on modelling and subtraction of artefacts. Finally, the persistent presence of signals in the data stream suggests that the detector noise properties will need to be estimated concurrently with the astrophysical signals [561], as no 'offsource' data might exist. Such considerations will become yet more pertinent with the advent of GW observatories in space, and the prospect of joint observations between Earth- and space-based interferometers [86,638,335,428] - pointing to a yet broader potential science scope, which will go hand in hand with these increased technical challenges.", '7.3 Conclusion': "The field of gravitational wave astronomy as an observational science has exploded in recent years, with compact binaries now being observed on a daily basis. Detecting and analysing CBC signals is the foundation of performing astrophysics with this wealth of new observations, and CBCs will continue to dominate the sources observed in future detectors.  \nAdjacent areas such as detector development[559,317,34], binary astrophysics [273,480], theoretical understanding of CBC sources [319,131,180,122, 625], and their potential electromagnetic counterparts [501,441], strongly interact with the observation and analysis of CBCs, and deserve their own reviews. However, in this article we have aimed to describe the foundations of practical CBC analysis, from the modelling of the signals, through detection and parameter estimation, to performing population inference with a large  \nnumber of signals. We hope the reader finds this comprehensive review useful as an entry point into the field, or as a reference for advanced research addressing the challenges that the future will hold.  \nAcknowledgements The authors acknowledge many helpful and enlightening interactions with members of the LSC, Virgo and KAGRA Compact Binary Coalescence working group on the topics of this work, as well as with the wider community; we thank Jolien Creighton, Fr'ed'erique Marion, and Ben Farr for comments on a previous version of the manuscript. K.C. was supported by NSF Grant PHY-2110111 and the Department of Energy under award number DE-SC0023101. This work has received financial support from the Xunta de Galicia (CIGUS Network of research centers), the European Union, and the 'Mar'ıa de Maeztu' Units of Excellence program CEX2020-001035-M. TD is supported by research grant PID2020-118635GB-I00 from the Spanish Ministerio de Ciencia e Innovaci'on. MF acknowledges support from the Natural Sciences and Engineering Research Council of Canada (NSERC) RGPIN-2023-05511. VR was supported by the UK Science and Technology Facilities Council grant ST/V005618/1. JV was supported by the UK Science and Technology Facilities Council grant ST/V005634/1.  \nThis research has made use of data or software obtained from the Gravitational Wave Open Science Center (gwosc.org), a service of the LIGO Scientific Collaboration, the Virgo Collaboration, and KAGRA. This material is based upon work supported by NSF's LIGO Laboratory which is a major facility fully funded by the National Science Foundation, as well as the Science and Technology Facilities Council (STFC) of the United Kingdom, the Max-Planck-Society (MPS), and the State of Niedersachsen/Germany for support of the construction of Advanced LIGO and construction and operation of the GEO600 detector. Additional support for Advanced LIGO was provided by the Australian Research Council. 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