Searches for continuous-wave gravitational radiation

Abstract

Now that detection of gravitational-wave signals from the coalescence of extra-galactic compact binary star mergers has become nearly routine, it is intriguing to consider other potential gravitational-wave signatures. Here we examine the prospects for discovery of continuous gravitational waves from fast-spinning neutron stars in our own galaxy and from more exotic sources. Potential continuous-wave sources are reviewed, search methodologies and results presented and prospects for imminent discovery discussed.

1 Introduction

The LIGO (Aasi et al. 2015a) and Virgo (Acernese et al. 2014) gravitational wave detectors have made historic discoveries over the last seven years. The first direct detection in September 2015 of gravitational waves marked a milestone in fundamental science (Abbott et al. 2016b), confirming a longstanding prediction of Einstein’s General Theory of Relativity (Einstein 1916, 1918). That the detection came from the first observation of a binary black hole merger provided a bonus not only in verifying detailed predictions of General Relativity, but in establishing unambiguously that stellar-mass black holes exist in the Universe. More than 80 binary black hole (BBH) systems have been observed since GW150914 (Abbott et al. 2016d, 2017h, i, j, 2019b, 2021e, f). Merging binary neutron star (BNS) systems (Abbott et al. 2017k, 2020a) have also been observed, including GW170817 (Abbott et al. 2017k), which was accompanied by a multitude of electromagnetic observations (Abbott et al. 2017l). Those observations confirmed the association of at least some short gamma ray bursts with binary neutron star mergers (Abbott et al. 2017g) and the onset of kilonovae in BNS mergers that contribute substantially to the heavy element production in the Universe (Abbott et al. 2017l). More recently came detections of merging neutron star—black hole (NSBH) systems (Abbott et al. 2021g). These discoveries of transient gravitational wave signals have ignited the field of gravitational wave astronomy.

This review concerns a quite different and as-yet-undiscovered gravitational wave signal type, one defined by stability and near-monochromaticity over long time scales, namely continuous waves. CW signals with strengths detectable by current and imminent ground-based gravitational wave interferometers could originate from relatively nearby galactic sources, such as fast-spinning neutron stars exhibiting non-axisymmetry (Thorne 1989), or more exotically, from strong extra-galactic sources, such as super-radiant Bose–Einstein clouds surrounding black holes (Arvanitaki et al. 2010).

We already know from prior LIGO and Virgo searches that the strengths of CW signals must be exceedingly weak [$\sim \,10^{-24}$ or less], which is consistent with theoretical expectation, from which we expect plausible CW strain amplitudes to be orders of magnitudes lower than the amplitudes of the transient signals detected to date [$\sim \,10^{-21}$]. This disparity in signal strength holds despite the much nearer distance of galactic neutron stars ($\sim$ kpc) compared to the compact binary mergers ($\sim$ 40 Mpc to multi-Gpc) seen to date. In fact, it is only their long-lived nature that gives us any hope of detecting CW signals through integration over long data spans, so as to achieve a statistically viable signal-to-noise (SNR) ratio. As discussed below, however, that SNR increases, at best, as the square root of observation time, but for most CW searches, increases with an even lower power of observation time, while computational cost increases with much higher powers. These different scalings of signal sensitivity and cost have led to a variety of approaches in targeting signals, depending on the size of signal parameter space searched.

The search for continuous gravitational radiation has been under way since the 1970’s, using data from interferometers (Levine and Stebbins 1972) and bars (Hirakawa et al. 1978; Suzuki 1995), including from early prototypes (Livas 1989) for the large gravitational wave detectors to come later. This review focuses primarily on the most recent searches from the Advanced LIGO and Virgo detectors, although summaries of search algorithm developments in the initial LIGO and Virgo era (and before) provide some historical context. For reference, the Advanced LIGO and Virgo runs to date comprise (with selected highlighted detections):

• The O1 observing run (LIGO only): September 12, 2015–January 12, 2016—First detection of gravitational waves from a BBH merger: GW150914 (Abbott et al. 2016b).

• The O2 observing run (LIGO joined by Virgo in last month): November 30, 2016–August 25, 2017—First detection of gravitational waves from a BNS merger: GW170817 (Abbott et al. 2017k).

• The O3 observing run (LIGO and Virgo): April 1, 2019–March 27, 2020—First detection of gravitational waves from the formation of an intermediate-mass black hole: GW190521 (Abbott et al. 2020d) and the first detection of NSBH mergers. The run was divided into a 6-month “O3a” epoch (April 1, 2019–October 1, 2019) and “O3b” (November 1–March 27, 2020) by a 1-month commissioning break. Many initial publications focused on results from the O3a data.

In the following, Sect. 2 reviews both conventional and exotic potential sources of CW gravitational radiation. Section 3 describes a wide variety of search methodologies being used to address the challenges of detection. Section 4 presents results (so far only upper limits) from searches based on these algorithms, with an emphasis on the most recent results from the Advanced LIGO and Virgo detectors. Finally, Sect. 5 discusses the outlook for discovery in the coming years, including the prospects for electromagnetic observations of the continuous gravitational-wave sources. This review focuses on CW radiation potentially detectable with current-generation and next-generation ground-based gravitational-wave interferometers, which are sensitive to gravitational frequencies in the human-audible band for sound. Past and future searches for lower-frequency CW radiation from supermassive black hole binaries at $\sim$ nHz frequencies using pulsar timing arrays (Manchester 2012) or from stellar-mass galactic binaries at $\sim$ mHz frequencies using the space-based LISA (Bender et al. 1996) are not discussed here.

Textbooks addressing gravitational waves, their detection and their analysis include (Misner et al. 1972; Schutz 1985; Maggiore 2008, 2018; Saulson 2017; Creighton and Anderson 2011; Jaranowski and Królak 2009; Andersson 2019). Review articles and volumes on gravitational-wave science include (Thorne 1989; Blair et al. 1991; Sathyaprakash and Schutz 2009; Pitkin et al. 2011; Freise and Strain 2010; Blair et al. 2012; Riles 2013; Romano and Cornish 2017). This review is a substantial expansion upon a briefer previous article (Riles 2017). Other reviews of CW search methodology include (Prix 2009; Palomba 2012; Lasky 2015; Sieniawska and Bejger 2019; Tenorio et al. 2021b; Piccinni 2022).

2 Potential sources of CW radiation

In the frequency band of present ground-based detectors, the canonical sources of continuous gravitational waves are galactic, non-axisymmetric neutron stars spinning fast enough to produce gravitational waves in the LIGO and Virgo detectable band (at 1$\times$, $\sim$ 4/3$\times$ or 2$\times$ rotation frequency, depending on the generation mechanism). These nearby neutron stars offer a “conventional” source of CW radiation—as astrophysically extreme as such objects are.

A truly exotic postulated source is a “cloud” of bosons, such as QCD axions, surrounding a fast-spinning black hole, bosons that can condense in gargantuan numbers to a small number of discrete energy levels, enabling coherent gravitation wave emission from boson annihilation or from level transitions. Attention here focuses mainly on the conventional neutron stars, but the exotic boson cloud scenario is also discussed.

2.1 Fast-spinning neutron stars

The following subsections give an overview of neutron star formation, structure, observables and populations, present the phenomenology of neutron-star spin-down, discuss potential sources of non-axisymmetry in neutron stars, and consider a number of particular GW search targets of interest. Although neutron stars were first postulated by Baade and Zwicky (1934) and their basic properties worked out by Oppenheimer and Volkoff (1939), the first definitive establishment of their existence came with the discovery of the first radio pulsar (Hewish et al. 1968) PSR B1919+21 in 1967 with prior theoretical support for neutron star radiation contributing to supernova remnant shell energetics (Pacini 1967) and rapid theoretical follow-up to explain the pulsation mechanism (Gold 1968; Goldreich and Julian 1969; Ruderman and Sutherland 1975).

2.1.1 Neutron star formation, structure, observables and populations

As background, this section surveys at a basic level the fundamentals of neutron star formation, structure, observables and populations. Much more detailed information can be found in the following review articles or volumes on neutron stars (Lattimer and Prakash 2001; Chamel and Haensel 2008; Becker 2009; Özel and Freire 2016), pulsars (Lorimer and Kramer 2005; Lyne and Graham-Smith 2006; Lorimer 2008), and rotating relativistic stars (Paschalidis and Stergioulas 2017).

Neutron stars are the final states of stars too massive to form white dwarfs upon collapse after fuel consumption and too light to form black holes, having progenitor masses in the approximate range 6–15 $M_\odot$ (Lyne and Graham-Smith 2006; Cerda-Duran and Elias-Rosa 2018; Stockinger et al. 2020). These remarkably dense objects, supported by neutron degeneracy pressure, boast near-nuclear densities in their crusts and well-beyond-nuclear densities in their cores. The range of densities and associated total stellar masses and radii depend on an equation of state that is not experimentally accessible in terrestrial laboratories because of the combination of high density and (relatively) low temperature. A variety of equations of state have been proposed (Lattimer and Prakash 2001), with a small subset disfavored by the measurement of neutron stars greater than two solar masses (Buballa et al. 2014), by radii of approximately ten kilometers (Miller et al. 2019b, 2021; Riley et al. 2019, 2021) and by the absence of severe tidal deformation effects in the gravitational waveforms measured for the BNS merger GW170817 (Abbott et al. 2017k, 2018c; Lim and Holt 2019; Essick et al. 2020). The detection of a $\sim$ 2.6-$M_\odot$ object in the GW190814 merger (Abbott et al. 2020e) poses a challenge to the nuclear equation of state if the object is indeed a neutron star instead of a light black hole.

In broad summary, a neutron star is thought to have a crust with outer radius between 10 and 15 km and a thickness of $\sim$ 1 km (Shapiro and Teukolsky 1983), composed near the top of a tight lattice of neutron-rich heavy nuclei, permeated by neutron superfluid. Deeper in the star, as pressure and density increase, the nuclei may become distorted and elongated, forming a “nuclear pasta” of ordered nuclei and gaps (Ravenhall et al. 1983; Caplan and Horowitz 2017). Still deeper, the pasta gives way to a hyperdense neutron fluid and perhaps undergoes phase transitions involving hyperons, perhaps to a quark-gluon plasma, or even perhaps to a solid strange-quark core (Shapiro and Teukolsky 1983; Lattimer and Prakash 2001).

Uncertainties in equation of state lead directly to uncertainties in the expected maximum mass and radius of a neutron star (Lattimer and Prakash 2001), but theoretical prejudice is consistent with the absence of observation in binary systems of neutron star masses much higher than two solar masses (Özel and Freire 2016; Demorest et al. 2010; Arzoumanian et al. 2018; Antoniadis et al. 2013; Cromartie et al. 2020; Fonseca et al. 2021). Neutron star radii are especially challenging to measure directly, with older measurements coming from X-ray measurements, where inferences are drawn from brightness of the radiation, its temperature and distance to the source, assuming black-body radiation, with corrections for the strong space-time curvature affecting the visible surface area (Özel and Freire 2016; Degenaar and Suleimanov 2018). New measurements from the NICER X-ray satellite are improving upon the precision with which mass and radius can be determined simultaneously from individual stars, constraining more tightly the allowed equations of state (Miller et al. 2019b, 2021; Bogdanov et al. 2019a, b, 2021; Raaijmakers et al. 2019, 2021; Riley et al. 2019, 2021).

Measurements of the gravitational waveform from the binary neutron star merger GW170817 have also provided new constraints and disfavor very stiff equations of state that lead to large neutron star radii (Abbott et al. 2018c). Detection of additional binary neutron star mergers in the coming years should improve these constraints. Broadly, one expects average neutron star densities of $\sim$ $7\times 10^{14}$ g $\hbox {cm}^{-3}$, well above the density of nuclear matter ($\sim$ $3\times 10^{14}$) (Lorimer and Kramer 2005), with densities at the core likely above $10^{15}$ g $\hbox {cm}^{-3}$ (Shapiro and Teukolsky 1983). See Yunes et al. (2022) for a recent review of what has been learned about the neutron star equation of state from gravitational-wave and X-ray observations. A recent Bayesian combined analysis (Huth et al. 2022) of predictions from chiral effective field theory of QCD, measured BNS gravitational waveforms, NICER X-ray observations and measurements from heavy ion (gold) collisions indicate a somewhat stiffer equation of state than previously favored and hence larger allowed radii of neutron stars.

Given the immense pressure on the nuclear matter, one expects a neutron star to assume a highly spherical shape in the limit of no rotation and, with rotation, to become an axisymmetric oblate spheroid. True axisymmetry would preclude emission of quadrupolar gravitational waves from rotation alone. Hence CW searchers count upon a small but detectable mass (or mass current) non-axisymmetry, discussed in detail in Sect. 2.1.3.

During the collapse of their slow-spinning stellar progenitors, neutron stars can acquire an impressive rotational speed as angular momentum conservation spins up the infalling matter. Even the two slowest-rotating known pulsars spin on their axes every 76 s (Caleb et al. 2022) and 24 s (Tan et al. 2018; Manchester and Hobbs 2005), implying rotational kinetic energies greater than $\sim \,10^{35}$ J, and other young pulsars with spin frequencies of tens of Hz have rotational energies of $\sim \,10^{43}$ J. Recycled millisecond pulsars acquire even higher spins via accretion from a binary companion star, leading to measured spin frequencies above 700 Hz (Hessels et al. 2006; Bassa et al. 2017) and a rotational energy of $\sim \,10^{45}$ J, or several percent of the magnitude of the gravitational bound energy of the star. This immense reservoir of rotational energy might appear to bode well for supporting detectable gravitational-wave emission, but vast energy is required to create appreciable distortions in highly rigid space-time. From the perspective of gravitational-wave energy density (Misner et al. 1972), one can define an effective, frequency-dependent Young’s modulus $Y_{\text{eff}} \sim \frac{c^2f_{\text{GW}}^2}{G}$ ($\sim \,10^{31}$ Pa for $f_{\text{GW}}\approx 100$ Hz, or 20 orders of magnitude higher than steel). As a result, one must tap a significant fraction of the reservoir’s energy loss rate in order to produce detectable radiation, as quantified below.

Most of the $\sim$ 3300 known neutron stars in the galaxy are pulsars, detected via pulsed electromagnetic emission, primarily in the radio band, but also in X-rays and $\gamma$-rays (with a small number detected optically) (Lyne and Graham-Smith 2006; Manchester and Hobbs 2005). Pulses are typically observed at the rotation frequency of the star, as a beam of radiation created by curvature radiation (Buschauer and Benford 1976) from particles that are flung out in a plasma from the magnetic poles (misaligned with the spin axis) and accelerated transversely by the magnetic field, sweeps across the Earth once per rotation (see Melrose et al. 2021, however, for a critique of this model). A subset of neutron stars presumed to have magnetic poles tilted nearly 90 degrees from the spin axis display two distinct pulses.

Other neutron stars are known from detection of X-rays from thermal emission (heat from formation and perhaps from magnetic field decay), particularly at sites consistent with the birth locations and times of supernova remnants (Lyne and Graham-Smith 2006). Still other neutron stars are inferred from accretion X-rays observed in binary systems, particularly low-mass X-ray binaries with accretion disks (Lyne and Graham-Smith 2006), although some accreting binaries with compact stars contain black holes, such as the high-mass X-ray binary Cygnus X-1. Figure 1 shows nearly the entire population of currently known pulsars (Manchester and Hobbs 2005) with spin period P shorter than 20 s¹ in the P–$\dot{P}$ plane, where $\dot{P}$ is the first time derivative of the period. Red triangles show isolated pulsars, and blue circles show binary pulsars.

Fig. 1

Measured rotational periods and period derivatives for known pulsars. Closed red triangles indicate isolated stars. Open blue circles indicate binary stars. The vertical dotted line denotes the approximate sensitivity band for Advanced LIGO at design sensitivity ($f_{\text{GW}}>10$ Hz, assuming $f_{\text{GW}}=2f_{\text{rot}}$). A similar band applies to design sensitivities of the Advanced Virgo and KAGRA detectors (Abbott et al. 2020b)

Neutron stars have strong magnetic field intensities as a natural result of their collapse. If the magnetic flux is approximately conserved, the reduction of the outer surface of the star to a radius of $\sim$ 10 km ensures a static surface field far higher than achievable in a terrestrial laboratory (Pacini 1967), with inferred values (see below) ranging from $10^8$ G to more than $10^{15}$ G (Lyne and Graham-Smith 2006). The strongest fields are seen in so-called “magnetars,” young neutron stars with extremly rapid spin-down, for which dynamo generation is also likely relevant (Guilet and Müller 2015; Mösta et al. 2015). Both in young pulsars and in binary millisecond pulsars, there is reason to believe that stronger magnetic fields are “buried” in the star from accreting plasma (Payne and Melatos 2004), although the burial mechanism is not confidently understood (Chevalier 1989; Geppert et al. 1999; Lyne and Graham-Smith 2006; Bernal et al. 2010). It has been suggested there is evidence in at least some pulsars for slowly re-emerging (strengthening) magnetic field (Ho 2011; Espinoza et al. 2011). Energy density deformation from a potentially non-axisymmetric buried field is another potential source of GW emission (Bonazzola and Gourgoulhon 1996). See Cruces et al. (2019) for a discussion of magnetic field decay preceding the accretion stage.

In principle, there should be $\sim \,10^{8-9}$ neutron stars in our galaxy (Narayan 1987; Treves et al. 2000). That only a small fraction have been detected is expected, for several reasons. Radio pulsations require high magnetic field and rotation frequency. Early studies (Goldreich and Julian 1969; Sturrock 1970; Ruderman and Sutherland 1975; Lorimer and Kramer 2005) implied the relation

$$\begin{aligned} B\cdot f_{\text{rot}}^2>1.7\times 10^{11}\,\text{G}\cdot (\text{Hz})^2, \end{aligned}$$

(1)

based on a model of radiation dominated by electron-positron pair creation in the stellar magnetosphere, a model broadly consistent with empirical observation, although the resulting “death line” (see Fig. 1) in the plane of period and period derivative is perhaps better understood to be a valley (Chen and Ruderman 1993; Zhang et al. 2000; Beskin and Litvinov 2022; Beskin and Istomin 2022).

The death line can be understood qualitatively from the following argument. The rotating magnetic field of a neutron star creates a strong electric field that pulls charged particles from the star, forming a plasma with charge density $\rho _0$ that satisfies (SI units): (Chen and Ruderman 1993)

$$ \rho _{0} = - \epsilon _{0} \nabla \cdot \left[ {(\vec{\Omega } \times \vec{r}) \times \vec{B}(\vec{r})} \right] $$

(2)

$$\approx - 2\epsilon{}_{0} {\mkern 1mu} \vec{\Omega } \cdot \vec{B}(\vec{r}), $$

(3)

where ${\vec{\Omega }}$ is the angular velocity of the star, and $\vec{B}$ is the local magnetic field at location $\vec{r}$ with respect to the star’s center. In steady-state equilibrium, one expects $\vec{E}\cdot \vec{B} \approx 0$ since free charges can move along B-field lines. In so-called “gaps,” however, where the plasma density is low, a potential difference large enough to produce spontaneous electron-positron pair production can lead to radio-frequency synchrotron radiation as the accelerated particles encounter curved magnetic fields. This emission is thought to account for most radio pulsations (Lyne and Graham-Smith 2006), where an “inner gap” refers to a region just outside the magnetic poles above the star’s surface, and an “outer gap” refers to a region where a nominally dipolar magnetic field is approximately perpendicular to the rotation direction, separating regions of proton and electron flow from the star to the region beyond the “light cylinder,” defined by the cylindrical radius at which a co-rotating particle in the magnetosphere must travel at the speed of light. For the inner gap to have a voltage drop high enough to induce an amplifying cascade of pair production leading to coherent radio wave emission imposes a minimum value on the gap potential difference $\varDelta V$ which, in general, can be approximated by (SI units): (Goldreich and Julian 1969; Sturrock 1970; Ruderman and Sutherland 1975; Chen and Ruderman 1993)

$$\begin{aligned} \varDelta V \sim {B\varOmega ^2R^3\over 2c}, \end{aligned}$$

(4)

where R is the neutron star radius, leading (in a more detailed calculation) to Eq. (1) and via magnetic dipole emission assumptions (see Sect. 2.1.2) to the death line shown in Fig. 1 (but see Smith et al. 2019 for evidence of selection effects and Pétri 2019 for a discussion of potentially important effects from higher order multipoles). Presumably, the vast majority of neutron stars created in the galaxy’s existence to date are now to the right of the line. Additional negative-sloped dashed lines in the figure indicate different nominal magnetic dipole field strengths and positive-sloped dashed lines indicate different nominal ages, based on observed present-day periods and period derivatives $P/(2\dot{P})$ (see Sect. 2.1.2).

Two distinct major pulsar populations are apparent in Fig. 1, defined by location in the diagram. The bulk of the population lies above and to the right of the line corresponding to $B\sim \,10^{11}$ G. The bulk also lies above and to the left of the line corresponding to ages younger than $\sim \,10^8$ years. Assuming a star’s magnetic field strength is stable, stars are expected to migrate down to the right along the B-field contours. Isolated pulsars seem to have typical pulsation lifetimes of $\sim \,10^7$ years (Lyne and Graham-Smith 2006), after which they become increasingly difficult to observe in radio. On this timescale, they also cool to where thermal X-ray emission is difficult to detect (Potekhin et al. 2015). There remains the possibility of X-ray emission from steady accretion of interstellar medium (ISM) (Ostriker et al. 1970; Blaes and Madau 1993), but it appears that the kick velocities from birth highly suppress such accretion (Hoyle and Lyttleton 1939; Bondi and Hoyle 1944) which depends on the inverse cube of the star’s velocity through the ISM, and steady-state X-ray emission from accretion onto even slow-moving neutron stars can be highly suppressed, consistent with non-observation to date of such accretion (Popov et al. 2015).

The remaining population, in the lower left of the figure, is characterized by shorter periods and smaller period derivatives. These are so-called “millisecond pulsars” (MSPs), thought to arise from “recycling” of rotation speed due to accretion of matter from a binary companion. MSPs are stellar zombies, brought back from the dead with immense rotational energies imparted by infalling matter (Alpar et al. 1982; Radhakrishnan and Srinivasan 1982). The rotation frequencies achievable through this spin-up are impressive—the fastest known rotator is PSR J1748−2446ad at 716 Hz (Hessels et al. 2006). One progenitor class for MSPs is the set of low mass X-ray binaries (LMXBs) in which the neutron star ($\sim$ 1.4 $M_\odot$) has a much lighter companion ($\sim$ 0.3 $M_\odot$) (Lyne and Graham-Smith 2006) that overfills its Roche lobe, spilling material onto an accretion disk surrounding the neutron star or possibly spilling material directly onto the star, near its magnetic polar caps. When the donor companion star eventually shrinks and decouples from the neutron star, the neutron star can retain a large fraction of its maximum angular momentum and rotational energy. Because the neutron star’s magnetic field decreases during accretion (through processes that are not well understood), the spin-down rate after decoupling can be very small. The minority of MSPs that are isolated are thought to have lost their one-time companions via consumption and ablation. A bridging class called “black widows” and “redbacks” refer to binary systems with actively ablating companions, such as B1957+20 (Fruchter et al. 1988; Strader et al. 2019; Roberts and van Leeuwen 2013), where black widows denote the extreme subclass with companion masses below 0.1 $M_\odot$ (Roberts and van Leeuwen 2013).

A nice confirmation of the link between LMXBs and recycled MSPs comes from “transitional millisecond pulsars” (tMSPs) in which accreting LMXB behavior alternates with detectable radio pulsations. The first tMSP found was PSR J1023$+$0038 (Bond et al. 2002; Thorstensen and Armstrong 2005; Archibald et al. 2009), with two more systems since detected (Weltevrede et al. 2018). The nominal ages of MSPs extend beyond 10$^{10}$ years, that is, some have apparent ages greater than that of the galaxy (or even that of the Universe). One possible explanation of this anomaly is reverse-torque spin-down during the Roche decoupling phase (Tauris 2012), although a recent numerical study suggests a more complex frequency evolution before and during the decoupling (Bhattacharyya 2021).

An obvious pattern in Fig. 1, consistent with the recycling model, is the higher fraction of binary systems at lower periods. For example, binary systems account for 3/4 of the lowest 200 pulsar periods (below $\sim$ 4 ms).

Aside from the disappearance of stars from this diagram as they evolve toward the lower right and cease pulsations, there are also strong selection effects that suppress the visible population. We observe pulsars only if their radiation beams cross the Earth, only if that radiation is bright enough to be seen in the observing band, and only if the radiation is not sufficiently absorbed, scattered or frequency-dispersed to prevent detection with current radio telescopes. When the Square Kilometer Array project comes to fruition in the late 2020’s, it is estimated that the current known population of pulsars will grow tenfold (Kramer and Stappers 2015).

2.1.2 Neutron star spin-down phenomenology and mechanisms

Nearly every known pulsar is observed to be spinning down, that is, to have a negative rotational frequency time derivative, implying loss of rotational kinetic energy. As discussed below in detail, there are many physical mechanisms, electromagnetic and gravitational, that can lead to this energy loss. For CW signal detection we want a gravitational-wave component, but there is good reason to believe that electromagnetic processes dominate for nearly every known pulsar.

A convenient and commonly used phenomenological model for spin-down is a power law:

$$\begin{aligned} \dot{f}= K f^n, \end{aligned}$$

(5)

where f is the star’s instantaneous frequency (rotational $f_{\text{rot}}$ or gravitational: $f_{\text{GW}}\propto f_{\text{rot}}$), $\dot{f}$ is the first time derivative, and K is a negative constant for all but a handful of stars (thought to be experiencing large acceleration toward us because of nearness to a deep gravitational well, such as in the core of a globular cluster). The exponent n depends on the spin-down mechanism and is known as the braking index. The four most common theoretical braking indices discussed in the literature are the following:

• $n=1$—“Pulsar wind” (extreme model)

• $n=3$—Magnetic dipole radiation

• $n=5$—Gravitational mass quadrupole radiation (“mountain”)

• $n=7$—Gravitational mass current quadrupole radiation ($r$-modes).

In principle, other oscillation modes that can generate gravitational waves are also possible, but the $n\!=\!5$ and $n\!=\!7$ modes discussed below are thought to be the most promising.

Assuming the same power law has applied since the birth of the star, the age $\tau$ of the star can be related to its birth rotation frequency $f_0$ and current frequency f by ($n\ne 1$):

$$\begin{aligned} \tau \, = \, -\left[ {f\over (n-1)\,\dot{f}}\right] \,\left[ 1-\left( {f\over f_0}\right) ^{(n-1)}\right] , \end{aligned}$$

(6)

and in the case that $f\ll f_0$,

$$\begin{aligned} \tau \, \approx \, -\left[ {f\over (n-1)\,\dot{f}}\right] . \end{aligned}$$

(7)

A common baseline assumption in radio pulsar astronomy is that the braking index is $n=3$ from which the nominal magnetic dipole age of a star can be defined

$$\begin{aligned} \tau _{\text{mag}} \equiv -{f\over 2\dot{f}}, \end{aligned}$$

(8)

again, under the assumption $f\ll f_0$.

From the more generic power-law spin-down model (Eq. (5)), the 2nd frequency derivative can be written:

$$\begin{aligned} \ddot{f} = nKf^{n-1}\dot{f} = nK^2f^{2n-1}, \end{aligned}$$

(9)

from which the current braking index can be determined if the spin frequency’s 2nd time derivative can be measured reliably:

$$\begin{aligned} n \, = \, {f\ddot{f} \over \dot{f}^2}. \end{aligned}$$

(10)

Before examining the empirical measurements of the braking indices, which are mostly inconsistent with $n=3$, let’s briefly review spin-down mechanisms with well defined braking indices, when dominant. For GW radiation spin-down dominance, related “spin-down” limits on strain amplitude will also be presented.

2.1.2.1 “Pulsar wind” ($n=1$)

Early on in pulsar astronomy (Michel 1969; Michel and Tucker 1969) it was recognized that the streaming of relativistic particles (electrons and positrons mainly, with some ions) away from the magnetosphere of a fast-spinning neutron star would lead to a spin-down torque that could, in principle, rival that from magnetic dipole radiation, in addition to distorting the shape of the magnetic field lines and affecting the dipole radiation (Gaensler and Slane 2006). In this perhaps too-simple model, the spin-down is dominated by a braking torque from a return current (predominantly counter-flowing electrons and positrons) crossing magnetic field lines in the polar cap regions of the star (Contopoulos et al. 1999), leading to a braking index of one. A more recent study of magnetar spin-down (Harding et al. 1999) considered a model with sporadic high winds following bursts, with magnetic dipole emission dominating spin-down between bursts. In the steady state, however, considering the interaction of the magnetic field and the plasma of the magnetosphere, both magnetic dipole emission and pulsar wind contributions tend to yield a braking index of about three (Michel and Li 1999; Spitkovsky 2004), discussed next. A phenomenological model (Melatos 1997) that is a variant of the vacuum dipole mode, featuring an inner magnetosphere strongly coupled to the star, accounts successfully for the braking indices of the Crab and other young pulsars with $n<1$.

2.1.2.2 Magnetic dipole ($n=3$)

The radiation energy loss due to a rotating magnetic dipole moment is (Pacini 1968)

$$\begin{aligned} \left( {dE\over dt}\right) _{\text{mag}} \, = \, -{\mu _0M_\perp ^2\omega _{\text{rot}}^4\over 6\pi c^3}, \end{aligned}$$

(11)

where $\omega _{\text{rot}}$ is the rotational angular speed and $M_\perp$ is the component of the star’s magnetic dipole moment perpendicular to the rotation axis (taken to be the z axis): $M_\perp =M\sin (\alpha )$, with $\alpha$ the angle between the axis and north magnetic pole.

In a pure dipole moment model, the magnetic pole field strength at the surface is $B_0 = \mu _0M\,/\,2\pi R^3$. Equating the radiation energy loss to that of the (Newtonian) rotational energy ${1\over 2}I_{\text{zz}}\omega _{\text{rot}}^2$ leads to the prediction:

$$\begin{aligned} {d\omega _{\text{rot}}\over dt} \, = \, -{2\pi \over 3} {R^6 \over \mu _0c^3I_{\text{zz}}}B_\perp ^2\omega _{\text{rot}}^3. \end{aligned}$$

(12)

Hence the magnetic dipole spin-down rate is proportional to the square of $B_\perp =B_0\sin (\alpha )$ and to the cube of the rotation frequency, giving $n=3$.

2.1.2.3 Gravitational mass quadrupole (“mountain”, $n=5$)

Let’s now consider the gravitational radiation one might expect from these stars. It is conventional to characterize a star’s mass quadrupole asymmetry by its equatorial ellipticity:

$$\begin{aligned} \epsilon \, \equiv \, {|I_{xx}-I_{yy}|\over I_{\text{zz}}}. \end{aligned}$$

(13)

An oblate spheroid naturally has a polar ellipticity, but in the absence of precession,² such a deformation does not lead to GW emission. Henceforth “ellipticity” will refer to equatorial ellipticity, often attributed to a “mountain”. For a star at a distance d away and spinning about the approximate symmetry axis of rotation (z), (assumed optimal—pointing toward the Earth), then the expected intrinsic strain amplitude $h_0$ is

$$\begin{aligned} h_0= & {} {4\,\pi ^2G\epsilon I_{\text{zz}}f_{\text{GW}}^2\over c^4d} \end{aligned}$$

(14)

$$\begin{aligned}= & {} (1.1\times 10^{-24})\left( {\epsilon \over 10^{-6}}\right) \left( {I_{\text{zz}}\over I_0}\right) \left( {f_{\text{GW}}\over 1\,\text{kHz}}\right) ^2 \left( {1\,\text{kpc}\over d}\right) , \end{aligned}$$

(15)

where $I_0=10^{38}\, \text{kg}\cdot \text{m}^2 (10^{45}\ {\text{g}}\cdot \text{cm}^2$) is a nominal moment of inertia of a neutron star used throughout this article, and the gravitational radiation is emitted at frequency $f_{\text{GW}}=2\,f_{\text{rot}}$. The total power emission in gravitational waves from the star (integrated over all angles) is

$$\begin{aligned} {dE\over dt}= & {} - {32\over 5} {G\over c^5}\,I_{\text{zz}}^2\, \epsilon ^2\, \omega _{\text{rot}}^6 \end{aligned}$$

(16)

$$\begin{aligned}= & {} - (1.7\times 10^{33}\,\mathrm{J/s})\left( {I_{\text{zz}}\over I_0}\right) ^2 \left( {\epsilon \over 10^{-6}}\right) ^2 \left( {f_{\text{GW}}\over 1\,\text{kHz}}\right) ^6. \end{aligned}$$

(17)

Equating this loss to the reduction of rotational kinetic energy ${1\over 2}I_{\text{zz}}\omega _{\text{rot}}^2$ leads to the spin-down relation:

$$\begin{aligned} {\dot{f}}_{\rm{GW}}= & {} -{32\,\pi ^4\over 5}{G\over c^5}I_{\rm{zz}}\epsilon ^2f_{\rm{GW}}^5 \end{aligned}$$

(18)

$$\begin{aligned}= & {} -(1.7\times 10^{-9}\, \mathrm{Hz/s}) \left( {\epsilon \over 10^{-6}}\right) ^2 \left( {f_{\rm{GW}}\over 1\mathrm{\,kHz}}\right) ^5, \end{aligned}$$

(19)

in which the braking index of 5 is apparent.

For an observed neutron star of measured f and $\dot{f}$, one can define the “spin-down limit” on maximum allowed strain amplitude by equating the power loss in Eq. (16) to the time derivative of the (Newtonian) rotational kinetic energy: ${1\over 2}I_{\rm{zz}}\omega _{\rm{rot}}^2$, as above for magnetic dipole radiation. One finds:

$$\begin{aligned} h_{\mathrm{spin-down}}= & {} {1\over d}\sqrt{-{5\over 2}{G\over c^3}I_{\rm{zz}}{{\dot{f_{\rm{GW}}}}\over f_{\rm{GW}}}} \nonumber \\= & {} (2.6\times 10^{-25}) \left[ {1\,\rm{kpc}\over d} \right] \! \left[ \left( {1\,\rm{kHz}\over f_{\rm{GW}}}\right) \! \left( {-{\dot{f_{\rm{GW}}}} \over 10^{-10}\,\mathrm{Hz/s}}\right) \! \left( {I_{\rm{zz}}\over I_0}\right) \! \right] ^{1\over 2}\!\!.\, \,\,\, \end{aligned}$$

(20)

Hence for each observed pulsar with a measured frequency, spin-down and distance d, one can determine whether or not energy conservation even permits detection of gravitational waves in an optimistic scenario. Unfortunately, nearly all known pulsars have strain spin-down limits below what can be detected by the LIGO and Virgo detectors at current sensitivities, as detailed below.

2.1.2.4 Gravitational mass current quadrupole ($r$-modes, $n=7$)

Different frequency scalings apply to mass quadrupole and mass current quadrupole emission. The most promising source of the mass current non-axisymmetry in neutron stars is thought to be “$r$-modes,” due to fluid motion of neutrons (or protons) in the crust or core of the star. Like jet streams in the Earth’s atmosphere that manifest Rossby waves, these currents are deflected by Coriolis forces, giving rise to spatial oscillations (Andersson 1998; Bildsten 1998; Friedman and Morsink 1998; Owen et al. 1998). These $r$-modes can be inherently unstable, arising from azimuthal interior currents that are retrograde in the star’s rotating frame, but which are prograde in an external reference frame. As a result, the quadrupolar gravitational-wave emission due to these currents leads to an increase in the strength of the current. This positive-feedback loop leads to a potential intrinsic (Chandrasekhar–Friedman–Schutz; Chandrasekhar 1970; Friedman and Schutz 1978) instability. The frequency of such emission is expected to be a bit more than approximately 4/3 the rotation frequency (Andersson 1998; Bildsten 1998; Friedman and Morsink 1998; Owen et al. 1998; Kojima 1998; Caride et al. 2019).

Following the notation of Owen (Owen 2010; Caride et al. 2019), the mass current can be treated as due to a velocity field perturbation $\delta v_j$, integration over which leads to the following expression for the intrinsic strain amplitude seen at a distance d:

$$\begin{aligned} h_0= & {} \sqrt{512\,\pi ^7\over 5} {G\over c^5}{1\over d}f_{\rm{GW}}^3\alpha MR^3{\tilde{J}} \end{aligned}$$

(21)

$$\begin{aligned}= & {} 3.6\times 10^{-26} \left( {1\,\rm{kpc}\over d}\right) \left( {f_{\rm{GW}}\over 100\,\rm{Hz}}\right) ^3 \left( {\alpha \over 10^{-3}}\right) \left( {R\over 11.7\,\rm{km}}\right) ^3, \end{aligned}$$

(22)

where $\alpha$ is the dimensionless $r$-mode amplitude, M is the stellar mass, R its radius, and ${\tilde{J}}$ is a dimensionless functional of the stellar equation of state, which for a Newtonian polytrope with index 1 gives ${\tilde{J}}\approx .0164$ (Owen 2010), assumed in the fiducial Eq. (22).

The energy loss in this model is (Thorne 1980; Owen 2010)

$$\begin{aligned} {dE\over dt}= & {} -{1024\,\pi ^9\over 25} {G\over c^7} f_{\rm{GW}}^8 \alpha ^2 M^2 R^6 {\tilde{J}}^2. \end{aligned}$$

(23)

Equating this loss to the reduction of rotational kinetic energy ${1\over 2}I_{\rm{zz}}\omega _{\rm{rot}}^2$, as above, leads to the spin-down relation:

$$\begin{aligned} {\dot{f}}_{\rm{GW}}= & {} -{4096\,\pi ^7\over 225} {G\over c^7} {M^2R^6{\tilde{J}}^2\over I_{\rm{zz}}} \alpha ^2f_{\rm{GW}}^7 \end{aligned}$$

(24)

$$\begin{aligned}= & {} -9.0\times 10^{-14}\,\mathrm{Hz/s}\left( {R\over 11.7\,\rm{km}}\right) ^6 \left( {\alpha \over 10^{-3}}\right) ^2 \left( {f_{\rm{GW}}\over 100\,\rm{Hz}}\right) ^7, \end{aligned}$$

(25)

in which the braking index of 7 is apparent.

As before, one can define a spin-down limit, but one based on pure $r$-mode radiation:

$$\begin{aligned} h_{\mathrm{spin-down}} = {1\over r}\sqrt{-{45\over 8}{G\over c^3}I_{\rm{zz}}{{\dot{f}}_{\rm{GW}}\over f_{\rm{GW}}}}, \end{aligned}$$

(26)

where the ratio of this spin-down limit to the one given in Eq. (20) is 3/2, which arises simply from the different ratios of GW signal frequency to spin frequency for mass quadrupole vs. mass current quadrupole radiation.³

2.1.2.5 Measured braking indices

Figure 2 shows the distribution of 12 reliably measured braking indices from a recent snapshot of the $\sim$ 3300 pulsars listed in the ATNF catalog [release V1.66—January 10, 2022 (Manchester and Hobbs 2005)]. Nearly all have values below the nominal value of 3 for a magnetic dipole radiator, although several have large uncertainties.

Fig. 2

Measured braking indices inferred from frequency derivatives of young pulsars with rotation frequencies greater than 10 Hz. For frequently glitching pulsars, such as Vela, the braking index is computed as a long-term average (Espinoza et al. 2017). Horizontal bars indicate uncertainties and are smaller than the plot markers for several pulsars. Vertical lines at braking indices of 3 and 5 denote the nominal expectations for magnetic dipole and gravitational quadrupole emission, respectively. References: 1 (Lyne et al. 2015), 2 (Ferdman et al. 2015), 3 (Espinoza et al. 2017), 4 (Weltevrede et al. 2011), 5 (Clark et al. 2016), 6 (Livingstone and Kaspi 2011), 7 (Archibald et al. 2016), 8 (Espinoza et al. 2011), 9 (Roy et al. 2012), 10 (Livingstone et al. 2007)

Fig. 3

Measured braking indices inferred from frequency derivatives of pulsars compiled in [Lower et al. (2021)—“this work”, including from Parthasarathy et al. (2020)—PJS]. An ensemble of glitching (open green circles) and non-glitching pulsars are included. For most of the glitching stars, the braking indices are representative of their average inter-glitch braking, not their long-term evolution

This distribution suggests that the model of a neutron star spinning down with constant magnetic field is, most often, inaccurate (Lyne and Graham-Smith 2006). All measured values for this collection lie below 3.0, except X-ray pulsar PSR J1640−4631 with a measured index of 3.15 ± 0.03 (Archibald et al. 2016). It is possible that for many stars the departure of the measured braking index from the nominal value is due to an admixture of magnetic dipole radiation and other steady-state processes (Melatos 1997), although secular mechanisms may also play a role. See Palomba (2000, 2005) for discussions of spin-down evolution in the presence of both gravitational-wave and electromagnetic torques. Other suggested mechanisms for less-than-3 braking indices are decaying magnetic fields (Romani 1990), re-emerging buried magnetic fields (Ho 2011), a changing inclination angle between the magnetic dipole axis the spin axis (Middleditch et al. 2006; Tauris and Konar 2001; Ho 2015; Lyne et al. 2015; Johnston and Karastergiou 2017), and a changing superfluid moment of inertia (Ho and Andersson 2012).

An interesting observation of the aftermath of two short GRBs noted indirectly inferred braking indices near or equal to three (Lasky et al. 2017a), suggesting the rapid spin-down of millisecond magnetars, possibly born from neutron star mergers. (No direct gravitational-wave evidence of a such a post-merger remnant has been observed from GW170817 (Abbott et al. 2017o, 2019d).) Similarly, a recent analysis of X-ray afterglows of gamma-ray bursts (Sarin et al. 2020) argues that at least some have millisecond magnetar remnants powering their emission, with GRB 061121 yielding a braking index $n=4.85^{+0.11}_{-0.15}$, consistent with gravitational radiation dominance (albeit with large required ellipticity, Ho 2016; Kashiyama et al. 2016). See Strang et al. (2021), however, for an alternative study in which radiation driven from a millisecond magnetar can account for short GRB X-ray afterglows. See Dall’Osso and Stella (2022) for a recent brief review of millisecond magnetars, including evidence of their serving as central engines to create GRBs, and see Jordana-Mitjans et al. (2022) for evidence of a protomagnetar remnant in the aftermath of GRB 180618A.

It has been argued that the inter-glitch evolution of spin for the X-ray pulsar PSR J0537−6910 displays behavior consistent with a braking index of 7, (Andersson et al. 2018; Ho et al. 2020) consistent with $r$-mode emission, while the long-term trends points to an underlying braking index of −1.25±0.01 (Ho et al. 2020). When intepreting the generally low values of well measured braking indices, one must bear in mind the potential for selection bias. Baysesian analysis of the spin evolution of 19 young pulsars (Parthasarathy et al. 2019, 2020), taking into account timing noise and extracting the long-term behavior from short-term, glitch-driven fluctuations, leads to braking indices much larger than 3. A similar follow-up analysis of an ensemble of glitching and non-glitching pulsars (Lower et al. 2021) confirmed that braking indices exceeding 100 are observed (see Fig. 3), albeit for stars in which a simple power-law spin-down is clearly inappropriate.

2.1.2.6 The gravitar model and associated figures of merit

Gravitars refer to neutron stars with spin-down dominated by gravitational-wave energy loss (Palomba 2005). Although there is good reason to believe that most known pulsars are not gravitars, nonetheless the model is useful in bounding expectation on what is possibly detectable. Figure 4 shows a subset of the pulsars from Fig. 1, now graphed in the $f_{\rm{GW}}$–${\dot{f}}_{\rm{GW}}$ plane, under the assumption that $f_{\rm{GW}}= 2\,f_{\rm{rot}}$. Again, isolated and binary stars are denoted by closed circles and open triangles, respectively. A vertical dashed line bounds the approximate detection bandwidth for Advanced LIGO at design sensitivity ($\sim$ 10 Hz and above). The same approximate frequency boundary applies to the design sensitivities of the Advanced Virgo and KAGRA detectors (Abbott et al. 2020b). As in Fig. 1, contours are shown for constant magnetic field, assuming spin-down dominated by magnetic dipole emission ($n=3$). In addition, contours of higher slope are shown for constant ellipticity. An intriguing deficit of millisecond pulsars with extremely low period derivatives appears consistent (Woan et al. 2018) with a population of sources with a minimum ellipticity of about $\sim \,10^{-9}$ with additional spin-down losses from magnetic dipole radiation (see near absence of sources in Fig. 4 to the right of the $\epsilon =10^{-9}$ line). At the other extreme are lower-frequency, younger pulsars with high spin-downs, the highest of which is $7.6\times 10^{-10}$ Hz/s (Crab pulsar).

Fig. 4

Nominal expected GW frequencies and frequency derivatives for known pulsars. Closed triangles indicate isolated stars. Open circles indicate binary stars. Contours are shown for constant magnetic fields (ellipticities) for spin-down dominated by magnetic dipole (gravitational mass quadrupole) emissions. In this figure and in Figs. 5, 6, 7 and 8, the frequency derivatives have been corrected for the Shklovskii effect (Shklovskii 1970) (apparent negative frequency derivative due to proper motion orthogonal to the line of sight). The vertical dotted line denotes the approximate sensitivity band for Advanced LIGO at design sensitivity. A similar band applies to design sensitivities of the Advanced Virgo and KAGRA detectors (Abbott et al. 2020b)

Using Eq. (20), these known pulsars can be mapped onto a plane of $f_{\rm{GW}}$–$h_0$ under the gravitar assumption, indicated in Fig. 5. That is, the spin-down strain limit (for $n=5$) is shown on the vertical axis. Also shown are corresponding contours of constant implied values of $\epsilon /d$, under the gravitar assumption, where d is the distance to the star. In addition, detector network sensitivities are shown for advanced detectors at design sensitivity (Abbott et al. 2020b) and for two proposed configurations of the “3rd-generation” Einstein Telescope (ET) (Maggiore et al. 2020) (ETB and ETC, for three detectors for five observing years). Another 3rd-generation proposal is for the “Cosmic Explorer” (Abbott et al. 2017c) which would have performance comparable to that of ET, being more sensitive at frequencies above $\sim$ 10 Hz and less sensitive at lower frequencies. To avoid clutter in these figures, only the ET sensitivities are shown.

In Fig. 5 and in succeeding figures, the “advanced detector” sensitivities are represented by those computed for two Advanced LIGO detectors running continuously for two observing years, henceforth designated as the “O4/O5 run”. Although the O4 run scheduled to start near the start of 2023 will likely run for only $\sim$ 1 year (Abbott et al. 2020b) and may not quite reach the original Advanced LIGO design sensitivity, the succeeding O5 run in the “$\hbox {A}^+$” configuration is expected to exceed Advanced LIGO sensitivity significantly and to last for more than a year, making the detector sensitivities assumed here conservative, in principle. Including Advanced Virgo and KAGRA into the network sensitivity would improve these sensitivities still further. On the other hand, the O4/O5 observing time assumed here does not account for realistic deadtime losses, which can be substantial ($\sim$ 25% per detector, Davis et al. 2021). The detection sensitivities shown in Fig. 5 assume a targeted search (discussed below) using known pulsar ephemerides. If a star is marked above a sensitivity curve, then it is at least possible to detect it if its spin-down makes it a gravitar. Note, however, that Eq. (20) has been applied with a nominal moment of inertia $I_{\rm{zz}}$, but the uncertainty in $I_{\rm{zz}}$ is of order a factor of two, depending on equation of state and stellar mass (Worley et al. 2008).

Fig. 5

Nominal expected GW frequencies and nominal strain spin-down limits for known pulsars. Closed triangles indicate isolated stars. Open circles indicate binary stars. The solid curves indicate the nominal (idealized) strain noise sensitivity for the O3 observing run (black), and expected sensitivities for 2-year advanced detector data run at design sensitivity (magenta) and a 5-year Einstein Telescope data run for two different detector designs: ETB (blue) and ETC (green). Dashed diagonal lines correspond to particular quotients of ellipticity over distance. A subset of pulsars of particular interest are labeled on the figure

Another take on the pulsars with accessible spin-down limits is shown in Fig. 6, where accessible ellipticity $\epsilon$ values are shown for advanced detector and Einstein Telescope (ETC) sensitivities. Each vertical bar represents a range of ellipticities detectable for that star (red = accessible to advanced detectors, green = accessible to Einstein Telescope), where the asterisk at the top of the each bar is the ellipticity corresponding to that star’s spin-down limit, given its $f_{\rm{GW}}$, ${\dot{f}}_{\rm{GW}}$ and distance d values, while the depth to which the bar falls indicates the lowest detectable ellipticity. Straight dashed lines of negative slope depict corresponding ${\dot{f}}_{\rm{GW}}$ values under the mass quadrupole gravitar model. The actual ${\dot{f}}_{\rm{GW}}$ may be significantly higher because of the spin-down mechanisms discussed earlier. A striking feature of this figure is that sensitivities to very low ellipticities come almost entirely from the highest-frequency stars (as a reminder from Eq. (14), $h_0\propto \epsilon f_{\rm{GW}}^2$). For example, no known pulsar with an ellipticity below $10^{-6}$ and that is accessible to advanced detectors has a $f_{\rm{GW}}$ value lower than 70 Hz, and no ellipticity below $10^{-8}$ is accessible to advanced detectors below 300 Hz.

Fig. 6

Nominal expected GW frequencies and maximum allowed ellipticities for known pulsars. Black or blue asterisks indicate ellipticities accessible with advanced detectors or ETC sensitivities (3 detectors, 5 years), respectively, using targeted searches, where red vertical lines terminated by red asterisks indicate ellipticity sensitivity range for advanced detectors, and green vertical lines and green asterisks indicate additional ellipticity sensitivity range for ETC. A selection of pulsars accessible with advanced detector sensitivity are labeled in red. Diagonal dashed lines correspond to corresponding ${\dot{f}}_{\rm{GW}}$ values under the gravitar model

Another figure of merit is the distance to which searches can detect sources of a particular ellipticity. Figure 7 shows the estimated distances to known pulsars over the detection frequency band. Also shown are solid contours of advanced detector sensitivity range for different ellipticity values and dashed contours for Einstein Telescope. Pulsars with spin-down limits accessible to advanced detectors are shown in red, and those accessible to Einstein Telescope are shown in green. Only a handful of pulsars within 500 pc are accessible to advanced detectors with ellipticities below 10$^{-8}$. On the other hand, to reach the galactic center ($\sim$ 8.5 kpc) at a signal frequency of 1 kHz requires an ellipticity larger than $\sim$ $3\times 10^{-8}$, and at 100 Hz requires an ellipticity greater than $\sim$ $3\times 10^{-6}$.

Fig. 7

Maximum allowed targeted-search ranges for gravitars versus GW frequencies for different assumed ellipticities for advanced detector sensitivity (solid magenta curves) and corresponding ranges for ETC sensitivity (dashed blue curves). Known pulsar distances are shown versus the expected GW frequencies, where red dots indicate pulsars with accessible spin-down limits for advanced detector sensitivity, and smaller green dots indicated pulsars with accessible spin-down limits for ETC sensitivity. Known pulsars in distinct horizontal bands (common distance) arise from stars in clusters or from distance capping in the galactic electron density model (Yao et al. 2017) used in the ATNF catalog (Manchester and Hobbs 2005)

As discussed in detail below, all-sky searches for unknown neutron stars necessarily have reduced sensitivity, such that the ranges shown for targeted searches using known pulsar timing do not apply. Figure 8 shows another range versus frequency plot, but for which (optimistic) advanced detector and Einstein Telescope all-sky sensitivities are assumed. For reference, the all-sky strain sensitivity is taken to be about 20 times worse than its targeted-search sensitivity for advanced detector and the corresponding ratio about 40 times worse for Einstein Telescope.⁴ Consequently, the all-sky range contours corresponding to those in Fig. 7 would be reduced by the same ratios. Alternatively, to obtain the same ranges in the all-sky search would require ellipticities higher by the same ratios. The all-sky ranges in Fig. 8, in contrast, are shown as contours for different assumed ${\dot{f}}_{\rm{GW}}$ values under the gravitar assumption. These contours are useful in assessing all-sky searches, since those searches are defined, in part, by their maximum spin-down range, which affects computational cost. Once again, known pulsars for which this search technique can reach the spin-down limit are shown in red for advanced detector and in green for Einstein Telescope. We see that for the advanced detectors to reach the galactic center at a signal frequency of 1 kHz requires a minimum spin-down magnitude greater than $10^{-9}$ Hz/s (minimum because another mechanism, such as magnetic dipole emission, may contribute to a higher spin-down magnitude), and at 100 Hz requires a minimum spin-down magnitude just less than $10^{-10}$ Hz/s. The corresponding required ellipticities at those frequencies are $\sim 8\times 10^{-5}$ and $\sim 8\times 10^{-7}$, respectively.

Fig. 8

Maximum allowed (optimistic) all-sky-search ranges for gravitars versus GW frequencies for different assumed spin-down derivatives for advanced detector sensitivity (solid magenta curves) and corresponding ranges for ETC sensitivity (dashed green curves). Known pulsar distances are shown versus the expected GW frequencies, where red dots indicate pulsars with accessible spin-down limits for advanced detector sensitivity, and smaller green dots indicated pulsars with accessible spin-down limits for ETC sensitivity. These all-sky search ranges assume an optimistic sensitivity depth of 50 $\hbox {Hz}^{-1/2}$ (see Sect. 3.8)

A simple steady-state argument by Blandford (Thorne 1989) led to an early estimate of the maximum detectable strain amplitude expected from a population of isolated gravitars of a few times 10$^{-24}$, independent of typical ellipticity values, in the optimistic scenario that most neutron stars become gravitars. A later detailed numerical simulation (Knispel and Allen 2008) revealed, however, that the steady-state assumption does not generally hold for mass quadrupole radiation, leading to ellipticity-dependent expected maximum amplitudes that can be 2–3 orders of magnitude lower in the LIGO/Virgo/KAGRA band for ellipticities as low as 10$^{-9}$ and a few times lower for ellipticity of about $10^{-6}$. Mass current quadrupole ($r$-mode) emission, however, would spin stars down faster, leading back to more optimistic maximum amplitudes (Owen 2010). A more detailed simulation including both electromagnetic and gravitational wave spin-down demonstrated the potential for setting joint constraints on natal neutron star magnetic fields and ellipticities (Wade et al. 2012). A recent population simulation study (Reed et al. 2021) estimated fractions of neutron stars probed by previous CW searches for different assumed ellipticities and concluded that the greatest potential gain from improving detector sensitity in accessing more neutron stars of plausible ellipticity comes at higher frequencies.

The spin-down limit on strain defined in Eq. (20) for known pulsars requires knowing the frequency $f_{\rm{GW}}$, its first derivative ${\dot{f}}_{\rm{GW}}$ and the distance d to the star. There are other neutron stars for which no pulsations are observed, hence for which neither $f_{\rm{GW}}$ nor ${\dot{f}}_{\rm{GW}}$ is known, but for which the distance and the age of the star are known with some precision. For such stars one can define an “age-based” limit—under the assumption of gravitar behavior since the neutron star’s birth in a supernova event. Using Eq. (7) and a braking index of 5 for mass quadrupole radiation gives the gravitar age:

$$\begin{aligned} \tau _{\rm{gravitar}} = -{f_{\rm{rot}}\over 4\,{\dot{f_{\rm{rot}}}}}. \end{aligned}$$

(27)

Therefore, if one knows the distance and the age of the star, e.g., from the expansion rate of its visible nebula, then under the assumption that the star has been losing rotational energy since birth primarily due to gravitational-wave emission, then one has the following frequency-independent age-based limit on strain (Wette et al. 2008):

$$\begin{aligned} h_{\rm{age}} = (2.3\times 10^{-24})\left( {1\,\rm{kpc}\over r}\right) \sqrt{\left( {1000\,\rm{yr}\over \tau }\right) \left( {I_{\rm{zz}}\over I_0}\right) }, \end{aligned}$$

(28)

along with a corresponding frequency-dependent but distance-independent ellipticity upper limit (Wette et al. 2008):

$$\begin{aligned} \epsilon _{\rm{age}} = (2.2\times 10^{-4}) \left( {100\,\rm{Hz}\over f_{\rm{GW}}}\right) ^2 \sqrt{ \left( {1000\,\rm{yr}\over \tau }\right) \left( {I_0\over I_{\rm{zz}}}\right) }. \end{aligned}$$

(29)

The corresponding calculation for $r$-mode emission leads to the age-based strain limit relation (Owen 2010):

$$\begin{aligned} h_{\rm{age}}^{r\mathrm{-mode}} = (1.9\times 10^{-24})\left( {1\,\rm{kpc}\over r}\right) \sqrt{\left( {1000\,\rm{yr}\over \tau }\right) \left( {I_{\rm{zz}}\over I_0}\right) }, \end{aligned}$$

(30)

along with a corresponding frequency-dependent but distance-independent $r$-mode amplitude upper limit (Wette et al. 2008):

$$\begin{aligned} \alpha _{\rm{age}} = 0.076 \left( {1000\,\rm{yr}\over \tau }\right) ^{1/2} \left( {100\,\rm{Hz}\over f_{\rm{GW}}}\right) ^2. \end{aligned}$$

(31)

Yet another empirically determined strain upper limit can be defined for accreting neutron stars in binary systems, such as Scorpius X-1. The X-ray luminosity from the accretion is a measure of mass accumulation rate at the surface. As the material rains down on the surface it can add angular momentum to the star, which in equilibrium may be radiated away in gravitational waves. Hence one can derive a torque-balance limit (Wagoner 1984; Papaloizou and Pringle 1978; Bildsten 1998) in the form (Watts et al. 2008):

$$\begin{aligned} h_{\rm{torque}}\sim & {} (3\times 10^{-27}) \left( {R\over \mathrm {10\,km}}\right) ^{3\over 4} \left( {M_\odot \over M}\right) ^{1\over 4} \nonumber \\{} & {} \times \left( {1000\,\rm{Hz}\over f_{\rm{rot}}}\right) ^{1\over 2} \left( {\mathcal {F}_{\rm{x}}\over 10^{-8}\,\mathrm{erg/cm}^2/\rm{s}}\right) ^{1\over 2}, \end{aligned}$$

(32)

where $\mathcal {F}_{\rm{x}}$ is the observed energy flux at the Earth of X-rays from accretion, M is the neutron star mass and R its radius. Taking nominal values of R = 10 km, $M = 1.4 M_\odot$ and reformulating in terms of the gravitational-wave frequency $f_{\rm{GW}}$ (benchmarked to 600 Hz), one obtains:

$$\begin{aligned} h_{\text{torque}}\, \sim \, (5\times 10^{-27}) \left( {600\,\text{Hz}\over f_{\text{GW}}}\right) ^{1\over 2} \left( {\mathcal {F}_\text{x}\over 10^{-8}\,\mathrm{erg/cm}^2/\text{s}}\right) ^{1\over 2}. \end{aligned}$$

(33)

Equations 32 and 33 assume the radius at which the accretion torque is applied is the stellar surface. If one assumes the torque lever arm is the Alfvén radius because of the coupling between the stellar rotation and the magnetosphere, then the implied equilibrium strain is $\sim$ 2.4 times higher (Abbott et al. 2019e). This limit is independent of the distance to the star. In general, variations in accretion inferred from X-ray flux fluctuations suggest similar (slower) fluctuations in the equilibrium frequency, which could degrade GW detection sensitivity for coherent searches that assume exact equilibrium. A first attempt to address these potential frequency fluctuations for Scorpius X-1 may be found in (Mukherjee et al. 2018). See Serim et al. (2022) for a recent compilation of timing fluctuations of seven accretion-powered pulsars, providing evidence that accretion fluctuations indeed dominate timing noise.

2.1.3 Assessing potential sources of neutron star non-axisymmetry

From the above, it is clearly possible for neutron stars in our galaxy to produce continuous gravitational waves detectable by current ground-based detectors, but is it likely that putative emission mechanisms are strong enough to give us a detection in the next few years. Let’s look more critically at those mechanisms.⁵

Isolated neutron stars may exhibit intrinsic non-axisymmetry from residual crustal deformation—e.g., from “starquakes” due to cooling and cracking of the crust (Pandharipande et al. 1976; Kerin and Melatos 2022) or due to changing centrifugal stress induced by stellar spin-down (Ruderman 1969; Baym et al. 1969; Fattoyev et al. 2018; Giliberti and Cambiotti 2022)—from non-axisymmetric distribution of magnetic field energy trapped beneath the crust (Zimmermann 1978; Cutler 2002) or from a pinned neutron superfluid component in the star’s interior (Jones 2010; Melatos et al. 2015; Haskell et al. 2022). See Haskell et al. (2015); Singh et al. (2020) for a discussion of emission from magnetic and thermal “mountains” and Lasky (2015); Glampedakis and Gualtieri (2018) for recent, comprehensive reviews of GW emission mechanisms from neutron stars.

Maximum allowed asymmetries depend on the neutron star equation of state (Johnson-McDaniel and Owen 2013; Krastev et al. 2008) and on the breaking strain of the crust. Detailed molecular dynamics simulations borrowed from condensed matter theory have suggested in recent years that the breaking strain may be an order of magnitude higher than previously thought feasible (Horowitz and Kadau 2009; Caplan and Horowitz 2017). Analytic treatments (Baiko and Chugunov 2018) indicate, however, that anisotropy may be important and caution that simulations based on relatively small numbers of nuclei may not capture effects due to a polycrystalline structure in the crust. A recent cellular automaton-based simulation (Kerin and Melatos 2022) of a spinning-down neutron star used nearest-neighbour tectonic interactions involving strain redistribution and thermal dissipation. That study found the resulting annealing led to emitted gravitational strain amplitudes too low to be detected by present-generation detectors.

A recent revisiting of the mountain-building scenario (Gittins et al. 2020) finds systematically lower ellipticities to be realistic. It is argued in Woan et al. (2018) that a possible minimum ellipticity in millisecond pulsars may arise from asymmetries of buried internal magnetic field $B_i$ (Cutler 2002; Lander et al. 2011; Lander 2014) of order of Woan et al. (2018)

$$\begin{aligned} \epsilon \sim 10^{-8}\,\left( {<\!B_i\!>\over 10^{12}\,\rm{G}}\right) \left( {<\!H_c\!>\over 10^{16}\,\rm{G}}\right) , \end{aligned}$$

(34)

where $H_c$ is the lower critical field for superconductivity (protons in the stellar core are assumed to form a Type II superconductor). Hence, a buried toroidal (equatorial) field of $\sim$ 10$^{11}$ G could yield an ellipticity at the 10$^{-9}$ level. It has been argued, on the other hand, that an explicit model of braking dynamics with non-axisymmetry due to magnetic field non-axisymmetry leads to still smaller ellipticities, based on observed braking indices of younger pulsars (de Araujo et al. 2016, 2017), where the magnetic contribution to the ellipticity depends quadratically on the field strength (Bonazzola and Gourgoulhon 1996; Konno et al. 1999; Regimbau and de Freitas Pacheco 2006). An analysis (Osborne and Jones 2020) of internal magnetic field contributions to non-axisymmetric temperature distributions in the neutron star crust finds that high field strengths ($>10^{13}$ G) are needed in an accreting system for GW emission to halt spin-up from the accretion, four orders of magnitude higher than is expected for surface fields in LMXBs. A follow-up study (Hutchins and Jones 2022) finds more optimistically large thermal asymmetries to be possible deeper in a star, and another study (Morales and Horowitz 2022) finds a maximum allowed ellipticity of $\sim 7.4\times 10^{-6}$.

$r$-modes (mass current quadrupole, see Sect. 2.1.2) offer an intriguing alternative GW emission source (Mytidis et al. 2015). Serious concerns have been raised, (Arras et al. 2003; Glampedakis and Gualtieri 2018) however, about the detectability of the emitted radiation for young isolated neutron stars, for which mode saturation appears to occur at low $r$-mode amplitudes because of various dissipative effects (Owen 2010). Another study, (Alford and Schwenzer 2014) though, is more optimistic about newborn neutron stars. The same authors, on the other hand, find that $r$-mode emission from millisecond pulsars is likely to be undetectable by advanced detectors (Alford and Schwenzer 2015).

The notion of a runaway rotational instability was first appreciated for high-frequency f-modes, (Chandrasekhar 1970; Friedman and Schutz 1978) (Chandrasekhar–Friedman–Schutz instability), but realistic viscosity effects seem likely to suppress the effect in conventional neutron star production (Lindblom and Detweiler 1977; Lindblom and Mendell 1995). Moreover, (Ho et al. 2019) set limits on the $r$-modes amplitude $\alpha$ for J0952−0607 below $10^{-9}$ based on the absence of heating observed in its X-ray spectrum, despite its high rotation frequency (707 Hz) which places it in the nominal $r$-modes instability window. Similarly, (Boztepe et al. 2020) set limits on $\alpha$ as low as $3\times 10^{-9}$, based on observations of two other millisecond pulsars (PSR J1810$+$1744 and PSR J2241−5236) which also sit in the instability window. Another potential source of $r$-modes dissipation is from the interaction of “ordinary” and superfluid modes, leading to a stabilization window for LMXB stars (Gusakov et al. 2014; Kantor et al. 2020). The f-mode stability could play an important role, however, for a supramassive neutron star formed as the remnant of a binary neutron star merger (Doneva et al. 2015) (spinning too fast to collapse immediately despite exceeding the nominal maximum allowed neutron star mass).

In addition, as discussed below, a binary neutron star may experience direct non-axisymmetry from non-isotropic accretion (Owen 2005; Ushomirsky et al. 2000; Melatos and Payne 2005) (also possible for an isolated young neutron star that has experienced fallback accretion shortly after birth), or may exhibit $r$-modes induced by accretion spin-up.

Given the various potential mechanisms for generating continuous gravitational waves from a spinning neutron star, detection of the waves should yield valuable information on neutron star structure and on the equation of state of nuclear matter at extreme pressures, especially when combined with electromagnetic observations of the same star.

The notion of gravitational-wave torque equilibrium is potentially important, given that the maximum observed rotation frequency of neutron stars in LMXBs is substantially lower than one might expect from calculations of neutron star breakup rotation speeds ($\sim$ 1400 Hz) (Cook et al. 1994). It has been suggested (Chakrabarty et al. 2003) that there is a “speed limit” due to gravitational-wave emission that governs the maximum rotation rate of an accreting star. In principle, the distribution of frequencies could have a quite sharp upper frequency cutoff, since the angular momentum emission is proportional to the 5th power of the frequency for mass quadrupole radiation. For example, for an equilibrium frequency corresponding to a particular accretion rate, doubling the accretion rate would increase the equilibrium frequency by only about 15%. For $r$-mode GW emission, with a braking index of 7, the cutoff would be still sharper.

Note, however, that a non-GW speed limit may well arise from interaction between the neutron star’s magnetosphere and an accretion disk (Ghosh and Lamb 1979; Haskell and Patruno 2011; Patruno et al. 2012). It has also been argued (Ertan and Alpar 2021) that correlation between the accretion rate and the frozen surface dipole magnetic field resulting from Ohmic diffusion through the neutron star crust in the initial stages of accretion in low mass X-ray binaries can explain a minimum rotation period well above the naive expectation.

A number of mechanisms have been proposed by which the accretion leads to gravitational-wave emission in binary systems. The simplest is localized accumulation of matter, e.g., at the magnetic poles (assumed offset from the rotation axis), leading to a non-axisymmetry. One must remember, however, that matter can and will diffuse into the crust under the star’s enormous gravitational field. This diffusion of charged matter can be slowed by the also-enormous magnetic fields in the crust, but detailed calculations (Vigelius and Melatos 2010) indicate the slowing is not dramatic. Relaxation via thermal conduction is considered in (Suvorov and Melatos 2019).

Another proposed mechanism is excitation of $r$-modes in the fluid interior of the star, (Andersson 1998; Bildsten 1998; Friedman and Morsink 1998; Owen et al. 1998) with both steady-state emission and cyclic spin-up/spin-down possible (Levin 1999; Heyl 2002; Arras et al. 2003). Intriguing, sharp lines consistent with expected $r$-mode frequencies were reported in the accreting millisecond X-ray pulsar XTE J1751−305 (Strohmayer and Mahmoodifar 2014a) and in a thermonuclear burst of neutron star 4U 1636−536 (Strohmayer and Mahmoodifar 2014b). The inconsistency of the observed stellar spin-downs for these sources with ordinary $r$-mode emission, however, suggests that a different type of oscillation is being observed (Andersson et al. 2014) or that the putative r-modes are restricted to the neutron star crust and hence gravitationally much weaker than core r-modes (Lee 2014). Another recent study (Patruno et al. 2017) suggests that spin frequencies observed in accreting LMXB’s are consistent with two sub-populations, where the narrow higher-frequency component ($\sim$ 575 Hz with standard deviation of $\sim$ 30 Hz) may signal an equilibrium driven by gravitational-wave emission. It has been suggested (Haskell and Patruno 2017) that the transitional millisecond pulsar PSR J1023$+$0038 (for which spin-down has been measured in both accreting and non-accreting states) shows evidence for mountain building (or $r$-modes) during the accretion state, based on different spin-downs observed in accreting vs. non-accreting states. It has also been argued (Bhattacharyya 2020) that J1023$+$0038 shows evidence for a permanent ellipticity in the range $0.48-0.93\times 10^{-9}$. An analysis (Chen 2020) of three transitional millisecond pulsars and ten redbacks concluded their ellipticities ranged over $0.9-23.4\times 10^{-9}$.

A recent analysis (De Lillo et al. 2022) based on the absence of evidence of a stochastic gravitational-wave background emitted by a population of neutron stars with a rotational frequency distribution similar to that of known pulsars inferred that the average ellipticity of the galactic population is less than $\sim 2\times 10^{-8}$.

2.1.4 Particular GW targets

In the following, particular neutron star targets for gravitational-wave searches are discussed in the following categories: known young pulsars with high spin-down rates; known high-frequency millisecond pulsars; neutron stars in supernova remnants, neutron stars in low-mass X-ray binary systems; and particular directions on the sky.

2.1.4.1 Known young pulsars with high spin-down rates

A young pulsar with a high spin-down rate presents an attractive target. Its age offers the hope of a star not yet annealed into smooth axisymmetry, a hope strengthened by the prevalence of observed timing glitches among young stars. A high spin-down rate not only makes it more likely that the spin-down limit is accessible, but also suggests a star with a reservoir of magnetic energy, some of which could give rise to non-axisymmetry. From the Advanced LIGO/Virgo O1, O2 and O3 data sets more than 20 pulsars were spin-down accessible (Abbott et al. 2019b, 2022j) (see Sect. 4.1), but most correspond to ellipticities of $\sim 10^{-4}$–$10^{-3}$. A small number are highlighted here, for which ellipticities below $10^{-5}$ are accessible already or with a 2-year data run at advanced detector design sensitivity (“O4/O5 run”).

• Crab (PSR J0534+2200)—This pulsar, created in a 1054 A.D. supernova observed by Chinese astronomers and discovered in 1968 (Staelin and Reifenstein 1968), has received more attention from LIGO / Virgo analysts than any other. Its spin-down limit was first beaten in the initial LIGO data set S5 (Abbott et al. 2008b), and now has been beaten (O3 data) by a factor of $\sim$ 100 (Abbott et al. 2022j) (see Sect. 4.1), leading to a 95% upper limit on ellipticity of $1.0\times 10^{-5}$. For a 2-year O4/O5 run, this sensitivity reaches $\sim 2\times 10^{-6}$. Spinning at just below 30 Hz, its nominal $f_{\rm{GW}}$ is just below 60 Hz, making the detector spectrum susceptible to power mains contamination (including non-linear upconversion, see Sect. 3.7) in the LIGO and KAGRA interferometers, but not in the Virgo interferometer, which uses 50 Hz power mains. Its inferred rotational kinetic energy loss rate based on its spin-down is $dE/dt \sim -5\times 10^{38}$ erg $\hbox {s}^{-1}$, assuming the nominal $I_{\rm{zz}}= 10^{38}$ kg $\hbox {m}^2$ (10$^{45}$ g $\hbox {cm}^{2}$).

• Vela (PSR J0835-4510)—Although older and lower in frequency than the Crab with a higher ellipticity spin-down limit ($1.9\times 10^{-3}$), the Vela pulsar, discovered in 1968 (Large et al. 1968), is nonetheless interesting, given its frequent glitches (Manchester 2018; Ashton et al. 2019). Its O4/O5 ellipticity sensitivity reaches $~\sim 8\times 10^{-6}$. Spinning at just above 11 Hz, its nominal $f_{\rm{GW}}$ is about 22 Hz, where detector noise is several times higher than at the Crab frequency. Its inferred $dE/dt \sim -7\times 10^{36}$ erg $\hbox {s}^{-1}$.

• PSR J0537-6910—This pulsar, observed to pulse only in X-rays, is distant ($\sim$ 50 kpc in the Large Magellanic Cloud). With a rotation frequency of $\sim$ 62 Hz, its nominal GW frequency of 124 Hz is quite high for a young pulsar (magnetic dipole spin-down age $\sim$ 5000 years), and its spin-down energy loss is comparable to the Crab’s. It is also extremely glitchy ($\sim$ 1 per 100 days) (Antonopoulou et al. 2018; Ferdman et al. 2018) and as noted above, may show evidence of $r$-mode emission between glitches (Andersson et al. 2018; Ho et al. 2020) (which would imply a GW frequency at $\sim$ 90 Hz). Its inferred $dE/dt \sim -5\times 10^{38}$ erg $\hbox {s}^{-1}$.

• PSR J1400-6325—This relatively recently discovered X-ray pulsar (Renaud et al. 2010) lies in a supernova remnant 7–10 kpc away and displays a spin-down energy about 1/10 of the Crab pulsar’s, but may be younger than 1000 years. With a spin frequency of $\sim$ 32 Hz, its nominal $f_{\rm{GW}}$ is 64 Hz, comparable to the Crab’s, but farther from the troublesome 60 Hz power mains. Its inferred $dE/dt \sim -5\times 10^{37}$ erg $\hbox {s}^{-1}$.

• PSR J1813-1749—First detected as a TeV $\gamma$-ray source (Aharonian et al. 2005), this star was found to exhibit non-thermal X-ray emission and to have a tentative association with a radio supernova remnant G12.8$-$0.0 (Brogan et al. 2005) suggesting a distance greater than 4 kpc and an age perhaps younger than 1000 years. X-ray pulsations detected still later with a period of 44 ms confirmed a pulsar source and posited an association with a young star cluster at 4.7 kpc (Gotthelf and Halpern 2009), while yielding a nominal pulsar spin-down age of 3.3$-$7.5 kyr. A more recent detection of highly dispersed radio pulsations, however, suggest a distance of 6 or 12 kpc (Camilo et al. 2021), depending on electron dispersion model, casting doubt on the association with the star cluster. The spin frequency of 22 Hz yields a nominal GW frequency of $\sim$ 45 Hz, and the frequency derivative imply $dE/dt \sim -6\times 10^{37}$ erg $\hbox {s}^{-1}$.

2.1.4.2 Known high-frequency millisecond pulsars

Because nearly all millisecond pulsars are old, with some characteristic ages greater than 10 billion years, they can be assumed to retain little asymmetry from their initial formation or from the accretion that spun them up. Thus one sees low spin-down for this population in Fig. 5 and hence low inferred maximum ellipticities in Fig. 6. On the other hand, the vast energy reservoirs in their rotation and the quadratic dependence of $h_0$ on frequency still makes these stars potentially intriguing. As noted above, there may be empirical evidence for a minimum ellipticity of order $\sim 10^{-9}$ (Woan et al. 2018). Highlighted below are particular millisecond pulsars of interest in the coming years.

• PSR J0711-6830 This isolated star at a distance of 0.11 kpc, with a nominal $f_{\rm{GW}}\sim$ 364 Hz, a spin-down upper limit of $1.2\times 10^{-26}$ and corresponding maximum ellipticity of $9.4\times 10^{-9}$, is the first MSP to have its spin-down limit beaten (in early O3 data, see Sect. 4.1).

• PSR J0437–4715 This binary star at a distance of 0.16 kpc, with a nominal $f_{\rm{GW}}\sim$ 347 Hz, a spin-down upper limit of $7.8\times 10^{-27}$ and corresponding maximum ellipticity of $9.7\times 10^{-9}$ also had its spin-down limit beaten (in the full O3 data).

• PSR J1737–0811 This binary star at a distance of 0.21 kpc, with a nominal $f_{\rm{GW}}\sim$ 479 Hz, a spin-down upper limit of $5.3\times 10^{-27}$ and corresponding maximum ellipticity of $4.6\times 10^{-9}$, will likely have its spin-down limit beaten by the O4/O5 data set.

• PSR J1231–1411 This binary star at a distance of 0.42 kpc, with a nominal $f_{\rm{GW}}\sim$ 543 Hz, a spin-down upper limit of $2.8\times 10^{-27}$ and corresponding maximum ellipticity of $3.8\times 10^{-9}$, will likely have its spin-down limit beaten by the O4/O5 data set.

• PSR J2124–3358 This binary star at a distance of $\sim$ 0.4 kpc, with a nominal $f_{\rm{GW}}\sim$ 406 Hz, a spin-down upper limit of $2.3\times 10^{-27}$ and corresponding maximum ellipticity of $5.6\times 10^{-9}$, will likely have its spin-down limit beaten by the O4/O5 data set.

• PSR J1643–1224 This binary star at a distance of 0.79 kpc,⁶ with a nominal $f_{\rm{GW}}\sim$ 433 Hz, a spin-down upper limit of $2.1\times 10^{-27}$ and corresponding maximum ellipticity of $8.0\times 10^{-9}$, may not have its spin-down limit beaten by the O4/O5 data set, but as noted by Woan et al. (2018), would have the highest GW SNR of any known star if its ellipticity were $10^{-9}$.

2.1.4.3 Central compact objects and Fomalhaut b

Not every neutron star of interest has been detected to pulsate. Central compact objects (CCOs) at the heart of supernova remnants present especially intriguing targets, especially those in remnants inferred from their size and expansion rate to be young (De Luca 2008). There may be direct evidence of a neutron star, such as from thermal X-rays emitted from a hot surface or from X-rays due to interstellar accretion, or there may be indirect evidence from a pulsar wind nebula driven by a fast-spinning star at the core. Most GW searches to date for a CCO lacking detected pulsations have focused on the particularly promising source, Cassiopeia A, but in recent years, such searches have also been carried out for as many as 15 supernova remnants (Abbott et al. 2019a, 2021i). Highlighted below are particular supernova remnants (“G” naming terminology based on the Green Catalog (Green 2014), see also Ferrand and Safi-Harb 2012) with known or suspected central compact objects, in addition to an object, Fomalhaut b, originally thought to be an exoplanet, but which may be a nearby neutron star. Results from searches for these targets are presented further below in Sect. 4.2.

• Cassiopeia A—Cas A (G111.7−2.1) is perhaps the most promising example of gravitational-wave CCO source in a supernova remnant. Its birth aftermath may have been observed by Flamsteed (Hughes 1980) $\sim$ 340 years ago in 1680, and the expansion of the visible shell is consistent with that date (Fesen et al. 2006). Hence Cas A, which is visible in X-rays (Tananbaum 1999; Ho et al. 2021) but shows no pulsations (Halpern and Gotthelf 2009), is almost certainly a very young neutron star at a distance of about 3.3 kpc (Reed et al. 1995; Alarie et al. 2014). From Eq. (28), one finds an age-based strain limit of $\sim$ $1.2\times 10^{-24}$, which is readily accessible to LIGO and Virgo detectors in their most sensitive band.

• Vela Jr.—This star (G266.2−1.2) is observed in X-rays (Pavlov et al. 2001; Kargaltsev et al. 2002; Becker et al. 2006) and is potentially quite close ($\sim$ 0.2 kpc) and young (690 years) (Iyudin et al. 1998), but searches have also conservatively assumed more a more pessimistic distance (0.9 kpc) and age (5100 years), based on other measurements (Allen et al. 2015). The optimistic age and distance assumptions lead to an age-based strain limit of $\sim$ $1.4\times 10^{-23}$, even more accessible than the Cas A limit. Even the pessimistic age-base limit of $1.1\times 10^{-24}$ is only slightly lower than that of Cas A. It has been argued (Ming et al. 2016) that a search over multiple CCOs, optimized for most likely detection success given fixed computing resources, favors focusing those resources on Vela Jr. over other CCOs, including Cas A.

• G347.3–0.5—An X-ray source (Slane et al. 1999; Ho et al. 2021) is consistent with the core of this supernova remnant, the nearness (~ 0.9 kpc) and youth (1600 years) of which make a search aimed at the remnant’s center intriguing, as they yield an age-based strain limit of $\sim$ $2.0\times 10^{-24}$—higher than that of Cas A.

• G1.9+0.3 This supernova remnant, the youngest in the galaxy at 100 years (Reynolds et al. 2008), has no detected CCO at its core, which is consistent with a Type IA supernova’s having left no neutron star behind. Nonetheless, its youth make it interesting despite this doubt and its distance (8.5 kpc), yielding an age-based strain limit of $\sim$ $8.4\times 10^{-25}$.

• Fomalhaut b This object was assumed to be an extrasolar planet (Kalas et al. 2008) until (Neuhäuser et al. 2015) noted that the absence of detected infrared radiation could indicate the object is a remarkably nearby neutron star ($\sim$ 0.01 kpc). The absence of attempted X-ray detection with Chandra observations (Poppenhaeger et al. 2017), however, disfavors its being a young, hot neutron star. More recent evidence (Gáspár and Rieke 2020) argues, in fact, that the optical observations point to a planetesimal collision.

2.1.4.4 Neutron stars in low-mass X-ray binary systems

Because of its high X-ray flux ($\mathcal {F}_{\rm{x}}\sim 3.9\times 10^{-7}\, \mathrm{erg\ cm}^{-2}\, \rm{s}^{-1}$, Watts et al. 2008) and the torque-balance relation for low-mass X-ray binaries (Eq. 32), Scorpius X-1 is thought to be an especially promising search target for advanced detectors and has been the subject of multiple searches in initial and Advanced gravitational-wave detector data. From Eq. (33), one expects a strain amplitude limited by Abbott et al. (2007a) and Messenger et al. (2015)

$$\begin{aligned} h \, \sim \, (3\times 10^{-26})\,\left( {600\,\rm{Hz}\over f_{\rm{GW}}}\right) ^{1\over 2}. \end{aligned}$$

(35)

While Sco X-1’s rotation frequency remains unknown (Galaudage et al. 2021), its orbital period is well measured, (Gottlieb et al. 1975; Wang et al. 2018) which allows substantial reduction in search space. A similar but less bright LMXB system is Cygnus X-2 (Premachandra et al. 2016) at a distance of 7 kpc and an average flux $\mathcal {F}_{\rm{x}} = 11\times 10^{-9}$ erg/$\hbox {cm}^2$ $\hbox {s}^{-1}$ (Galloway et al. 2008), yielding a torque-balance strain limit about 20 times lower than that of Sco X-1. Unlike Sco X-1 which is assumed but not known to contain a neutron star (as opposed to a black hole with an accretion disk), Cyg X-2 has displayed thermonuclear bursts, confirming the presence of a neutron surface.

Another interesting class contains “accreting X-ray millisecond pulsars” (AXMPs) which are fast-spinning neutron stars in LMXBs that show sporadic outbursts during accretion episodes (when “active”) from which rotation frequencies can be determined. When active, the frequencies can increase or decrease, while frequencies between outbursts (when “quiescent”) generally decrease. One could hope to detect CW radiation from either active or quiescent phases. Although the limited durations of bursts and their stochastic nature constrain potential search sensitivity, it is during such outbursts when one might expect the largest generation of non-axisymmetries or excitation of $r$-modes. The fastest-spinning stars, such as IGR J00291$+$5934 at $f_{\rm{rot}}\sim$ 599 Hz and a distance of $\sim$ 4 kpc (Torres et al. 2008; Patruno 2017), offer deeper probing of equatorial ellipticity and $r$-mode amplitude. Current search sensitivities to strain amplitude (Abbott et al. 2022g) remain an order of magnitude or more away from inferred spin-down limits ($\sim$ 10$^{-28}$–10$^{-27}$), but improvements in detector sensitivity, search methodology and potential future electromagnetic observations make this type of source potentially intriguing in the coming years.

2.1.4.5 Particular sky directions

In addition to known (or suspected) neutron stars, there are other localized sky regions or points where a directed search might yield a continuous gravitational-wave detection. Listed below are possibilities that have attracted attention in recent years.

• Galactic center—The vicinity of the galactic center (Sgr A*) is particularly interesting (Aasi et al. 2013a), as an active, star-forming region with known pulsars (Deneva et al. 2009). Moreover, it is highly likely that only a small fraction of pulsars near the galactic center have been detected to date, since there is extreme dispersion and scattering of radio signals along the propagation line to the Earth (Lazio and Cordes 1998). The inference of there being many hidden pulsars is supported by $\sim$ 20 pulsar wind nebula candidates detected within 20 pc of Sgr A* (Muno et al. 2008). In addition, searches for dark-matter annihilation signals have detected an excess of high-energy gamma ray emission from the galactic center region above what is expected from conventional models of diffuse gamma-ray emission and catalogs of known gamma-ray sources (Ackermann et al. 2017), a tension which may be resolved by the existence of a hidden population of millisecond pulsars (Abazajian 2011). A systematic radio survey of the central 1 parsec of Sgr A* at a frequency of 15 GHz (Macquart and Kanekar 2015), high enough to reduce dispersion and scattering substantially, yielded no detections, but the rapidly falling spectrum of most pulsars makes detection at 15 GHz at that distance difficult. This survey obtained a 90% CL upper limit of 90 on the number of pulsars within 1 parsec of Sgr A*, assuming the population there is similar to known pulsars. Unfortunately, the $\sim$ 8.5 kpc distance to the galactic center makes CW searches challenging with present detector sensitivities. Only stars with extreme ellipticities are accessible to advanced detectors at design sensitivity (see Fig. 8). At the same time, however, young neutron stars are those most likely to exhibit such ellipticities.

• Globular cluster cores—One normally associates globular clusters with ancient stellar populations and might expect, at best, to see only pulsars that are themselves ancient – recycled and well annealed millisecond pulsars. Indeed many MSPs are seen in globular clusters (Freire 2012). For example, Tucanae 47 is known to host at least 25 MSPs (Freire et al. 2017). Nonetheless, not all observed pulsars in globular clusters seem to be old (Freire 2012). A plausible explanation is that the dense core of a globular cluster leads to multibody exchange interactions in which a previously recycled but decoupled neutron star acquires a close new companion that proceeds to overflow its Roche lobe, leading to new accretion. Another, related mechanism is possible debris accretion triggered by multibody interactions, given that some pulsars are known to host debris disks and even planets (Abbott et al. 2017m). The well localized core of a globular cluster makes a deep, directed search tractable.

• High-latitude Fermi sources The Fermi satellite’s LAT experiment has detected $\sim$ 100 previously unknown gamma ray pulsars since observing began in 2008. Gamma ray pulsars tend to be sources with low variability and relatively low spectral cutoffs, and most lie near the galactic plane, as expected. Fermi-LAT point sources well outside the galactic plane tend to be extagalactic, e.g., active galactic nuclei, but an intriguing possiblity is that a source with high galactic latitude could be a galactic neutron star, in which case the high latitude favors a nearby source (Sanders 2016), consistent with a scale height of $\sim$ 600 pc with respect to the galactic plane observed for known pulsars (Lyne and Graham-Smith 2006). Arguing against this possibility, however, are extensive searches for gamma-ray pulsations from pulsar-like Fermi-LAT sources (see, e.g., Clark et al. 2018), based in part on algorithms developed for CW gravitational-wave searches (Pletsch et al. 2011). On the other hand, such searches are challenged to probe binary sources with large accelerations, suggesting that CW searches directed at such sources include algorithms sensitive to binary sources (Neunzert 2019).

In between all-sky searches and directed searches for single sky points reside “spotlight” searches, in which a patch of sky is searched more deeply than in all-sky searches (with increased computational cost), but less deeply than is computationally feasible for a single sky location. Such spotlights have been applied in searches for a broad star-forming region along two directions of the Orion spur of the local galactic spiral arm (Aasi et al. 2016b) and toward the galactic center region, including the globular cluster Terzan 5 (Dergachev et al. 2019).

2.2 Axion clouds bound to black holes

An intriguing potential connection between gravitational waves and the still-unknown missing dark matter of the Universe comes from the possibility that the dark matter is composed of ultralight, electromagnetically invisible bosons, such as axions. One novel idea is that these bosons could be disproportionately found in the vicinity of rapidly spinning black holes (Arvanitaki et al. 2010; Arvanitaki and Dubovsky 2011). The ultralight particles could, in principle, be spontaneously created via energy extraction from the black hole’s rotation (Penrose 1969; Christodoulou 1970) and form a Bose–Einstein “cloud” with nearly all of the quanta occupying a relatively small number of energy levels. For a cloud bound to a black hole, the approximate inverse-square law attraction outside the Schwarzchild radius ($r_{\mathrm{Schwarz.}}\equiv {2\,GM_{\rm{BH}}\over c^2}$) leads to an energy level spacing directly analogous to that of the hydrogen atom (Arvanitaki et al. 2010; Baumann et al. 2019). The number of quanta occupying the low-lying levels can be amplified enormously by the phenomenon of superradiance in the vicinity of a rapidly spinning black hole (with angular momentum that is a signficant fraction of the maximum value allowed in General Relativity). The bosons in a non-s ($\ell >0$) negative-energy state can be thought of as propagating in a well formed between an $\ell$-dependent centrifugal barrier at $r>r_{\mathrm{Schwarz.}}$ and a potential rising toward zero as $r\rightarrow \infty$; wave function penetration into the black hole ergosphere permits transfer of energy from the black hole spin (Zel’dovich 1971; Misner 1972; Starobinskiǐ 1973) into the creation of new quanta.

Two particular gravitational-wave emission modes of interest here can arise in the axion scenario, both potentially leading to intense coherent radiation (Arvanitaki et al. 2015). In one mode, axions can annihilate with each other to produce gravitons with frequency double that corresponding to the axion mass: $f_{\rm{graviton}} = 2m_{\rm{axion}}c^2/h$. In another mode, emission occurs from level transitions of quanta in the cloud. This Bose condensation is most pronounced when the reduced Compton wavelength of the axion is comparable to but larger than the scale of the black hole’s Schwarzchild radius:

(36)

$$\begin{aligned} \Rightarrow \quad m_{\rm{axion}}&\,\lessapprox \,(7\times 10^{-11} {\,\mathrm{eV/c^2}}) {M_\odot \over M_{\rm{BH}}}, \end{aligned}$$

(37)

where $\hbar$ is the reduced Planck constant and G is Newton’s gravitational constant. A key parameter governing detectability is a parameter analogous to the electromagnetic fine structure constant:

$$\begin{aligned} \alpha \equiv {Gm_{\rm{axion}}M_{\rm{BH}}\over \hbar c}, \end{aligned}$$

(38)

where both the growth rate of a cloud upon black hole formation and the amplitude of gravitational-wave emission due to axion annihilation depend on high powers of $\alpha$. Hence small $\alpha$ impedes detection; at the same time, superradiance itself requires (Isi et al. 2019):

$$\begin{aligned} \alpha< {1\over 2}m\chi \left( 1+\sqrt{1-\chi ^2}\right) ^{-1} < {m\over 2}, \end{aligned}$$

(39)

where $\chi$ is the dimensionless black hole spin proportional to its total angular momentum magnitude J: $\chi = {cJ\over GM_{\rm{BH}}^2}$, and m is the quantum number corresponding to the axion’s orbital angular momentum projection along the spin axis of the black hole (the first level to be populated in a newborn black hole is $m=1$, Isi et al. 2019). Hence the range of $\alpha$ (and therefore axion mass) for which a particular black hole produces superradiance may be narrow. In general, more massive black holes produce stronger signals over wider ranges in axion mass. Clouds composed of ultralight vector or tensor bosons would lead to stronger, but shorter-lived signals (Siemonsen and East 2020; Brito et al. 2020). Nominal limits on axion masses can be placed based on the existence of high-spin binary black holes in our galaxy (Arvanitaki et al. 2017; Cardoso et al. 2018), but those limits are subject to uncertainties in inferred black hole spins (Reynolds 2014; McClintock et al. 2014) and may be invalidated by tidal disruption effects from the companion star (Cardoso et al. 2020). Constraints have also been inferred from spin measurements in the population of binary black hole merger detections (Ng et al. 2021).

Given the many orders of magnitude of uncertainty in, for example, axion masses that could account for dark matter (Bertone 2010), the relatively narrow mass window accessible to currently feasible CW searches (1–2 orders of magnitude) makes searching for such an emission a classic example of “lamppost” physics, where one can only hope that nature places the axion in this lighted area of a vast parameter space.

In principle, searching for these potential CW sources requires no fundamental change in the search methods described below, but search optimization can be refined for the potentially very slow (and positive) frequency evolution expected during annihilation emission (as the relative magnitude of the axion field’s binding energy decreases). In addition, for a known black hole location, a directed search can achieve better sensitivity than an all-sky search. For string axiverse models, however, the axion cloud (Arvanitaki et al. 2010; Arvanitaki and Dubovsky 2011; Yoshino and Kodama 2014, 2015) can experience significant self-interactions which can lead to appreciable frequency evolution of the signal and to uncertainty in that evolution, a complication less important for the postulated QCD axion (Arvanitaki et al. 2015). In an optimistic scenario with many galactic black holes producing individually detectable signals, (Zhu et al. 2020) points out that the signals would all lie in a very narrow band, complicating CW searches, which typically implicitly assume no more than one detectable signal in narrow bands. A later study (Pierini et al. 2022), however, finds that semi-coherent searches can be robust with respect to potential signal confusion.

Until recently, most published searches have not been tailored for a black hole axion cloud source, but instead existing (non-optimized) limits on neutron star CW emission could be reinterpreted as limits on such emission (Arvanitaki et al. 2015; Dergachev and Papa 2019; Palomba et al. 2019). More recently, though, searches have been carried out that exploit the narrow spin-up parameter space expected for such sources (Sun et al. 2020; Abbott et al. 2022c).

One interesting suggestion includes the possibility that a black hole formed from the detected merger of binary black holes or neutron stars could provide a natural target for follow-up CW searches (Arvanitaki et al. 2017; Ghosh et al. 2019; Isi et al. 2019). Recent studies (Brito et al. 2017a, b; Tsukada et al. 2019, 2021) argue that the lack of detection of a stochastic gravitational radiation background from the superposition of extragalactic black holes already places significant limits on axion masses relevant to CW searches. Another recent study (Isi et al. 2019) examined in detail the prospects for detecting superradiance from both post-merger black hole remnants and known black holes in galactic X-ray binaries, such as Cygnus X-1.

3 Continuous Wave Search Methods

Being realistic, we must acknowledge that the first discovered CW signal will be exceedingly weak compared to the transient signals detected to date, an assumption borne out by many unsuccessful CW searches to date. One must integrate the signal over a long duration to observe it with statistical significance. Those long integrations in noise that is instantaneously much higher in amplitude require application of assumed signal templates to the data. In general, the more restrictive is the model, the better is the achievable signal-to-noise ratio, as one can search over a smaller volume of source parameter space. The following sections discuss the challenges faced in searching over larger parameter space volumes, a common classification of general search methods, and specific algorithms devised to meet the challenges.

3.1 Challenges in CW signal detection and types of searches

At first glance, it may seem puzzling that a signal due to a rapidly spinning neutron star is challenging to find. One might expect a simple discrete Fourier transform of the data stream to reveal a sharp spike at the nominal frequency. There are several severe complications, however, for most CW searches. For concreteness, imagine that a signal is weak enough to require a coherent, phase-preserving 1-year integration time $T_{\text{coh}}$. The nominal frequency resolution from a discrete Fourier transform (DFT⁷) is then 1/year $\sim$ 30 nHz. In order for the signal’s central frequency to remain in the same DFT bin (integer index into the transform result, see Eq. (52) below) during that year, its first derivative $\dot{f}$ would need to satisfy $\dot{f}T_{\text{coh}}\lesssim 1/T_{\text{coh}}$, or $\dot{f}\lesssim 10^{-15}$ Hz/s and its second derivative $\ddot{f}\lesssim 6\times 10^{-23}$ Hz/$\hbox {s}^2$. In practice, not only are Doppler modulations of detected frequency due to the Earth’s motion much larger than these values, as discussed below, but the frequency derivative of a detectable source is typically also much larger, in order for its rotational kinetic energy loss to be compatible with detection (detectable spin-down limit). If the precise frequency evolution of the source is known already from radio or gamma-ray pulsar timing (assuming a fixed EM/GW phase relation), then one can make corrections for that evolution via barycentering, discussed below, without SNR degradation as long as the uncertainties in frequency derivatives are well below the above constraints.

For sources with large frequency uncertainties, however, especially those with unknown frequencies, correcting for intrinsic source frequency evolution and for modulations due to the Earth’s motion incurs a substantial computing cost for searching over parameter space. Because of these costs, it is useful to categorize CW searches broadly into three categories (Prix 2009) (while recognizing there are special cases that fall near the boundaries).

• 1. Targeted searches in which the star’s position and rotation frequency are known, i.e., known radio, X-ray or $\gamma$-ray pulsars;

• 2. Directed searches in which the star’s position is known, but rotation frequency is unknown, e.g., a non-pulsating X-ray source at the center of a supernova remnant; and

• 3. All-sky searches for unknown neutron stars.

The volume of parameter space over which to search increases in large steps as one progresses through these categories. In each category a star can be isolated or binary. For 2) and 3) any unknown binary orbital parameters further increase the search volume, making a subclassification helpful, as discussed below. In general, the greater the a priori knowledge of sources parameters, the more computationally feasible it is to integrate data coherently for longer time periods in order to improve strain amplitude sensitivity.

To illustrate, consider a directed search for a source of known location but with unknown frequency and unknown frequency derivatives, where the signal phase is expanded in truncated Taylor form in the source frame time $\tau$ with respect to a reference time $\tau _0$:

$$\begin{aligned} \varPhi (\tau ) \, \approx \, \varPhi _0 + 2\,\pi \left[ f_s(\tau -\tau _0) + {1\over 2}\dot{f}_s(\tau -\tau _0)^2 + {1\over 6}\ddot{f}_s(\tau -\tau _0)^3\right] . \end{aligned}$$

(40)

Using the phase evolution model of Eq. (40), if we wish to preserve phase fidelity to a tolerance $\varDelta \varPhi$ over a coherence time $T_{\text{coh}}\approx \tau -\tau _0$, then we need (in a naive estimate) to know the frequency and derivatives to a tolerance better than

$$\begin{aligned} \varDelta f_{\rm{GW}}\approx & {} {\varDelta \varPhi \over 2\,\pi }{1\over T_{\rm{coh}}}, \end{aligned}$$

(41)

$$\begin{aligned} \varDelta {\dot{f}}_{\rm{GW}}\approx & {} {\varDelta \varPhi \over 2\,\pi }{2\over T_{\rm{coh}}^2}, \end{aligned}$$

(42)

$$\begin{aligned} \varDelta \ddot{f}_{\rm{GW}}\approx & {} {\varDelta \varPhi \over 2\,\pi }{6\over T_{\rm{coh}}^3}. \end{aligned}$$

(43)

Hence the numbers of steps to take in $f_{\rm{GW}}$, ${\dot{f}}_{\rm{GW}}$, and $\ddot{f}_{\rm{GW}}$ to cover a given range in the parameters are proportional to $T_{\rm{coh}}$, $T_{\rm{coh}}^2$ and $T_{\rm{coh}}^3$, respectively—if it’s necessary to step at all in those derivatives. Naively, for a search over a long enough coherence time to require multiple steps in $\ddot{f}_{\rm{GW}}$, one has a template count proportional to $T_{\rm{coh}}^6$ and, presumably, pays a price proportional to another factor of $T_{\rm{coh}}$ in computational cost in processing the associated data volume. In principle, then, the computational cost of a coherent search scales as the 7th power of the coherence time used, although, in practice the scaling tends not to be as extreme because the numbers of steps needed for $\ddot{f}_{\rm{GW}}$ can be small integers that take on new discrete values only slowly with increased $T_{\rm{coh}}$. In practice, these considerations for a 2nd frequency derivative come into play for only directed searches or for the deep follow-up of outliers from all-sky searches, when segment coherence times exceed several days. Section 3.6 will discuss more quantitatively the placement of search templates in parameter space to maintain acceptable phase tolerance.

In carrying out all-sky searches for unknown neutron stars, the computational considerations grow worse. The corrections for Doppler modulations and antenna pattern modulation due to the Earth’s motion must be included, as for the targeted and directed searches, but the corrections are sky-dependent, and the spacing of the demodulation templates is dependent upon the inverse of the coherence time of the search. Specifically, for a coherence time $T_{\rm{coh}}$ the required angular resolution is (Abbott et al. 2008a)

$$\begin{aligned} \delta \theta \, \approx \, {0.5\, \rm{c}\, \delta f\over f\,[v\sin (\theta )]_{\max }}, \end{aligned}$$

(44)

where $\theta$ is the angle between the detector’s velocity relative to a nominal source direction, where the maximum relative frequency shift $[v\sin (\theta )]_{\max }/c\approx 10^{-4}$, and where $\delta f$ is the size of the frequency bins in the search. For $\delta f=1/T_{\rm{coh}}$, one obtains:

$$\begin{aligned} \delta \theta \, \approx \, 9\times 10^{-3}\,\rm{rad}\,\left( {30\,\rm{minutes}\over T_{\rm{coh}}}\right) \left( {300\,\rm{Hz}\over f_s}\right) , \end{aligned}$$

(45)

where $T_{\rm{coh}}$ = 30 min has been used in several all-sky searches to date. Because the number of required distinct points on the sky scales like $1/(\delta \theta )^2$, the number of search templates scales like $(T_{\rm{coh}})^2(f_s)^2$ for a fixed signal frequency $f_s$. Now consider attempting a search with a coherence time of 1 year for a signal frequency $f_s=1$ kHz. One obtains $\delta \theta \sim 0.3$ $\mu$rad and a total number of sky points to search of $\sim \,10^{14}$—again, for a fixed frequency. Adding in the degrees of freedom to search over ranges in $f_s$, $\dot{f}_s$ and $\ddot{f}_s$ (and higher-order derivatives, as needed) makes a brute-force, fully coherent 1-year all-sky search hopelessly impractical, given the Earth’s present total computing capacity.

As a result, tradeoffs in sensitivity must be made to achieve tractability in all-sky searches. The simplest tradeoff is to reduce the observation time to a computationally acceptable coherence time. It can be more attractive, however, to reduce the coherence time still further to the point where the total observation time is divided into $N=T_{\text{obs}}/T_{\text{coh}}$, segments, each of which is analyzed coherently and the results added incoherently to form a detection statistic. One sacrifices intrinsic sensitivity per segment in the hope of compensating (partially) with the increased statistics from being able to use more total data. In practice, for realistic data observation spans (weeks or longer), the semi-coherent approach gives better sensitivity for fixed computational cost and hence has been used extensively in both all-sky and directed searches (Prix and Shaltev 2012). One finds a strain sensitivity (threshold for detection) that scales approximately as the inverse fourth root of N (Abbott et al. 2005a). Hence, for a fixed observation time, the strain sensitivity degrades roughly as $N^{1\over 4}$ as $T_\text{coh}$ decreases (see Wette 2012) for a discussion of variations from this scaling). This degradation is a price one pays for not preserving phase coherence over the full observation time, in order to make the search computationally tractable. An important virtue of semi-coherent searches methods, however, is robustness with respect to deviations of a signal from an assumed coherent model.

In general, fully coherent search methods are potentially the most sensitive, but their applicability depends on several considerations, perhaps the most important being sheer computational tractability. Even when tractable for a particular search, moreover, a fully coherent initial search stage may incur a statistical trials factor large enough to make a putative detection questionable, because of the necessarily ultra-fine search needed to probe coherently a multi-dimensional signal parameter space. That is, the statistical significance of a nominally “loud” detection statistic must account for the number of independent trials carried out in the search. Applying a priori constraints instead, when available from electromagnetic observations or theoretical expectation, can reduce the parameter space volume and hence trials factor, making a detection more convincing. For example, a “5 $\sigma$” detection of a signal from a known pulsar in a targeted search might not qualify as even a weak outlier in an all-sky search, much less as a discovery. (Dergachev 2010b) discusses the tradeoff between fully templated and “loose coherence” methods (see Sect. 3.3.4) in a broad parameter space, arguing against brute-force template application.

In the next sections, a variety of general approaches and specific algorithms will be presented, methods that attempt to achieve tradeoffs best suited to particular CW search types.

3.2 Signal model

CW searches must account for large phase modulations (or, equivalently, frequency modulations) of the source signal due to detector motion and potentially due to source motion (expecially for binary sources). The precision of the applied modulation corrections must be high in the case of targeted searches, which use measured ephemerides from radio, optical, X-ray or $\gamma$-ray observations valid over the gravitational-wave observation time. The precision must also be high in following up outliers from directed or all-sky searches, while much less precision is needed in the first stage of hierarchical searches. This section describes the intrinsic signal model assumed, along with the expected modulations due to detector motion.

For the Earth’s motion, one has a daily relative frequency modulation of $v_{\rm{rot}}/c\approx 10^{-6}$ and a much larger annual relative frequency modulation of $v_{\text{orb}}/c\approx 10^{-4}$. The pulsar astronomy community has developed a powerful and mature software infrastructure for measuring ephemerides and applying them in measurements, using the TEMPO 2 program (Hobbs et al. 2006). The same physical corrections for the Sun’s, Earth’s and Moon’s motions (and for the motion of other planets), along with general relativistic effects including gravitational redshift in the Sun’s potential and Shapiro delay for waves passing near the Sun, have been incorporated into the LIGO and Virgo software libraries (LIGO Scientific Collaboration 2018; Astone et al. 2002a).

Consider an isolated, rotating rigid triaxial ellipsoid (conventional model for a GW-emitting neutron star), for which the strain waveform detected by an interferometer can be written as

$$\begin{aligned} h(t) \, = \, F_+(t,\psi )\,h_0{1+\cos ^2(\iota )\over 2}\,\cos (\varPhi (t)) \,+\, F_\times (t,\psi )\,h_0\,\cos (\iota )\,\sin (\varPhi (t)), \end{aligned}$$

(46)

where $\iota$ is the angle between the star’s spin direction and the propagation direction ${\hat{k}}$ of the waves (pointing toward the Earth). $F_+$ and $F_\times$ are the (real) detector antenna pattern response factors ($-1 \le F_+,F_\times \le 1)$ to the $+$ and $\times$ polarizations. $F_+$ and $F_\times$ depend on the orientation of the detector and the source, and on the polarization angle $\psi$ (Abbott et al. 2004). Here, $\varPhi (t)$ is the phase of the signal, which can often usefully be Taylor-expanded as in Eq. (40), in the solar system barycenter (SSB) time $\tau$ with apparent frequency derivatives with respect to detector-frame time arising from source motion. A more general signal model with GW emission at both once and twice the rotation frequency is considered in Jaranowski et al. (1998), with effects of free precession addressed in Jones and Andersson (2001), Jones and Andersson (2002), and Van Den Broeck (2005); Gao et al. (2020), and a convenient reparametrization is presented in Jones (2015).

Explicitly, the time of arrival of a signal at the solar system barycenter, $\tau (t)$, can be written in terms of the signal time of arrival t at the detector:

$$\begin{aligned} \tau (t) \, \equiv \, t + \delta t \, = \, t - {\vec{r}_d\cdot {\hat{k}}\over c}+ \varDelta _{E\odot }+\varDelta _{S\odot }, \end{aligned}$$

(47)

where $\vec{r}_d$ is the position of the detector with respect to the SSB, and $\varDelta _{E\odot }$ and $\varDelta _{S\odot }$ are solar system Einstein and Shapiro time delays, respectively (Taylor 1992; Hobbs et al. 2006).

Equation 47 implicitly assumes planar gravitational wavefronts and neglects proper motion of the source (transverse to the line of sight), corrections for which are common in radio pulsar astronomy (Lorimer and Kramer 2005; Lyne and Graham-Smith 2006). In principle, long-duration (multi-year) fully coherent observations of a near-enough ($\sim$ 100 pc), high-frequency ($\sim$ 1 kHz) CW source would allow inference of its distance from determination of the wavefront curvature (Seto 2005). Similarly, multi-year coherent observations of a high-frequency source would need to account for significant proper motions ($\sim$ 50 mas/year, typical of known pulsars) (Covas 2020). Both wavefront curvature and proper motion have been neglected in CW searches for unknown sources to date because the coherence times used in the searches don’t require those corrections, but in the happy event of a future detection and subsequent extended observations, these corrections may become relevant.

Existing gravitational-wave detectors are far from isotropic in their response functions. In the long-wavelength limit, Michelson interferometers have an antenna pattern sensitivity with polarization-dependent maxima normal to their planes and nodes along the bisectors of the arms. As the Earth rotates at angular velocity $\varOmega _r$ with respect to a fixed source, the antenna pattern modulation is quite large and polarization dependent via the functions $F_+(t)$ and $F_\times (t)$ which depend on the orientation of the detector and the source.

A commonly used parametrization of these amplitude response modulations is defined in (Jaranowski et al. 1998).

$$\begin{aligned} F_+(t)= & {} \sin (\zeta )\left[ a(t)\cos (2\psi )+b(t)\sin (2\psi )\right] , \end{aligned}$$

(48)

$$\begin{aligned} F_\times (t)= & {} \sin (\zeta )\left[ b(t)\cos (2\psi )-a(t)\sin (2\psi )\right] , \end{aligned}$$

(49)

where $\zeta$ is the angle between the arms of the interferometer (nearly or precisely 90 degrees for all major ground-based interferometers), and where $\psi$ defines the polarization angle of the source wave frame (e.g., angle between neutron star spin axis projected onto the plane of the sky and local Cartesian coordinates aligned with its right ascension and declination directions). The antenna pattern functions a(t) and b(t) depend on the position and orientation of the interferometer on the Earth’s surface, the source location and sidereal time:

$$\begin{aligned} a(t)= & {} {1\over 16}\sin (2\gamma )(3-\cos (2\lambda ))(3-\cos (2\delta ))\cos [2(\alpha -\phi _r-\varOmega _rt)] \nonumber \\{} & {} -{1\over 4}\cos (2\gamma )\sin (\lambda )(3-\cos (2\delta ))\sin [2(\alpha -\phi _r-\varOmega _rt)] \nonumber \\{} & {} +{1\over 4}\sin (2\gamma )\sin (2\lambda )\sin (2\delta )\cos [(\alpha -\phi _r-\varOmega _rt] \nonumber \\{} & {} -{1\over 2}\cos (2\gamma )\cos (\lambda )\sin (2\delta )\sin [(\alpha -\phi _r-\varOmega _rt] \nonumber \\{} & {} +{3\over 4}\sin (2\gamma )\cos ^2(\lambda )\cos ^2(\delta ), \end{aligned}$$

(50)

$$\begin{aligned} b(t)= & {} \cos (2\gamma )\sin (\lambda )\sin (\delta ) \cos [2(\alpha -\phi _r-\varOmega _rt)] \nonumber \\{} & {} + {1\over 4}\sin (2\gamma )(3-\cos (2\lambda ))\sin (\delta )\sin [2(\alpha -\phi _r-\varOmega _rt)] \nonumber \\{} & {} + \cos (2\gamma )\cos (\lambda )\cos (\delta )\cos [\alpha -\phi _r-\varOmega _rt] \nonumber \\{} & {} + {1\over 2}\sin (2\gamma )\sin (2\lambda )\cos (\delta )\sin [\alpha -\phi _r-\varOmega _rt]. \end{aligned}$$

(51)

Specifically, in these equations, $\lambda$ is the interferometer’s latitude, and $\gamma$ is the counterclockwise angle between the bisector of its arms and the eastward direction. The source direction is specified by right ascension $\alpha$ and declination $\delta$, while $\phi _r$ is a deterministic phase defined implicitly by the interferometer’s longitude. These functions reveal amplitude modulations with periods of 1/2 and 1 sidereal day, and in the case of a(t), a constant term independent of time. As a result, the interferometer’s response to a monochromatic source in the Earth center’s reference frame will, in general, display five distinct frequency components, corresponding to the “carrier” frequency and two pairs of positive and negative sidebands, with a splitting between adjacent frequencies of ${\varOmega _r\over 2\,\pi } \approx 1.16\times 10^{-5}\, \rm{Hz}$.

Searches for CW signals must take into account the phase/frequency modulations embodied in Eq. (47) due to detector translational motion and the antenna pattern modulations embodied in Eqs. (48)–(51) due to detector orientation changes.

Figure 9 shows a sample spectrogram of a pure signal simulation using one of the so-called “hardware injections” used in LIGO data runs. These signal simulations are used to verify end-to-end the detector’s response to a CW signal, including sustained phase coherence over long durations. The simulations are injected via “photon calibrators,” which are auxiliary lasers shining on mirrors with a modulated intensity. The imposed relative motion of the mirror mimics (in the long-wavelength regime) the response of the interferometer to a gravitational wave. Various such signals, ranging in nominal frequency from 12 to 2991 Hz were injected into the LIGO detectors over the O1, O2 and O3 observation runs (Biwer et al. 2017). In the example shown, a signal (“Pulsar 2”) with source frequency 575.163573 Hz (reference time = November 1, 2003 00:00 UTC) and spin-down $-1.37\times 10^{-13}$ Hz/s is simulated at a sky location of right ascension $\alpha$ = 3.75692884 radians (14 h 21 m 1.48 s) and declination $\delta$ = 0.060108958 radians ($3^\circ$ 26’ 38.36”). Its orientation is defined by inclination angle $\iota$ = 2.76 radians (158. deg), and polarization angle $\psi$ = −0.222 radians (−12.7 deg). The simulation shown (with negligible noise for clarity) in Fig. 9 applies over a duration of the calendar year 2019 UTC. One can see the annual modulation from the Earth’s orbit imposed on an imperceptible decrease in the intrinsic frequency. A zoom of 100-h duration is also shown in Fig. 10, to indicate the much smaller frequency modulation (by $\sim$ 2 orders of magnitude), along with the intensity modulation.

As seen from the spectrogram, the frequency modulations lead to stationary bands at the turning points of the modulation. As a result, the spectrum averaged over the signal duration peaks at the turning points, as shown in Fig. 11. These “horns” are a characteristic spectral signature of expected signals, where the relative heights of the horns depend on the duration of the observation and on the Earth’s orbital phase at the start. For a signal with negligible spin-down and a duration equal to a multiple of a year, the horns are approximately symmetric, but in the general case that includes observations of a few months or less, one or both horns may not be apparent. A more detailed analysis of the Fourier transform of a CW signal can be found in Valluri et al. (2021).

Fig. 9

Sample signal spectrogram for a LIGO “hardware injection” (negligible noise), where the pixel dimensions are 0.5 h by 0.556 mHz

Fig. 10

Zoomed-in sample signal spectrogram for a LIGO “hardware injection” (100 h from spectrogram in Fig. 9). The sidereal Doppler modulations ($\sim$ 24 h) of frequency and amplitude modulations ($\sim$ 12 and 24 h) are more apparent

Fig. 11

Sample signal amplitude spectral density for a LIGO “hardware injection” (same injection as for spectrograms in Fig. 9)

In the following, we discuss in more detail the implementations of a selection of these methods developed in searches of the initial LIGO and Virgo data sets (2001–2011) and that have been further refined for searches of advanced detector data. Section 4 presents the results of each type, from searches in the data of Advanced LIGO’s and Virgo’s observing runs O1, O2 and O3.

The volume of parameter space over which to search increases in large steps as one progresses through these categories. In each category a star can be isolated or binary. Any unknown binary orbital parameters further increase the search volume. In all cases we expect (and have now verified from unsuccessful searches to date) that source strengths are very small. Hence one must integrate data over long observation times to have any chance of signal detection. How much one knows about the source governs the nature of that integration. In general, the greater that knowledge, the more computationally feasible it is to integrate data coherently (preserving phase information) over long observation times, for reasons explained below.

3.3 Broad approaches in CW searches

Computational cost depends critically upon the search method used, which in turn, depends on the a priori knowledge one has about the source. In the following, a broad overview is given of a few key search methods used in published searches to date. More details of implementation are presented in Sect. 4, where a selection of these methods is applied to particular classes of potential CW sources.

In this overview, a simplified “toy model” will be used to illustrate scaling relations. Methods specific to correcting for modulations will be addressed further below in the presentation of particular search implementations. For now, amplitude modulation of the signal strength due to rotation of the GW detectors with respect to the source is ignored in the following, along with Doppler modulations.

3.3.1 Fully coherent methods

When applicable, as discussed in Sect. 3.1, fully coherent methods provide the best sensitivity. Explicit search methods, taking into account source frequency and amplitude evolution along with detector noise non-stationarity, are discussed in Sect. 4.1. Here, though, let’s consider the highly simplified problem of detecting a sinuosidal signal of amplitude $h_0$ and known frequency $f_{\rm{sig}}$, but with unknown phase constant, in random Gaussian noise.

Imagine the data observation is continuous of duration $T_{\rm{obs}}$ and has a one-sided power spectral noise density function $S_h(f_{\rm{GW}})$, and for convenience, assume $f_{\rm{sig}}$ is an integer multiple of $1/T_{\rm{obs}}$ (see (Allen et al. 2002) for a didactic treatment of the more general case). The DFT bin ${\tilde{D}}_i$ is defined by the following sum over a real time series of length $N_{\rm{sample}}= f_{\rm{sample}}T_{\rm{obs}}$ with values $d_j$ (j = 0...$N_{\rm{sample}}$) where $f_{\rm{sample}}$ is the sampling frequency of the data stream:

$$\begin{aligned} {\tilde{D}}_i= \sum _{j=0}^{N_{\rm{sample}}-1}d_je^{-\rm{i}2\pi ji/N_{\rm{sample}}}. \end{aligned}$$

(52)

From the DFT, one can define the one-sided power spectral noise density estimate $S_{h_{i}}$:

$$\begin{aligned} S_{h_{i}} = {2 \langle \!\left[ ( \Re \{{\tilde{D}}_i\})^2 + (\Im \{{\tilde{D}}_i\})^2 \right] \!\rangle T_{\rm{obs}}\over N_{\rm{sample}}^2} = {2\langle \!|{\tilde{D}}_i|^2\!\rangle T_{\rm{obs}}\over N_{\rm{sample}}^2}, \end{aligned}$$

(53)

for $0<i<N_{\rm{sample}}/2$ and where “$\langle$ $\rangle$” indicates an expectation value in the absence of signal (e.g., determined from an average over many nearby bins and excluding the bin i itself). Then one can construct a dimensionless detection statistic $\rho _i^2$ using the measured strain power in the DFT bin ${\tilde{D}}_i$ corresponding to a signal frequency $f_{\rm{sig}}$:

$$\begin{aligned} \rho _i^2 = 4{|{\tilde{D}}_i|^2T_{\rm{obs}}\over N_{\rm{sample}}^2S_{h_{i}}}, \end{aligned}$$

(54)

which follows a non-central $\chi ^2$ distribution with two degrees of freedom and a non-centrality parameter $\lambda (h_0) = {h_0^2T_{\rm{obs}}\over S_{h_{i}}}$, which implies an expectation value $2+{h_0^2T_{\rm{obs}}\over S_{h_{i}}}$ and variance $4+4{h_0^2T_{\rm{obs}}\over S_{h_{i}}}$. In Gaussian noise one expects a $\chi ^2$ distribution with two degrees of freedom from summing the squares of the normally distributed real and imaginary DFT coefficients.

In the absence of a signal, one can define a threshold value $\rho _i^*$ corresponding to a false alarm probability $\alpha$ such that the cumulative density probability function satisfies:

$$\begin{aligned} \rm{CDF}_{\rm{noise}}[{\rho _i^*}^2] \equiv \int _0^{{\rho _i^*}^2} {p_{\rm{noise}}(\rho _i^2;2)}\,d(\rho _i^2) \equiv 1-\alpha , \end{aligned}$$

(55)

where the probability density function is ($\chi ^2$ with two degrees of freedom)

$$\begin{aligned} p_{\rm{noise}}(x;2) = {1\over 2}e^{-x/2}. \end{aligned}$$

(56)

From this threshold and a desired false dismissal rate $\beta$, one can then determine the corresponding signal amplitude $h_0^{1-\beta }$ from

$$\begin{aligned} \rm{CDF}_{\mathrm{signal+noise}}[{\rho _i^*}^2] \equiv \int _0^{{\rho _i^*}^2} {p_{\mathrm{signal+noise}}(\rho _i^2;2,\lambda (h_0^{1-\beta }))}\,d(\rho _i^2) = \beta , \end{aligned}$$

(57)

where the probability density function is (non-central $\chi ^2$ with two degrees of freedom)

$$\begin{aligned} p_{\mathrm{signal+noise}}(x;2,\lambda ) = {1\over 2}e^{-(x+\lambda )/2}I_0(\sqrt{\lambda x}), \end{aligned}$$

(58)

and where $I_0(y)$ is a modified Bessel function of the first kind:

$$\begin{aligned} I_0(y) = \sum _{j=0}^\infty {(y^2/4)^j\over (j!)^2}. \end{aligned}$$

(59)

Choosing a 1% false alarm probability ($\alpha$ = 0.01) leads to ${\rho _i^*}^2\approx 9.21$, from which numerical evaluation of Eq. 57 for a false dismissal probability $\beta$ = 5% leads to an expected sensitivity $h_0^{95\%}$ of

$$\begin{aligned} h_0^{95\%}\approx 4.54\, \sqrt{S_{h_{i}}\over T_{\rm{coh}}}, \end{aligned}$$

(60)

which can be taken as a proxy for the expected 95% confidence level upper limit on signal amplitude based simply on an observation that an observed $\rho _i$ does not exceed $\rho _i^*$. In practice, many CW search upper limits are based on the loudest statistic found in a search, (e.g., largest $\rho ^2$ value for multiple computations at different frequencies), regardless of whether or not a pre-defined threshold has been exceeded. (Tenorio et al. 2022) discusses the statistics of loudest candidates using extreme value theory.

The sensitivity expression in Eq. (60) is shown as the leftmost point of the lower blue curve in Fig. 12, where it can be compared to sensitivities from other methods, discussed below.

If simultaneous data sets from two independent detectors of identical sensitivity are added coherently (and a phase correction applied, to account for detector separation and source direction), one can construct a combined averaged detection statistic $\rho _{i,\rm{comb}}^2$ using the measured power from the square of the sum of the simultaneous DFT bin coefficients ${\tilde{D}}_{1,i}$ and ${\tilde{D}}_{2,i}$ containing $f_{\rm{sig}}$:

$$\begin{aligned} \rho _{i,\rm{comb}}^2 = {1\over 2}\left[ 4{|{\tilde{D}}_{1,i}+{\tilde{D}}_{2,i}|^2\,T_{\rm{obs}}\over N_{\rm{sample}}^2S_{h_{i}}}\right] \, \end{aligned}$$

(61)

which has an expectation value of $2+{2\,h_0^2T_{\rm{obs}}\over S_{h_{i}}}$ and a variance of $4+{8\,h_0^2T_{\rm{obs}}\over S_{h_{i}}}$, that is, it follows a non-central $\chi ^2$ distribution with two degrees of freedom and a non-centrality parameter $\lambda (h_0) = {2\,h_0^2T_{\rm{obs}}\over S_{h_{i}}}$. Applying the same methodology as above for a single detector, one obtains an expected sensitivity of

$$\begin{aligned} h_0^{95\%}\approx {1\over \sqrt{2}} \times 4.54\, \sqrt{S_{h_{i}}\over T_{\rm{coh}}}, \end{aligned}$$

(62)

indicating an improvement by $\sqrt{2}$ with respect to a single detector. This sensitivity is shown as the leftmost point of the green curve in Fig. 12. More generally, N identical detectors with simultaneous data sets,⁸ for which phase corrections are known, gain a sensitivity improvement of $\sqrt{N}$ with respect to a single detector, as one would naively expect. Put another way, combining the N sets gives a sensitivity equal to that of a single detector with an amplitude spectral noise density of $\sqrt{S_{h_{i}}/N}$. Again, this is a simplified model. In practice, detectors have different, frequency-dependent $S_{h_{i}}$ noise levels and unequal live times of observing, in addition to different orientations affecting antenna pattern sensitivity, ignored here.

3.3.2 Semi-coherent methods

Fully coherent methods are computationally costly when covering a large parameter space because of the fine steps needed to avoid missing a signal as coherence times increase. A crude solution is simply to reduce the coherence time and suffer the reduction in strain sensitivity, by approximately $\sqrt{T_{\rm{coh}}}$. A better solution is to apply a semi-coherent method in which the observation time is divided into $N_{\rm{seg}}$ segments of equal length $T_{\rm{coh}}= T_{\rm{obs}}/N_{\rm{seg}}$, where the detection statistic is constructed from an incoherent sum of powers from the individual segments. This method sacrifices the constraint of phase consistency among the different segments and hence is less sensitive than a fully coherent method, but is more sensitive than analysis of a single segment alone. As shown below, the strain sensitivity scales with $1/N_{\rm{seg}}^{1\over 4}$ for fixed coherence time and large $N_{\rm{seg}}$.

For illustration, consider a detection statistic constructed from the sum of $N_{\rm{seg}}$ individual DFT powers covering the observation time $T_{\rm{obs}}$, and once again, each normalized by its power spectral density (taken to be stationary here, for convenience):

$$\begin{aligned} R_i\equiv \sum _{k=1}^{N_{\rm{seg}}} 4{|{\tilde{D}}^{(k)}_i|^2T_{\rm{coh}}\over N_{\rm{sample}}^2S_{h_{i}}}, \end{aligned}$$

(63)

where $N_{\rm{seg}}=1$ yields $R_i=\rho _i^2$ in Eq. (54). The underlying statistical distribution of $R_i$ is that of a non-central $\chi ^2$ with $2N_{\rm{seg}}$ degrees of freedom and a non-centrality parameter of $\lambda (h_0) = N_{\rm{seg}}{h_0^2T_{\rm{coh}}\over S_{h_{i}}}$.

In the absence of signal a false alarm probability $\alpha$ implies a threshold $R_i^*$, found from requiring that the cumulative probability density function satisfy:

$$ {\text{CDF}}_{{{\text{noise}}}} [R_{i}^{*} ] \equiv \int_{0}^{{R_{i}^{*} }} {p_{{{\text{noise}}}} (R_{i} ;2)} {\mkern 1mu} dR_{i} \equiv 1 - \alpha , $$

(64)

where the probability density function is

$$\begin{aligned} p_{\rm{noise}}(x;2N_{\rm{seg}}) = {x^{N_{\rm{seg}}-1}e^{-x/2}\over 2^{N_{\rm{seg}}}\varGamma (N_{\rm{seg}})}, \end{aligned}$$

(65)

which reduces to Eq. (56) for $N_{\rm{seg}}=1$. $R_i^*$ can be determined numerically for arbitrary $N_{\rm{seg}}$, but in the limit of large $N_{\rm{seg}}$, $p_{\rm{noise}}$ reduces to a normal distribution with a mean of $2N_{\rm{seg}}$ and variance $4N_{\rm{seg}}$, in which approximation $R_i^*(\alpha =0.01) \approx 2N_{\rm{seg}}+ 4.65\sqrt{N_{\rm{seg}}}$.

In the presence of a signal, the $h_0^{95\%}$ value can be obtained numerically from the cumulative probability density function:

$$ {\text{CDF}}_{{{\text{signal + noise}}}} [R_{i}^{*} ] \equiv \int_{0}^{{R_{i}^{*} }} {p_{{{\text{signal + noise}}}} (R_{i} ;2,h_{0} )} {\mkern 1mu} dR_{i} = \beta , $$

(66)

where the probability density function is

$$\begin{aligned} p_{\mathrm{signal+noise}}(x;2N_{\rm{seg}},h_0)= & {} {1\over 2} e^{-{1\over 2}\left( x+{h_0^2T_{\rm{obs}}\over S_h}\right) } \left( {xS_h\over h_0^2T_{\rm{obs}}}\right) ^{{N_{\rm{seg}}-1\over 2}} \nonumber \\{} & {} I_{N_{\rm{seg}}-1}\left( h_0\sqrt{{xT_{\rm{obs}}\over S_h}}\right) \end{aligned}$$

(67)

which reduces to Eq. (58) for $N_{\rm{seg}}=1$, where $T_{\rm{obs}}=N_{\rm{seg}}T_{\rm{coh}}$ has been used.

In the limit of large $N_{\rm{seg}}$ and weak signal, however, the distribution approaches that of a Gaussian with variance $4N_{\rm{seg}}$, from which an approximate expression for $h_0^{95\%}$ can be obtained:

$$\begin{aligned} h_0^{1-\beta } \approx \sqrt{2}\,\left[ \sqrt{2}\,(\rm{erfc}^{-1}(2\alpha )+\rm{erfc}^{-1}(2\beta ))\right] ^{1/2} N_{\rm{seg}}^{1\over 4} \sqrt{{S_h\over T_{\rm{obs}}}}, \end{aligned}$$

(68)

where $\rm{erfc}$ is the inverse complementary error function. This scaling of sensitivity with $N_{\rm{seg}}^{1\over 4}$ for fixed observation time is a universal result in semi-coherent searches with large $N_{\rm{seg}}$ (Krishnan et al. 2004; Mendell and Landry 2005; Prix and Shaltev 2012; Wette 2012). It can be understood qualitatively from the SNR of an approximately Gaussian detection statistic (large $N_{\rm{seg}}$) scaling with $\sqrt{N_{\rm{seg}}T_{\rm{coh}}} = \sqrt{T_{\rm{obs}}/N_{\rm{seg}}}$ and the direct dependence of that detection statistic on the squared signal amplitude.

For $\alpha =0.01$ and $\beta =0.05$, Eq. 68 yields $h_0^{95\%}\approx 2.82N_{\rm{seg}}^{1\over 4}\sqrt{{S_h/T_{\rm{obs}}}}$. This expression does not agree with Eq. (60) when $N_{\rm{seg}}=1$ because the Gaussian approximation breaks down for small $N_{\rm{seg}}$. For the much lower false alarm probability $\alpha =10^{-10}$, Eq. (68) yields $h_0^{95\%}\approx 4.00N_{\rm{seg}}^{1\over 4}\sqrt{{S_h/T_{\rm{obs}}}}$. These asymptotic approximations are shown as dashed lines in Fig. 12, together with numerically evaluated values from Eq. (66).

Fig. 12

Strain sensitivities (defined by $\alpha =0.01$ or $\alpha =10^{-10}$; and $\beta =0.05$) of various search methods to a bin-centered sinusoidal signal in noise of power spectral density $S_h$ versus the number of DFT segments into which a fixed observing time $T_{\rm{obs}}$ is divided. Curves are shown for $h_0^{95\%}$ for $\alpha =0.01$ and $\alpha = 10^{-10}$ for semi-coherent searches in a single detector’s data (blue) and for $\alpha =0.01$ in two-detector searches (green). Asymptotic expressions based on the large-$N_{\rm{seg}}$ Gaussian approximation are shown as dashed lines. The points at the bottom left of each curve represent the fully coherent search sensitivities for 1 and 2 identical detectors. Semi-coherent curves for two detectors assumed coherent summing of simultaneous DFTs for the 2 detectors. Sensitivities for cross-correlation of simultaneous data from two detectors are also shown (red)

As is the case for combining data in a fully coherent search from two identical detectors with simultaneous data sets, there is a gain of $\sqrt{2}$ in sensitivity for combining simultaneous DFTs from two detectors in a semi-coherent search—as long as the DFT coefficients are combined coherently for each segment (otherwise, the gain is only $2^{1/4}$ from semi-coherent combination of DFT powers). Figure 12 shows exact (solid green curve) and asymptotic (dashed green line) results for two detectors.

An important variation on semi-coherent methods, used in Hough transform methods described below, applies thresholding to DFT powers prior to summing. By applying a threshold corresponding to a relatively high false alarm rate (and relatively high false dismissal rate for weak signals), one can reduce the data volume in processing, to reduce computational cost. In addition, by adding integer counts (or, optimally, pre-computed weights) instead of measured DFT powers, one’s detection statistic is less susceptible to distortions from transient, non-Gaussian outliers from instrumental contamination. The optimum false alarm probability for thresholding in this idealized sine-wave detection analysis (weak-signal limit) is $\approx$20% (Allen et al. 2002).

3.3.3 Cross-correlation methods

Another attractive approach uses cross correlation between independent (and ideally, simultaneous) data streams. The canonical example is cross correlation between coincident data sets taken with the nearly aligned Hanford and Livingston interferometers, but cross correlation can also be used with poorly aligned detector pairs and with non-coincident data streams—if sufficient signal coherence can be established over longer time scales. In this section, only truly coincident data sets will be considered, for simplicity.

Once again, let’s use the artificial but informative toy model of a bin-centered, constant-amplitude sinusoidal signal in Gaussian noise. Let’s also assume two identically oriented detectors (approximation to Hanford-Livingston, which have normal vectors to their planes only 27.3 degrees apart (Althouse et al. 1998), in addition to a 90-degree relative rotation about the normal, leading to a sign flip in GW response). Also assume that the relative positions of the detectors can be accounted for via a signal phase correction for any given source direction. In the following, that phase correction is assumed to have been applied.

Using the notation from Sect. 3.3.2, we can combine the two independent data streams via DFT coefficients in a narrow band of interest to define a new detection statistic:

$$\begin{aligned} \rho _{\rm{CC}}= {2\,\sqrt{2}\,\Re \left\{ {\tilde{D}}_{1,i}{\tilde{D}}_{2,i}^*\right\} T_{\rm{obs}}\over N_{\rm{sample}}^2S_{h_{i}}}. \end{aligned}$$

(69)

In the absence of a signal, this statistic has an expectation value of zero and (by construction) a variance of one. The underlying statistical distribution is far from Gaussian, however. In the absence of a signal, one has the sum of two normal product distributions (from $\Re \left\{ {\tilde{D}}_{1,i}{\tilde{D}}_{2,i}^*\right\} = \Re \left\{ {\tilde{D}}_{1,i}\right\} \Re \left\{ {\tilde{D}}_{2,i}\right\} + \Im \left\{ {\tilde{D}}_{1,i}\right\} \Im \left\{ {\tilde{D}}_{2,i}\right\}$), which can be obtained analytically, using characteristic functions.

Specifically, a single normal production distribution for the product of two zero-mean normal distributions of variance $\sigma _1^2$ and $\sigma _2^2$ is

$$\begin{aligned} p_{\mathrm{1\, normal\, product}}(x) = {1\over \pi \sigma _1\sigma _2}K_0\left( {|x|\over \sigma _1\sigma _2}\right) , \end{aligned}$$

(70)

where $K_0$ is a modified Bessel function of the second kind, and for which the characteristic function is (McNolty 1973)

$$\begin{aligned} \rm{CF}[K_0\left( {|x|\over \sigma _1\sigma _2}\right) ] = {1\over \sqrt{1+\sigma _1^2\sigma _2^2t^2}}. \end{aligned}$$

(71)

Inverting the product of characteristic functions gives the following probability distribution for the sum of two such (signal-free) normal product distributions:

$$\begin{aligned} p_{\mathrm{2\, normal\, products}}(x) = {1\over 2\sigma _1\sigma _2}e^{-{|x|\over \sigma _1\sigma _2}}. \end{aligned}$$

(72)

For identical detectors ($\sigma _1=\sigma _2\equiv \sigma$) of power spectral density $S_{h_{i}}$ with a signal present of amplitude $h_0$, the expectation value of $\rho _{\rm{CC}}$ is ${h_0^2T_{\rm{obs}}\over \sqrt{2}\,S_{h_{i}}}$, and the variance is $1 + {h_0^2T_{\rm{obs}}\over S_{h_{i}}}$.

In the presence of a common signal $h_0$ in both data streams, one can evaluate numerically the value $h_0^{1-\beta }$ for which the false dismissal rate is $\beta$ for a threshold on the detection statistic corresponding to the signal-free false alarm probability $\alpha$. In the absence of a signal, the threshold on $\rho _{\rm{CC}}$ for a false alarm probability $\alpha = 0.01$ is approximately 2.766. For $\beta =0.05$ one then finds $h_0^{95\%}\approx 3.335\sqrt{{S_{h_{i}}/T_{\rm{obs}}}}$, which is only slightly higher than that obtained for a fully coherent search using data from two identical detectors (see Fig. 12).

As with semi-coherent searches, discussed in Sect. 3.3.2, one typically finds it necessary in wide-parameter searches to segment the data. As before, consider dividing the observation time $T_{\rm{obs}}$ into $N_{\rm{seg}}$ equal-duration segments of coherence time $T_{\rm{coh}}$. In the presence of a signal of amplitude $h_0$, the following detection statistic,

$$\begin{aligned} \rho _{\rm{CC}}^{N_{\rm{seg}}} = {1\over N_{\rm{seg}}}\sum _{i=1}^{N_{\rm{seg}}} \rho _{\rm{CC}}^i, \end{aligned}$$

(73)

has a mean value of ${h_0^2T_{\rm{coh}}\over \sqrt{2}S_{h_{i}}} = {1\over N_{\rm{seg}}}{h_0^2T_{\rm{obs}}\over \sqrt{2}S_{h_{i}}}$ and variance ${1\over N_{\rm{seg}}}\left[ 1 + {h_0^2T_{\rm{obs}}\over S_{h_{i}}}\right]$.

In the regime of large $N_{\rm{seg}}$ and weak signal, the underlying probability distribution approaches that of a Gaussian for which one expects:

$$\begin{aligned} h_0^{1-\beta } \approx \sqrt{2}\,\left[ \sqrt{2}\,(\rm{erfc}^{-1}(2\alpha )+\rm{erfc}^{-1}(2\beta ))\right] ^{1/2} \left[ {N_{\rm{seg}}\over 2}\right] ^{1\over 4} \sqrt{{S_h\over T_{\rm{obs}}}}. \end{aligned}$$

(74)

This asymptotic expectation is shown as a dotted red line in Fig. 12, together with results from numerical simulation over a range of $N_{\rm{seg}}$ values (solid red curve). This detection statistic is $2^{1/4}$ more sensitive than the asymptotic 1-detector semicoherent sensitivity (lower dashed blue curve), equally sensitive to the asymptotic 2-detector semicoherent sensitivity for which powers from separate detectors are added (not shown), and $2^{1/4}$ less sensitive than the asymptotic 2-detector semicoherent behavior with coherent summing of simultaneous DFTs from the 2 detectors before computing power, as in Eq. (61) (green dash-dotted curve).

One practical consideration to keep in mind for these comparisons is that while coherent summing or cross-correlation of simultaneous DFTs provides improved sensitivity where possible, those gains are limited by achievable livetimes of interferometers that operate near their technological limits.

One can compute nominal signal-to-noise ratios for given signal strengths for the coherent, semi-coherent and cross-correlations methods from the noise-only variances and the expectation value dependences on signal $h_0$ presented above. Those SNRs allow sensible direct comparisons among semi-coherent and cross-correlation methods when $N_{\rm{seg}}$ is large enough for the noise-only detection statistics to exhibit approximate Gaussian behavior over the range of interest, but for small $N_{\rm{seg}}$, including especially the fully coherent case of $N_{\rm{seg}}$ = 1, the underlying statistics are highly non-Gaussian, requiring care in making comparisons.

3.3.4 Long-lag cross-correlation and loose coherence

Fully coherent, long integrations seem distinctly different from the multiple-short-segment searches based on semi-coherent and cross-correlation described (in simplified form) above, and in fact, using fully coherent methods to follow up on outliers produced by the latter methods is challenging because of the typical mismatch in parameter space fineness. Nonetheless, between these extremes exist bridges that offer the possibility of both systematic follow-up of outliers and more sensitive initial stages for multi-segment searches. The first method is long-lag correlation (Dhurandhar et al. 2008), and the second method is known as loose coherence (Dergachev 2010b).

Each method benefits from coherently summing DFT coefficients from data segments offset in time, which can be motivated by considering the segmentation of a single (continuous) Fourier transform of a strain signal h(t) into $N_{\rm{seg}}$ segments:

$$\begin{aligned} F(\omega ;[t_A,t_B])\equiv & {} {1\over T} \int _{t_A}^{t_B} h(t) e^{-i\omega (t-t_A)}dt \end{aligned}$$

(75)

$$\begin{aligned}= & {} {e^{i\omega t_A}\over N_{\rm{seg}}} \sum _{i=1}^{N_{\rm{seg}}}{e^{-i\omega t_{i-1}}\over T_{\rm{seg}}}\int _{t_{i-1}}^{t_i} h(t)e^{-i\omega (t-t_{i-1})}dt \end{aligned}$$

(76)

$$\begin{aligned}= & {} {e^{i\omega t_A}\over N_{\rm{seg}}} \sum _{i=1}^{N_{\rm{seg}}} e^{-i\omega t_{i-1}} F(\omega ;[t_{i-1},t_i]) \end{aligned}$$

(77)

$$\begin{aligned}= & {} {e^{i\omega t_A}\over N_{\rm{seg}}} \sum _{i=1}^{N_{\rm{seg}}} e^{-i\phi _{i}} F_i(\omega ), \end{aligned}$$

(78)

where $T_{\rm{seg}}= (t_B-t_A)/N_{\rm{seg}}$, $t_i = t_A + i{T_{\rm{seg}}}$ and $\phi _i = \omega t_{i-1}$. Hence the Fourier transform for the full data span is proportional to the sum of transforms for the individual segment transforms with phase corrections $e^{-i\phi _i}$.

Now consider once again the artificial but informative special case of a monochromatic signal detected via its strength in DFT bins from two detectors 1 and 2 with identical observation periods segmented into $N_{\rm{DFT}}$ epochs for which DFT are computed. For simplicity, assume the signal is bin-centered for each epoch’s DFT and that the detector noise is both stationary and identical for the two detectors. Guided by the relation above, one can then define a detection statistic based on the coherent sum of all DFTs from both detectors:

$$\begin{aligned} P= & {} \left| \sum _{i=1}^{N_{\rm{DFT}}} \left[ {\tilde{D}}_{1,i}e^{-i\phi _{1,i}}+{\tilde{D}}_{2,i}e^{-i\phi _{2,i}}\right] \right| ^2 \end{aligned}$$

(79)

$$\begin{aligned}= & {} \sum _{I,J=1}^2\sum _{i,j=1}^{N_{\rm{DFT}}} {\tilde{D}}_{I,i}{\tilde{D}}_{J,j}^*e^{-i(\phi _{I,i}-\phi _{J,j})}, \end{aligned}$$

(80)

where $\phi _{1,i}$ and $\phi _{2,i}$ account for the signal phase evolution for each detector for each DFT i and for any geometric offset between the detectors relative to the source direction (see Eq. 47). This full double sum is computationally costly to evaluate explicitly, not only because of the additional operations, but more important, because the implicit full coherence requires a fine stepping in parameter space. The form, however, makes more clear the relations between full coherence and both semi-coherence and cross-correlations (Dhurandhar et al. 2008), which can be viewed as subsets of the double sum. A semi-coherent sum of powers from individual detectors can be represented by

$$\begin{aligned} P_{\mathrm{semi-coherent}} = \sum _{I=1}^2\sum _{i=1}^{N_{\rm{DFT}}} {\tilde{D}}_{I,i}{\tilde{D}}_{I,i}^*, \end{aligned}$$

(81)

while cross-correlation of simultaneous amplitudes (see Eq. 73) from the two detectors is proportional to

$$\begin{aligned} P_{\mathrm{cross-correlation}} = \sum _{(I\ne J)=1}^2\sum _{i=1}^{N_{\rm{DFT}}} {\tilde{D}}_{I,i}{\tilde{D}}_{J,i}^*e^{-i(\phi _{I,i}-\phi _{J,j})}. \end{aligned}$$

(82)

This last relation suggests the possibility of following up an interesting search outlier from the first stage of a simultaneous-segment cross-correlation search by increasing the number of terms kept from the full double sum, allowing non-simultaneous cross terms and allowing self-correlation terms. For example, allowing an offset of up to $N_{\rm{lag}}$ segment durations would yield:

$$\begin{aligned} P_{\mathrm{cross-correlation(N_{\rm{lag}})}} = \sum _{I,J=1}^2\sum _{\begin{array}{c}i,j=1;\\ |i-j|\le N_{\rm{lag}}\end{array}}^{N_{\rm{DFT}}} {\tilde{D}}_{I,i}{\tilde{D}}_{J,j}^*e^{-i(\phi _{I,i}-\phi _{J,j})}. \end{aligned}$$

(83)

This approach can also be used at the first stage if rapid frequency evolution of the source, such as in a short-period binary system or in young object, argues for short DFT coherence times.

Another, related approach is to define a “loosely coherent” detection statistic as a subset of the original sum for which phase correlation between nearby (small-lag) DFT coefficients is favored (as opposed to the completely random relation allowed by semi-coherent sums). To illustrate,⁹ consider for simplicity a sum restricted to a single detector. Assume the phase correction applied to a product of DFT coefficients for segments separated by a single segment lag is taken to be unknown but uniformly distributed in probability between $-\delta$ and $+\delta$. A useful detection statistic can then be formed by (Dergachev 2010b)

$$\begin{aligned} P_{\mathrm{loose-coherence}}= & {} {1\over (2\delta )^{N_{\rm{seg}}-1}}\int _{-\delta }^{+\delta }d\varDelta \phi _{1,2}\int _{-\delta }^{+\delta }d\varDelta \phi _{2,3}...\int _{-\delta }^{+\delta }d\varDelta \phi _{N_{\rm{seg}}-2,N_{\rm{seg}}-1} \nonumber \\{} & {} \sum _{i,j}^{N_{\rm{seg}}}{\tilde{D}}_{I,i}{\tilde{D}}_{I,j}^*e^{-i(\varDelta \phi _{i',i'+1}+\varDelta \phi _{i'+1,i'+2}+...+\varDelta \phi _{j'-1,j'})}, \end{aligned}$$

(84)

where $i' (j') = \rm{min} ({\max }) \{i,j\}$ and $\varDelta \phi _{m,n}=\phi _n-\phi _m$. Evaluation of the integrals leads to

$$\begin{aligned} P_{\mathrm{loose-coherence}} = \sum _{i,j}^{N_{\rm{seg}}}{\tilde{D}}_{I,i}{\tilde{D}}_{I,j}^*\left( {\sin (\delta )\over \delta }\right) ^{|i-j|}, \end{aligned}$$

(85)

where the factor $\left( {\sin (\delta )\over \delta }\right) ^{|i-j|}$ weights adjacent-lag products more heavily than those with longer lags (and yields unity for $i=j$). The single-detector semicoherent power sum is recovered by setting $\delta =\pi$. In practice, to reduce computational cost, the factor is replaced by a discrete kernel function that truncates terms for which the weight contribution is too small to warrant the additional operations. A generalized version of this detection statistic, including more than one detector and non-integer segment lags, has been used in multi-stage searches with decreasing $\delta$ at each stage, e.g., $\delta =\pi \rightarrow \pi /2 \rightarrow \pi /4 \rightarrow \pi /8$. Each reduction in $\delta$ brings an increase in computational cost per parameter space volume as more terms are retained in the sum and the parameter space is searched more finely (“zooming in”).

3.4 Barycentering and coherent signal demodulation

Taking into account the phase/frequency modulations of Eq. (47) due to detector translational motion and the antenna pattern modulations embodied in Eqs. (48)–(51) due to detector orientation changes requires an accurate model of relevant solar system motions. As noted above, the gravitational-wave community has adopted techniques of pulsar astronomy researchers, with many LIGO data searches using the TEMPO 2 program (Hobbs et al. 2006) as a guide and for cross-checking (LIGO Scientific Collaboration 2018; Abbott et al. 2004). Correction for the Earth’s and a detector’s motions with respect to the solar system barycenter is called barycentering. Independently, Virgo analysts developed another barycentering software package (Astone et al. 2008), also checked against TEMPO 2 and the LIGO software. These packages choose steps in time fine enough to allow reliable interpolation of detector motion between sampled times.

Several approaches to incorporating the corrections have been developed for continuous gravitational-wave searches. These include time-domain heterodyning, Fourier-domain decomposition and hybrid techniques, as discussed below. More recently, techniques have been developed for more computationally efficient barycentering for use in targeted searches (Pitkin et al. 2018) and all-sky searches (Sauter et al. 2019).

3.4.1 Heterodyne method

Since CW signals are inherently quite narrowband with respect to deviations from idealized models, a heterodyning procedure using a base frequency near the nominal signal frequency, followed by a low-pass filter allows a large reduction in the number of data samples required to capture the signal modulations. Conceptually, if one has a pure signal h(t) that can be expressed as a slowly varying amplitude function A(t) times a sinusoid of base frequency $f_{\rm{base}}$, namely, $h(t) = \Re \left\{ A(t)e^{\rm{i}(2\,\pi f_{\rm{base}}\tau (t)+\phi _0)}\right\}$, one can apply the following heterodyne for a base frequency $f_{\rm{base}}$:

$$\begin{aligned} H_{f_{\rm{base}}}(t) \equiv h(t)\,e^{-\rm{i}2\,\pi f_{\rm{base}}\tau (t)} = A(t)\,e^{\rm{i}\phi _0}, \end{aligned}$$

(86)

where $\tau (t$) relates the SSB time to detector time. This approach allows the heterodyned function to have a low effective bandwidth. Applying a low-pass filter and then downsampling allows a large reduction in data volume while preserving signal fidelity.

In practice, the heterodyne used in targeted CW searches applies not a pure sinusoid factor, but rather a slowly modulated sinusoidal phase $\phi _{\rm{model}}(t)$ dependent on the topocentric (observatory-centric) time t, a model that includes the effects of Eqs. (40) and 47 on signal frequency evolution and propagation delays:

$$\begin{aligned} H_{\rm{model}}(t) \equiv d(t)\,e^{-\phi _{\rm{model}}(t)}, \end{aligned}$$

(87)

where it is assumed the signal is well approximated by the model: $h(t) = \Re \left\{ f(t)e^{\rm{i}(\phi _{\rm{model}}(t))}\right\}$, and the data stream d(t) contains h(t) and a (much larger-amplitude) random noise n(t). The resulting heterodyne product $H_{\rm{model}}(t)$ can then be interrogated for consistency with noise in addition to a signal amplitude function subject to antenna pattern modulations. Small residual deviations from the model (“timing noise”) measured empirically from electromagnetic pulsation observations can also be taken into account straightforwardly.

This technique is well suited to searches for known pulsars, for which the nominal frequency is precisely known from ephemeris measurements. For example, it has been customary in many LIGO and Virgo targeted searches to heterodyne, low-pass filter and then downsample to 1 data sample per minute, starting from a raw data stream of 16384 Hz. This technique assumes the residual intrinsic bandwidth of the signal following the heterodyne is no greater than the Nyquist frequency of 8.3 mHz, which is an excellent approximation for effects due to Earth / detector motion. This specific implementation is not as well suited to wide-parameter searches, for which the bandwidth must be increased or many distinct heterodynes be carried out.

The resulting heterodyned data samples have had frequency / phase modulations due to detector motion removed, but they retain antenna pattern due to detector rotation about the Earth’s spin axis. Section 4.1 below presents a Bayesian analysis method for such samples (Dupuis and Woan 2005).

3.4.2 Resampling methods

An alternative barycentering technique to heterodyning is to “resample” the detector data in order to transform it into SSB time (Schutz 1991; Jaranowski et al. 1998; Brady et al. 1998). A mundane but difficult nuisance is that data samples uniformly in detector frame time is not uniformly sampled in SSB time, making it difficult to apply conventional discrete Fourier transforms to the SSB samples. Two distinct methods have been used to date in CW analysis, to make the SSB samples uniform in time. The first (Patel et al. 2010; Meadors et al. 2018) uses spline interpolation of the non-uniformly sampled data to create uniformly sampled data. In practice, this interpolation is carried out on heterodyned subbands, much wider than those used in targeted searches, but much narrower than the full bandwidth of the original data collected. Another method (Astone et al. 2014b; Singhal et al. 2019) is based on selective data sample deletions and duplications, where narrow bands of data are temporarily upsampled to much higher frequencies, allowing smaller errors when extra samples are deleted or duplicated as SSB time appears to run faster or slower than detector-frame time (as defined by successive gravitational wavefronts), depending on the relative velocity of the detector with respect to the source. As for the heterodyne method, the result in both methods is a data stream for which detector translational motion has been corrected, but which still contains antenna pattern modulations from daily detector rotation.

3.4.3 Dirichlet kernel method

An alternative method can be applied in the Fourier domain by breaking the observation time into segments of short-enough duration that the signal frequency has negligible evolution during that duration, that is, the frequency change during the time $T_{\rm{seg}}$ is small relative to intrinsic frequency resolution of a discrete Fourier transform over that duration: ${1\over T_{\rm{seg}}}$. In a templated search for a particular signal, the frequency for that segment is known, and a Dirichlet filter (Dirichlet 1829) can be applied to the DFT coefficients in a narrow band surrounding the nominal frequency (e.g., ±4 DFT bins), using the expected weights for those bins for the nominal central frequency.

For a bin-centered signal and rectangular windowing, the filter would be a Kronecker delta, but in general, spectral leakage favors use of a handful of neighboring bins, to recover the full signal strength. By coherently combining the resulting extracted complex coefficients from the observed segments, one can achieve a full, coherent demodulation of the signal.¹⁰ Table 1 shows example power fractions in adjacent DFT bins (using rectangular windowing) for a monochromatic signal that is bin-centered or offset positively from the bin center by bin fractions of 0.1$-$0.5 in increments of 0.1. One sees that the bulk of signal power can be recovered by a modest number of neighboring bins. Figure 13 shows a visual representation of the values in Table 1.

Table 1 Fractional powers in neighboring DFT bins (rectangularly windowed) for a monochromatic signal with a frequency that ranges from bin-centered (bin 0 of the 10 bins shown) to a positive offset of a half-bin

Fig. 13

Visual representation of the fractional power values listed in Table 1

3.4.4 Time-domain parameter extraction

After frequency demodulation for detector translational motion, one has a highly reduced band-limited, data stream (time-domain for heterodyne or resampling, Fourier-domain for the Dirichlet filter) implicitly containing the amplitude modulations embodied in Eqs. (48)–(51), which can be considered deterministic functions of time, dependent upon the putative signal parameters. In the case of a source of known position and phase evolution but unknown orientation, as for many known pulsars, the unknown source parameters can be taken to be the strain amplitude $h_0$, the signal phase constant $\phi _0$, the inclination angle $\iota$ and the polarization angle $\psi$. The data stream can then be analyzed to extract those parameters.

For direct time-domain analysis, the principal method used to date for LIGO and Virgo data analysis has been a Bayesian inference (Abbott et al. 2004; Dupuis and Woan 2005). In brief, the heterodyned data samples $\{B_k\}$ can be expressed as complex quantities, with signal template expectations: (adopting the notation of Abbott et al. 2004):

$$\begin{aligned} y(t_k,\vec{a})= & {} {1\over 4}F_+(t_k;\psi )h_0(1+\cos ^2(\iota ))e^{\rm{i}2\phi _0} \nonumber \\{} & {} -{\rm{i}\over 2}F_\times (t_k;\psi )h_0\cos (\iota )e^{\rm{i}2\phi _0}, \end{aligned}$$

(88)

where $\vec{a}$ is a vector with components $(h_0,\iota ,\psi ,\phi _0)$ and $t_k$ is the time stamp of the kth sample.

With a set of priors on the $\vec{a}$ parameters one can extract a joint posterior probability density function for these parameters:

$$\begin{aligned} p(\vec{a}|\{B_k\})\propto & {} p(\vec{a}) \exp \left[ -\sum _k {\Re \left\{ B_k-y(t_k;\vec{a})\right\} ^2\over 2\,\sigma _{\Re \{B_k\}}^2}\right] \nonumber \\{} & {} \times \exp \left[ -\sum _k {\Im \left\{ B_k-y(t_k;\vec{a})\right\} ^2\over 2\,\sigma _{\Im \{B_k\}}^2}\right] , \end{aligned}$$

(89)

where $p(\vec{a})$ is the prior on $\vec{a}$ (uniform for $\cos (\iota )$, $\psi$ and $\phi _0$ and $h_0$), and $\sigma _{\Re \{B_k\}}^2$ and $\sigma _{\Im \{B_k\}}^2$ are the variances on the real and imaginary parts of $B_k$. This posterior distribution can be examined for evidence of a signal present. In the absence of a signal, an upper limit on strain amplitude $h_0$ can be found via marginalization over the other three signal parameters to obtain a marginalized posterior:

$$\begin{aligned} p(h_0|\{B_k\}) \propto \int \!\!\!\!\int \!\!\!\!\int p(\vec{a}|\{B_k\})\,d\iota \, d\psi \, d\phi _0, \end{aligned}$$

(90)

normalized so that $\int _0^\infty p(h_0|\{B_k\})dh_0 = 1$. Unlike a frequentist confidence level, the resulting curve versus $h_0$ represents the distribution of degree of belief in any particular value of $h_0$, given the signal model, the parameter priors and the data observations $\{B_k\}$. One can derive a 95% credible Bayesian upper limit $h_0^{95\%\,\rm{UL}}$ for which the probability lies below $h_0^{95\%\,\rm{UL}}$ via

$$\begin{aligned} 0.95 = \int _0^{h_0^{95\%\,\rm{UL}}} p(h_0|\{B_k\})dh_0. \end{aligned}$$

(91)

The combined posterior distribution from multiple, independent detectors can be obtained via the product of the individual likelihoods (Dupuis and Woan 2005). In the event that estimates of $\iota$ and $\psi$ can be inferred from electromagnetic measurements of the source, e.g., from images of jets assumed to be emitted along the spin axis of a star, then the precision on $h_0$ can be improved by assigning much narrower priors to the parameters.

3.4.5 Five-vector method

The so-called “Five-vector” method exploits the property that the complexity in Eqs. (46) and 48-51 can be distilled down to five terms (Astone et al. 2010b)

$$\begin{aligned} h(t) = h_0\vec{A}\cdot \vec{W}e^{\rm{i}(\omega _0 t+\phi _0)}, \end{aligned}$$

(92)

where $\omega _0$ is the signal frequency in the SSB frame, where $\vec{A}$ can be decomposed into plus- and cross-polarized terms that depend on complex amplitudes $H_+$ and $H_\times$:

$$\begin{aligned} \vec{A} = H_+\vec{A}^+ + H_\times \vec{A}^\times , \end{aligned}$$

(93)

and where $\vec{A}^+$ and $\vec{A}^\times$ can be expressed in terms of trigonometic functions, using Eqs. (48)–(51) (see Astone et al. 2010b for detailed expressions). The vector $\vec{W}$ is a five-component set of basis functions, indexed by $k = [-2, -1, 0, 1, 2]$:

$$\begin{aligned} \vec{W}_k = e^{-\rm{i}k\varTheta }, \end{aligned}$$

(94)

where $\varTheta$ is the detector’s local sidereal time in radians.

The data stream x(t) too can be decomposed using these basis functions:

$$\begin{aligned} \vec{X} = \int _Tx(t)\vec{W} e^{-\rm{i}\omega _0 t}dt. \end{aligned}$$

(95)

One can then construct a detection statistic using a weighted sum of the squared projections:

$$\begin{aligned} S = c_+|{\hat{h}}_+|^2 +c_\times |{\hat{h}}_\times |^2, \end{aligned}$$

(96)

where the projections are defined by

$$\begin{aligned} {\hat{h}}_+ = {\vec{X}\cdot \vec{A}^+\over |\vec{A}^+|^2}; \qquad {\hat{h}}_\times = {\vec{X}\cdot \vec{A}^\times \over |\vec{A}^\times |^2}. \end{aligned}$$

(97)

Empirically (Astone et al. 2010b), it is found that best performance for known $\iota$, $\psi$ can be obtained with the weightings: $c_{+,\times } = |\vec{A}^{+,\times }|^4$, while estimation of signal amplitude can be obtained from

$$\begin{aligned} {\hat{h}}_0 = \sqrt{|{\hat{h}}_+|^2+|{\hat{h}}_\times |^2}. \end{aligned}$$

(98)

3.4.6 The $\mathcal {F}$-statistic

The most pervasive detection statistic used in broadband CW searches can also be used for targeted searches, namely the $\mathcal {F}$-statistic (Jaranowski et al. 1998). As above, the $\mathcal {F}$-statistic is constructed to take into account not only the frequency / phase modulation of the detector’s translational motion (using time-domain or frequency-domain techniques), but also the amplitude modulation from daily detector rotation.

It is constructed from a general maximum likelihood approach, where the data is taken to be a sum of random noise n(t) and a signal h(t):

$$\begin{aligned} x(t) = n(t) + h(t), \end{aligned}$$

(99)

where h(t) from Eqs. (48)–(51) can be written¹¹

$$\begin{aligned} h(t) = \sum _{i=1}^4 A_i h_i(t), \end{aligned}$$

(100)

where the coefficients $A_i$ are inferred from Eqs. (48)–(51):

$$\begin{aligned} A_1= & {} h_0\sin (\zeta )\Biggl [{1\over 2}(1+\cos ^2(\iota )\cos (2\psi )\cos (2\,\varPhi _0) \nonumber \\{} & {} \qquad \qquad -\cos (\iota )\sin (2\psi )\sin (2\varPhi _0)\Biggr ], \end{aligned}$$

(101)

$$\begin{aligned} A_2= & {} h_0\sin (\zeta )\Biggl [{1\over 2}(1+\cos ^2(\iota )\sin (2\psi )\cos (2\,\varPhi _0) \nonumber \\{} & {} \qquad \qquad +\cos (\iota )\cos (2\psi )\sin (2\varPhi _0)\Biggr ], \end{aligned}$$

(102)

$$\begin{aligned} A_3= & {} h_0\sin (\zeta )\Biggl [-{1\over 2}(1+\cos ^2(\iota )\cos (2\psi )\sin (2\,\varPhi _0) \nonumber \\{} & {} \qquad \qquad -\cos (\iota )\sin (2\psi )\cos (2\varPhi _0)\Biggr ], \end{aligned}$$

(103)

$$\begin{aligned} A_4= & {} h_0\sin (\zeta )\Biggl [-{1\over 2}(1+\cos ^2(\iota )\sin (2\psi )\sin (2\,\varPhi _0) \nonumber \\{} & {} \qquad \qquad +\cos (\iota )\cos (2\psi )\cos (2\varPhi _0)\Biggr ], \end{aligned}$$

(104)

and the time-dependent functions $h_i$ have the form:

$$\begin{aligned} h_1 = a(t)\cos (2\varPhi (t)),\quad\,h_2(t) = b(t)\cos (2\varPhi (t)) \end{aligned}$$

(105)

$$\begin{aligned} h_3 = a(t)\sin (2\varPhi (t)),\quad\,h_4(t) = b(t)\sin (2\varPhi (t)), \end{aligned}$$

(106)

where $\varPhi (t)$ is the phase of the signal, including modulations.

A log-likelihood function $\log (\varLambda )$ is constructed via:

$$\begin{aligned} \log (\varLambda ) = (x|h) - {1\over 2}(h|h), \end{aligned}$$

(107)

where the scalar product $(\,|\,)$ is defined by a filter matched to the detection noise spectrum:

$$\begin{aligned} (x|y):= 4\Re \left\{ \int _0^\infty {{\tilde{x}}(f){\tilde{y}}^*(f) \over S_h(f) }df \right\} , \end{aligned}$$

(108)

where $\tilde{ }$ denotes a Fourier transform, $*$ is the complex conjugation, and $S_h$ is the one-sided power spectral density.

Following Jaranowski et al. (1998), the narrowband signal allows, in principle, conversion of the scalar product to a time-domain expression:

$$\begin{aligned} (x|h) \approx {2\over S_h(f_0)}\int _0^{T_{\rm{obs}}} x(t)h(t)dt, \end{aligned}$$

(109)

where stationarity of the noise over the observation period $T_{\rm{obs}}$ is implicitly assumed, which unfortunately, is rarely a good assumption for interferometers at the frontier of technology. Nonetheless, practical implementations of the $\mathcal {F}$-statistic are not limited by this assumption. Defining a time-domain scalar product:

$$\begin{aligned} (x||y):= {2\over T_{\rm{obs}}}\int _0^{T_{\rm{obs}}}x(t)y(t)dt, \end{aligned}$$

(110)

the log-likelihood function can be approximated via

$$\begin{aligned} \log (\varLambda ) \approx {T_{\rm{obs}}\over S_h(f_0)}\left[ (x||h)-{1\over 2}(h||h)\right] , \end{aligned}$$

(111)

which is proportional to a normalized log-likelihood $\log (\varLambda ')$:

$$\begin{aligned} \log (\varLambda ') = (x||h) - {1\over 2}(h||h), \end{aligned}$$

(112)

which does not depend explicitly on the spectral noise density. The signal depends linearly on the four amplitudes $A_i$ and can, in principle, be extracted from a likelihood maximization:

$$\begin{aligned} {\partial \log \varLambda '\over \partial A_i} = 0, \end{aligned}$$

(113)

from which a set of linear algebraic equations can be derived:

$$\begin{aligned} \sum _{i=1}^4\mathcal {M}_{ij}A_j = (x||h_i), \end{aligned}$$

(114)

where the components of the matrix $\mathcal {M}_{ij}$ are given by

$$\begin{aligned} \mathcal {M}_{ij}:= (h_i||h_j). \end{aligned}$$

(115)

Cross-terms of the $\cos (\varPhi (t))$ and $\sin (\varPhi (t))$ terms in Eqs. (105)–106 can be neglected in the time integrations. The surviving terms can be expressed:

$$\begin{aligned} (h_1|h_1) \approx&(h_3|h_3)&\approx {1\over 2}A, \nonumber \\ (h_2|h_2) \approx&(h_1|h_4)&\approx {1\over 2}B, \nonumber \\ (h_1|h_2) \approx&(h_3|h_4)&\approx {1\over 2}C, \end{aligned}$$

(116)

where $A:=(a||a)$, $B:=(b||b)$ and $C:=(a||b)$. With these approximations, the matrix $\mathcal {M}$ becomes

$$\begin{aligned} \mathcal {M}= \left( \begin{array}{cc} \mathcal {C}&{} \mathcal {O}\\ \mathcal {O}&{} \mathcal {C}\\ \end{array} \right) , \end{aligned}$$

(117)

where $\mathcal {O}$ is a zero 2 $\times$ 2 matrix, and $\mathcal {C}$ is

$$\begin{aligned} \mathcal {C}= {1\over 2}\left( \begin{array}{cc} A &{} C \\ C &{} B \\ \end{array} \right) , \end{aligned}$$

(118)

from which maximum-likelihood estimators ${\tilde{A}}_i$ of the true amplitudes $A_i$ can be obtained:

$$\begin{aligned} {\tilde{A}}_1= & {} 2{B(x||h_1)-C(x|h_2)\over D}, \nonumber \\ {\tilde{A}}_2= & {} 2{A(x||h_2)-C(x|h_1)\over D}, \nonumber \\ {\tilde{A}}_3= & {} 2{B(x||h_3)-C(x|h_4)\over D}, \nonumber \\ {\tilde{A}}_4= & {} 2{A(x||h_4)-C(x|h_3)\over D}, \end{aligned}$$

(119)

where $D = AB-C^2$. Substituting these expressions into Eqs. (111) leads to the $\mathcal {F}$-statistic (denoted by $2\mathcal {F}$):

$$\begin{aligned} 2\mathcal {F}&= {T_{\rm{obs}}\over S_h(f_0)} \Biggl [{B(x||h_1)^2+A(x||h_2)^2-2C(x||h_1)(x||h_2)\over D} \nonumber \\{} & {} + {B(x||h_3)^2+A(x||h_4)^2-2C(x||h_3)(x||h_4) \over D} \Biggr ]. \end{aligned}$$

(120)

The quantity $2\mathcal {F}$ has a probability distribution of a chi-squared with four degrees of freedom in the absence of a signal and that of a non-central chi-squared with a non-centrality parameter:

$$\begin{aligned} \lambda \equiv d^2 = (h|h) \end{aligned}$$

(121)

where d is proportional to signal amplitude (Jaranowski et al. 1998). The probability distributions $p_{\rm{noise}}(2\mathcal {F})$ and $p_{\mathrm{signal+noise}}(2\mathcal {F};\,d)$ are hence:

$$\begin{aligned} p_{\rm{noise}}(2\mathcal {F})= & {} {1\over 4}(2\mathcal {F}) e^{-(2\mathcal {F})/2}, \end{aligned}$$

(122)

$$\begin{aligned} p_{\mathrm{signal+noise}}(2\mathcal {F};\,d)= & {} {\left( 2\mathcal {F}\right) ^{1\over 2}\over d}I_1\left( d\sqrt{(2\mathcal {F})}\right) e^{-{1\over 2}(2\mathcal {F})-{1\over 2}d^2}, \end{aligned}$$

(123)

where $I_1$ is a modified Bessel function of the first kind (order 1).

As discussed in Sect. 3.3, $2\mathcal {F}$ can be used as a detection statistic, where a threshold $2\mathcal {F}_0$ can be chosen to satisfy a desired false alarm probability:

$$\begin{aligned} \rm{CDF}_{\rm{noise}}[2\mathcal {F}_0]= & {} \int _0^{2\mathcal {F}_0} {p_{\rm{noise}}(2\mathcal {F})}\,d(2\mathcal {F}) = 1-\alpha , \end{aligned}$$

(124)

$$\begin{aligned}= & {} 1 - \left( 1+2\mathcal {F}_0+{1\over 2}2\mathcal {F}_0^2+{1\over 6}2\mathcal {F}_0^3\right) e^{-2\mathcal {F}_0}, \end{aligned}$$

(125)

and where the probability of detection for a given d is

$$\begin{aligned} P_{\rm{detection}}(d,2\mathcal {F}_0) = \int _{2\mathcal {F}_0}^\infty p_{\mathrm{signal+noise}}(2\mathcal {F};\,d)\,d(2\mathcal {F}). \end{aligned}$$

(126)

The formalism above describes a time-domain implementation (Jaranowski et al. 1998; Astone et al. 2010a), but a narrowband frequency implementation (Prix 2018) has been used extensively in LIGO searches.

In searches for known pulsars for which optical or X-ray observations of pulsar wind nebulae allow inference of $\iota$ and $\psi$, a modified version of the $\mathcal {F}$-statistic known as the $\mathcal {G}$-statistic can be applied to gain slightly in sensitivity, depending on the stellar orientation (Jaranowski and Królak 2010).

Although originally derived in a frequentist, log-likelihood framework, the $\mathcal {F}$-statistic can also be obtained in a Bayesian approach (Prix and Krishnan 2009) with an unphysical prior (non-isotropic in stellar orientation), an alternative framework that has received additional study (Prix et al. 2011; Keitel et al. 2014; Whelan et al. 2014; Dhurandhar et al. 2017; Bero and Whelan 2019; Wette 2021).

3.5 Semi-coherent signal demodulation

Let’s now consider a coarser demodulation, in which phase fidelity is not required for the full observation time. Instead, the observation is broken into discrete segments of coherence time $T_{\rm{coh}}$ which need not be contiguous with each other. The segmentation reduces the fineness with which the parameter space (e.g., frequency, frequency derivatives, sky location) must be sampled, leading to often dramatic reduction in computing cost to search a given parameter space volume, albeit with a degradation of achievable strain sensitivity.

3.5.1 The stack-slide method

For short-enough $T_{\rm{coh}}$, no frequency demodulation need be applied within a single segment. One can simply sum the strain power from each bin in an DFT containing the frequency of the signal for that time interval. Figure 14 illustrates the simplest version of this approach, known as “stack-slide” (Brady et al. 1998; Mendell and Landry 2005). In a spectrogram where each column represents DFT powers for a given $T_{\rm{coh}}$, the bin containing the signal frequency (indicated by the green square) varies in frequency from one column to the next. Correcting for the frequency modulation by shifting columns up or down leads to the signal’s power being contained in a horizontal track in the demodulated spectrogram. For a relatively narrow frequency band, the amount of vertical shift for a given column is nearly the same for all frequencies in the band, for a given set of source parameters, including sky location. Hence by stacking powers across rows in the demodulated spectrogram, one can look for an outlier indicating a signal.

Fig. 14

Conceptual illustration of the “stack-slide” method in which rows of a spectrogram are shifted up or down in frequency to account for Doppler modulations

To be concrete, define the power ${\tilde{P}}^{(k)}_i$ to be the strain power spectral density measured in bin i of DFT k, where the bin i is the appropriate bin after “sliding”:

$$\begin{aligned} {\tilde{P}}^{(k)}_i= {2|{\tilde{D}}^{(k)}_{i(\text{demod})}|^2\over T_{\rm{coh}}}. \end{aligned}$$

(127)

Following Abbott et al. (2008a), Mendell and Landry (2005), this power is renormalized to form a dimensionless quantity $\eta _{i}^k$

$$\begin{aligned} \eta _{i}^k= {{\tilde{P}}^{(k)}_i\over S_{h_{i}}^k}, \end{aligned}$$

(128)

where $S_{h_{i}}^k$ is the one-sided power spectral density expected in the absence of signal. This quantity differs from the $\rho _i^2$ defined in Eq. (54), both in the implicit demodulation associated with bin i and in a factor of 2. Here $\eta _{i}^k$ has an expectation value of 1 in the absence of signal.

The stack-slide detection statistic $P^{(k)}_{i(\mathrm SS)}$ then is the average value of $\eta _{i}^k$ over the $N_{\rm{DFT}}$ DFT’s used in the analysis:

$$\begin{aligned} P^{(k)}_{i(\mathrm SS)}= {1\over N_{\rm{DFT}}} \sum _{k=1}^{N_{\rm{DFT}}} \eta _{i}^k. \end{aligned}$$

(129)

This quantity has an expectation value of 1 in the absence of signal and a variance of $1/N_{\rm{DFT}}$. Signal candidates are chosen based on exceeding a threshold corresponding to a false alarm probability $\alpha$, from which detection sensitivity is determined from a desired false dismissal probability $\beta$. Appendix B of Abbott et al. (2008a) details the statistical behavior. In brief, the quantity (similar to $R_i$ of Eq. (63) above)

$$\begin{aligned} \mathcal {P}_{i(\mathrm SS)}= 2N_{\rm{DFT}}\eta _{i}^k\end{aligned}$$

(130)

has the probability density distribution of a non-central $\chi ^2$ with $2N_{\rm{DFT}}$ degrees of freedom and a non-centrality parameter $2N_{\rm{DFT}}<\!\!d^2\!\!>$ which is the expectation value of the estimator in Eq. (121) when evaluated over a single DFT. Hence the probability density distribution for $\mathcal {P}_{i(\mathrm SS)}$ follows:

$$\begin{aligned} p_{\mathrm{signal+noise}}(\mathcal {P}_{i(\mathrm SS)};N_{\rm{DFT}},d)= & {} {I_{N_{\rm{DFT}}-1}\left( \sqrt{\mathcal {P}_{i(\mathrm SS)}N_{\rm{DFT}}<\!d^2\!>}\right) \over \left( N_{\rm{DFT}}<\!d^2\!>\right) ^{N_{\rm{DFT}}-1}} \nonumber \\&\quad \times \mathcal {P}_{i(\mathrm SS)}^{{N_{\rm{DFT}}-1\over 2}} e^{-\left( N_{\rm{DFT}}+<\!{d^2\over 2}\!>\right) }. \end{aligned}$$

(131)

Numerical evaluation (Abbott et al. 2008a) for $\alpha =0.01$ and $\beta =0.10$ leads (in the large $N_{\rm{DFT}}$ limit) to a sensitivity $<\!d^2\!>^{(90)} \approx 7.385 / \sqrt{N_{\rm{DFT}}}$ and to a strain sensitivity for a single template search of $h_0^{(90)} \approx 7.7\sqrt{S_h}/\left( T_{\rm{coh}}T_{\rm{obs}}\right) ^{1/4}$, where $T_{\rm{obs}}$ refers here to the total observing time analyzed and where stationary data is implicitly assumed. In practice, however, this method is applied to wide-parameter searches for which trials factors lead to much worse strain sensitivities (Tenorio et al. 2022). Prix and Shaltev (2012) carry out a detailed analysis of maximizing sensitivity at fixed computational cost for different stack-slide search configurations.

3.5.2 The powerflux method

The PowerFlux method (Abbott et al. 2008a), in its simplest form, is similar to the stack-slide method, with the following refinements: (1) an explicit polarization is assumed for each signal template searched, with an antenna pattern correction applied; (2) detection statistic variance is minimized in the presence of non-stationary noise; and the detection statistic itself is a direct measure of strain amplitude.

Using the same notation as above (see Eq. 127), the PowerFlux detection statistic $R_{\rm{PF}}$ for a given set of orientation parameters $\iota$ and $\psi$ is written:

$$\begin{aligned} R_{\rm{PF}}= {2\over T_{\rm{coh}}} { \sum _{i=1}^{N_{\rm{DFT}}} W_i{\tilde{P}}^{(k)}_i/ (F_i(\iota ,\psi ))^2 \over \sum _{i=1}^{N_{\rm{DFT}}} W_i}, \end{aligned}$$

(132)

where the weights are defined as

$$\begin{aligned} W_i= [(F_i(\iota ,\psi ))^2]^2/S_{h_{i}}^2, \end{aligned}$$

(133)

and where $F_i(\iota ,\psi )$ is the antenna pattern weight calculated for the midpoint of the time segment i for the assumed polarization such that the detector amplitude response can be written as $h_{\rm{det},i} = h_0F_i(\iota ,\psi )$. In practice, searches have been carried out for circular polarization ($\iota =0$ or $\pi$) and for particular linear polarization angles $\psi$ (with $\iota =\pi /2$) to define “best-case” and “worst-case” orientations, respectively.

The choice of weight definition comes from minimizing the variance of the strain amplitude estimator ${\tilde{P}}^{(k)}_i/(F_i(\iota ,\psi ))^2$, where the noise (in the weak signal regime) is assumed to be dominated in each time segment i by a power spectral density $S_{h_{i}}$ with underlying Gaussian distributions for real and imaginary DFT components. Under that assumption, the variance of the noise is proportional to $(S_{h_{i}})^2$. As a result, each term in the numerator of Eq. (132) is proportional to $(F_i(\iota ,\psi ))^2{\tilde{P}}^{(k)}_i/S_{h_{i}}^2$, which gives higher weight to segments with higher $F_i(\iota ,\psi )$ magnitude and lower noise $S_{h_{i}}$, as one would wish. For a given polarization choice defined by $(\iota ,\psi )$ the detection statistic $R_{\rm{PF}}$ is a direct measure of total strain power such that subtracting the expectation value based on neighboring bin yields a direct estimator for signal power.

3.5.3 Hough transform methods

Hough transform methods refer, in practice, to an application of a pattern recognition algorithm first developed for use in the 1960’s by high energy particle physicists (Hough 1959, 1962) to reconstruct a charged particle’s trajectory from discrete positions (“hits”), measured by a tracking detector. The method is best suited to data that is “sparse” and for which a simple transformation from the raw measurements to the signal parameter space can amplify the detection statistic. In the original application to particle tracking, the hits were two-dimensional projections for which looking for straight lines built out of all hit combinations was computationally intensive (especially in the 1960’s!). To represent a straight line, instead of offset and slope, the vector of its minimum distance to the origin, in polar coordinates $(r,\theta )$, is used. A point (x, y) belonging to that line sets the relation $r=x\cos \theta +y\sin \theta$ which is a sinusoidal curve in the $\theta$-r plane. Cells in that plane count how many curves pass within their boundaries, and the most occupied cell identifies $(r,\theta )$ of the original track.

In the case of CW searches, two different Hough transform methods (“Sky Hough” and “Frequency Hough”) have been used in recent years, both of which accumulate excess power from frequency-demodulated DFTs. In the Sky Hough method (Krishnan et al. 2004), the transformation is from a narrow frequency band and frequency derivative to right ascension and declination, where broad patches of sky are searched collectively. In the Frequency Hough method (Antonucci et al. 2008; Astone et al. 2014a), the transform is from a time-frequency plane to a plane of frequency and frequency derivative. In each case, one searches for a statistically significant excess among the pixels and applies a thresholding to individual accumulated powers, in order to reduce computational cost in the accumulation.

The Hough number count is defined as a weighted sum of binary counts $n_i$:

$$\begin{aligned} n = \sum _{i=1}^{N_{\rm{DFT}}} w_in_i, \end{aligned}$$

(134)

where $n_i= 1$ if the normalized segment power $\eta _{i}^k$ exceeds a threshold $\eta ^*$ and zero otherwise, and where the weights favor low-noise times and are optimized for circular polarization (Antonucci et al. 2008; Abbott et al. 2008a):

$$\begin{aligned} w_i\propto {1\over S_{h_{i}}} \left[ (F_i^+)^2+(F_i^\times )^2\right] , \end{aligned}$$

(135)

with a normalization chosen to satisfy:

$$\begin{aligned} \sum _{i=1}^{N_{\rm{DFT}}} w_i= N_{\rm{DFT}}. \end{aligned}$$

(136)

In the Sky Hough method (Krishnan et al. 2004), so-called “Hough maps” in right ascension and declination are created for each assumed frequency and frequency derivative, where signal outliers produce “hot” pixels in the sky patch for which the map applies. In the Frequency Hough method (Antonucci et al. 2008; Astone et al. 2014a), the Hough map is created instead in the plane of frequency and frequency derivative for each localized sky point. The primary motivations for this alternative mapping to parameter space are reduction of inaccuracies arising from approximations and non-linearities in the mapping to the sky; avoidance of artifact “pileup” in which certain regions of the sky are contaminated over subbands by particular narrowband artifacts; and the possibility to use over-resolution in frequency, at negligible additional computational cost (Antonucci et al. 2008).

Regardless of the choice of parameter space mapping, the statistical character of the Hough number counts is governed by the value of the threshold used to define the binary counts $n_i$. The mean number count in the absence of a signal is ${\bar{n}}= N_{\rm{DFT}}p$, where p is the probability that the normalized power $\eta _{i}^k$ exceeds a threshold value $\eta ^*$. For unity weighting, the standard deviation is $\sigma _{{\bar{n}}} = N_{\rm{DFT}}p(1-p)$. For the more general weighting, this becomes:

$$\begin{aligned} \sigma = \left[ p(1-p)\sum _{i=1}^{N_{\rm{DFT}}}w_i^2\right] ^{1/2}. \end{aligned}$$

(137)

For $N_{\rm{DFT}}\gg 1$, the underlying distribution can be approximated as Gaussian, in which case a threshold $n_{\rm{th}}(\alpha )$ corresponding to a false alarm rate $\alpha$ is given by (Krishnan et al. 2004)

$$\begin{aligned} n_{\rm{th}}= N_{\rm{DFT}}\,p+\sqrt{2}\,\sigma \, \rm{erfc}^{-1}(2\alpha ), \end{aligned}$$

(138)

where it is natural to regard the significance of a given measured n to be

$$\begin{aligned} s = {n-{\bar{n}}\over \sigma }. \end{aligned}$$

(139)

In Krishnan et al. (2004), Abbott et al. (2005a), and Abbott et al. (2008a) an optimal choice of the normalized power threshold parameter is found to be $\eta ^*\approx 1.6$, for which $p=e^{-\eta ^*} \approx 0.2$.

One can compute (Abbott et al. 2008a) a sensitivity $h_0^{1-\beta }(\alpha )$ for a false dismissal probability $\beta$ and false alarm probability $\alpha$:

$$\begin{aligned} h_0^{1-\beta }(\alpha ) \approx 3.38\mathcal (S)^{1/2} \left( {||\vec{w}||\over \mathbf{w\cdot X}}\right) ^{1/2}\,\sqrt{1\over T_{\rm{coh}}}, \end{aligned}$$

(140)

where $||\vec{w}|| = \sum _{i=1}^{N_{\rm{DFT}}}w_i^2$ and

$$\begin{aligned} \mathcal {S}= & {} \rm{erfc}^{-1}(2\alpha )+\rm{erfc}^{-1}(2\beta ), \end{aligned}$$

(141)

$$\begin{aligned} X_i= & {} {1\over S_{h_{i}}} \left[ \left( F_+^i\right) ^2+\left( F_\times ^i\right) ^2\right] , \end{aligned}$$

(142)

and where $F_{+/\times }^i$ refer to the antenna pattern functions for the $+$ and $\times$ polarizations evaluated at the midpoint of time segment i.

Other improvements to the Sky Hough method have included incorporating a hierarchical approach (Sancho de la Jordana 2010), adaptation to a search for stars in binary systems (Covas and Sintes 2019) (see Sect. 4.5), clustering of outliers (Tenorio et al. 2021c) and systematic outlier follow-up (Tenorio et al. 2021a).

3.5.4 The stacked $\mathcal {F}$-statistic method

The semi-coherent approach used above (in various approaches) with DFT coefficients can also be applied to longer segments of time for each of which the coherent $\mathcal {F}$-statistic is computed. This approach permits deeper sensitivity since the $\mathcal {F}$-statistic can be computed without degradation of signal coherence for arbitrarily long periods of time. The disadvantage is that the much finer resolution in parameter space associated with such sensitivity leads to much greater computational cost, coming from the fine stepping needed within each segment and from the mapping with negligible signal loss from one segment to the next. A variety of $\mathcal {F}$-statistic “stacking” methods¹² have been implemented over the years, both inside and outside of the framework of the Einstein@Home distributed computing system (see Sect. 4.4). When computing the $\mathcal {F}$-statistic over short time segments, a modified variation, the $\mathcal {F}_{\rm{AB}}$-statistic, which avoids degeneracy due to minimal antenna pattern modulation can be more effective (Covas and Prix 2022b).

Many of the considerations discussed in semi-coherent summing of DFT power have analogs in $\mathcal {F}$-statistic summing, including the use of thresholding and the use of Hough transform mapping. Particular implementations will be discussed below in Sects. 3.6.1 and 4.4. One critical issue in these computationally costly searches is the optimum placement of signal templates in parameter space, to be discussed next, more generally. Another important consideration is clustering of initial outlier candidates (Steltner et al. 2022a) to reduce computational cost in hierarchical searches prior to follow up with deeper search algorithms.

3.6 Template placement

Computationally demanding searches must choose step sizes in signal parameter space, with finer spacing leading to greater cost, in general. The choices are typically governed by what is considered an acceptable maximum “mismatch”, normally parametrized by the fractional decrease in detection statistic for a given offset in parameter space.

For an n-dimensional, hypercubic grid defined by n search parameters, one can regard the mismatch parameter $\mu$ as governing the maximum half-length of the diagonal of the n-dimensional cell containing the correct signal parameters. Conceptually, we imagine having made the least optimum choice of grid offset such that the true parameters lie at the center of the cell, and no matter which of the $2^n$ corners of the cell is sampled, the value of the detection statistic is no smaller than $1-\mu$ of the value obtained, had the center of the cell been sampled. Figure 15 illustrates the concept with a detection statistic “surface” above a plane in two signal parameters, where the contours correspond to mismatch values of 20%, 40%, 60% and 80%.

Fig. 15

Illustration of mismatch for a generic detection statistic. The upper panel shows a “surface” of height equal to the detection statistic for a pure signal above a plane defined by two signal-defining parameters (with zero covariance for simplicity). The green cross marks the true location for the two parameters and the maximum possible detection statistic. The lower panel shows detection statistic contours in the two-parameter space, where the contours correspond to mismatch values of 20%, 40%, 60% and 80%. The red crosses define a search template grid chosen to be least optimal for this signal location in that the true signal location is centered in a 2-dimensional cell, which maximizes the possible minimum mismatch (20%) between the detection statistics for the true signal and the closest template. The dashed diagonal line defines the “distance” in the 2-dimensional parameter space between the true signal location and the closest search template

In the following, general considerations of template placement are considered, first for directed searches for particular points on the sky, for which placement is relatively straightforward, and then for all-sky searches, where template placement is quite subtle and remains an active research front.

3.6.1 Template placement in directed searches

For coherent directed searches, the phase evolution Eq. (40) governs template placement, where for multi-day analyses, the effects of amplitude modulation can be safely neglected in choosing template spacing (Prix 2007a, b). Consider for a moment a highly simplified detection statistic based on multiplying in the time domain an assumed sinusoidal signal template having a particular phase constant $\phi _0$ and frequency $f_0$ against the raw data x(t), assumed to be a sum of random Gaussian noise n(t) and a sinusoid signal having amplitude $h_0$, phase constant $\phi _0'$ and frequency $f_0'$:

$$\begin{aligned} F(\phi _0',f_0')= & {} \left| {2\over T}\int _0^T e^{-i(\phi _0+2\pi f_0t)}\,x(t)dt\right| ^2 \end{aligned}$$

(143)

$$\begin{aligned}= & {} \left| {2\over T}\int _0^T e^{-i(\phi _0+2\pi f_0t)}\,[n(t)+h_0\cos (\phi _0'+2\pi f_0't)]dt\right| ^2. \end{aligned}$$

(144)

In the limit of large T and strong signal (neglecting n(t)), the expectation value of F when maximized over possible template values for $f_0$ is simply $h_0^2$, independent of $\phi _0$, $\phi _0'$ and $f_0'$, where F is maximized for $\varDelta f\equiv f_0'-f_0 = 0$. To understand how rapidly F decreases as $|\varDelta f|$ departs from zero, it’s helpful to rewrite $\cos (\phi _0'+2\pi f_0't) = {1\over 2}(e^{i(\phi _0'+2\pi f_0't)}+e^{-i(\phi _0'+2\pi f_0't)})$, where in the strong-signal limit of large T and for small $|\varDelta f|$ such that the second term of the cosine expansion can be neglected, F approaches

$$\begin{aligned} F\approx\, & {} \left| {h_0\over T}\int _0^T e^{i[\varDelta \phi +2\pi \varDelta ft]}\,dt\right| ^2 \end{aligned}$$

(145)

$$\begin{aligned}=\, & {} h_0^2 \left| \rm{sinc}(\pi \varDelta fT)\right| ^2 \end{aligned}$$

(146)

$$\begin{aligned}\approx \,&\, {} h_0^2\left[ 1-{1\over 3}\left( \pi \varDelta fT\right) ^2\right] , \end{aligned}$$

(147)

where $\varDelta \phi \equiv \phi _0'-\phi _0$ drops out and where the convention $\rm{sinc}(x) \equiv {\sin (x)\over x}$ is chosen. If we rewrite this last result as $F\approx h_0^2\cos ^2(\varDelta \phi _{\rm{mismatch}})$, then the tolerance in $\varDelta f$ for a phase mismatch value $\varDelta \phi _{\rm{mismatch}}$ is

$$\begin{aligned} \varDelta f_{\rm{mismatch}} \approx {\sqrt{3}\over \pi T}\varDelta \phi _{\rm{mismatch}}, \end{aligned}$$

(148)

which is $2\sqrt{3}$ larger than the naive underestimate of Eq. (41). Consequently, in a search that automatically maximizes F over the unknown phase constant, one need not search as finely in frequency as suggested by Eq. (41), which implies reduced computational costs in large-scale searches.

Given the importance of template placement to those costs, in fact, a systematic approach is merited. Following methodology developed originally for template placement in compact binary merger searches (Sathyaprakash and Dhurandhar 1991; Owen 1996; Balasubramanian et al. 1996), one can rewrite and generalize the simplified detection statistic in Eq. (143), replacing the data with another template and address the reduction in F’s value due to mismatch of template parameters

$$\begin{aligned} F(\vec {\lambda },\vec{\lambda }')= & {} \left| {1\over T}\int _0^T e^{-i\varPhi (t;\vec {\lambda })}e^{i\varPhi (t;\vec {\lambda }')}\,dt\right| ^2 \end{aligned}$$

(149)

$$\begin{aligned}= & {} \left| {1\over T}\int _0^T e^{i\varDelta \varPhi (t;\vec {\lambda },\varDelta \vec{\lambda })}\,dt\right| ^2, \end{aligned}$$

(150)

where $\vec{\lambda }$ and $\vec{\lambda }'$ refer to a set of N parameters, such as phase and frequency derivatives, and where $\varDelta \vec{\lambda }\equiv \vec{\lambda }'-\vec{\lambda }$ is taken small enough that 2nd-order $\varDelta \vec{\lambda }$ corrections in $\varDelta \varPhi \equiv \varPhi (t;\vec{\lambda }+\varDelta \vec{\lambda })-\varPhi (t;\vec{\lambda })$ can be neglected. Clearly, for $\varDelta \vec{\lambda }= 0$, $F = 1$ and is maximum, with vanishing first partial derivatives. Hence we expect F to have the following form in the vicinity of $\varDelta \vec{\lambda }=0$:

$$\begin{aligned} F \approx 1 + {1\over 2}\sum _{k,\ell =1}^N {\partial ^2 F\over \partial \varDelta \lambda _k\partial \varDelta \lambda _\ell }\biggr |_{\varDelta \vec{\lambda }=0}\varDelta \lambda _k\varDelta \lambda _\ell , \end{aligned}$$

(151)

where the diagonal 2nd-partial derivatives are negative and which leads to the definition of a metric:

$$\begin{aligned} g_{k\ell }= - {1\over 2}{\partial ^2 F\over \partial \varDelta \lambda _k\partial \varDelta \lambda _\ell }\biggr |_{\varDelta \vec{\lambda }=0}, \end{aligned}$$

(152)

such that the mismatch $\mu$ of a template deviation is $\mu = \sum _{k,\ell }g_{k\ell }\varDelta \lambda _k\varDelta \lambda _\ell$. Hence the appropriate spacing of templates in parameter space to avoid excessive mismatch is governed by the form of $g_{k\ell }$.

A general treatment of finding $g_{k\ell }$ (Owen 1996) can be approached by Taylor-expanding the exponential in Eq. 150: $e^{i\varDelta \varPhi } \approx 1 + i\varDelta \varPhi - {1\over 2}\varDelta \varPhi ^2$ and evaluating the second derivatives of F with respect to $\varDelta \lambda _k$ and $\varDelta \lambda _\ell$. In the limit $\varDelta \vec{\lambda }\rightarrow 0$, one finds:

$$\begin{aligned} -{1\over 2} {\partial ^2 F\over \partial \varDelta \lambda _k\partial \varDelta \lambda _\ell }\biggr |_{\varDelta \mathbf {\lambda }=0} = \left[ \left\langle\!{\partial \varDelta \varPhi \over \partial \varDelta \lambda _k}{\partial \varDelta \varPhi \over \partial \varDelta \lambda _\ell }\!\right\rangle - \left\langle\!{\partial \varDelta \varPhi \over \partial \varDelta \lambda _k}\!\right\rangle \left\langle\!{\partial \varDelta \varPhi \over \partial \varDelta \lambda _\ell }\!\right\rangle \right] _{\varDelta \mathbf {\lambda }=0}, \end{aligned}$$

(153)

where

$$\begin{aligned} \left\langle f(t) \right\rangle \equiv {1\over T}\int _0^T f(t)\,dt. \end{aligned}$$

(154)

More specifically, in the context of the Taylor Nth-order expansion of the phase function (henceforth omitting $\vec{\lambda }$ dependence in $\varDelta \varPhi$):

$$\begin{aligned} \varDelta \varPhi (t;\varDelta \vec{\lambda }) \approx \varDelta \phi _0 + 2\pi \sum _{m=0}^N {\varDelta f^{(m)}t^{m+1}\over (m+1)!}, \end{aligned}$$

(155)

where $f^{(m)} = {d^mf\over dt^m}\bigr |_{t=0}$, and the set of frequency derivatives can be treated as a parameter vector ${\textbf{f}} \equiv [f^{(0)},f^{(1)},...,f^{(N)}]$. The detection statistic F can be expanded:

$$\begin{aligned} F\approx & {} \left| {1\over T}\int _0^T e^{i\left( \varDelta \phi _0 + 2\pi \sum _{m=0}^N {\varDelta f^{(m)}t^{m+1}\over (m+1)!}\right) }\,dt\right| ^2 \end{aligned}$$

(156)

$$\begin{aligned}\approx & {} |e^{i\varDelta \phi _0} |^2\Biggl |{1\over T}\int _0^T \biggl [1 + i\,2\pi \sum _{m=0}^N{\varDelta f^{(m)}t^{m+1}\over (m+1)!} \nonumber \\{} & {} - {1\over 2} (2\pi )^2\sum _{m,n=0}^N{\varDelta f^{(m)}\varDelta f^{(n)}t^{m+n+2}\over (m+1)!(n+1)!}\biggr ] \,dt \Biggr |^2 \end{aligned}$$

(157)

$$\begin{aligned}\approx & {} \Biggl |{1\over T} \biggl [T + i\,2\pi \sum _{m=0}^N{\varDelta f^{(m)}T^{m+2}\over (m+2)!} \nonumber \\{} & {} - {1\over 2} (2\pi )^2\sum _{m,n=0}^N{\varDelta f^{(m)}\varDelta f^{(n)}T^{m+n+3}\over (m+1)!(n+1)!(m+n+3)}\biggr ] \Biggr |^2 \end{aligned}$$

(158)

$$\begin{aligned}\approx & {} \biggl [1 + (2\pi )^2\sum _{m,n=0}^N{\varDelta f^{(m)}\varDelta f^{(n)}T^{m+n+2}\over (m+2)!(n+2)!} \nonumber \\{} & {} - (2\pi )^2\sum _{m,n=0}^N{\varDelta f^{(m)}\varDelta f^{(n)}T^{m+n+2}\over (m+1)!(n+1)!(m+n+3)}\biggr ] \end{aligned}$$

(159)

$$\begin{aligned}= & {} 1 - (2\pi )^2\sum _{m,n=0}^N{\varDelta f^{(m)}\varDelta f^{(n)}T^{m+n+2}(m+1)(n+1)\over (m+2)!(n+2)!(m+n+3)}. \end{aligned}$$

(160)

Terms higher in order than $\varDelta f^{(m)}\varDelta f^{(n)}$ have been neglected in the above. From this last expression, we conclude that the metric $g_{k\ell }$ can be written:

$$\begin{aligned} g_{k\ell }= (2\pi )^2{T^{k+\ell +2}(k+1)(\ell +1)\over (k+2)!(\ell +2)!(k+\ell +3)}. \end{aligned}$$

(161)

See Wette et al. (2008) for the same expression for the metric for the $\mathcal {F}$-statistic(Jaranowski et al. 1998) in a directed search.

As examples, consider the 2-parameter metric with respect to frequency $f_0$ and its first derivative $f_1$:

$$\begin{aligned} g_{00}= & {} {1\over 3}\left( \pi T\right) ^2, \end{aligned}$$

(162)

$$\begin{aligned} g_{01}= & {} {1\over 6}\left( \pi T^{3/2}\right) ^2, \end{aligned}$$

(163)

$$\begin{aligned} g_{11}= & {} {4\over 45}\left( \pi T^2\right) ^2. \end{aligned}$$

(164)

For a given desired mismatch $\varDelta M$, define nominal offsets $\varDelta f_0^*$ and $\varDelta f_1^*$, using only the diagonal metric elements: ($\varDelta f_k^* \equiv \sqrt{\varDelta M}/g_{kk}$)

$$\begin{aligned} \varDelta f_0^*= & {} {\sqrt{3\varDelta M}\over \pi T}, \end{aligned}$$

(165)

$$\begin{aligned} \varDelta f_1^*= & {} {3\sqrt{5\varDelta M}\over \pi T^2}. \end{aligned}$$

(166)

Since off-diagonal terms in the metric are non-zero, a rectangular grid using only diagonal terms will, in general, be inefficient. Figure 16 illustrates for a 2-dimensional slice of $\varDelta f_0$ vs. $\varDelta f_1$ (=$\varDelta {\dot{f}}_{\rm{GW}}$) a template grid that accounts for these correlations in mismatch. A grid placement based on only the diagonal metric elements would lead to inefficient coverage, as shown. Prix (2007b) and Wette (2014) discuss more generally and in more detail template grid placement for CW searches, with special focus on searches over the three-dimensional parameter space ($f_{\rm{GW}}$,${\dot{f}}_{\rm{GW}}$,$\ddot{f}_{\rm{GW}}$). As noted above, however, for short coherence times, the range of $\ddot{f}_{\rm{GW}}$ searches may be smaller than $\varDelta \ddot{f}_{\rm{GW}}^*$ in regions of parameter space, depending on braking-index assumptions and the value of ${\dot{f}}_{\rm{GW}}$.

Fig. 16

Illustration of a ($\varDelta f_0$, $\varDelta f_1$) template grid (black stars) and constant-mismatch elliptical contours for which the grid point placement gives complete coverage. The values of the frequency and frequency derivative are given in normalized units of $\varDelta f_0^*$ and $\varDelta f_1^*$ defined in Eqs. (165)–166. The magenta diamonds indicate a rectangular grid with full coverage when the off-diagonal metric term is ignored

3.6.2 Template placement in all-sky searches

Template placement in all-sky searches is relatively straightforward for semi-coherent searches using short coherence times $T_{\rm{coh}}$ of $\sim$ h or less, but is quite subtle for coherent searches using much longer coherence times (days) and for semi-coherent searches using long coherence times for each data segment.

Short-$T_{\rm{coh}}$ template grids can be factorized over sky location ($\alpha$, $\delta$) and over ($f_{\rm{GW}}$, ${\dot{f}}_{\rm{GW}}$), using isotropic grid point placement, e.g., density proportional to $\cos (\delta$) and uniform in $\alpha$, with a rectangular grid in ($f_{\rm{GW}}$, ${\dot{f}}_{\rm{GW}}$), with spacings determined empirically or semi-analytically for a given data run. For example, the rule of thumb given in Eq. (44) overestimates the density needed for short observation times of $\sim$few months because of correlations (Prix and Itoh 2005) in the dependence of a semi-coherent power sum on sky location and frequency parameters. For a data set collected over 1–2 months of the Earth’s orbit, the average acceleration of the detector toward the Sun creates an apparent offset in the spin-down of a putative source. Hence a search over a band of frequencies and 1st derivatives may detect a signal with nearly as high an SNR as the nominal maximum, but with correlated offsets in the four parameters ($\alpha$, $\delta$, $f_{\rm{GW}}$, ${\dot{f}}_{\rm{GW}}$). For longer observation times, these near degeneracies in parameter space become less helpful; signal templates must be placed more densely. At additional computational cost, these semi-coherent searches may also search explicitly over source polarizations, or may choose to apply a circular polarization weighting and sacrifice some sensitivity to near-linear polarizations (Abbott et al. 2008a).

Template placement for much longer coherence times is more challenging because analytic approximations break down for long coherence times and because naive grid spacings depend on the specific region of the Earth’s orbit covered by a particular coherence time, making the systematic matching of signal candidates across different time segments non-trivial in semi-coherent searches. Template placement for the $\mathcal {F}$-statistic has received much attention in the last decade and a half (Whitbeck 2006; Prix 2007a, b; Wette and Prix 2013; Wette 2014), in part because of its use in the Einstein@Home (see Sect. 4.4) distributed computing platform. From Eq. (121), one can define a mismatch analogous to that of Sect. 3.6.1:

$$\begin{aligned} 1 - \mu \equiv {(h_{\vec{\lambda }+\varDelta \vec{\lambda }}|h_{\vec {\lambda }+\varDelta \vec {\lambda }})\over (h_{\vec {\lambda }}|h_{\vec{\lambda }})}, \end{aligned}$$

(167)

where we expect an approximately quadratic falloff from unity for small $|\varDelta \vec{\lambda }|$ (but see Allen 2019 for a discussion of template placement for larger $|\varDelta \vec{\lambda }|$ and see Allen 2021 which distinguishes between optimality for setting rigorous upper limits and optimality for signal detection).

The complexity of the definition of (h|h) (see Eqs. 50–51 and 100–106) do not yield a definition of (h|h) in the convenient form of Eq. (149). In particular, the sidereal antenna pattern modulations due to the Earth’s rotation are not accommodated by the phase-only dependence of the simplified form. For long observation times, however, amplitude modulation effects can be averaged with sufficient accuracy (Prix 2007a). Phase modulation from the Earth’s motion is captured by Eq. (149), allowing use of Eq. (153) to determine the $\mathcal {F}$-statistic metric with respect to frequency parameters and sky location.

Following the treatment of Prix (2007a), a more explicit phase evolution can be written:

$$\begin{aligned} \varPhi (t) = \phi _0 + 2\pi \sum _{m=0}^N{f^{(m)}(\tau _{\rm{ref}})\,\tau (t)^{m+1}\over (m+1)!}, \end{aligned}$$

(168)

where $\tau (t)$ is the SSB arrival time of the signal. Ignoring the Shapiro and Einstein delays in Eq. (47) for metric definition, one can write:

$$\begin{aligned} \tau (t) = t + {\vec{r}(t)\cdot {\hat{n}}\over c} - \tau _{\rm{ref}}, \end{aligned}$$

(169)

where $\vec{r}(t)$ is the position of the detector at time t, ${\hat{n}}$ is the unit vector pointing from the detector to the source, and $\tau _{\rm{ref}}$ is the reference time in the SSB frame at which the frequency and its derivatives are defined.

The phase derivatives entering Eq. (153) can then be written:

$$\begin{aligned} {\partial \varPhi \over \partial f^{(k)}}= & {} 2\pi {\tau (t)^{k+1}\over (k+1)!}, \end{aligned}$$

(170)

$$\begin{aligned} {\partial \varPhi \over \partial n_i}= & {} 2\pi {r_i(t)\over c}\sum _{m=0}^N{f^{(m)}(\tau _{\rm{ref}})\,\tau (t)^m\over m!}, \end{aligned}$$

(171)

where ${\vec{r}}(t)\cdot \hat{n} = \sum _i r_i(t) n_i$, from which the metric terms for a particular point in parameter space ($\textbf{f}$, ${\hat{n}}$) can be computed via numerical integration of Eq. (153) over the observation span, with precise description of ${\vec{r}}(t)$, accounting for the non-zero eccentricity of the Earth’s orbit. It is convenient in some studies, though, to make the “Ptolemaic” approximation (Jones et al. 2005; Whitbeck 2006) in which the Earth’s orbit is treated as circular, for which analytic but quite lengthy trigonometric expressions can be obtained (Whitbeck 2006).

As shown in Jaranowski et al. (1998), the GW phase described by Eqs. (168)–(169) can be well approximated (setting the reference time $\tau _{\rm{ref}}=0$ for convenience) by

$$\begin{aligned} \varPhi (t) = \phi _0 + 2\pi \sum _{m=0}^N{f^{(m)}t^{m+1}\over (m+1)!} + 2\pi {\vec{r}(t)\cdot {\hat{n}}\over c} \left( \sum _{k=0}^N {f^{(k)} t^k\over k!}\right) . \end{aligned}$$

(172)

The last term in Eq. 172 can be usefully decomposed into the orbital motion of the Earth’s center and the spin of the detector about the Earth’s center with orbital and spin phases ($\vec{r}(t) = \vec{r}_{\rm{orb}}(t) + \vec{r}_{\rm{spin}}(t)$):

$$\begin{aligned} \varPhi _{\rm{orb}}(t)= & {} 2\pi {\vec{r}_{\rm{orb}}(t)\cdot {\hat{n}}\over c} \left( \sum _{k=0}^N {f^{(k)} t^k\over k!}\right) , \end{aligned}$$

(173)

$$\begin{aligned} \varPhi _{\rm{spin}}(t)= & {} 2\pi {\vec{r}_{\rm{spin}}(t)\cdot {\hat{n}}\over c} \left( \sum _{k=0}^N {f^{(k)} t^k\over k!}\right) . \end{aligned}$$

(174)

An inconvenient property of the metric defined above using the parameters ($\vec{f}$, ${\hat{n}}$) is that converting the 3-D Cartesian ${\hat{n}}$ components to the 2-D sky coordinates $\alpha$ and $\delta$ leads to a sky spacing that depends on the parameter themselves. A metric more convenient for large-scale CW searches over the entire sky can be obtained by using global correlations in parameter space (Pletsch 2008; Pletsch and Allen 2009). Some searches exploiting these correlations are known as “GCT” searches for “Global Correlation Transform.” For multi-day coherence times short compared to one orbital year, one can Taylor-expand the rescaled position of the Earth’s center $\vec{\xi }(t) \equiv {\vec{r}}_{\rm{orb}}(t)/c$ in Eq. (169) about the midpoint $t_0$ of the coherence time span $T_{\rm{coh}}$:

$$\begin{aligned} \vec{\xi }(t) = \vec{\xi }(t_0) + \sum _{n=1}^\infty {\vec{\xi }^{(n)}(t_0)(t-t_0)^n\over n!}. \end{aligned}$$

(175)

The Earth’s orbital motion contribution to signal phase (Eq. 173) can then be rewritten:

$$\begin{aligned} \varPhi _{\rm{orb}}(t)= & {} 2\pi \left( \sum _{k=0}^N {f^{(k)} (t-t_0)^k\over k!}\right) \left( \sum _{\ell =0}^\infty {(t-t_0)^\ell \over \ell !}\vec{\xi }^{(\ell )}\cdot {\hat{n}}\right) \end{aligned}$$

(176)

$$\begin{aligned}= & {} 2\pi \sum _{m=0}^\infty (t-t_0)^m\left( \sum _{n=0}^{m'} {f^{(n)}\vec{\xi }^{(m'-n)}\over n!(m'-n)!}\cdot {\hat{n}}\right) , \end{aligned}$$

(177)

where $m'$ = min(m,N).

It is also convenient to define new sky coordinates that capture the vector difference in signal phase (radians) between the source direction ($\alpha$, $\delta$) and the detector’s direction from the Earth’s center ($\alpha _D(t_0)$, $\delta _D$) at time $t_0$ (Pletsch