Gravitational Wave Timing Array

Home , Big Bang Observer, Pulsar timing array

Mar´ıaJoséBustamante-Rosell,1 Joel Meyers,2 Noah Pearson,2 Cynthia Trendafilova,2 and Aaron Zimmerman1 1Center for Gravitational Physics, University of Texas at Austin, 2515 Speedway, C1600, Austin, TX 78712, USA 2Department of Physics, Southern Methodist University, 3215 Daniel Ave, Dallas, TX 75275, USA (Dated: July 7, 2021) We describe the design of a gravitational wave timing array, a novel scheme that can be used to search for low-frequency gravitational waves by monitoring continuous gravitational waves at higher frequencies. We show that observations of gravitational waves produced by Galactic binaries using a space-based detector like LISA provide sensitivity in the nanohertz to microhertz band. While the expected sensitivity of this proposal is not competitive with other methods, it fills a gap in frequency space around the microhertz regime, which is above the range probed by current pulsar timing arrays and below the expected direct frequency coverage of LISA. The low-frequency extension of sensitivity does not require any experimental design change to space-based gravitational wave detectors, and can be achieved with the data products that would already be collected by them.

I. INTRODUCTION the sensitivity of LISA and pulsar timing arrays has been targeted by at least one proposal [21] but remains un- likely to be filled in the next few decades by currently Current and future observatories probe gravitational funded gravitational wave observatories. Additionally, an waves in several regimes of the frequency spectrum. interesting recent proposal showed that high-cadence as- Ground-based gravitational wave detectors, such as the trometric measurements obtained from photometric sur- Advanced LIGO [1], Advanced Virgo [2], and KAGRA [3] veys can be used to search for gravitational waves with detectors, cover the ∼ 20 − 2000 Hz range by measur- −22 frequencies ranging from nanohertz to microhertz [22]. ing induced gravitational wave strains of order 10 In this paper, we describe a new method that can be in their kilometer-scale Fabry-Pérotinterferometers. In used to search for gravitational waves in the microhertz the near future, the Laser Interferometer Space Antenna band. Gravitational waves emitted from Galactic bina- (LISA) will probe the millihertz regime, measuring grav- ries observed with space-based interferometers like LISA itational waves using million kilometer-scale arms [4]. act as stable oscillators that can be monitored to con- At nanohertz frequencies, pulsar timing arrays indirectly struct a gravitational wave timing array. Within the measure gravitational waves by monitoring an array of LISA band, the most common source of gravitational pulsars which serve as standard clocks [5–9]. Strongly waves will be the millions of white dwarf binaries present lensed, repeating fast radio bursts [10] can also be uti- in the Milky Way. About ten thousand of them should lized to detect gravitational waves in a similar manner be well resolved by the LISA mission within its planned to pulsar timing arrays, by monitoring changes to the four-year lifetime. Additionally, it is possible that LISA arrival times of the lensed burst images [11]. Astromet- will detect other classes of binaries emitting in the mHz ric observations of distant objects can be used to search regime, such as mixed binaries of white dwarfs and neu- for photon deflection caused by gravitational waves in the tron stars or black holes [23, 24]. Most of these Galac- nanohertz regime, along with an integrated constraint on tic binaries will emit nearly monochromatic gravitational a background of lower frequency gravitational waves [12– waves during LISA’s lifetime, with the evolution of the 14]. A key goal of current and future cosmic microwave binaries dominated by gravitational wave radiation. background surveys is the search for primordial gravi- A gravitational wave background, either stochastic or tational waves with frequencies in the attohertz range coherent, imprints correlated phase modulations on the arXiv:2107.02788v1 [gr-qc] 6 Jul 2021 through measurements of B-mode polarization [15, 16]. gravitational waves from the binaries, similar to the tim- These techniques, together with current and future ob- ing delays measured in pulsar timing arrays. A gravi- servatories, cover a wide swath of the gravitational wave tational wave timing array like the one described here spectrum. Gaps remain between the lowest frequencies benefits from the large number of sources and continuous accessible on the ground, ∼ 1 Hz, and the highest fre- monitoring of the whole sky, and it provides sensitivity to quencies accessible to LISA. Several proposed observato- gravitational waves in a frequency range 10−9–10−5 Hz. ries plan to cover the gap between ground-based interfer- The frequency coverage of such a gravitational wave tim- ometers and LISA, including the DECi-hertz Interferom- ing array is illustrated in Fig.1, along with several ex- eter Gravitational wave Observatory (DECIGO) [17, 18] isting experiments and future proposals for gravitational and the Big Bang Observer (BBO) [19]. Meanwhile, an- wave detection. Constructing such an array with the ob- other gap exists between the highest frequencies accessi- servations from a space-based interferometer like LISA ble to pulsar timing arrays, limited by the cadence with requires no experimental design changes nor a special- which pulsars in the network can be observed, and the ized observing campaign. Gravitational waves are not lowest frequencies accessible to LISA, limited by acceler- subject to any plasma dispersion effects in the interstel- ation performance [20]. This microhertz regime between lar medium or radio interference at Earth that can com- 2

the carrier wave is dϕ ≈ ω (1 − z) , (1) dt c where ϕ is the phase of the carrier, and z is a redshift. It is given by [14]

i j ˆ nˆ nˆ î ˆ ˆ ó z(t, k) = hij(t, k) − hij(tc, k) , (2) 2(1 + kˆ · nˆ)

where kˆ is the direction of propagation of the modulating wave andn ˆ is the unit vector pointing from the ob- FIG. 1. Frequency coverage of several existing and pro- server towards the source of the carrier waves, as shown posed gravitational wave detection methods. Direct detec- in Fig.2. tion methods (red) include the existing ground-based detec- ˆ ˆ tors LIGO [1], Virgo [2], and Kagra [3] (LVK), the space-based The two terms hij(t, k) and hij(tc, k) are the metric LISA, and proposed DECIGO [18] detectors. Indirect meth- perturbations from the modulating wave at the observer ods (blue) include existing pulsar timing arrays [5], astrom- and at the source of the carrier wave, respectively. In the etry with Gaia observations [25], and proposed high-cadence context of pulsar timing arrays, the first term is called the astrometry with the future Nancy Grace Roman Space Tele- Earth term, and the second the pulsar term. The time ˆ scope [22, 26]. The gravitational wave timing array proposed tc = t−d(1+ˆn·k)/c incorporates the relative propagation here (purple) would use LISA observations to detect gravita- times of the carrier wave and modulating waves, where tional waves indirectly. d is the distance between the observer and the source of the carrier waves. When considering the timing residuals of pulsars, the plicate gravitational wave searches with pulsar timing ar- redshift of Eq. (2) is derived by considering the propa- rays [27]. On the other hand, a gravitational wave tim- gation of a null geodesic between the source of the pulse ing array relies on intrinsically weak gravitational waves and the observer [14, 29]. The same result is arrived at as the primary signal to be monitored, and this leads in our case by noting that for ωm ωc, the carrier wave to lower sensitivity than direct LISA measurements and is in the geometric optics limit, so that the wavefronts pulsar timing arrays in the regimes where those strategies propagate along geodesic null rays, as discussed further are sensitive. in AppendixA. The phase of the carrier wave is then This paper is organized as follows. In Sec.II we discuss given by the phase modulation induced on a gravitational wave propagating in a flat spacetime which is perturbed by an Z t ˆ 0 ˆ 0 additional background gravitational wave. In Sec.III we ϕ(t, k) ≈ ωct − ωc z(t , k)dt . (3) calculate the sensitivity of a gravitational wave timing 0 array using multiple methods. SectionIIIA presents a When searching for a stochastic background of gravita- timing estimate approach leveraging existing results for tional waves with pulsar timing arrays, the pulsar term pulsar timing arrays [28], Sec.IIIB provides a matched in Eq. (2) is a nuisance parameter that contributes to the filter sensitivity estimate, and Sec.IIIC describes our timing noise for each pulsar. It can often be neglected most complete frequency-domain Fisher estimates. Sec- when estimating the sensitivity of the array. Since we tionIV uses a mock Galaxy catalog and the results from consider a binary source of modulating waves, we find Sec.III to get realistic estimates of the sensitivity of a that this pulsar term potentially contributes to the de- gravitational wave timing array constructed from LISA tected signal. Whether we can neglect this term, which observations. Lastly, Sec.V discusses the viability of such we call the carrier term, will thus depend on the evolution an array, the limitations of our approximations, and fu- of the modulating wave. ture prospects.

A. Source of modulating waves II. MODULATION OF A CARRIER WAVE BY A BACKGROUND WAVE We assume that the source of the modulating wave is a supermassive black hole binary, which produces a We consider an approximately monochromatic gravi- transverse-traceless gravitational wave tational wave with frequency ωc, which we refer to as the ˆ + + ˆ × × ˆ carrier wave. This wave propagates in the presence of hij(t, k) = AmHij (k) cos Φ(t) + AmHij (k) sin Φ(t) , (4) a lower frequency, modulating gravitational plane wave, with frequency ωm ωc. To leading order in the mod- where Φ(t) is the phase of the modulating wave. The ulating wave amplitude, the observed phase evolution of binary has an inclination angle ι relative to the line of 3

of the secondary. The modulating wave is approximately monochromatic with frequency ωm over the timescale of the observation, which is of the order of years (the lifetime of the LISA mission). Equations (2)–(8) show that the phase of the carrier wave is modulated sinusoidally by the background wave, potentially at two distinct frequencies, one associated with the local term and one with the carrier term. The rate of change of ωm is a steep function of ωm, and so at a fixed chirp mass we see that there are two cases. Either the frequency evolves very slowly, even on timescales sufficiently long for light to propagate across the Galaxy (∼ 10 kyr), or else the frequency evolves rapidly over such timescales. In the first case, we cannot neglect the carrier term, while in the second we can.

B. Slow evolution of modulating source

First consider the case where we neglect the evolution of the modulating binary. Then both the local (Earth) term and the carrier (pulsar) term of the modulation have the same frequency, but a relative phase difference due to propagation effects. The various sinusoidal terms com- FIG. 2. Illustration of the proposed gravitational wave tim- bine to give a redshift ing array. An array of Galactic binaries within the Milky Way produce gravitational waves whose frequencies are mod- z = AmF sin γ cos(ωmt − δ) , (9) ulated by a longer-wavelength gravitational wave, exempli- fied here by one produced by a black hole binary. Unit where vectorn ˆc points from the observer (such as LISA) to each » + + 2 × × 2 (AmHnn) + (AmHnn) galactic binary, and kˆ denotes the propagation direction of F = , (10) ˆ the modulating gravitational wave. The inclination of the Am(1 + k · nˆ) binary black hole is ι, such that the binary is face-on for ω d γ = m (1 + kˆ · nˆ) , (11) ι = 0. Ac,Am, ωc, ωm, αc, αm, are the amplitudes, angular 2c frequencies, and phases of the carrier and modulating gravi- π tational waves, respectively. δ = α + γ + β − , (12) m 2 × × AmHnn tan β = + + , (13) sight, so that the amplitudes of the plus- and cross- AmHnn polarized waves are related to a characteristic amplitude and where H+ = H+nînˆj is the plus-polarized com- A through nn ij m ponent of the wave projected onton ˆ, and similarly for 1 + cos2 ι H× . We have introduced a phase α , which is a refer- A+ = A ,A× = A cos ι . (5) nn m m m 2 m m ence phase for the modulating binary, a phase β which varies withn ˆ at fixed kˆ, and a phase γ which also depends For later reference, we define two preferred transverse + ˆ × ˆ on the distance to the source of the carrier. polarization tensors ij(k) and ij(k), and a polarization Overall this gives a modulated carrier wave phase angle ψ. Then ˆ AmF ωc H+ = cos 2ψ + + sin 2ψ × , (6) ϕ(t, k) ≈ ωct − αc − sin γ sin(ωmt − δ) , (14) ij ij ij ωm H× = − sin 2ψ + + cos 2ψ × . (7) ij ij ij where we have introduced another reference phase αc to Assuming that the modulating binary evolves only due absorb the integration constant. Equation (14) shows to the emission of gravitational waves, its frequency evo- that the modulation of the phase is directly proportional lution is given by to number of cycles of the carrier that occur per half a cycle of the modulating wave. 12 ÅGMã5/3 We mostly work in the frequency domain, where the ω˙ = ω11/3 (8) m 5 c3 m carrier signal is a sharp peak at a frequency f = fc, and another peak mirrored around zero at f = −fc as re- 3/5 1/5 where M = (m1m2) /(m1 + m2) is the chirp mass quired for a real signal. The modulation produces a side- of the binary, m1 is the mass of the primary, and m2 that band peak to each side of these carrier peaks, separated 4

In order to treat the local and carrier sidebands as well separated, we require that the change in the modulation δ=π frequency over the propagation time of the carrier wave δ= 0 be larger than the full width at half maximum (FWHM) of the sideband peak in the frequency domain, ˙ FWHM [sinc(πfmT )] < fmd/c . (15) Combining with Eq. (8), we see that the modulating fre- 1 quencies for which the local and carrier terms are well T separated satisfy

Å ã3/11 Å 6 ã5/11 Å ã3/11 0.1 kpc 10 M 4 yr fm > 0.46 µHz . f -f f f +f d M T c m c c m (16) FIG. 3. Schematic representation of the signal in the fre- We have chosen a small fiducial chirp mass and distance quency domain. The full width at half maximum of each to display the largest frequency for which the carrier (pul- peak is ∼ 1/T (highlighted with a red line), where T is the sar) terms can be neglected. total observation time. For modulating frequencies higher than this, the phase of the Galactic binary wave in the presence of a modulating wave simplifies to by fm as shown in Fig.3. The redshift phase δ intro- duces an asymmetry in the two sideband peaks around ˆ AmF ωc ϕ(t, k) ≈ ωct − αc − sin(ωmt − αm − β) . (17) each carrier peak. Since we deal with a finite observation 2ωm time T , these peaks are sinc functions rather than delta functions, with finite widths in frequency space of order Comparing to the case where we include the carrier term, 1/T . we recover Eq. (17) from Eq. (14) by setting γ = π/2 and In the limit that the modulating binary does not adding a factor of 1/2 to the modulating part. Further, evolve, both the phase and amplitude of the modulat- if we fix the location of the modulating binary along with ing signal depend on d, through γ. Since the distance d its inclination and polarization angles, F and β depend to any Galactic binary is unknown to the precision of a only on the sky location of the white dwarf binary and not wavelength of the modulating wave, γ is a nuisance pa- its distance from Earth. In practice, the sky position of a rameter that degrades our ability to detect the influence white dwarf binary detected by LISA can be determined of a modulating wave on the carrier waves coming from to a few square degrees (see Sec.IV), and so we treat F a network of Galactic binaries, all at different unknown and β as approximately known in what follows. distances. For later use, we define a convenient coordinate system However, if the frequency of the modulating wave in which to evaluate the geometric quantities F and β. Let the modulating wave propagate in the z direction, evolves over typical timescales ∼ d/c, then the local + x x y y term and the carrier term each produce distinct side- and set the polarization tensors to be ij =e î eˆj − eî eˆj + x y y x x y bands around the carrier. When the frequency of the and ij =e î eˆj +e î eˆj , wheree ˆ ande ˆ are unit vectors modulating wave evolves fast enough that the sidebands in the x and y directions. In this basis, the redshift will produced by the local term and the carrier term are well- be in terms of θ and φ, the colatitude and longitude of separated in frequency space, the local term will provide the carrier source, a pair of sidebands around each carrier peak whose am- θ Å ã1/2 plitude does not depend on d/c. We turn to this case F = sin2 cos2(2φ − 2ψ) sin4 ι + 4 cos2 ι , (18) next. 2 ï2 cos ι tan(2φ − 2ψ)ò δ = α + tan−1 . (19) m 1 + cos2 ι C. Fast evolution of modulating source We can further specialize to the two extremes of the inclination. When the binary is face-on, ι = 0, the waves For the range of frequencies that we are interested in, are circularly polarized, and we can determine the parameter space where the modulating frequency evolves enough to separate the local θ F = 2 sin2 , β = 2(φ − ψ) . (20) and carrier sidebands for each Galactic binary. The lo- 2 cal term would then be common to all the carrier waves, When the binary is edge-on, the modulating wave is lin- up to a phase and geometric factors that do not depend early polarized, and upon the distance d to any binary. Meanwhile the car- θ rier terms differ in frequency and amplitude among the F = sin2 cos(2ψ − 2φ) , β = 0 . (21) Galactic binaries, and can be neglected as extra noise. 2 5

The amplitude of the modulation is maximized in the this approach using frequency-domain methods, employ- face-on case, and each carrier’s longitude φ contributes ing matched ﬁltering to create simple signal-to-noise ratio to the eﬀective phase of the modulation. For the edge-on (SNR) estimates of the sensitivity of the array. Finally, case, the amplitude depends on the longitude of the car- we use a Fisher matrix approach to arrive at a more com- rier, but the phase of the modulation is common among plete frequency-domain estimate for the sensitivity of a the galactic binaries. gravitational wave timing array, applicable at both high and low modulating frequencies.

D. Total gravitational wave signal A. Timing sensitivity estimates Considering the results so far, we can write the total gravitational wave signal produced by a network of Although not as accurate as the frequency-based sen- Galactic binaries. Each binary produces a carrier wave sitivity approaches presented in subsequent sections, a contribution hi to the total measured strain h. Here i timing estimate of the gravitational wave timing array indexes over the N Galactic binaries. If we account for sensitivity provides a good approximation to the more both the local and carrier terms of the modulation, in the complete results, and illustrates the function of the array limit that the modulating frequency fm remains constant in a way most analogous to pulsar timing array methods. during timescales ∼ di/c for all the Galactic binaries, we We break up the full length of the observation period T have into a set of segments each with duration δt. Within each observation period of length δt, we consider the ques- ï ò AmFifi tion of how well we can measure a time (or phase) delay hi(t) =Ai cos 2πfit − αi − sin γi sin(2πfmt − δi) fm in the arrival of gravitational waves from a Galactic bi- ≈Ai cos(2πfit − αi) nary. This time delay could be caused, for example, by a long-wavelength gravitational wave that is modulating AiAmFifi + sin γ sin(2πf t − α ) sin(2πf t − δ ) , the frequency of the carrier wave. Within a given time f i i i m i m interval, this would appear as a small phase shift in the (22) carrier signal, which would not be present if the carrier frequency were fixed. Let σ be the error with which we and the total signal arising from the sum of carriers is td can measure such a time delay. X We model the gravitational wave signal from a single h(t) = hi(t) . (23) Galactic binary as a sinusoid with amplitude Ai and fre- i quency fi, and we denote the induced time delay by td. In the opposite extreme, where fm evolves rapidly (but Thus the signal waveform from the binary is given by is still approximately constant over the timescale T of the observations), we can neglect the effect of the carrier hi(t) = Ai cos [2πfi(t − td)] . (25) (pulsar) terms. Then we can make the simple substitu- We assume that δt is small enough that any drift in the tion γi → π/2 and divide Am by an additional factor of frequency f over the duration is negligible, and we treat two to get h(t). Specifically, i the amplitude Ai as constant during this window as well. We approximate the noise on the observed gravita- h (t) ≈A cos(2πf t − α ) i i i i tional wave strain as white noise in the neighborhood AiAmFifi of the carrier frequency, since we treat fi as being nearly + sin(2πfit − αi) sin(2πfmt − αm − βi) 2fm constant over the period δt. Using Sn(f) to denote the (24) one-sided power spectral density of the noise on the observed strain (e.g. from the LISA mission [30]), we take in this case. Sn(fi) to denote the power spectral density of white noise with a fixed value of Sn(f = fi) for all f. From here, we take a Fisher matrix approach, where III. GRAVITATIONAL WAVE TIMING ARRAY the parameters θa that we would like to estimate from SENSITIVITY our observations are td and fi. Then the elements of the Fisher matrix for a waveform given in the time domain We now turn to the problem of estimating the sensitiv- are [31] ity of a network of nearly monochromatic carrier sources to modulation by a background, low-frequency gravita- 2 Z δt/2 ∂h (t, t , f ) ∂h (t, t , f ) Γ = i d i i d i dt . (26) tional wave. We carry out the sensitivity estimate in ab a b Sn(fi) −δt/2 ∂θ ∂θ three ways. First, we use time-domain techniques bor- rowed from the familiar method of time-delay measure- The lower bound on the variance of any unbiased esti- ments in pulsar timing arrays to estimate the sensitiv- mator of these parameters can be found from the covari- ity of the gravitational wave timing array. We confirm ance matrix given by the inverse of the Fisher matrix 6

−1 Σab = Γab . The Fisher matrix elements are straightfor- array, using the assumption of rapidly evolving sources ward to compute and are given by so that we can neglect the pulsar term as in [28]. How-

ever, in our case the timing uncertainties σtd can vary 1 Sn(fi) from binary to binary. In Sec.IV we evaluate this sen- Σtdtd = 2 2 2 , (27) (2π) Ai fi δt sitivity estimate using a mock catalog of LISA sources, where σ varies significantly among the binaries. To ac- Σt f = 0 , (28) td d i count for this, we introduce a modification to Eqs. (31) 3 S (f ) n i and (32), making the replacement Σfifi = 2 2 3 , (29) π Ai δt " #−1/2 where we have assumed that the time delay t and the σt X d d → σ−2 . (33) period of the Galactic binary carrier wave are small com- [N(N − 1)]1/4 td,i pared to δt. From here, we get the error in measuring i the time delay, This replacement agrees in the limit of many identical binaries and weights the contribution of each binary ac- p 1 Sn(fi) cording to its SNR in the expected manner. However, it σtd = Σtdtd = 2 2 , (30) 2π Ai fi δt replaces the cross correlation among the binaries with the autocorrelation of each binary and should not be taken given some observing “cadence” 1/δt. as a rigorous limit at large N. To determine the sensitivity of a gravitational wave We emphasize that due to differences in conventions timing array to the presence of a modulating wave, we and definitions between our work and [28], we do not must combine the information provided by timing mea- intend for this to be a precise mapping onto their re- surements from N binaries, each with a timing measure- sults. Our timing estimate of the sensitivity is meant ment error of σtd , while the time-based subdivision of the to illustrate the concept of the gravitational wave tim- full data set provides a cadence 1/δt. This is analogous ing array, and particularly, in a way that is analogous to to finding the sensitivity of a pulsar timing array whose the measurements taken by pulsar timing arrays. It does measurements have a given timing error and cadence, and not match precisely with our results from the more ac- thus we can apply existing estimates for pulsar timing ar- curate frequency-domain approaches given in subsequent ray sensitivity curves. sections. We compare this timing estimate to frequency The total strain sensitivity of a pulsar timing array domain estimates using a mock catalog of LISA detec- may be approximated as the sum of two power laws, de- tions in Sec.IV. scribing the low- and high-frequency limit, and given respectively by [28] √ B. Matched filtering sensitivity estimate 3 % Å 13 ã1/4 σ …δt hLOW(f ) ≈ th td sec ξ , c m 27/4χπ3 N(N − 1) f 2 T 3 T m A common approach to the detection of gravitational (31) waves is the use of matched filtering. We assume that and the noise in the gravitational wave detector is approxi- Å 16%2 ã1/4 …δt mately colored Gaussian noise, characterized by a one- hHIGH(f ) ≈ th σ f , (32) sided spectral noise density S (f). It is then natural c m 3χ4N(N − 1) td m T n to define a noise-weighted inner product between two frequency-domain signalsg ˜(f) and h˜(f), where %th is the threshold SNR value above which a detection is claimed, N is the number of pulsars√ in the Z fh g˜∗h˜ array, χ is a geometric factor which is 1/ 3 for pulsar hg˜|h˜i = 4 Re df . (34) timing arrays, T is the total time span of the observa- fl Sn(f) LOW HIGH tions, and ξ is chosen such that hc = hc at a frequency of 2/T [28]. The spatially averaged geometric Here fl is a lower frequency cutoff for the integral, and factor for the gravitational wave timing array is χ = hF i fh is an upper frequency cutoff which must be below the given by 4/3 for face-on binaries and 1/6 for edge-on bi- Nyquist frequency (half the sampling rate of the time se- naries. The characteristic strain hc corresponds, up to ries). We assume we work with the Fourier transforms of factors of order unity, with the amplitude of the mod- real time-domain signals, so that the signals at negative ulating wave Am defined in Eqs. (4) and (5). We can frequencies are given by the complex conjugate of the sig- see from Eq. (30) that under the approximations made nals at positive frequencies. The prefactors of Eq. (34) in deriving that result, the resulting gravitational wave account for these facts, and in what follows we write only timing array sensitivity curve does not depend explicitly the positive-frequency parts of the signals h˜, with the on δt. negative parts implied. We apply this estimate to the gravitational wave tim- The optimal linear detection statistic for a signal h˜ in ing array results with N Galactic binaries included in the the presence of such noisen ˜ is the matched filter between 7 data, d˜ = h˜ +n ˜, and h˜. Its expectation value is the In going to the second line we made a coordinate trans- squared SNR (e.g. [32]), formation x = π(f − fi)T , which centers the integral on the location of the carrier wave peak and stretches out a 2 ˜ ˜ ˜ ˜ ρ = hd|hi = hh|hi , (35) small region around that peak to a large domain. This where the overbar denotes expectation value. Thus the allows us to approximate Sn(f) in the integral with its product of the signal with itself gives the (squared) SNR constant value at the peak, and extend the integral over that might be achieved by matched filtering. an infinite range of x. We then used an integral identity We estimate the sensitivity of the gravitational wave for the square of a sinc function. timing array by isolating the sideband contributions to Using the same approximations, we compute the gravitational waves of the array of monochromatic hδT (f − fi + fm)|δT (f − fi + fm)i signals, and squaring this contribution. We first treat 4T Z ∞ 4T the case where we include the carrier (pulsar) terms in ≈ sinc2(x + x )dx = . (42) πS (f ) m S (f ) the modulation. Let h˜(f) be the total gravitational wave n i −∞ n i signal; a finite-time Fourier transform over the period of The overlap of two factors of δT (f − fi − fm) gives the observation applied to Eqs. (22) and (23) gives same result, but the overlap of neighboring sidebands is ˜ X î˜ ó hδ (f − f + f )|δ (f − f − f )i h(f) = hi(f) +s ˜i(f) , (36) T i m T i m ∞ i 4T Z ≈ sinc(x + xm) sinc(x − xm)dx . (43) where the carrier wave contributions are πSn(fi) −∞ A eiαi To resolve this we need the identity h˜ = i δ (f − f ) , (37) i 2 T i Z ∞ sinc(x + a) sinc(x + b)dx = π sinc(a − b) . (44) and around each carrier wave is a pair of sidebands whose −∞ contributions are With these results we find A A F f sin γ eiαi i m i i i −iδi 2 s˜i = [e δT (f − fi + fm) 2 Am X 2 4fm ρside ≈ 2 (ρiFifi sin γi) 2fm − eiδi δ (f − f − f )] . (38) i T i m 2 Am X 2 Here we write δ (f) for the finite-time delta function − sinc(2πfmT ) (ρiFifi sin γi) cos(2δi) . T 2f 2 following [28], m i (45) δT (f) = T sinc(πfT ) , (39) This shows that the squared SNR is the weighted square- with T the total observation time. When πfT 1, δ T sum of the individual carrier√ wave SNRs, so that the side- is highly peaked around f = 0. We emphasize again band SNR grows with N. The terms are enhanced by that we write only the positive-frequency parts of these 2 2 the factors fi /fm, which count the number of cycles of signals. the carrier wave over which we accumulate the modulat- The squared SNR in the sidebands is ing signal. At high modulating frequencies, πfmT 1, 2 X we can neglect the second term in Eq. (45), and we see ρside = hs˜i|s˜ji . (40) that ρside decreases as 1/fm, limiting sensitivity at high ij modulating frequencies. For a generic modulating wave, In evaluating this product, we consider observation times we also expect that δi varies randomly among the Galac- T long compared to the periods of any of the carrier tic binaries, and so this second sum also tends to be sup- waves, so πfiT 1. Then the products of sidebands pressed by contributions with different signs. around distinct carrier signals have negligible overlaps, We also expect the gravitational wave timing array so that the product vanishes when i 6= j. Similarly, when loses sensitivity at very low modulating frequencies, a πfmT 1 we expect each of the two sidebands around behavior that is not captured in Eq. (45). We return to each carrier to be well separated, and the products of this issue in Sec.IIIC and use only the high-frequency different sidebands vanish. However, when we consider approximation to Eq. (45) for the moment. modulating waves with very long periods of order T , the To get a sensitivity estimate, we can set a threshold two sidebands around each carrier can have nonzero over- SNR ρth above which we claim a detection of the side- lap with one another and with the carrier peak. band power. Then the required modulating amplitude It is convenient to first compute the squared SNR of a for detection is single carrier wave peak, 2 fm 2 Am & ρth √ A N ρRMSFRMSfRMS ρ2 = hh˜ |h˜ i = i heiαi δ (f − f )|eiαi δ (f − f )i i i i 4 T i T i Åρ ãÅ f ãÅ 100 ãÅ 1 ãÅ mHz ãÅ 102 ã −7 th m √ A2T Z ∞ A2T ∼ 10 ≈ i sinc2 x dx = i . (41) 1 µHz ρRMS FRMS fRMS N πS (f ) S (f ) n i −∞ n i (fmT 1), (46) 8 where RMS is the root-mean-square value of the given carrier frequencies fi becomes important. This is true quantity over the array, assuming these variables are un- when estimating the sensitivities of pulsar timing arrays, correlated with each other. Here we averaged over γi where the low-frequency sensitivity of Eq. (31) is limited assuming a uniform distribution of angles, appropriate by the need to fit the pulsar spin down rate. In binaries for the case where we include the carrier (pulsar) contri- the frequency can change due to the slow gravitational- butions to the modulation. wave driven inspiral, and in the case of stellar binaries it Since we are in the high-frequency limit, it is more can also occur due to tidal interactions and mass trans- appropriate to assume that fm evolves rapidly enough fer [33]. To capture these effects, we allow for the slow ˙ that we can neglect the carrier term contributions to the evolution fi of the carrier frequencies and compute the sidebands. In the case where we neglect the carrier terms, Fisher matrix including these parameters, but we take ˙ the sideband signal is then only the leading (zeroth) order results in small fi for Γab. Our model for h(t) and h˜(f) when including nonzero f˙ A A F f eiαi i i m i i −i(αm+βi) s˜i = [e δT (f − fi + fm) is described in AppendixB. 8fm

i(αm+βi) − e δT (f − fi − fm)] . (47) Making the appropriate substitutions and taking the 1. Fast evolution of the modulating wave high-frequency limit,

A2 X ρ2 ≈ m (ρ F f )2 , (f T 1) . (48) We first treat the case where the evolution of fm is side 8f 2 i i i m m i fast enough that we can neglect the carrier terms, as discussed in Sec.IIC. In this case, the model parameters Overall, the threshold modulating amplitude is increased a ˙ √ are θ = {A1, α1, f1, f1,A2,...,Am, αm}. The first 4N by a factor 2 compared to the case including the car- entries correspond to the parameters of the N Galac- rier terms. The network is slightly less sensitive, because tic binaries, and the final two entries correspond to the the carrier terms do not (incoherently) contribute to the parameters of the modulating wave. For simplicity we measured signal in this regime. do not incorporate the Fi and βi terms into the Fisher analysis in this case, instead fixing the direction of propagation, polarization angle ψ, and inclination ι of the C. Fisher matrix sensitivity estimate source of the modulation, and assuming that the sky lo- cations of the binaries are relatively well measured. For The sensitivity estimate using the sideband SNR the modulating wave, we include only the modulating threshold, Eq. (46), gives a reasonable result for modu- amplitude Am and common phase term αm as parame- lating waves with high modulating frequencies. However, ters in the Fisher matrix, meaning that we marginalize our SNR estimate predicts ever-improving sensitivity at over the phase to get a sensitivity estimate of the gravi- lower fm, even for fm < 1/T . This is because the es- tational wave timing array at a fixed frequency fm. timate assumes more information can be extracted from We compute the entries of the Fisher matrix using the the signal than is actually available at these low frequen- same approximations as we did for ρside, but the compu- cies. In reality, uncertainties in the estimated parameters tations are more involved. Some details and all the en- of the binaries in the array and in the modulating wave tries of Γ are given in AppendixC. Our goal is to com- itself correlate with the uncertainty in the measured am- ab pute the component of the covariance matrix ΣAmAm , plitude Am, limiting our ability to distinguish Am from describing uncertainty in the measurement of Am. At zero. high modulating frequencies, we find To capture these effects, we turn to a Fisher-based estimate of the measurement uncertainties of the gravita- " #−1 tional wave timing array. In this language, we can detect X Σ ≈ 8f 2 (ρ F f )2 , (f T ) 1 , (50) the modulating wave if the posteriors for Am peak suffi- AmAm m i i i m ciently far from zero relative to their width. To estimate i the width, we need the relevant terms of the covariance matrix Σab for our signal model. As in Sec.IIIA the in agreement with our SNR threshold estimate. The de- covariance matrix is the inverse of the Fisher matrix Γab, pendence on the common phase αm vanishes in this limit, whose entries are given here by which is expected—for times long compared to the period of modulation, the δ terms become very sharp, and one T Æ ∂h˜ ∂h˜ ∏ can check that a global time shift will remove the α Γ = , (49) m ab a b phases from the gravitational-wave signal. In the case ∂θ ∂θ where fm is smaller, we cannot execute this global shift, where θa is the vector of model parameters for the array. since the center of our observing window is used as our In addition, when considering the sensitivity of this ar- origin of time, and the relative phase of the modulating ray at low frequencies fm, the effect of a slow drift in the wave can change its impact on the carrier signals. 9

At low frequencies, we ﬁnd 106

−1 " # α +β =0 396900 X 2 m i Σ ≈ [ρ F f csc(α + β )] , 105 AmAm (πT )8f 6 i i i m i m i αm+β i=π/2 - 1

(fmT ) 1 . (51) ) i 104 + β

We see that there is a steep loss in sensitivity towards m very small f , in agreement with expectations that the α m , T m array is insensitive to very low frequency waves. Note f 1000 that the apparent singularities at δi = αm + βi = 0 and ( π g π are artifacts of the expansion in small fmT , which does not hold for these values of δi. Instead, the power law 100 −4 −6 becomes fm , shallower than fm , for these cases.

The full expression for ΣAmAm is given by 10 " #−1 0.5 1 5 10 50 100 X 2 2 2 ΣAmAm = 4 ρi Fi (πfiT ) g(πfmT, αm + βi) , i (52) FIG. 4. Generic shape of the variance of the amplitude of a modulating wave as a function of modulating frequency af- where the definition of the function g is given in Ap- ter marginalizing over {Ai, αi, fi, f˙i, αm}, for the two limit- pendixE, Eq. (E1), and is non-singular for all δi. In ing cases in which all carrier phases are chosen such that −1 Fig.4 we plot [ g(πfmT, αm + βi)] , which controls the αm + βi = π/2 or αm + βi = 0. The functional form of g shape of the sensitivity curve of the array, as a function is given in Eq. (E1). The low- and high-frequency limits are of the dimensionless parameter πfmT . We plot the ex- shown with dashed lines. The low-frequency limit of the vari- −6 treme cases αm +βi = π/2, which gives the generic power ance has the generic form ∝ fm , with a different scaling of −4 −6 fm when αm + βi approaches 0 or π. The variance scales as law behavior ΣAmAm ∝ fm , and αm +βi = 0, which dis- f 2 in the high-frequency limit. plays the shallower power law behavior at low fm. This m sensitivity curve transitions from the high-frequency to low-frequency limit around the frequency where the sidebands plotted in Fig.3 begin to blend with the main where we have defined Ami = Am sin γi. So long as we treat the common modulating amplitude fm as a fixed peaks, which occurs when fm ∼ 1/T . We can repeat our Fisher analysis while also marginal- parameter, the resulting Fisher matrix is block diagonal, with a 6 × 6 block for each carrier. We can therefore izing over fm, which gives a conceptually different sensitivity estimate. Namely, rather than an estimate invert each block independently to find the uncertainties σ on each of the A . The resulting expression for of the sensitivity to a fixed frequency of gravitational Ami mi σ2 is similar to the expression for Σ in Eq. (52), waves, it incorporates the uncertainties inherent in a Ami AmAm search over modulating frequencies. In this case, the but without the sum over binaries. This means at high high-frequency sensitivity estimate of Eq. (51) remains frequencies, unchanged, while the low-frequency behavior becomes σ2 ≈ 2f 2 (ρ F f )−2 , (f T ) 1 , (53) −8 Ami m i i i m steeper, with ΣAmAm ∝ fm . and at low frequencies 2. Slow evolution of the modulating wave 99225 σ2 ≈ [ρ F f csc(δ )]−2 , (f T ) 1 . (54) Ami 8 6 i i i i m (πT ) fm When the modulating frequency evolves sufficiently slowly, the sidebands produced by the local term and The full expression, valid for all frequencies and values carrier terms overlap. In this case, the carrier terms also of the phases δi, is given in AppendixE, Eq. (E2). contribute to the modulating signal, and two complica- In order to compute the sensitivity of the whole ar- tions arise in the Fisher analysis. First, the phases δi are ray to a common modulating wave with amplitude Am, no longer determined by the common phase αm and a we must combine the sensitivities from each of the car- function of sky location βi. Instead, they depend on the riers. We have observations of N individual Ami, and unknown distance to each Galactic binary di through the we would like to determine the error on Am, given that phase γi. Secondly, the amplitude of the modulation also Ami = Am sin γi. However, this constitutes N measure- varies with di, due to the presence of the factors sin γi; ments with which to constrain N + 1 correlated param- see Eq. (14). eters, and it is straightforward to see that the Fisher In this case we must consider an expanded vector of pa- matrix including Am and all of the γi is singular. This a ˙ rameters given by θ = {A1, α1, f1, f1, δ1,Am1,A2,... }, suggests that there is no unbiased estimator with finite 10 variance for the quantity Am (without imposing addi- We find Eq. (59) to be a very good numerical fit to the tional constraints). This is due simply to the fact that threshold derived when using the χ2(N + 1) distribu- a very large value of Am could always be compensated tion for D(Am), even for small N. As such, we use this by some choice of the γi to produce the same data. On more transparent expression in Sec.IV when deriving the the other hand, it is still possible to estimate our sen- sensitivity of the array in the limit of a slowly evolving sitivity to detecting the presence of a modulating wave. modulation. One procedure to do so is to compute the Moore-Penrose Note that for an array of equally constraining bina- pseudo-inverse of the singular Fisher matrix, which would ries, where all σAmi are equal, the threshold Am,th scales allow us to find the constrained Cramer-Rao bound on roughly as N −1/4, as is apparent from Eq. (59) in the the variance of the modulating amplitude [34]. large-N limit. However, when the σAmi are taken to be We will proceed by a different route, and utilize the different for each binary, the addition of poorly measured likelihood ratio test to compare a model that accounts binaries to the array can lead to an increased detection for the presence of a modulating wave with amplitude threshold by increasing N (and thus the number of de- Am and a set of phases γi to the null hypothesis with grees of freedom of the modulation model) without a sig- no parameters. Under the null hypothesis, any apparent nificant change to P σ−2 . In practice, the maximum i Ami modulation results purely from noise. The log-likelihood sensitivity of the array in the case of slow evolution of ratio for these two models is given by the modulating wave is obtained by retaining only the Å ã binaries that provide the most significant contribution, L0 D = −2 ln , (55) those with the largest values of ρifi. maxθ L(θ) where we estimate the likelihood as a Gaussian IV. GRAVITATIONAL WAVE TIMING ARRAY Ç 2 å Y 1 (Ami + ni − Am sin γi) USING MOCK LISA CATALOG L = √ exp − , 2πσ 2σ2 i Ami Ami (56) With the analytic sensitivity estimates in hand, we can calculate the sensitivity of a gravitational wave timing ar- where n is the Gaussian noise with variance σ2 in ray based on LISA measurements of gravitational waves i Ami the measurement of each Ami. The null hypothesis has from Galactic white dwarf binaries. The binary white Am = 0. Plugging this in to the likelihood ratio, and dwarf population in the Milky Way has been estimated maximizing the likelihood in the modulation model, we to number on the order of millions. It is expected that find data from LISA will be sufficient to individually resolve 4 2 the gravitational waves from about 10 white dwarf bina- X (Ami + ni) D = , (57) ries with SNR > 7. Most of these binaries would be very σ2 nearly monochromatic for the extent of the LISA mis- i Ami sion. The main source for their frequency evolution is which, according to Wilks’ theorem, is asymptotically χ2 likely to be gravitational wave radiation, with tides and distributed with N + 1 degrees of freedom [35]. The ex- mass transfer being effects only dominant in the later pectation value of this likelihood ratio when a modulating stages of inspiral (at frequencies above 1 mHz [33]). wave is present is given by In order to have realistic inputs for white dwarf binary parameters and their SNR as measured by LISA * 2 + 2 X (Am sin γi + ni) X A over a four-year mission, we used the Radler dataset [36] hDi = = N + m , σ2 2σ2 created for the LISA Mock data challenges, in combina- i Ami i Ami (58) tion with the Gbfisher code [37]. Gbfisher computes the spacecraft ephemerides during the observation time where we have used the fact that the γi are uniformly and, in combination with input sources, creates a time distributed and taken the average over the network, delay interferometry data stream for the mission. Af- 2 hsin γii = 1/2. ter estimating the confusion noise, it proceeds to give a 2 Now since D(Am) ∼ χ (N +1), after choosing a detec- matched-filter SNR for each source. In this analysis, we tion threshold we can write the sensitivity of the array as only use white dwarf binaries that Gbfisher identifies the value of A that exceeds the threshold. In the limit of with SNR > 7 over a four-year LISA mission. The me- m p large N, the quantity 2D(Am) is approximately Gaus- dian error on the sky localization of these binaries is 3 p √ square degrees, which justifies our assumption that local- sian distributed, 2D(Am) ∼ N ( 2N + 1, 1). We can therefore approximate the threshold value of the modu- ization errors are negligible in our analysis. For each sensitivity estimate, we calculate the modulating amplitude Am,th for a j-σ detection as lating amplitude needed for a 3-σ detection. We denote √ 1/2 ñj2 + 2j 2N + 1 + 1ô this threshold as Am,th. The timing estimate of the grav- Am,th ≈ . (59) itational wave timing array sensitivity curve is calculated P σ−2 i Ami as the sum of Eqs. (31) and (32), using the timing resid- 11

-5.0 10-6 Slow Evolution -8 10 ) -5.5 Fast Evolution Hz

Slow Evolution - 6 10-10 10

Fast Evolution = m ,th m

A - f 6.0

-12 ( 10 Timing m ,th PTA (5 years) A 10-14 10 -

log 6.5

10-16

10-9 10-8 10-7 10-6 10-5 -7.0 1 10 100 1000 104 fm [Hz] N (Number of Galactic binaries in array) FIG. 5. Forecasted sensitivity curves for a gravitational wave timing array based on the mock LISA catalog. For FIG. 6. Dependence of the sensitivity of a gravitational wave the frequency-domain estimates, we show the case where the timing array on the number of Galactic binaries used in the modulation has a slow evolution in dark blue and the case array, using a mock catalog of LISA observations [36, 37]. The of fast evolution in light blue. The approximate sensitivity binaries are ordered by decreasing ρifi, which controls the in the fast-evolution case from the timing estimate is shown contribution of each binary to the sensitivity. We use a 3-σ in dashed light blue. For comparison, we also plot the sen- detection threshold at a characteristic frequency of 10−6 Hz, sitivity of a pulsar timing array (PTA) assuming 5 years of and average over {ι, ψ, kˆ}. In the slow-evolution case, adding monitoring 36 pulsars with timing error 100 ns in dashed light “noisy” observations degrades the sensitivity, and we ﬁnd that green. utilizing only the best 10 binaries gives the most sensitivity. In the fast-evolution case, the best 10 binaries (about 0.08% of the total resolved population) can account for ∼ 40% of uals given by Eq. (30), with ξ chosen to match the con- the constraining power on Am, but adding binaries always improves the sensitivity for the fast-evolution case. tributions at fm = 2/T . In order to correctly weight the binaries contained in the mock LISA catalog, we implement the substitution shown in Eq. (33) as described in Sec.IIIA. We use a detection threshold of %th = 3 to vations given by each treatment. We also include for roughly match the 3-σ criterion, and we average over the comparison an estimate of pulsar timing array sensitiv- two limits of face-on and edge-on inclinations. ity, which is calculated from Eqs. (31) and (32) assuming For the frequency-domain analysis, as stated in an array of 36 pulsars timed once per fortnight, with a Sec.IIIC, we consider two extreme cases for the evo- total observation baseline of 5 years, a timing precision√ of lution of the modulating wave: the fast case, when we 100 ns, and a sky-averaged geometric factor χ = 1/ 3; can neglect the carrier term, and the slow case, when the this oﬀers a direct comparison with results from [28]. In carrier term has the same frequency as the local term. In dashed light blue is the timing sensitivity estimate of the the case of fast evolution, we use the Fisher estimate for gravitational wave timing array. The light and dark blue correspond respectively to the fast- and slow-evolution σAm , given in full by Eq. (52), and set Am,th = 3σAm . We average over the phases δi, as well as the properties case evaluated with the mock catalog. of the modulating wave: its inclination ι, polarization The timing estimate of the gravitational wave timing angle ψ, and direction of propagation kˆ. For the case array shows a higher sensitivity than the more complete of slow evolution, we use the likelihood ratio threshold of frequency-domain estimates. This higher sensitivity is

Eq. (59), with j = 3 and the full expression for σAmi from to be expected, since the timing estimate is an order Eq. (E2). We average over the same parameters as for of magnitude calculation and includes fewer parameters the fast case. Furthermore, as discussed in Sec. IIIC2, that correlate with the amplitude of the modulation. In in the slow case we select only the binaries with the high- the regime where both the gravitational wave timing ar- est ρifi, since including poorly measured binaries would ray and pulsar timing arrays have sensitivity, the pulsar worsen the forecasted sensitivity of the array. We find timing array is about 7 orders of magnitude more sensi- that using the ten best binaries from the mock catalog tive. However, the sensitivity of the gravitational wave gives the optimal estimate, and discuss this choice further timing array extends above the maximum frequency of below. the pulsar timing array, all the way to the lower end of Figure5 shows the estimated sensitivity curve for a the LISA band (∼ 2 × 105 Hz). gravitational wave timing array based on LISA obser- The sensitivity of the gravitational wave timing array 12 can be improved in various ways, including via longer magnitude worse than pulsar timing arrays in the over- observation times or by monitoring a larger number of lapping region. Despite being a much lower sensitivity, Galactic binaries with lower detector noise. With a fixed an entirely independent probe into the nanohertz gravi- number of binaries in the array, the threshold Am,th tational wave sky may prove useful, since it is unaffected scales with observation time as T −1/2 due to the lin- by some of the major sources of noise for pulsar timing ar- 2 ear increase in ρi with mission time. In general this rays, including electromagnetic propagation effects in the provides a lower bound on the improvement, since ad- interstellar medium and radio interference at Earth. The ditional Galactic binaries will be resolved above the SNR gravitational wave timing array also requires no change threshold with increased observational time and can thus of design for the LISA mission, since it can be achieved be added to the array. by analyzing data that is already planned to be collected. Figure6 shows how Am,th changes with the number of It is worth considering whether LISA or pulsar tim- detected binaries in each of the frequency-domain esti- ing arrays could measure the microhertz regime directly, mates, starting from those with the highest ρifi. We since their strain sensitivity is many orders of magni- use the standard four-year mission duration and plot tude better than would be achieved by the indirect anal- −6 Am,th at a frequency fm = 10 Hz (the high-frequency ysis with a gravitational wave timing array. As cur- limit). For the fast-evolution case, we use the high- rently designed, the low-frequency sensitivity of LISA is frequency approximation for the measurement error, so significantly limited by acceleration performance, which P 2 2 2−1/2 effectively forms a frequency cutoff for the mission at that Am,th ∝ ρi Fi fi , and again average over i ∼ 10−4–10−5 Hz [20]. Pulsar timing arrays could poten- {ι, ψ, kˆ}. As can be seen in the figure, as few as 10 bina- tially extend their sensitivity into the microhertz regime ries (about 1/1000 of the total resolved population) can if their measurements were to be taken at a higher ca- account for almost half of the sensitivity of the gravi- dence than the current ∼ 2 weeks. However, this would tational wave timing array in this case. The threshold require a costly change to the current observing strate- continues to improve with N even when binaries with gies, especially as the number of monitored pulsars in- low ρ f are included, but those with the highest ρ f i i i i creases. An interesting recent proposal demonstrated contribute significantly more to the sensitivity of the ar- that high-cadence photometric observations may be use- ray. In the slow-evolution case, the sensitivity decreases ful for astrometric gravitational wave searches in the mi- when more than the 10 best binaries are included, and so crohertz regime [22]. Relative astrometry of stars in the we use only these best 10 in the corresponding sensitivity Galactic bulge with the Nancy Grace Roman Space Tele- curve of Fig.5. scope [26] may be capable of providing sensitivity to gravitational waves in the frequency range above the upper limit of pulsar timing arrays and below the lower limit of V. DISCUSSION LISA. In any case, the gravitational wave timing array provides a valuable independent probe in the microhertz In this paper we have discussed the concept of a grav- regime, using data products which the LISA mission al- itational wave timing array. The array is composed of a ready plans to produce. large number of Galactic binaries, observed continuously Our sensitivity estimates could be improved by includ- through their gravitational wave emission by future de- ing a more complete analysis of the effect of the carrier tectors. Lower frequency gravitational waves impact the term (the pulsar term in pulsar timing array literature) propagation of the signals from these binaries, modulat- on our sensitivity, possibly by modeling the frequency ing their phase. This effect can be used to search for evolution of the modulating wave explicitly. We do in- these low-frequency gravitational waves in a matter di- corporate the presence of a slow evolution of the fre- rectly analogous to pulsar timing array searches for grav- quency of each binary in the array, whether it comes from itational waves. gravitational-wave driven inspiral or from effects such as We have specialized our results to the gravitational mass transfer or tides. Improvements could also be made waves emitted by binary white dwarfs as measured by with a more realistic observing scenario for the LISA mis- LISA, using a mock galaxy catalog and a mock LISA sion, taking into account non-stationary noise [38] and pipeline to estimate their properties. In this case, the the mission’s duty cycle, as well as the need to demod- array provides sensitivity in the microhertz frequency ulate the orbital motion of the satellites from the raw regime, a region not probed by current and planned grav- data. itational wave detectors, as illustrated in Fig.1. Our While we focused on the sensitivity of the array to a main analysis is in the frequency domain, although we coherent source of low-frequency gravitational waves, it have also included an approximate timing estimate to al- would be natural to also provide sensitivity estimates for low for direct comparison with the standard pulsar tim- a stochastic background of low-frequency gravitational ing array approaches. Our results show that a gravita- waves, bursts of gravitational waves, and gravitational tional wave timing array based on the nominal four-year wave memory. It would also be interesting to consider the LISA mission lifetime would provide a sensitivity to long- impact of additional sources of sidebands in the binaries wavelength gravitational waves that is about 7 orders of observed by LISA, such as orbital eccentricity, or inter- 13 actions with a companion star in triple systems. These vector, or tensor fields. The amplitude of these waves effects would not generate sidebands at a common spac- decreases as the wavefront expands, in order to conserve ing for all the members of the gravitational wave timing quanta (see e.g. [40]). array, but they may contribute an important source of More precisely, we augment our gauge such that γµν noise for some of the binaries. is in Lorenz gauge and traceless with respect to gµν , so αβ α Although we have concentrated on the idea of a gravi- that g γαβ = 0 and ∇αγµ = 0. Then in vacuum γµν tational wave timing array using LISA observations, the obeys the wave equation idea may be applied to other situations. For example, α the detection of continuous wave sources by ground-based ∇ ∇αγµν = 0 . (A1) detectors (see e.g. [39]), such as pulsars with some small equatorial ellipticity, would open up an additional av- To implement the geometric optics limit we expand the enue. Searches for common modulations in a collection carrier wave as of such sources could benefit from the higher frequen- −iϕ/ γµν = Aeµν e , (A2) cies fi of the carrier waves, since the sensitivity scales as f /f , and could potentially access the entire range i m where the phase function ϕ varies rapidly over scales L, of sub-Hz frequencies not directly accessible to ground- the polarization tensor is normalized as e eαβ = 2, and based detectors. αβ is a bookkeeping parameter which tracks orders in the Gravitational wave astronomy is a rapidly evolving rapidly varying phase. The wave equation and gauge con- field that has already provided important insights that ditions are then solved order by order in the parameter would not have been possible with electromagnetic ob- . At leading order, the result is servations alone. New direct detection experiments and indirect detection schemes are under development that αβ αβ g (∂αϕ)(∂βϕ) = 0 , g (∂αϕ)∇β(∂µϕ) = 0 , (A3) will greatly expand the frequency coverage and sensitivity with which we can search for gravitational waves. where the second equation is arrived at by differentiating The gravitational wave timing array described here is a the first and commuting the derivatives on the scalar ϕ. If novel proposal that offers a nearly cost-free extension of we define the normal vector to the wavefronts of constant µ the sensitivity of future detectors to a currently uncon- phase as ζµ = ∂µϕ, we see that ζ is a null geodesic in strained regime of gravitational wave frequencies. the perturbed flat space, along which the carrier wave propagates. From here, we can quote the standard results for such ACKNOWLEDGMENTS a null geodesic, perturbed by the modulating wave hµν , see e.g. Refs. [14, 29]. We expand ϕ = ϕ0 + ϕ1 + ... µ µ µ MJBR thanks Tyson Littenberg for his introduction and so ζ = ζ0 + ζ1 + ... , counting orders in the small to Gbfisher and the Radler dataset. MJBR and AZ are modulating wave amplitude. If the spatial normal from supported by NSF Grant PHY-1912578. JM and CT are the observer to the carrier source isn î, we have supported by the US Department of Energy under grant µ i no. DE-SC0010129. NP was supported by the Hamilton ζ0 = ωc(1, −nˆ ) , (A4) Undergraduate Research Scholars Program at SMU. µ and ζ1 given by Eq. (16) of [14]. The observed frequency at the origin is given by con- µ µ Appendix A: Modulation of a continuous signal tracting the observer’s 4-velocity u = δt with the phase derivative In this appendix we briefly discuss the geometric op- dϕ ≈ −uαg (ζβ + ζβ) = ω (1 − z) , (A5) tics formalism required to derive the phase modulation αβ 0 1 c dt obs of a monochromatic gravitational wave propagating in the presence of another, lower frequency gravitational where the redshift z is quoted in Eq. (2). This expression wave. The high-frequency carrier wave propagates in a is used to compute the modulated phase ϕ. perturbed flat spacetime gµν = ηµν + hµν and is itself a tensor perturbation γµν . We use a gauge such that hµν is transverse and traceless with propagation vector Appendix B: Including the slow evolution of the kµ. The carrier wave is emitted at a stationary source, carrier frequencies and received by a stationary observer at the origin of our coordinate system. In this appendix we consider the possibility that the The curvature scale of the perturbation is given by gravitational waves from Galactic binaries are not purely the wavelength of the modulating wave L ∼ λm, and so monochromatic, and allow for a small linear time depen- λm λe. This is the geometric optics limit, where high- dence in their instantaneous frequency, so that the carrier ˙ 2 frequency waves propagate along null geodesics in the phase expands as ϕi = −αi + 2πfit + πfit . This con- curved spacetime, regardless of whether they are scalar, sideration is motivated both by the fact that we expect 14 some non-negligible frequency drift over the lifetime of same frequency offsets (but differing phases). We de- a our observations and also that we expect low frequency fine Ami = Am sin γi, and our parameter set is θ = ˙ modulating waves to be degenerate with the frequency {A1, f1, α1, f1, δ1,Am1,A2,... }. We hold fm, the sky lo- drift (as in the case of pulsar timing arrays). cations of the binaries, and the other properties of the When we include the slow evolution of the carrier fre- incident modulating wave fixed. The resulting Fisher quencies, the modulated signal from a single Galactic bi- matrix is block diagonal, with one 6 × 6 block for each nary is Galactic binary. To leading order in the small Ami, the entries in each block are given by ï ˙ 2 h(t) =Ai cos 2πfit + πfit − αi ρ2 ò Γ = i , (C1) AmFifi AiAi A2 − sin γi sin(2πfmt − δi) i fm 2 Γα α = ρ , (C2) ˙ 2 i i i ≈Ai cos(2πfit − αi) − πfit Ai sin(2πfit − αi) 2 2 (πT ) ρi AiAmFifi Γfifi = , (C3) + sin γi sin(2πfmt − δi) 3 fm 2 π 4 2 ï ò Γf˙ f˙ = (πT ) ρi , (C4) × sin(2πf t − α ) + πf˙ t2 cos(2πf t − α ) . i i 5 i i i i i 2 2 2 2 Amiρi Fi fi (B1) Γδiδi = 2 [1 + cos(2δi) sinc(2πfmT )] , 2fm (C5) In the frequency domain, the carrier wave and sideband 2 2 2 signals are then ρi Fi fi ΓAmiAmi = 2 [1 − cos(2δi) sinc(2πfmT )] , (C6) 2fm iαi ˜ Aie î ˙ 00 ó hi = δT (f − fi) + iπfiδT (f − fi) , (B2) 2 for the entries on the diagonal, and iαi AiAmFifi sin γie s˜i = 4fm f î Ä ä 2 i −iδi ˙ 00 Γαiδi = −ρi FiAmi cos(δi) sinc(πfmT ) , (C7) × e δT (f − fi + fm) + iπfiδT (f − fi + fm) fm

iδi Ä ˙ 00 äó 2 fi 0 −e δT (f − fi − fm) + iπfiδT (f − fi − fm) . Γ = −πT ρ F A sin(δ ) sinc (πf T ) , (C8) fiδi i i mi f i m (B3) m f Γ = −π(πT )2ρ2F A i cos(δ ) sinc00(πf T ) , f˙iδi i i mi i m Since we work to leading order in the slow frequency fm evolution, we generally set f˙ ≈ 0 in our ﬁnal expressions. (C9) This means we only need to consider the additional terms f ˙ Γ = −ρ2F i sin(δ ) sinc(πf T ) , (C10) involving fi when taking derivatives with respect to those αiAmi i i f i m parameters, m 2 fi 0 ΓfiAmi = πT ρi Fi cos(δi) sinc (πfmT ) , (C11) iαi f ˜ Aie 00 m ∂f˙ hi = iπ δT (f − fi) . (B4) i 2 2 2 fi 00 Γ ˙ = −π(πT ) ρ Fi sin(δi) sinc (πfmT ) , (C12) fiAmi i f These results are needed for approximating entries in the m

Fisher matrix Γaf˙ , as described below. i for the oﬀ-diagonal entries in each block. The remaining terms are zero to leading order. Appendix C: Details on the Fisher matrix approach Computing these entries requires the application of the various approximations stated in the text, in particular In this appendix we give more detail on the Fisher ma- that fiT 1 in all cases, resulting in narrow carrier wave trix approach for estimating the sensitivity of the gravi- peaks, all well separated from each other. For example, tational wave timing array. we have

˜ ˜ ˜ ˜ ˜ ˜ hhi|hji δijhhi|hii 1. Slow evolution of modulating wave ΓAiAj = h∂Ai h|∂Aj hi = ≈ 2 + O(Am) AiAj Ai ρ2 We treat first the case where both the local term i = δij 2 + O(Am) . (C13) (the Earth term in the pulsar timing array literature) Ai and carrier term (the pulsar term in the pulsar timing array literature) contribute to the sidebands with the For those terms which involve derivatives of the finite- 15 time delta functions, the following identities are useful: eters with Am and αm are f Z ∞ d sinc(x + a) Γ = −ρ2F i sin(δ ) sinc(πf T ) , (C21) sinc(x + b)dx =π sinc0 x| , αiAm i i 2f i m dx x=a−b m −∞ f (C14) 2 i 0 ΓfiAm = πT ρi Fi cos(δi) sinc (πfmT ) , (C22) Z ∞ 2fm d sinc(x + a) d sinc(x + b) 00 dx = − π sinc x|x=a−b . 2 2 fi 00 dx dx Γ ˙ = −π(πT ) ρ Fi sin(δi) sinc (πfmT ) , (C23) −∞ fiAm i 2f (C15) m f Γ = −ρ2F A i cos(δ ) sinc(πf T ) , (C24) αiαm i i m 2f i m These can be derived by differentiating Eq. (44) under m f the integral. 2 i 0 Γfiαm = −πT ρi FiAm sin(δi) sinc (πfmT ) , (C25) 2fm f Γ = −π(πT )2ρ2F A i cos(δ ) sinc00(πf T ) . f˙iαm i i m i m 2fm 2. Fast evolution of the modulating wave (C26) Finally, the terms that involve only the modulating In the case where we neglect the carrier term of the source are modulation (the pulsar term in the pulsar timing array 2 2 2 literature), the parameters γi which differentiate the Ami X ρi Fi fi ΓAmAm = 2 [1 − cos(2δi) sinc(2πfmT )] , and add an unmeasurable phase term to the δi are ab- 8f i m sent. While the amplitude of the sideband varies from (C27) carrier to carrier, this variation depends only on the F , i 2 2 2 X Amρ F f which can be determined from the sky location and which Γ = i i i sin(2δ ) sinc(2πf T ) , (C28) Amαm 8f 2 i m we therefore neglect from our analysis. Meanwhile, the i m phase of the modulation encoded in δ depends only on i X A2 ρ2F 2f 2 a common phase α and the phases β , which depend Γ = m i i i [1 + cos(2δ ) sinc(2πf T )] . m i αmαm 8f 2 i m on sky location and polarization content of the mod- i m ulating wave. Thus for our analysis, the sidebands of (C29) all the carriers share a common unknown amplitude parameter Am and phase term αm. Our parameter set is a ˙ θ = {A1, α1, f1, f1,A2,...,Am, αm}. Appendix D: Inverting the Fisher matrix The Fisher matrix for our total signal h˜ breaks into a 4N × 4N block describing the Galactic binary param- In AppendixC we provide the Fisher matrix entries for eters, a rectangular 4N × 2 matrix mixing the binary the measurement of the gravitational wave timing array parameters with those of the modulating wave, and a fi- signal h˜. There we treat two extreme cases, the fast and nal 2 × 2 block with the modulating wave parameters. slow cases for the evolution of the modulating wave. Here We need to keep terms only to leading order in Am. The we discuss the aspects of inverting these matrices needed 4N × 4N block is block diagonal with a 4 × 4 sub-block for our sensitivity estimates. for each carrier. These sub-blocks have diagonal entries In the slow case, our goal is to compute the en- −1 tries ΣAmiAmi of the covariance matrix Σab = (Γ )ab. These entries are the squared measurement errors σ ρ2 Ami Γ = i , (C16) of the amplitudes A . Since the Fisher matrix is block- AiAi A2 mi i diagonal in this case, inversion can be carried out block- 2 Γαiαi = ρi , (C17) wise, and it is straightforward to get the entries ΣAmiAmi . 2 2 In the fast case, the common parameters αm and Am (πT ) ρi Γfifi = , (C18) couple together the N blocks which correspond to each 3 Galactic binary, and the inversion of Γ to get Σ π2 ab AmAm Γ = (πT )4ρ2 , (C19) is more involved. f˙if˙i i 5 In the fast case, the Fisher matrix breaks into pieces as follows. The parameters describing the individual bi- and non-vanishing off-diagonal entries naries form a 4N × 4N block a at the upper-left of Γ, and this sub-matrix is itself block-diagonal, since the in- π Γ = − (πT )2ρ2 . (C20) dividual binaries do not correlate with each other in our αif˙i i 3 approximation. Denote each of the N 4 × 4 blocks as ai, with i indexing the Galactic binaries. Next, the last M The off-diagonal terms that couple the 4N binary param- columns of the first 4N rows of Γ form a 4N × M matrix 16 b, which couples the binaries into the parameters of the decomposition of the inverse into sums over the contri- modulating wave. This array itself breaks into N 4 × M bution from each Galactic binary makes the inversion of blocks bi, with entries such as (b1)A1Am = ΓA1Am . Since the Fisher matrix straightforward: the Fisher matrix can Γ is symmetric, the first 4N columns of the final M rows be built using a single Galactic binary and inverted, and > > are b , made up of N arrays bi . Finally, in the lower in the final solution we need only to sum over the binary right we have an M × M matrix c which covers only the indices. modulating wave parameters. For example, consider a Fisher matrix where we include two parameters of the modulating wave, A and Now ΣAmAm sits in the lower-right M×M block of Σ = m −1 ˙ Γ . Denote these M × M entries of Σ as s. A standard αm, and for brevity remove fi from our parameter list. matrix identity, when applied to our decomposition of Γ, Then, recalling that ΓAiAm = 0 = ΓAiαm at leading or- yields der, the covariance ΣAmAm is given by

−1 −1 ñ d2 ô s = d , (D1) Amαm ΣAmAm = dAmAm − , (D3) N dαmαm X −1 > −1 > 2 d = c − b a b = c − bi ai bi , (D2) ÇΓ2 Γ å X Amαi Amfi i=1 d = Γ − + , (D4) AmAm AmAm Γ Γ i αiαi fifi where we have deﬁned a useful auxiliary matrix d. The

−1 2 2 " Ç 2 2 å# " Å ã# dA α X Γα α Γα f X ΓA α Γα α ΓA f Γα f m m = Γ − m i + m i Γ − m i m i + m i m i . (D5) d αmαm Γ Γ Amαm Γ Γ αmαm i αiαi fifi i αiαi fifi

˙ Similar, but more involved expressions give the case we treat in the text, where we include fi, but the main point remains: the inversion can be carried out as if for a single Galactic binary and the common parameters, and then summing any term involving the binary parameters over all the binaries in the network. This approach gives us the full expressions for our measurement uncertainties given below.

Appendix E: Full expression for the variance of the modulating amplitude

In order to present the full expressions for the covariance matrix entry ΣAmAm (in the case of a source of modulating waves with fast evolution) and the variance σAmi (in the case of a source of modulating waves with slow evolution), it is useful to deﬁne an auxiliary function, " ! 6 4 2 4 2 4 2 g(xm, δi) = xm − 6xm − 15xm + 6xm − 75xm + 45 cos(2xm) + xm − 60xm + 180 xm sin(xm) cos(xm) − 45 !# 2 2 4 2 × − xm xm − 12 sin(2xm) − 6 xm − 1 cos(2xm) + 2 xm − 3xm − 3

" Å ãÅ ã 6 2 2 2 × 2xm 2 cos(2δi) xm − 3 sin(xm) + 3xm cos(xm) 3 5 − 2xm sin(xm) + xm xm − 15 cos(xm)

!#−1 2 2 2 2 4 2 + xm − 6 2xm + 3 xm + 6 15 − 4xm xm sin(2xm) + 3 xm − 24xm + 15 cos(2xm) − 45 (E1)

with xm = πfmT .

Then in the case of a source with fast evolution, our estimate for the measurement uncertainty comes from ΣAmAm , ˙ where we marginalize over {Ai, αi, fi, fi, αm} with a (4N + 2) × (4N + 2) Fisher matrix. It is given by Eq. (52). In the case where the source of modulating waves evolves slowly, we need the uncertainty on each Ami measurement, so ˙ we compute the entry ΣAmiAmi arising from a 6 × 6 Fisher matrix, where we marginalize over {Ai, αi, fi, fi, δi}, for each binary. The result is

−1 σ2 = Σ = ρ2F 2(πf T )2g(x , δ ) . (E2) Ami AmiAmi i i i m i 17

[1] LIGO Scientific Collaboration, J. Aasi et al., waves in polarization of the microwave background,” “Advanced LIGO,” Class. Quant. Grav. 32 (2015) Phys. Rev. Lett. 78 (1997) 2054–2057, 074001, arXiv:1411.4547 [gr-qc]. arXiv:astro-ph/9609169. [2] VIRGO Collaboration, F. Acernese et al., “Advanced [17] DECIGO Working Group Collaboration, M. Musha Virgo: a second-generation interferometric gravitational et al., “Space gravitational wave antenna DECIGO and wave detector,” Class. Quant. Grav. 32 no. 2, (2015) B-DECIGO,” CEAS Space Journal 9 no. 4, (Dec., 2017) 024001, arXiv:1408.3978 [gr-qc]. 371–377. [3] KAGRA Collaboration, T. Akutsu et al., “Overview of [18] DECIGO Working Group Collaboration, KAGRA: Detector design and construction history,” S. Kawamura et al., “Current status of space arXiv:2005.05574 [physics.ins-det]. gravitational wave antenna DECIGO and B-DECIGO,” [4] LISA Collaboration, P. Amaro-Seoane et al., “Laser arXiv e-prints (June, 2020) arXiv:2006.13545, Interferometer Space Antenna,” arXiv:1702.00786 arXiv:2006.13545 [gr-qc]. [astro-ph.IM]. [19] J. Crowder and N. J. Cornish, “Beyond LISA: [5] J. P. W. Verbiest, S. Oslowski, and S. Burke-Spolaor, Exploring future gravitational wave missions,” Phys. “Pulsar Timing Array Experiments,” Rev. D 72 (2005) 083005, arXiv:gr-qc/0506015. arXiv:2101.10081 [astro-ph.IM]. [20] S. Larson, “Lisa: A modern astrophysical observatory,” [6] NANOGrav Collaboration, M. F. Alam et al., “The in 33rd SLAC Summer Institute on Particle Physics. NANOGrav 12.5 yr Data Set: Wideband Timing of 47 Jan., 2005. Millisecond Pulsars,” Astrophys. J. Suppl. 252 no. 1, [21] A. Sesana et al., “Unveiling the Gravitational Universe (2021) 5, arXiv:2005.06495 [astro-ph.HE]. at \mu-Hz Frequencies,” arXiv:1908.11391 [7] M. Kerr et al., “The Parkes Pulsar Timing Array [astro-ph.IM]. project: second data release,” Publ. Astron. Soc. [22] Y. Wang, K. Pardo, T.-C. Chang, and O. Doré, Austral. 37 (2020) e020, arXiv:2003.09780 “Gravitational Wave Detection with Photometric [astro-ph.IM]. Surveys,” Phys. Rev. D 103 no. 8, (2021) 084007, [8] S. Babak et al., “European Pulsar Timing Array Limits arXiv:2010.02218 [gr-qc]. on Continuous Gravitational Waves from Individual [23] V. Korol, E. M. Rossi, P. J. Groot, G. Nelemans, Supermassive Black Hole Binaries,” Mon. Not. Roy. S. Toonen, and A. G. A. Brown, “Prospects for Astron. Soc. 455 no. 2, (2016) 1665–1679, detection of detached double white dwarf binaries with arXiv:1509.02165 [astro-ph.CO]. Gaia, LSST and LISA,” Mon. Not. Roy. Astron. Soc. [9] G. Hobbs et al., “The international pulsar timing array 470 no. 2, (2017) 1894–1910, arXiv:1703.02555 project: using pulsars as a gravitational wave detector,” [astro-ph.HE]. Class. Quant. Grav. 27 (2010) 084013, [24] K. Breivik et al., “COSMIC Variance in Binary arXiv:0911.5206 [astro-ph.SR]. Population Synthesis,” Astrophys. J. 898 no. 1, (2020) [10] E. Petroff, J. W. T. Hessels, and D. R. Lorimer, “Fast 71, arXiv:1911.00903 [astro-ph.HE]. radio bursts,” The Astronomy and Astrophysics Review [25] S. A. Klioner, “Gaia-like astrometry and gravitational 27 no. 1, (May, 2019). waves,” Classical and Quantum Gravity 35 no. 4, (Feb., http://dx.doi.org/10.1007/s00159-019-0116-6. 2018) 045005, arXiv:1710.11474 [astro-ph.HE]. [11] N. Pearson, C. Trendafilova, and J. Meyers, “Searching [26] WFIRST Astrometry Working Group for gravitational waves with strongly lensed repeating Collaboration, R. E. Sanderson et al., “Astrometry with fast radio bursts,” Phys. Rev. D 103 (Mar, 2021) the Wide-Field Infrared Space Telescope,” Journal of 063017. https: Astronomical Telescopes, Instruments, and Systems 5 //link.aps.org/doi/10.1103/PhysRevD.103.063017. (Oct., 2019) 044005, arXiv:1712.05420 [12] V. B. Braginsky, N. S. Kardashev, A. G. Polnarev, and [astro-ph.IM]. I. D. Novikov, “Propagation of electromagnetic [27] L. Levin, “Interstellar medium mitigation techniques in radiation in a random field of gravitational waves and pulsar timing arrays,” Journal of Physics: Conference space radio interferometry.,” Nuovo Cimento B Serie Series 610 (May, 2015) 012020. 105 no. 10, (Oct., 1990) 1141–1158. https://doi.org/10.1088/1742-6596/610/1/012020. [13] T. Pyne, C. R. Gwinn, M. Birkinshaw, T. M. Eubanks, [28] C. J. Moore, S. R. Taylor, and J. R. Gair, “Estimating and D. N. Matsakis, “Gravitational radiation and very the sensitivity of pulsar timing arrays,” Class. Quant. long baseline interferometry,” The Astrophysical Grav. 32 no. 5, (2015) 055004, arXiv:1406.5199 Journal 465 (Jul, 1996) 566. [astro-ph.IM]. http://dx.doi.org/10.1086/177443. [29] M. Anholm, S. Ballmer, J. D. E. Creighton, L. R. Price, [14] L. G. Book and E. E. Flanagan, “Astrometric Effects of and X. Siemens, “Optimal strategies for gravitational a Stochastic Gravitational Wave Background,” Phys. wave stochastic background searches in pulsar timing Rev. D 83 (2011) 024024, arXiv:1009.4192 data,” Physical Review D 79 no. 8, (Apr, 2009). [astro-ph.CO]. http://dx.doi.org/10.1103/PhysRevD.79.084030. [15] M. Kamionkowski, A. Kosowsky, and A. Stebbins, “A [30] LISA Science Study Team Collaboration, Probe of primordial gravity waves and vorticity,” Phys. J. Bellovary et al., “Lisa science requirements Rev. Lett. 78 (1997) 2058–2061, document.” https://www.cosmos.esa.int/documents/ arXiv:astro-ph/9609132. 678316/1700384/SciRD.pdf, 2018. [16] U. Seljak and M. Zaldarriaga, “Signature of gravity [31] P. Swerling, “Parameter estimation accuracy formulas,” 18

IEEE Transactions on Information Theory 10 no. 4, 60 – 62. https://doi.org/10.1214/aoms/1177732360. (1964) 302–314. [36] S. Babak, “Radler dataset.” https://lisa-ldc.lal. [32] M. Maggiore, Gravitational waves. Volume 1: Theory in2p3.fr/media/uploads/LDC1-4_GB_v2.hdf5, 2019. and experiments. Oxford University Press, 2008. [37] T. B. Littenberg, N. J. Cornish, K. Lackeos, and [33] B. Willems, A. Vecchio, and V. Kalogera, “Probing T. Robson, “Ldasoft.” Free software (gpl), 2020. white dwarf interiors with LISA: periastron precession [38] M. C. Edwards, P. Maturana-Russel, R. Meyer, J. Gair, in eccentric double white dwarfs,” Phys. Rev. Lett. 100 N. Korsakova, and N. Christensen, “Identifying and (2008) 041102, arXiv:0706.3700 [astro-ph]. addressing nonstationary LISA noise,” Phys. Rev. D [Erratum: Phys.Rev.Lett. 101, 219903 (2008)]. 102 no. 8, (Oct., 2020) 084062, arXiv:2004.07515 [34] Y.-H. Li and P.-C. Yeh, “An interpretation of the [gr-qc]. moore-penrose generalized inverse of a singular ﬁsher [39] K. Riles, “Recent searches for continuous gravitational information matrix,” IEEE Transactions on Signal waves,” Mod. Phys. Lett. A 32 no. 39, (2017) 1730035, Processing 60 no. 10, (Oct, 2012) 5532–5536. arXiv:1712.05897 [gr-qc]. http://dx.doi.org/10.1109/TSP.2012.2208105. [40] C. W. Misner, K. S. Thorne, and J. A. Wheeler, [35] S. S. Wilks, “The Large-Sample Distribution of the Gravitation. W. H. Freeman, San Francisco, 1973. Likelihood Ratio for Testing Composite Hypotheses,” The Annals of Mathematical Statistics 9 no. 1, (1938)