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Gravitational Wave Timing Array Gravitational Wave Timing Array Mar´ıaJos´eBustamante-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-P´erotinterferometers. 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 d' ≈ ! (1 − z) ; (1) dt c where ' is the phase of the carrier, and z is a redshift. It is given by [14] i j ^ n^ n^ î ^ ^ ó z(t; k) = hij(t; k) − hij(tc; k) ; (2) 2(1 + k^ · n^) where k^ is the direction of propagation of the modulat- ing wave andn ^ 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- ^ ^ 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 ^ scope [22, 26]. The gravitational wave timing array proposed tc = t−d(1+^n·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 !m !c, 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 ^ 0 ^ 0 additional background gravitational wave. In Sec. III we '(t; k) ≈ !ct − !c z(t ; k)dt : (3) calculate the sensitivity of a gravitational wave timing 0 array using multiple methods. Section IIIA 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 !c, which we refer to as the ^ + + ^ × × ^ 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 !m !c.
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