PHYSICAL REVIEW D 101, 102004 (2020) Limits on the stochastic gravitational wave background and prospects for single-source detection with GRACE Follow-On M. P. Ross ,* C. A. Hagedorn, E. A. Shaw, A. L. Lockwood, B. M. Iritani, J. G. Lee, K. Venkateswara, and J. H. Gundlach Center for Experimental Nuclear Physics and Astrophysics, University of Washington, Seattle, Washington 98195, USA (Received 26 November 2019; accepted 6 May 2020; published 21 May 2020) With a reinterpretation of recent results, the GRACE Follow-On mission can be applied to gravitational wave astronomy. Existing GRACE Follow-On data constrain the stochastic gravitational wave background Ω < 3 3 107 to GW . × at 100 mHz. With a dedicated analysis, GRACE Follow-On may be able to detect the inspiral of local neutron star binaries, inspiral of subgalactic stellar-mass black hole binaries, or mergers of intermediate-mass black hole binaries within the Milky Way. DOI: 10.1103/PhysRevD.101.102004 I. INTRODUCTION xðtÞ¼x0hxx cos ð2πft þ ψÞð1Þ GRACE Follow-On (GRACE-FO) [1] is a satellite h ¼ h ð 2 θ 2 ϕ − 2 ϕÞ mission currently in Earth orbit studying the terrestrial xx þ cos cos sin þ 2h θ ϕ ϕ ð Þ gravitational field. We show that by interpreting recent × cos sin cos 2 results from this mission as strain, one can (1) constrain the h gravitational wave background and (2) provide an oppor- where þ;× are the plus and cross-polarization components tunity for single-source gravitational wave (GW) searches. of the gravitational wave, θ and ϕ are the polar and Incoherent GW must be present throughout the universe azimuthal angles of the source, f is the wave frequency, at some amplitude. A broadband stochastic background is and ψ is an arbitrary phase. Here, the plus and cross one of the few observable consequences of cosmic inflation polarizations are defined in the source frame. [2] and a signature of proposed exotic phenomena such as The GRACE-FO satellite mission has produced sensitive cosmic strings [3]. Measurements of, and constraints on, relative-displacement measurements of an inertial mass pair such a background would yield insight into of a wide range formed by its two satellites separated by x0 ¼ 220 km. The of phenomena, from inflationary models and predictions of published displacement spectrum [1] can be converted to ˜ string theory to populations of black holes and neutron strain via hxxðfÞ¼x˜ðfÞ=x0. The inferred strain-noise stars. Existing limits on the stochastic background come spectral density is shown in Fig. 1. As noted in Ref. [1], from pulsar timing [4], satellite ranging [5,6], torsion balances [7,8], terrestrial interferometers [9], and geophysi- 10 0 cal observations [10–12]. The most prominent single-source GW are emitted either by binary systems or spinning asymmetric stars. While asymmetric stars emit GW with a quasi-stationary fre- 10 -5 quency, inspiraling binary systems radiate semimonotone GW before a rapid merger and ring-down phase. The only direct observations of GW have been from the merger of black hole binaries and binary neutron stars [13–15] by the 10 -10 LIGO [16] and Virgo [17] observatories. II. STRAIN MEASUREMENTS 10 -15 As a GW passes two inertial masses nominally separated 10 -6 10 -4 10 -2 10 0 by a distance x0 their relative displacement, x,is[18]: FIG. 1. Strain amplitude spectral density for the GRACE-FO mission. The increase in apparent noise below 30 mHz is due to *[email protected] terrestrial gravitational signals and not instrumental noise. 2470-0010=2020=101(10)=102004(6) 102004-1 © 2020 American Physical Society M. P. ROSS et al. PHYS. REV. D 101, 102004 (2020) the noise below 30 mHz may be significantly reduced by frequencies between 30 mHz and 5 Hz. Both stellar-mass subtracting models of the terrestrial gravitational field. black hole binaries and neutron star binaries emit GW in the band of interest during their inspiral phase, while inter- III. BACKGROUND CONSTRAINTS mediate-mass black hole binaries would merge within the band. Assuming an isotropic, unpolarized, stochastic gravita- Due to the orbit of the satellites, a GW seen by GRACE- tional wave background, this strain noise can be interpreted FO would be modulated as the antenna pattern sweeps as limits on the cosmological energy density of gravita- across the sky. This modulation can be approximated by Ω tional waves, GW [19], by integrating Eq. (2) over rotating ϕ at the orbit frequency, f ¼ 1=94.5 min [1]. incidence angle and polarization. orb f ρ 32π3 15 ϕðtÞ¼2πf t þ ϕ ð Þ Ω ðfÞ¼ d GW ¼ f3 h˜ ðfÞ2 ð Þ orb 0 4 GW ρ f 2 4 meas 3 crit d 3H0 where ρ is the critical energy density of the universe, ρ is Simulated antenna factors [the angular dependence of crit GW Eq. (2)] for GRACE-FO, at a collection of θ values, are the energy density of gravitational waves, H0 is Hubble h shown in Fig. 3. For sources in-line with the orbital axis constant, and meas is the measured displacement interpreted H ¼ 70 3 −1 −1 (θ ¼ 0) the instrument’s polarization sensitivity is modu- as strain. A Hubble constant of 0 . km s Mpc θ ¼ π=2 was used [20]. The factor of 15=4 corrects for the polarization lated, while for sources in the orbital plane ( ) the and angular sensitivity of the instruments in question, see the total observed power is modulated. Appendix A. Detailed analysis of the orbit of the satellites and Confidence intervals were extracted from the GRACE- backgrounds due to the terrestrial gravitational field would FO strain spectrum by separating the data into frequency be required for a definitive search. However, an estimate of bins which each encompassed 20 data points. Limits for the characteristic strain [21] sensitivity can be inferred from each bin were set at the 95th percentile to yield the results the strain spectral density. This is shown in Fig. 4 along shown in Fig. 2. We anticipate that a dedicated analysis with a collection of known and speculative sources. The 2 10−15 would improve these constraints further. best characteristic strain sensitivity is × at 0.5 Hz. The characteristic strain allows for the estimation of the signal-to-noise ratio (SNR) of an optimal search for a given IV. SINGLE-SOURCES source [21]. Figure 5 shows the estimated maximum In addition to setting stochastic GW limits, the GRACE- detectable distance for a given equal component-mass FO mission may allow for gravitational wave searches at system with a select SNR threshold. 10 20 10 10 10 0 10 -10 10 -8 10 -6 10 -4 10 -2 10 0 10 2 FIG. 2. Limits on the stochastic gravitational wave background set by GRACE-FO (blue) and LISA-PF (dashed). Analysis of LISA- PF is detailed in Appendix B. Additionally, limits set by NanoGrav [4], Cassini [5], ULYSSES [6], TOBA [7,8], LIGO [9], observations of the earth’s normal modes [10], terrestrial seismic motion [11], and lunar seismic motion [12] are shown. Gray shading indicates regions excluded at 95% confidence. 102004-2 LIMITS ON THE STOCHASTIC GRAVITATIONAL WAVE … PHYS. REV. D 101, 102004 (2020) 1 0 -1 1 0 -1 1 0 -1 1 0 -1 1 0 -1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 FIG. 3. Simulated antenna factors for hþ and h× where θ is the polar angle of the source. For sources in-line with the satellite’s orbital axis (θ ¼ 0) the instrument’s polarization sensitivity is modulated by the orbit, while for sources in the orbital plane (θ ¼ π=2) the total observed power is modulated. 10 -15 10 -20 10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3 FIG. 4. Estimates of the characteristic strain sensitivity of GRACE-FO along with the expected strain from known white dwarf binaries [22–25] and a sensitivity curve for LIGO’s second observing run [26]. Also shown are lines that correspond to the predicted characteristic strain of an equal component-mass 1.4 M⊙ binary at 150 pc, a 60 M⊙ at 2 kpc, and a 5000 M⊙ at 50 kpc [21]. The points at the end of each line note the merger frequency for the corresponding system. These approximations are only valid for the inspiral phase of binary evolution. The detection range peaks at component-masses of signals severely limit detection capability while below 2000–10000 M⊙ with a volume encompassing most of ∼100 M⊙ the decreased emission does not provide an the Milky Way. Above 10000 M⊙, terrestrial gravitational appreciable detection volume. 102004-3 M. P. ROSS et al. PHYS. REV. D 101, 102004 (2020) where hxx is the strain along the x-axis, hþ; are the strain 5 × 10 amplitudes respectively in the plus and cross polarization, and θ and ϕ are the polar and azimuth angles, respectively. 4 Assuming isotropic emission, the fraction of incident 10 power that is captured by a single-axis instrument can be found by: 3 Z 10 1 h˜ 2 ðfÞ¼ h˜ 2 ðfÞdΩ meas 4π xx Z 2 1 10 ¼ dΩ½h˜ 2 ðfÞð 2 θ 2 ϕ − 2 ϕÞ2 ð Þ 4π þ cos cos sin A2 10 1 þ 4h˜ 2 ðfÞð θ ϕ ϕÞ2 10 0 10 1 10 2 10 3 10 4 × cos sin cos þ 2h˜ ðfÞh˜ ðfÞð 2 θ 2 ϕ − 2 ϕÞ þ × cos cos sin FIG. 5. Maximum detectable distance for equal component- × cos θ sin ϕ cos ϕðA3Þ mass binaries with SNR thresholds of 2, 4, and 8.