Localizing Gravitational Wave Sources with Single-Baseline Atom Interferometers
LLocalizing Gravitational Wave Sources with Single-Baseline Atom Interferometers
Peter W. Graham ∗ and Sunghoon Jung
2, 3, † Stanford Institute for Theoretical Physics, Department of Physics, Stanford University, Stanford, CA 94305, USA Center for Theoretical Physics, Department of Physics and Astronomy, Seoul National University, Seoul 08826, Korea SLAC National Accelerator Laboratory, Stanford University, Menlo Park, CA 94025, USA
Localizing sources on the sky is crucial for realizing the full potential of gravitational waves forastronomy, astrophysics, and cosmology. We show that the mid-frequency band, roughly 0.03 to 10Hz, has significant potential for angular localization. The angular location is measured through thechanging Doppler shift as the detector orbits the Sun. This band maximizes the effect since theseare the highest frequencies in which sources live several months. Atom interferometer detectors canobserve in the mid-frequency band, and even with just a single baseline can exploit this effect forsensitive angular localization. The single baseline orbits around the Earth and the Sun, causing itto reorient and change position significantly during the lifetime of the source, and making it similarto having multiple baselines/detectors. For example, atomic detectors could predict the location ofupcoming black hole or neutron star merger events with sufficient accuracy to allow optical and otherelectromagnetic telescopes to observe these events simultaneously. Thus, mid-band atomic detectorsare complementary to other gravitational wave detectors and will help complete the observation ofa broad range of the gravitational spectrum. ∗ [email protected] † [email protected] a r X i v : . [ g r- q c ] O c t CONTENTS
I. Introduction 2II. Executive Summary 3III. Calculation of Angular Resolution and Other Parameter Uncertainties 3A. Detector Configurations 3B. Calculation of Source Parameter Uncertainties 5C. Benchmark Signals 6IV. Results for Angular Localization and Other Source Parameters 7A. Approximate Analytic Description 7B. Specific Results for Single Baseline Detectors 8C. Dependence on Detector and Source Parameters 11V. Conclusion 12Acknowledgments 13A. GW Waveform 13B. Details of Fisher Calculation 14C. Optimal separation of measurements 14References 14
I. INTRODUCTION
LIGO’s historic observation of gravitational waves has opened a new spectrum in which to view the universe [1–3]. We are already learning much from these first observations (e.g. [4, 5]) which raise many interesting questions(e.g. [6, 7]). The next step is to fully exploit the gravitational spectrum by gaining precision information on manygravitational wave sources such as black holes, neutron stars, and white dwarfs. As LIGO is improved and futuredetectors such as VIRGO [8] and KAGRA [9] turn on, we will gain ever greater sensitivity to gravitational waves inthe high frequency band above about 10 Hz. But it is also crucial to open up as much of the gravitational spectrumas possible. Observing different bands in the electromagnetic spectrum led to many important new discoveries, anddifferent bands in the gravitational spectrum seem just as promising.A major advantage to observing in different frequency bands is their ability to provide a wealth of complementaryinformation (see e.g. [10]). For example, an important step to improve gravitational wave astronomy is to improvethe ability to localize gravitational wave (GW) sources on the sky. In order for gravitational wave telescopes torealize their full impact on astronomy, astrophysics, and cosmology they will need to have good angular localization.This greatly increases the use of GW observations, for example improved angular localization will allow optical andother electromagnetic telescopes to accurately observe the same source, realizing the major goal of multi-messengerastronomy. As one example, this is necessary to make cosmological measurements with the standard siren program [11,12].LIGO will improve angular resolution in the high frequency band in the future, though it is a challenging measure-ment to make since the sources do not live a long time in this band [5, 13]. At lower frequencies, sources generallylive much longer. Over a long signal duration, a detector can reorient and the modulation of the observed signal canthen give the direction of GW source. The proposed LISA detector operates around mHz, but even with longer sourcelifetimes, angular localization is still a challenge for most sources [14, 15]. Other more complex, farther-future spacemissions such as BBO and DECIGO [16, 17] are expected to achieve better precision [12] with many more satellites.Atomic interferometry has great potential as a gravitational wave detector [18–29]. Detectors such as MAGIS [30]and MIGA [24, 25] are being planned now. These detectors are based on techniques similar to atomic clocks whichhave achieved impressive precision [31]. Unlike laser interferometry, atom interferometry (AI) allows sensitive single-baseline gravitational wave detection [21, 22]. Further, it allows observation in the mid-frequency band ∼ II. EXECUTIVE SUMMARY
We will show that the mid-frequencies are ideal for angular localization. In this band the sources live a long time(usually months at least). In this observation time, the detector reorients and changes position significantly. Thismodulates the observed GW signal in a direction-dependent manner which allows the direction of the GW to bemeasured with high precision. Just from observing the gravitational wave with detectors in different orientations wewould expect to be able to measure the angular location with an accuracy of roughly ρ − (in square root of solidangle), where ρ is the signal-to-noise ratio (SNR). However, with multiple detectors separated by a distance R , thisangular resolution is enhanced by the ratio ∼ Rλ where λ is the wavelength of the GW. This enhancement arisesbecause the dominant contributor to the localization accuracy is by measuring the phase lag of the GW betweenthe two detectors. And in fact, multiple detectors are not necessary, a single detector that orbits over a distance of R during the time of the measurement will have the same enhancement . Thus in the mid-band where sources liveseveral months, R can be ∼ AU and so this enhancement can be several orders of magnitude, allowing good angularlocalization.Although a single-baseline detector can have lower cost and risk, it is often questioned whether such a simpleconfiguration can have precision capabilities and whether a single baseline can provide a measurement of sky positionand polarization. In fact, we will show that mid-band atomic detectors with only a single baseline can provide highprecision measurements of the angular location of GW sources as well as other parameters such as polarization orluminosity distance. So long as the detector baseline reorients rapidly (faster than the source lifetime), it is in manycases equivalent to having multiple baselines in different orientations so the angular location can often be measured aswell with a single baseline as with many. A terrestrial detector clearly reorients rapidly. Because a satellite detectorcan be in an Earth orbit, it can also reorient rapidly with a period of several hours. Further, within the source lifetimethe detector can complete a significant fraction of its orbit around the Sun. This is equivalent to having multipledetectors spaced by ∼ AU and allows the enhanced angular localization discussed above.Since sources last a long time in the mid-frequency band, this also allows prediction of a merger event a significanttime in the future. Thus, events in this band can be observed and then passed off to LIGO and to electromagnetictelescopes with a prediction of both when and where the merger will occur. Many sources are localized by the atomicdetector with sub-degree resolution, well within the field of view of many optical telescopes. Thus atomic detectorscan provide useful information which is highly complementary to that from LIGO in the higher frequency band, evenwhen observing the same source.We give details of our calculation of angular localization and other parameters in Sec. III. Then in Sec. IV, we showour results and discuss the underlying physics. We generalize the discussion and conclude in Sec. V. Our main resultsare shown in Tables I and II and Figures 2 and 3.
III. CALCULATION OF ANGULAR RESOLUTION AND OTHER PARAMETER UNCERTAINTIES
In this section we give the details of our calculation. Section III A gives the details of the specific atomic detectorswe take as examples. The actual calculation is discussed in Section III B. The example benchmark signals we considerare discussed in Section III C.
A. Detector Configurations
Our main point is to demonstrate that atom interferometer detectors can provide angular localization of GWsources, even with only a single-baseline. And further, the mid-frequency band is an ideal band for achieving highangular resolution for many sources. To illustrate this point we consider two example detectors, a satellite-baseddetector and a terrestrial detector. In this paper we are not attempting to actually design detectors, so we simplychoose example parameters for these two detectors that are within the possible range.The satellite-based detector that we consider in this paper consists of two satellites orbiting the Earth at a radiusof 2 × km (orbital period T AI = 7 . ◦ . Each satellite has an AI (free-falling, cold atomclouds) driven by a common laser baseline between the two satellites. The two satellites form a long (single) baselineof 3 . × km which is the satellite-based detector. Since the GW is a plane wave of wavelength much longerthan the laser baseline, we simplify the calculation by assuming the GW is measured at the midpoint between the For the enhancement, it is necessary that the detector not move in a straight line with constant velocity since this would appear as aconstant Doppler shift. ●●●● ●●●● ●●●● ●●●● - - - - - - f [ Hz ] s t r a i n s en s i t i v i t y [ H z - / ] Satellite AI ( resonant ) AgressiveTerrestrial AI aLIGOeLISA - G W G W N S - N S FIG. 1. Strain sensitivities of the satellite-based resonant AI detector [23] and a single aggressive terrestrial AI detector aswell as GW strains of benchmarks in Table I: 140-140 (red solid), GW150914 (blue dashed), GW151226 (orange dot-dashed),NS-NS (green dotted). The four dots on each lifetime curve indicate remaining lifetimes of 1 hour, 1 week, 3 months and 1year until merger. Lifetime curves are bent at 1 year as we consider up to one-year of measurement. The satellite-AI noise isthe envelope of the resonant AI detector taken from [23]. The sharp increase of the terrestrial noise at low frequencies is dueto the GGN; the particular curve is taken from the Homestake mine result [32] reduced by a factor 10 as discussed in the text.Also shown are eLISA [14] and aLIGO future design sensitivities [33]. two satellites, orbiting at a radius R AI = 8440 km. This orbital plane around the Earth is inclined by θ inc from theecliptic.We consider the possibility of having either one or two terrestrial detectors. Each terrestrial detector operatestwo AIs vertically separated by ∼ km, interrogated by a common laser, and forming a vertical ∼ km-long baseline formeasurements of the GW. We take each detector to be on the equator (just for the sake of computation) and the twodetectors are taken separated by π/ T AI = 7 . R AI = 8440 km) as well as slowly by the Earth’s annual orbit around the Sun. Likewise, the terrestrial detectorrotates once every day ( T AI = 1 day) and orbits around the Sun every year. In both cases, the orbit around theEarth reorients more quickly, whereas the orbit around the Sun moves in a larger radius. Consequently, they containdifferent angular information and become important in different regimes, as will be discussed. The combined orbitaround the Sun and the Earth (with a relative angle θ inc ) allows detectors to efficiently measure a wide range ofdirections, helping pinpoint source location and other source parameters (e.g. polarization) as will be discussed.Detector noises that we use in this paper are shown in Fig. 1. Both curves are optimistic projections for futuredetectors. The satellite detector noise shown is an envelope of a resonant-mode detector discussed in [23], havinga Q -factor enhanced sensitivity at a certain frequency f with a narrow bandwidth ∼ f /Q . Naturally with thisresonant detector the Q factor starts low, O (1), at the lower frequencies and rises a higher frequencies. Changing theresonance frequency and width can be done in real time by just changing the laser pulse sequence. Thus, a resonant-mode can either be used to track a quickly-evolving, higher-frequency GW source or to simultaneously measure manyslowly-evolving, lower-frequency GWs within its narrowband. As an illustration of the angular resolution of AI, weassume that we track each GW source for its last 1 year in this paper; and we use the noise envelope in Fig. 1 forcalculation. For example, one simple measurement strategy might be to observe in a ‘detection mode’ at a lowerfrequency (e.g. around 0.1 Hz) where the resonance is naturally quite broad until a source is detected. Then thatsource could be tracked as it evolves upwards in frequency with ever narrowing bandwidth (which should be possiblesince the SNR is already built up at low frequencies where the frequency of the source did not need to be knownwell). We assume a one year measurement of each source just to have a simple calculation with a result that isunderstandable, but in reality the best way to observe a particular source may be to watch it for only a fraction ofeach month over a couple years for example, and then use the rest of the time to watch other sources. The resultingSNR should be about as high as with our calculation, and the angular resolution should be at least as good if notbetter than what we have calculated. For good angular resolution, we mainly need several measurements at well-spaced places along the orbit around the Sun, the measurements certainly do not need to be continuous for a year.See below for more discussion. In practice, a sophisticated detection strategy for a resonant-mode (switching betweentracking a small number of GWs with higher accuracies and observing many GWs) can be developed in future work.The terrestrial detector noise shown in Fig. 1 is based on a broadband-mode. It is an optimistic noise curve thatrequires advances in the atomic technology that appear possible but have not yet been realized. This can be takenas a goal, and we will present results for this as it is a benchmark that would be useful to reach. The sharp increaseof the noise at low frequencies is due to gravity-gradient noise (GGN). The GGN is mainly due to seismic activitiesand air density/pressure variations, causing variation of the gravitational potential at the location of the AIs. It islikely to be the dominant background in frequencies below a few Hz [34]. Although it is very challenging to suppresssignificantly [35], it can potentially be reduced by various techniques that are being developed, e.g. from an array ofAIs to utilize the fact that wavelengths of GW are longer than the characteristic coherence length of GGN [26]. It isalso strongly dependent on the location of the detector, so can be reduced by a suitable choice of site and potentiallyalso by other techniques being considered for LIGO [32]. We optimistically assume that the GGN can be improvedby a factor 10 from the measurement of Homestake mine noise [32]; the resulting smaller GGN is shown in Fig. 1 andwill be used in our numerical study. B. Calculation of Source Parameter Uncertainties
A single-baseline detector measures the gravitational stretching and contraction along its baseline direction. Wedefine the detector response tensor D ij from the baseline direction unit vector a i ( t ) as D ij ( t ) = a i ( t ) a j ( t ) . (1)The GW strain tensor is decomposed in terms of polarization tensors e + , × ij (in Eq. (A9)) as h ij ( t ) = h + ( t ) e + ij + h × ( t ) e × ij , (2)the observed waveform is given by h ( t ) ≡ D ij h ij = h + ( t ) D ij ( t ) e + ij + h × ( t ) D ij ( t ) e × ij ≡ h + ( t ) F + ( t ) + h × ( t ) F × ( t ) . (3)The angular information of the GW source (location, polarization and binary orbit inclination) is encoded in time-dependent antenna functions F + , × ( t ) and phases arg( h + , × ( t )). As a detector reorients and/or moves, the observedwaveform and phase are modulated and Doppler-shifted, yielding important angular information. Meanwhile, ampli-tudes | h + , × ( t ) | depend on the chirp mass, luminosity distance and binary orbit inclination. We collect and summarizethe GW waveforms h ( t ) in Appendix A.By assuming that the satellite detector orbits around the Earth along θ a = π/ θ is the Earth’s polarcoordinate) without loss of generality, we parameterize the detector location on the orbit by a unit vector (in Cartesiancoordinates) r AI , ( t ) = (cos φ a ( t ) , sin φ a ( t ) , , (4)where φ a ( t ) = 2 πt/T AI + φ is the azimuthal orbit angle around the Earth. For the aforementioned satellite-missionorbit, the baseline direction is given by a unit vector a ( t ) = ( − sin φ a ( t ) , cos φ a ( t ) ,
0) (satellite mission). (5)The vertical baseline of a terrestrial detector is a ( t ) = (cos φ a ( t ) , sin φ a ( t ) ,
0) (terrestrial mission). (6)
Satellite DetectorBenchmark masses distance D L lifetime √ Ω s [deg] SNR ∆ D L D L ∆ ψ [rad]GW150914 36-29 Ms 410 Mpc 9.6 months 0.16 67 0.21 0.85GW151226 14.2-7.5 Ms 440 Mpc 5.5 years 0.20 16 0.88 3.4NS-NS 1.5-1.5 Ms 140 Mpc 140 years 0.19 5.2 2.8 11140-140 140-140 Ms 410 Mpc 25 days 0.75 190 0.80 1.2TABLE I. Benchmark sources and results for the satellite mission. The most important source parameters are the massesand luminosity distance D L , which determine the overall signal strength and the lifetime spent in the AI frequency band f = 0 . − ρ and the uncertainties for angular resolution √ ∆Ω s in degrees, distance D L ,and polarization ψ (in radians) from the resonant satellite detector discussed in the text. Results are integrated for the lastone year up to 1 hour before merger, or up to the ISCO whichever is earlier. All uncertainties can be scaled almost linearlywith 1/distance. Unphysically large uncertainties (no priors are applied) would mean that the parameters will not be wellconstrained; but uncertainties linearly-scaled with 1/distance become meaningful and physical for close enough sources. These vectors defined in the Earth polar coordinate are transformed to the Sun’s polar coordinate as (again byassuming the ecliptic at θ Ea = π/ r AI ( t ) = cos φ Ea ( t ) − sin φ Ea ( t ) 0sin φ Ea ( t ) cos φ Ea ( t ) 00 0 1 · cos θ inc − sin θ inc θ inc θ inc · r AI , ( t ) , (7)and similarly for a ( t ). The azimuthal angle of the Earth’s orbit around the Sun is φ Ea ( t ) = 2 πt/ (1yr) + φ (cid:48) , and thelocation of the Earth in the Sun frame is r Ea ( t ) = (cos φ Ea ( t ) , sin φ Ea ( t ) , θ inc (of a detector orbitaround the Earth with respect to the ecliptic) is chosen to be π/ . ◦ for the satellite and terrestrial missions,respectively. (Of course, the satellite-orbit inclination is not a fixed number and its optimal value can be studied.)We estimate parameter measurement accuracies by calculating a Fisher matrix Γ. 10 free parameters that weconsider are binary masses, source direction n = n ( θ, φ ), polarization ψ , binary orbit inclination c ι ≡ cos ι , luminositydistance D L , coalescence time t c and phase φ c , and spin-orbit coupling β (we include β in the Fisher estimationalthough we set β = 0; neutron star spins are small, and spin-orbit dipole interactions are suppressed by large binaryseparations during the inspiral phase far from merger). The first three angle parameters are defined in the Sunframe. The covariance matrix Γ − is a theoretical estimation of experimental uncertainties. In particular, the angularresolution of the source location is defined as the solid-angle uncertainty [15]∆Ω s ≡ π sin θ (cid:113) Γ − θθ Γ − φφ − (Γ − θφ ) . (8)We use √ ∆Ω s (in degree) and the square-root of diagonal elements of Γ − as measurement accuracies in this paper.More details on Fisher matrix analysis can be found in, e.g. Refs. [15, 36], and its utilities and limitations are discussedin, e.g. Refs. [37]. Most of our benchmark results have SNR ρ (cid:38) ρ and Fisher elements Γ ij are integrated over the measurement time as ρ = 4 (cid:90) ˜ h ∗ ( f )˜ h ( f ) S n ( f ) df, (9)Γ ij = 4 Re (cid:90) ( ∂ i ˜ h ∗ ) ∂ j ˜ hS n ( f ) df, (10)where ˜ h ( f ) is the Fourier-transform of h ( t ) as in Eq. (A1). If the same data multiples, ∆Ω s · ρ is constant. C. Benchmark Signals
Since two disparate time scales – 7 . ∼
24 hours (earth orbit) and 1 year (solar orbit) – are present in satellite andterrestrial AI missions, detection strategies and measurement accuracies depend crucially on the GW lifetime in the
One Terrestrial Detector Two Terrestrial DetectorsBenchmark masses distance D L lifetime √ Ω s [deg] SNR ∆ D L D L ∆ ψ [rad] √ Ω s [deg] SNR ∆ D L D L ∆ ψ [rad]GW150914 36-29 Ms 410 Mpc 9.6 months 140 4.8 39 140 77 6.1 21 70GW151226 14.2-7.5 Ms 440 Mpc 5.5 years 150 1.7 45 180 110 2.4 31 120NS-NS 1.5-1.5 Ms 140 Mpc 140 years 2.6 1.1 22 74 1.8 1.6 15 52140-140 140-140 Ms 410 Mpc 25 days 370 14 94 330 70 14 20 71TABLE II. Benchmark sources are the same as in Table I, results shown are for one and two terrestrial detectors. Resultsare integrated for the last one year up to 10 minutes before merger, or up to the ISCO whichever is earlier. Unphysicallylarge uncertainties (no priors are applied) would mean that the parameters will not be well constrained; but uncertaintieslinearly-scaled with 1/distance become meaningful and physical for close enough sources. The results with SNR ρ (cid:28) AI frequency band f = 0.03 - 5 Hz. Although any compact binary mergers lighter than about a few 1000 solar masswill pass AI band, their lifetime in the AI band vary from a few seconds to several years depending on the masses.We choose four benchmark GW sources that span a wide range of lifetime in the AI band. Our point in this paper isnot to estimate what sources could definitely be seen with reasonable rates. So we simply chose some example sourceparameters. But it is easy to scale the results for any source to a different distance for example. Since the strain h of the source scales linearly with distance, then in a wide range of parameters, all the uncertainties we estimate (e.g.for angular resolution, polarization, etc) also simply scale linearly with distance.The four benchmark sources are shown in Table I and Fig. 1. The first two listed are LIGO’s first two discoveries(GW150914 [1], GW151226 [2]), spending at least about a year in the AI band so that both time scales are relevant.Their binary masses and luminosity distances are taken from LIGO measurements [2, 5], but other source parameterssuch as locations on the sky, orbit inclination, polarization in Eq. (A8) are chosen somewhat randomly (but wechecked that they do not lead to particularly good or bad results; see discussion in Sec. IV). However, the massesand distances are the most important factors determining signal strengths and frequencies. In addition to the LIGOsources spending at least about a year, we consider a neutron star binary (NS-NS). This is an important source ofGWs that is expected to produce electromagnetic signals during merger as well. We chose it at a distance whichLIGO could see as well, and in which it is reasonable to expect at least some events during a mission lifetime. Lastly,we consider a black hole (BH) binary with two 140 solar mass BH’s, “140-140”, which spends only 25 days in the AIband, representing the case where only the smaller time scale – 7 . ∼
24 hours – is relevant. This source would likelynot be visible at LIGO since it merges at too low a frequency. This type of source is not known to exist, but suchheavy BH’s would likely only be seen when we turn on a GW detector in this mid-frequency band anyway. So herewe have simply chosen as an example distance, the same distance as LIGO’s first source, but the results can be easilyscaled to any other distance as discussed. Results will be discussed in Sec. IV.
IV. RESULTS FOR ANGULAR LOCALIZATION AND OTHER SOURCE PARAMETERS
In this section we discuss the results of our calculation for the angular localization and uncertainties in otherparameters such as polarization and luminosity distance. We give an approximate analytic understanding in SectionIV A. Our specific numerical results are discussed in Section IV B. The robustness of our results against varyingdetector and source parameters is discussed in IV C.
A. Approximate Analytic Description
The single-baseline AI measurement contains directional information in the form of the modulation of GW signalstrength and phase as the AI detector reorients and orbits. When the detector baseline reorients, the relative anglebetween its baseline and source direction (and polarization) changes so that the observed signal strength (signal-to-noise ratio SNR ρ ) and polarization-phase (in Eq. (A5)) changes. Thus the angular resolution from such informationimproves roughly as √ ∆Ω s ∝ /ρ . Moreover, if a detector orbits (or, moves non-linearly, in general), the Doppler-shift of the GW phase (in Eq. (A4)) changes. As the phase accumulation changes linearly with frequency f and orbitradius R/c , such angular resolution improves approximately as √ ∆Ω s ∝ / ( ρ πf R/c ). By roughly comparing thetwo contributions, we expect that the Doppler effects would dominate when f R/c (cid:38) f (cid:38) − Hz for R = 1 AU (cid:39)
500 sec/c), improving angular resolutions in proportion to the GW frequency. Thus the interplay of the two effectswill determine overall performance and properties of atomic measurements, as will be discussed below. We will callthe two effects by “reorientation” and “Doppler” effects; but the latter effect is actually the change in the Dopplereffect over time. These give an approximate analytic description of the enhancement in angular resolution.The above estimations are valid only when full angular information is gained by measuring at a large range ofangles. Consider the Doppler effect from a circular orbit. The GW phase will be periodically Doppler-shifted andwill modulate along the orbit, giving important information about source direction. But a small portion of the orbitis close to linear motion so that Doppler effects do not change much. Since the linear Doppler shift is not measurable(redshift is not measurable from GW measurements since we do not know the source frequency [11]), exactly linearmotion with a constant velocity would not give us any angular or polarization information. So a small portion of anorbit alone does not provide good angular resolution. This means that, in order for the Doppler effect around theSun to be fully utilized, a GW should spend at least a few months in the AI band (and most benchmarks indeed do).We first demonstrate the underlying physics in Fig. 2 and Fig. 3. The results are obtained by integrating the lastone year of AI measurement (or as soon as the GW enters the AI band) up to 1 hour (for satellite mission) or 10minutes (for terrestrial mission) before merger. This time gap is to allow some time to warn LIGO and telescopefollow-ups; in later sections, we vary the time gap. Compact binaries with masses M and M are assumed to mergeat the innermost stable circular orbit (ISCO) of the orbital separation r = 6 M z = 6( M + M )(1 + z ) corresponding to f = (6 √ πM z ) − . Thus, we conservatively include only the inspiral stage but not the merger and ringdown (althoughthey can be important for very short signals merging in the AI band). Based on these and our Fisher estimations,Fig. 2 plots the accumulation of angular resolution √ ∆Ω s in Eq. (8), SNR ρ , the uncertainty of the luminosity distance∆ D L /D L and polarization ∆ ψ as functions of the measurement time. Fig. 3 decomposes the contributions to √ ∆Ω s and ∆ D L /D L from reorientation and Doppler effects for GW150914 benchmark.The first panel of Fig. 2 shows the discussed interplay of Doppler versus reorientation for the case of satelliteAI measurements. Angular resolutions generally improve quickly in the first few months, then reach a plateau,and improve again in the last few months. In the first few months, most angular information is gained from thequick reorientation around the Earth ( T AI = 7 . D L and ψ in Fig. 2; although we show only these two, we checked that most discussions here apply to otherimportant parameters such as orbit inclination c ι as well. The Fig. 2 shows that ∆ D L /D L and ∆ ψ accuracies growsignificantly only in the first few months because of reorientation effects but do not grow very much after that byDoppler effects. Supporting this, Fig. 3 right panel shows that the full result (red-solid) and the result without Dopplereffects (blue-dashed) almost coincide; but if reorientation is approximately ignored (green-dotted), the resolutions ofsuch parameters are degraded significantly. This result is understood because a Doppler phase in Eq. (A4) dependsonly on the flight-time difference or the sky location n projected along the displacement but not on other sourceparameters. This study can provide important feedback for designing a real satellite mission, for example in choosingorbital parameters to maximize the ability to measure all the source parameters.It is remarkable that a single-baseline detector can utilize all these physics. Although multi-baseline measurementscan add more information on various parameters, the essential part of the underlying physics – the change of Dopplereffect – is induced (regardless of detector details) by a non-linear trajectory of the detector, which is provided by therapid orbit around the earth and the annual orbit around the Sun. Our satellite mission is indeed designed to containa rapid orbit around the Earth ( T AI = 7 . B. Specific Results for Single Baseline Detectors
We now turn to a discussion of specific results for the satellite and terrestrial detectors. Our main results aretabulated in Tables I and II. Foremost, the satellite mission’s mid-frequency measurements can be extremely usefuland unique. Its angular resolution O (0 .
1) deg with SNR ∼ O (10) is comparable to typical field-of-views (FOV)of ground-based telescopes ∼ O (10) Mpc can even befollowed up by the Hubble telescope (FOV ∼ . − .
05 deg). The electromagnetic identification is the first [ days ] S q r t [ Δ Ω s ][ deg ] Satellite AI140 - - NS [ days ] S NR ρ [ days ] Δ D L / D L [ days ] Δ ψ [ r ad ] [ days ] f [ H z ] FIG. 2. Accumulation of measurement accuracies from the satellite resonant-AI detector for the benchmarks in Table I: 140-140 (red solid), GW150914 (blue dashed), GW151226 (orange dotdashed), NS-NS (green dotted). The observables shown arelocation angular resolution √ ∆Ω s , SNR ρ , errors on luminosity distance ∆ D L /D L and on polarization ∆ ψ as functions ofmeasurement time starting from 1 year before merger (or as soon as the GW enters the AI band) up to the last 1 hour or theISCO, whichever is earlier. No priors are added. Errors and SNR improve almost linearly with 1/distance for the given sourceparameters. The bottom figure is the source frequency as a function of time for reference. step of multi-messenger physics and standard-siren program [11]. It is also worthwhile to note that the sub-degreeresolution is achieved with relatively small SNRs (just big enough for discovery). For comparisons, LISA measuringlower frequencies with three detector baselines can achieve 0.5 deg resolutions with SNR ∼ − [ days ] S q r t [ Δ Ω s ][ deg ] f [ Hz ] GW150914 benchmarkSatellite AI f u ll r e s u l t w i t hou t D opp l e r w i t hou t r eo r i en t a t i on [ days ] Δ D L / D L f [ Hz ] FIG. 3. Decomposition of angular resolution √ ∆Ω s and ∆ D L /D L by reorientation and Doppler contributions. The source isthe GW150914 benchmark as seen in the satellite detector. Shown are full results (red-solid), Doppler effects ignored (blue-dashed), or detector baseline not reorienting in the Earth frame (green-dotted). The left panel illustrates that reorientation isimportant in the first few months, but Doppler effects can further improve angular resolution after a few months. The rightpanel shows that D L (and other source parameters) can only be well measured by detector reorientation; in the green-dottedcurve, the AI baseline is fixed in the Earth frame leading to poor resolution until the detector slowly reorients by orbitingaround the Sun. LIGO. The sub-degree resolution means that those discovered GWs could have been localized and warned in advancefrom the mid-frequency measurement.On the other hand, the terrestrial detector’s angular resolutions are about two orders of magnitude worse. Othersource parameters are also not so well measured . It is mainly because the low-frequency regime is swamped by GGNso that only the short measurement of the high-frequency regime becomes useful (such useful durations are from afew hours to a few days; see Fig. 1). Consequently, no full Doppler effects can be utilized; we recall that the changeof Doppler effects measured over several months is the one that enhances angular information. Although the resultscan perhaps be used by the Fermi satellite (FOV ∼
90 deg), reducing GGN in mid-frequency terrestrial detectors isone of the most important tasks for a terrestrial experiment [35].We also learn by comparing among benchmarks. A notable result of the NS-NS localization √ Ω s is that it can bedone much better than other benchmark localizations for the given SNR. It is because the NS-NS spends longer timein high-frequency band that subsequently enables larger Doppler effects. This contrast is most clearly seen in ourterrestrial results as only the high-frequency regime is useful there. By the same reason, on the other hand, the 140-140 spends too short time in the high-frequency band so that its localization is not significantly better in proportionto its high SNR, ρ . Thus, angular resolutions among different GW sources do not simply scale as √ Ω s ∝ /ρ , butdepend also on the frequency content of the GW, and the motion of the detector during the observation time.The uncertainties of D L and ψ (and c ι as well) from the satellite-mission results in Table I follow the 1/SNR scalingbetter among GW150914, GW151226 and NS-NS results (except the 140-140). For example, ∆ D L /D L of NS-NS isabout 20 times worse than that of GW150914 as its SNR is about 20 times smaller, and so on. Here, we find that thelocalization and measurements of other source parameters become decoupled. The former is dominantly determined byDoppler effect, whereas the latter is by reorientation. In addition, the reorientation effects for GW150914, GW151226and NS-NS are saturated. Therefore, measurements of other source parameters improve simply with data, i.e., 1/SNR.This also explains why terrestrial results of ∆ D L /D L , ∆ ψ (and ∆ c ι ) do not scale with 1/SNR among benchmarks.Doppler effects are not fully utilized in these cases so that uncertainties and correlations are general mixtures of(unsaturated) Doppler and reorientation effects, obscuring any simple scaling rule. Quantitatively, a dimensionlessquantity measuring the correlation Γ − ij / (cid:113) Γ − ii Γ − jj for i ∈ { θ, φ } , j ∈ { D L , ψ, c ι } is O (0 .
1) for the terrestrial results,which is 5-10 times larger than that of the satellite mission.From our results, we can see the use of having multiple detectors. For example, the results for one and for twoterrestrial detectors are shown in Table II. Note that for the very long-lived sources (GW151226 and NS-NS) all results Their uncertainties are unphysically large as we do not add physical priors to the Fisher matrix. But we still show them becauseuncertainties scale with 1/distance, and the scaled results can be meaningful for close enough sources. √ √ C. Dependence on Detector and Source Parameters
As a very preliminary step towards optimizing measurement protocols, we study the impact of varying the mea-surement time. As mentioned above, we take the measurement time for each source to be the last year of its lifetimefor simplicity and so our results are easily interpretable. But in reality the optimal observation strategy is likely to bequite different. As mentioned above, the optimal observing strategy may be to detect sources at the lower frequenciesand then keep coming back to observe them periodically as they rise to higher frequencies. This lets us observethem at different parts of the detector’s orbit around the Sun, maintaining the ∼ f R enhancement to the angularresolution, and helping in the final prediction for merger time, but still allows observing time to watch other sourcesas well. For example, our benchmark NS-NS binary source is seen with an SNR of 5.2 in the last year. However if thepreceding one year is used instead (i.e. the second to last year of the binary’s life), the SNR is 3.3. So not much SNRwould be lost by using some combination of observing times over the last few years of that source’s life. The preciseoptimal observing strategy including spacing of measurements for each source is beyond the scope of this paper (butsee Appendix C for a few more details). We simply note that in order to attain the angular resolution enhancement wehave discussed, it is necessary to observe the source from a few different points spaced by O (1) around the detector’sorbit around the Sun.As another example we consider the effect of taking the measurement of each source to end one day before themerger (allowing more time for follow-up observations to be ready) instead of one hour or 10 minutes. For satellite-mission results, the angular resolutions are the most affected and mildly worsen by a factor of 1.5-3, whereas the otherparameter measurements are affected even less. This pattern of impact can be understood from Fig. 2 by cuttingthe final one day of the measurement time. The main change of the angular resolution results from the reduction ofDoppler effects that are still growing rapidly at the end of the measurement time; in contrast, reorientation effects arewell saturated by that time, and so do not lose much information from losing the final day of observation. These mildsensitivities to measurement time can be helpful in designing detection protocols. For terrestrial-mission results, onthe other hand, all measurement accuracies degrade by an order of magnitude. The degradation is well captured bythe loss of SNR by a factor 6 or so (satellite-mission’s SNRs reduce only by 10%). As the useful measurement times(which are not swamped by GGN) are short, the reduction of measurement time causes a bigger loss of measurementaccuracies. Thus, reducing GGN will again improve this situation.So far, we have chosen and fixed one particular set of source parameters. How would the results change withdifferent source parameters? We varied n , ψ and c ι to obtain the possible range of measurement accuracies. It turnsout that the 140-140 result and terrestrial results are most sensitive to the choice of source parameters; accuraciesvary generally by a factor 10 up or down. On the contrary, results for the other sources in the satellite-missionvary generally only by 20-30% for GW150914, GW151226 and NS-NS. The big sensitivity stems mainly from theshort measurement time. The short measurement time means the detector can essentially span only a 2-dimensionalplane. Whereas in the satellite mission, the long measurements allow the detector to cover a significant part of itsorbit around the Sun meaning the single baseline sweeps out a 3-dimensional volume. Thus, for short measurementtimes, particular source angles with respect to the 2-dimensional plane play a crucial role in the observed signalstrength. Though it is possible that different choices of orbit around the earth could improve this situation for the2short measurements. If it were not for rapid reorientation, the parameter dependencies would have been terrible; butthese single-baseline AI detectors naturally reorient as discussed. In general, the assumption we have made that thesatellite detector orbits the earth can play a crucial role for several types of sources by allowing rapid reorientation ofthe detector. V. CONCLUSION
In order to fully realize the promise of gravitational wave astronomy, we will need the ability to accurately localizedetected objects on the sky. Angular localization is a crucial feature of any telescope, and for example will allowoptical and other electromagnetic telescopes to observe the same source, greatly increasing the information gained.We have demonstrated that the mid-frequency band, roughly 0.03 to 10 Hz, is in many ways the ideal band forthe angular localization of many gravitational wave sources. Even sources that are not observed with very high SNRcan nevertheless be localized to high precision on the sky (see Tables I and II). The angular resolution is enhanced(approximately) by the ratio of the distance over which the detector moves during the measurement of the source(or the distance between multiple detectors) to the wavelength of the GW. For the mid-frequency band this can bean enhancement of some orders of magnitude. In this band, most sources live at least a few months, allowing thedetector to move over roughly an AU. Of course, sources live even longer at lower frequencies but this does not helpsince the earth never moves farther than 2 AU . And at lower frequencies the wavelength becomes longer, reducingthe effect. For example in the mHz band (LISA’s band) the GW wavelength is about an AU, significantly reducingthis enhancement to the angular resolution. So the mid-frequency band appears optimal for angular localization sincethese are the highest frequencies in which sources live several months.This is a general point that applies to any type of gravitational wave detector, not just the atomic detectors wehave discussed. The power of the measurement comes from the large change of the Doppler shift over a measurementtime of several months. This is induced not by multiple detectors but by the Earth’s (and hence the detector’s) orbitaround the Sun. Therefore any detector with good sensitivity in the mid-band ( ∼ . Multiple baselines or multiple detectorsare certainly not required for angular localization or measurement of other parameters. This allows atomic detectorsto provide useful information that is complementary to other detectors such as LIGO or LISA. In fact, having severalof these detectors observe the same source across a wide range of frequencies may allow measurements significantlybetter than any one of them could achieve on its own. It would be interesting to study the gain from observing asource by multiple detectors (see e.g. [10]), particularly if the same source can be observed by both atomic detectorsand LIGO or LISA.The satellite atomic detector considered here has excellent angular resolution for a gravitational wave telescope. Infact a discovery almost guarantees sub-degree angular resolution. And many sources can be localized down to O (0 . The motion of the solar system through our galaxy, or of our galaxy through the universe, does not help with localization or other sourceparameter estimation. This motion is in a straight line with constant velocity and so is a constant Doppler shift. This kind of motionis not equivalent to having simultaneous, multiple detectors spread over that distance and does not improve angular localization. So long as it does not move in a straight line with constant velocity, as discussed above.
ACKNOWLEDGMENTS
We thank Jason M. Hogan for many useful discussions on detector configuration and noise curves and Rana Adhikariand Leo Singer for various comments and checking our antenna functions for single-baseline detectors. We also thankDouglas Finkbeiner, Mark Kasevich, Peter Michelson, Surjeet Rajendran, and TJ Wilkason. PWG acknowledgesthe support of NSF grant PHY-1720397, DOE Early Career Award de-sc0012012 and the W.M. Keck Foundation.The work of SJ is supported by the US Department of Energy contract DE-AC02-76SF00515, NRF Korea grant2017R1D1A1B03030820, and Research Settlement Fund for the new faculty of Seoul National University.
Appendix A: GW Waveform
We collect the formula used in this paper here. The waveform in the frequency domain with spin-orbit coupling β = 0 [11, 15, 36] is the Fourier-transform of the time-domain one in Eq. (3): (cid:101) h ( f ) = (cid:90) + ∞−∞ h ( t ) e πift dt (A1)= (cid:114) (cid:113) A F + A × F × D L π − / M / z f − / exp[ i Ψ( f )] , (A2)where the phase is Ψ( f ) = 2 πf t c − φ c − π π M z f ) − / − φ P ( t ) − φ D ( t ) + · · · , (A3)and the chirp mass M = ( M M ) / / ( M + M ) / with M z ≡ M (1+ z ). We do not write next-order Post-Newtoniancorrections here, but we include them for phase in our numerical calculation [43]; see the above references for thesenext-order terms. The Doppler phase measured by AI detectors contains important angular information as φ D = 2 πf (cid:0) R AI r AI · n /c + R AU r Ea · n /c (cid:1) (A4)with R AU = 1 AU, and the polarization phase is φ P = arctan[( A + F + ) / ( A × F × )] . (A5)Each polarization amplitude is A + = 1 + c ι and A × = − c ι . The monotonic GW frequency evolution dfdt = 965 π / M / z f / + · · · (A6)allows the one-to-one correspondence between GW frequency and the time before merger t ( f ) = t c − M z ( π M z f ) − / + · · · . (A7)We use the following source parameters for all benchmarks in this paper: θ = π . , φ = π . , ψ = π , c ι = cos(150 ◦ ) . (A8)They are somewhat randomly chosen, but not leading to particularly good or bad angular resolution.Polarization tensors used in Eq. (2) are e + ij = ˆ X i ˆ X j − ˆ Y i ˆ Y j , e × ij = ˆ X i ˆ Y j + ˆ Y i ˆ X j , (A9)where the basis is [11, 36] ˆX = ( sin φ cos ψ − sin ψ cos φ cos θ, − cos φ cos ψ − sin ψ sin φ cos θ, sin ψ sin θ ) , (A10) ˆY = ( − sin φ sin ψ − cos ψ cos φ cos θ, cos φ sin ψ − cos ψ sin φ cos θ, cos ψ sin θ ) . (A11)4 Appendix B: Details of Fisher Calculation
Our Fisher matrix calculation involves a highly oscillatory integration due to the quick reorientation around theEarth. To facilitate Mathematica
NIntegrate computation, we (1) divide integration frequency range (mostly ac-cording to the number of GW cycles), (2) keep relative errors of every subregion integral small and similar in size, and(3) use high-precision numerical variables. As a result, the inversion of the Fisher matrix becomes relatively stable(even though matrix condition numbers are very large). But the calculation is more than 10 times slower than thecalculation without the reorientation around the Earth. For an almost monochromatic GW (whose frequency evolvesvery slowly, e.g. in the inspiral phase far from merger), we use the following approximation to cross-check the fullresult of frequency-domain ρ (cid:39) S n ( f ) (cid:90) | h ( t ) | dt, Γ (cid:39) S n ( f ) (cid:90) ( ∂ i h )( ∂ j h ) dt. (B1)Using our AI calculation, we could approximately reproduce the LISA [15] and BBO [12, 17] Fisher estimations of √ Ω s and ∆ D L /D L normalized to a certain SNR from the last 1-yr observation. We take this as one cross-check ofour calculation. This may also imply that, for long enough measurements, details of detector configuration and orbitare not so important in an order-of-magnitude estimation. Appendix C: Optimal separation of measurements
Given that the change of Doppler shift contains measurable angular information, which two angles from a circularorbit can yield maximum angular information? By solving the 2 × × composed of θ and φ (thus,ignoring any uncertainties correlated with other parameters), we obtain∆Ω s ≈ π sin θ (det Γ × ) − / . (C1)From the two measurements of δ -duration( δ (cid:28) α , the above 2 × − s ∝ ( f R ) sin 2 θ (cid:112) δ + cos 2 δ − − δ cos 2 α ≈ ( f R ) sin 2 θ (cid:112) δ (1 − cos 2 α ) . (C2)Thus, Doppler effects are maximized for α (cid:39) π/
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