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Coherent observations of gravitational radiation with LISA and gLISA

The geosynchronous Laser Interferometer Space Antenna (gLISA) is a space-based gravitational wave (GW) mission that, for the past five years, has been under joint study at the Jet Propulsion Laboratory, Stanford University, the National Institute for Space Research (I.N.P.E., Brazil), and Space Systems Loral. If flown at the same time as the LISA mission, the two arrays will deliver a joint sensitivity that accounts for the best performance of both missions in their respective parts of the mHz band. This simultaneous operation will result in an optimally combined sensitivity curve that is"white"from a few mHz to 1 Hz, making the two antennas capable of detecting, with high signal-to-noise ratios (SNRs), coalescing black-hole binaries (BHBs) with masses in the range (10 - 100 million) solar masses. Their ability of jointly tracking, with enhanced SNR, signals similar to that observed by the Advanced Laser Interferometer Gravitational Wave Observatory (aLIGO) on September 14, 2015 (the GW150914 event) will result in a larger number of observable small-mass binary black-holes and an improved precision of the parameters characterizing these sources. Together, LISA, gLISA and aLIGO will cover, with good sensitivity, a frequency band from a tenth of a mHz to a kHz frequency band.

The first direct observation of a GW signal announced by the LIGO project [1] on February 11, 2016 [2], represents one of the most important achievements in experimental physics today, and marks the beginning of GW astronomy [3]. By simultaneously measuring and recording strain data with two interferometers at Hanford (Washington) and Livingston (Louisiana), scientists were able to reach an extremely high level of detection confidence and infer unequivocally the GW source of the observed signal to be a coalescing binary system containing two black-holes of masses M 1 = 36 +5 −4 M and M 2 = 29 +4 −4 M out to a luminosity distance of 410 +160 −180 M pc corresponding to a red-shift z = 0.09 +0.03 −0.04 (the above uncertainties being at the 90 percent confidence level). The network of the two LIGO interferometers could constrain the direction to the binary system only to a broad region of the sky because the French-Italian VIRGO detector [4] was not operational at the time of the detection and no electromagnetic counterparts could be uniquely associated with the observed signal.
Ground-based observations inherently require use of multiple detectors widely separated on Earth and operating in coincidence. This is because a network of GW interferometers operating at the same time can (i) very effectively discriminate a GW signal from random noise and (ii) provide enough information for reconstructing the parameters characterizing the wave's astrophysical source (such as its sky-location, luminosity distance, mass(es), dynamical time scale, etc.) [5,6]. Space-based interferometers instead, with their six links along their three-arms, have enough data redundancy to validate their measurements and uniquely reconstruct an observed signal [7]. A mission such as LISA, with its three operating arms, will be able to assess its measurements' noise level and statistical properties over its entire observational frequency band. By relying on a time-delay interferometric (TDI) measurement that is insensitive to gravitational waves [8], LISA will assess its in-flight noise characteristics in the lower part of the band, i.e. at frequencies smaller than the inverse of the round-trip-light time. At higher frequencies instead, where it can synthesize three independent interferometric measurements, it will be able to perform a data consistency test by relying on the null-stream technique [6,9,10], i.e. a parametric non-linear combination of the TDI measurements that achieves a pronounced minimum at a unique point in the search parameter space when a signal is present. In addition, by taking advantage of the Doppler and amplitude modulations introduced by the motion of the array around the Sun on long-lived GW signals, LISA will measure the values of the parameters associated with the GW source of the observed signal [7].
Although a single space-based array such as LISA can synthesize the equivalent of four interferometric (TDI) combinations (such as the Sagnac TDI combinations (α, β, γ, ζ)) [11], its best sensitivity level is achieved only over a relatively narrow region of the mHz frequency band. At frequencies lower than the inverse of the round-trip light time, the sensitivity of a space-based GW interferometer is determined by the level of residual acceleration noise associated with the nearly free-floating proof-masses of the onboard gravitational reference sensor and the size of the arm-length. This is because in this region of the accessible frequency band the magnitude of a gravitational wave signal in the interferometer's data scales linearly with the arm-length. At frequencies higher than the inverse of the round-trip light time instead, the sensitivity is primarily determined by the photon-counting statistics [12].
Note that this noise grows linearly with the arm-length through its inverse proportionality to the square-root of the received optical power and because, in this region of the accessible frequency band, the gravitational wave signal no-longer scales with the arm-length. In summary, for a given performance of the onboard science instrument and optical configuration, the best sensitivity level and the corresponding wideness of the bandwidth over which it is reached are uniquely determined by the size of the array.
The best sensitivity level and the corresponding bandwidth over which it is reached by a space-based interferometer become particularly important when considering signals sweeping up-wards in frequency such as those emitted by coalescing binary systems containing blackholes. As recently pointed out by Sesana [13], it is theoretically expected that a very large ensemble of coalescing binary systems, with masses comparable to those of GW150914, will have characteristic wave's amplitudes that could be observable by LISA over an accessible frequency region from about 1.5 × 10 −2 Hz to about 7.6 × 10 −2 Hz. These frequency limits correspond to (i) the assumption of observing a GW150914-like signal for a period of five years (approximately equal to its coalescing time), and (ii) the value at which the signal's amplitude is equal to the LISA sensitivity. Although one could in principle increase the size of the LISA's optical telescopes and rely on more powerful lasers so as to increase the upper frequency cut-off to enlarge the observational bandwidth, in practice pointing accuracy and stability requirements together with the finiteness of the onboard available power would result in a negligible gain.
A natural way to broaden the mHz band, so as to fill the gap between the region accessible by LISA and that by the ground interferometers, is to simultaneously fly an additional interferometer with a smaller arm-length. The geosynchronous Laser Interferometer Space Antenna (gLISA) [12,15,16], which has been analyzed during the past five years by a collaboration of scientists and engineers at the Jet Propulsion Laboratory, Stanford University, the National Institute for Space Research (I.N.P.E., Brazil), and Space Systems Loral, would allow this [17]. With an arm-length of about 7.3 × 10 4 km, it could achieve a shot-noise limited sensitivity in the higher region of its accessible frequency band that is about a factor of 70 better than that of LISA [18]; gLISA will display a minimum of its sensitivity in a frequency region that perfectly complements those of LISA and aLIGO, resulting into an overall accessible GW frequency band equal to (10 −4 − 10 3 ) Hz.
To derive the expression of the joint LISA-gLISA sensitivity, we first note that the noises in the TDI measurements made by the two arrays will be independent. This means that the joint signal-to-noise ratio squared, averaged over an ensemble of signals randomly distributed over the celestial sphere and random polarization states, SN R 2 eL+gL , can be written in the following form where the functions S i h (f ), i = eL , gL correspond to the squared optimal sensitivities (as defined through the A, E, and T modes [11,21]) of the LISA and gLISA arrays respectively.
Note that the factor 4 in front of the integral in Eq.(1) reflects the adopted convention throughout this article of using one-sided power spectral densities of the noises [22,23].
In Fig. 1  In our analysis we will use the following expressions (valid for circular orbits) of the Fourier transform of the amplitude of the gravitational wave signal emitted by such systems, h(f ), and the time, t c , it takes them to coalesce [25] h(f ) = 5 24 In Eqs.(3, 4) G is the gravitational constant, c is the speed of light, z is the cosmological red-shift, D L is the corresponding luminosity distance, M c ≡ (M 1 M 2 ) 3/5 /(M 1 + M 2 ) 1/5 is the chirp mass associated with the binary system whose components have masses M 1 and M 2 , f is the Fourier frequency and f gw the instantaneous frequency of the emitted GW.
By using the above signal amplitude and time to coalescence, and by further assuming an integration time of 5 years (which, in the case of the GW150914 signal, defines the lower-limit of integration in the integral of the SNR to be equal to 0.0151 Hz), we have estimated the SNRs achievable by the two interferometers when operating either as stand-alone or jointly.
We find LISA can observe a GW150914-like signal with a SNR equal to 10.7, while gLISA with an SNR of 14.4 because of its better sensitivity over a larger part of its observable band (see Fig. 1). As expected, the joint LISA-gLISA network further improves upon the SNR of gLISA-alone by reaching a value of about 18.0. From these results we can further infer that gLISA will achieve a sufficiently high SNR to warrant the detection of a GW150914-like signal by integrating for a shorter time. We find that, by integrating for a period of 135 days prior to the moment of coalesce of a GW150914-like system, gLISA can achieve a SNR of 10.7.
The level of SNR achievable by the LISA-gLISA network over a five-year integration is about 80 percent higher than that of LISA alone. This implies, from the estimated parameter precisions derived by Sesana [13] and their dependence on the value of the SNR [14], that the LISA-gLISA network will estimate the parameters associated with the GW source of the observed signal with a precision that is 1.8 better than that obtainable by LISA alone. It should be said, however, that this is a lower bound on the improved precision by which the parameters can be estimated since it is based only on SNR considerations. As pointed out by McWilliams in an unpublished document [27], the Doppler frequency shift together with the larger diurnal amplitude modulations experienced by the GW signal in the gLISA TDI measurements will further improve the precision of the reconstructed parameters beyond that due to only the enhanced SNR. We will analyze and quantify this point in a follow-up article as this is beyond the scope of this letter.
From the expressions of the SNR and of the Fourier amplitude of the GW signal,h(f ) (Eqs. 1, 3), and by fixing the SNR to a specific value for each operational configuration (stand-alone vs. network), it is possible to infer the corresponding average luminosity distance to a BHB in terms of its chirp mass parameter [26]. In Fig. 2  In addition to derive the luminosity distance as a function of the chirp-mass (with a chosen value of the SNR), we have also calculated the SNR as a function of the chirp-mass by assuming the SNR of the LISA mission to be constant and equal to 10 (see Fig. 3).
This of course implies that the luminosity distance of the system is uniquely determined by the chirp-mass through the assumed SNR value of 10 for LISA. Since we are focusing more than four times larger than the luminosity distance to the Virgo cluster of galaxies. This letter has shown the scientific advantages of simultaneously flying with the LISA mission an additional, smaller size, space-based interferometer such as gLISA. Because of its smaller arm-length, gLISA will be able to survey the region of the GW frequency band that is in between those accessible by LISA and aLIGO. By covering the entire mHz and kHz GW frequency band, LISA, gLISA and aLIGO will detect all known sources emitting in this broad frequency region, and observe signals requiring multi-band detection for understanding the physical nature of their sources.