Probing soft signals of gravitational-wave memory with space-based interferometers

The gist Gravitational-wave displacement memory is a remarkable and ubiquitous phenomenon predicted by general relativity, which has not yet been detected, and this paper investigates its detection prospects with future space-based laser interferometers.

How it works

Displacement memory is associated with soft gravitons, making it the only observable signal of its parent event at sufficiently low frequencies. This memory can serve as an important test of general relativity. It may have an “ordinary” or “null” origin, and is associated with the supertranslation charge balance laws of asymptotically flat spacetimes as well as with Weinberg’s soft graviton theorem. Displacement memory can therefore serve as an important test of general relativity.

The paper introduces three types of soft waveforms corresponding respectively to the displacement memory, velocity memory and integrated-displacement memory. These are defined based on the asymptotic difference in a waveform observable in the time domain as a soft waveform. The first type is displacement memory, which refers to a DC (direct current) shift of h(t) between the asymptotic past and future, i.e., ∆h ≡ h(∞) − h(−∞) ≠ 0.

Waveform Analysis

The soft waveform of displacement memory depends solely on the amplitude ∆h of the memory jump. The models considered include the arctan model, the tanh model, and a linear growth in t. In all three models above, 1/f∗ measures the time scale of the jump. In the limit f∗ → ∞, the waveform approaches a step function at t = t∗ with the step size ∆h, corresponding to h˜∞(f) = [i∆h/(2πf) + δ(f)/2]e2πif t∗.

Velocity memory is a DC shift of h˙(t): ∆h˙ ≡ h˙(∞) − h˙(−∞) ≠ 0. The soft waveform is given by h˜∞(f) → -h˜˙ ∞(f)/2e2πif t∗. Integrated-displacement memory is defined by an asymptotic change in the observable O(t) ≡ R t−∞ dt′ h(t′), and has the zero-frequency limit: h˜∞(f) → O e2πif t∗.

Realistic Examples and Corrections

The paper examines realistic examples, such as the infrared spectral features of gravitational waves from moderately relativistic compact binary scattering and nearly equal-mass quasi-circular, non-precessing black hole mergers. The results of simulated Bayesian parameter estimation demonstrate that independent measurement of a soft displacement-memory signal with a single LISA-like detector is achievable at signal-to-noise ratios ≳ 10.

The soft waveform is an idealized zero-frequency limit, and the exact correction factor to be C(f) is defined as C(f) ≡ [C(−f)]∗. For all three models in Sec. II A, C(f) is analytic at f = 0.

Detector Response and Detectability

The paper investigates the detector responses and the parameter estimation precision for both a single LISA-like detector and the proposed LISA-Taiji network. In optimal configurations, the signal-to-noise ratio in LISA, Taiji and TianQin can reach 10 for a displacement memory of amplitude ∼ 10−20, or a velocity memory of amplitude ∼ 10−22 Hz.

The joint observations by LISA and Taiji can achieve significantly more precise measurements. Soft memory signals in BBO (occurring at f ≲ 1 Hz) appear to suffer from less degeneracy and can achieve much higher sensitivity, with the signal-to-noise ratio reaching 10 for a displacement memory of amplitude ∼ 10−24, or a velocity memory of amplitude ∼ 10−23 Hz.

Stochastic Backgrounds and Constraints

The paper simulates an idealized background of soft displacement-memory signals, and evaluates its detectability at the considered detectors. The stochastic background of soft displacement-memory signals and its detectability in space-based detectors is discussed in Sec. VI. A constraint on the source can be derived from ∆h, Q ≡ Z domega2 ∆Q = X l≥2 X m (∆Q)l,m2 ≥ ∆h2 I(Θ, Φ).

Parameter Estimation Simulation

Once a soft memory signal is identified through matched filtering, further measurements through Bayesian parameter estimation can be performed. The root-mean-squared error of ξi given by σi = √Σii. Joint observations with LISA and Taiji can again greatly improve parameter estimation.

The posterior distributions of log10 H and Ψ for a compact binary scattering event measured at Taiji are shown in Fig. 19. The optimal SNR is 16.

Conclusion

Soft signals from bursts with memory can be promising targets for space-based gravitational-wave detection. The results indicate that soft signals from bursts with memory can be promising targets for space-based gravitational-wave detectors. The possible soft memory signals associated with a GW event observed by a ground-based detector can also be searched for with space-based detectors.

How it works

The paper examines the prospects for detecting and measuring the soft memory signals with future space-based interferometers, including Laser Interferometer Space Antenna (LISA) [63, 64], Taiji [65], TianQin [65], and Big Bang Observer (BBO) Stage I [66].

The TDI response is further simplified in the static-detector approximation (SEA), in which we neglect the motion of SC and use Lrs ≈ L = const. The SNR of T channel is significantly larger than that in the SEA, which, however, is still much smaller than in the A and E channels.

The angular dependence of the SNR exhibits the following symmetry: ρA+E(ϕ, θ, ψ) = ρA+E(ϕ + π, θ, ψ) = ρA+E(θ, ψ) ≈ ρA+E(π − θ, ψ). The result is roughly the same for all considered detectors and is shown for Taiji in Fig. 13.

The SNR is maximized at θ ∈ 0, π, where it is independent of ψ. The SNR is minimized at θ = π/2, ψ ∈ π/4, 3π/4, where it is approximately 1.6% of the maximum SNR. For a given θ, the SNR is maximized at ψ ∈ 0, π/2, π, and minimized at ψ ∈ π/4, 3π/4.

The values of H10 and H˙10 of the same order of magnitude for LISA, Taiji and TianQin are found. However, BBO can achieve significantly higher sensitivity to the soft displacement-memory signal, provided it occurs at f ≲ 1 Hz.

The low-frequency part of the spectrum is more important for the velocity memory. The situation is reversed in the case of integrated-displacement memory, as shown in Fig. 12.

The paper concludes that soft signals from bursts with memory can be promising targets for space-based gravitational-wave detection. The possible soft memory signals associated with a GW event observed by a ground-based detector can also be searched for with space-based detectors.

1. Improved Bayesian parameter estimation precision for displacement memory signals by utilizing joint observations of space-based interferometers like LISA and Taiji, as demonstrated in Section II B: The measurement precision can be significantly improved by joint observations with a LISA-Taiji network. The improved AI system can perform more accurate parameter estimation, achieving higher signal-to-noise ratios for detecting memory signals.

2. Enhanced template matching capability for complex astrophysical sources by employing corrected soft waveforms, as discussed in Section III B: "We examine the correction to the soft waveform in two realistic examples: hyperbolic encounter of compact binary and nearly equal-mass quasi-circular binary black hole (BBH) merger, and assess its impact on displacement memory measurements using the uncorrected template." This allows the AI system to better constrain parameters for complex scattering events.

3. Optimized detection strategies for different memory types by leveraging detector-specific sensitivities, as shown in Section IV A: The results are shown in Fig. 10 and Fig. 12, which indicate that the low-frequency part of the spectrum is more important for the velocity memory. The improved AI system can dynamically select the optimal TDI channel or detector configuration based on whether it is searching for displacement or velocity memory signals.

4. Robust stochastic background detection by simulating an ensemble of independent soft displacement-memory signals, as described in Section VI: We consider an ensemble of mutually-independent soft displacement-memory signals with a uniform distribution of these parameters, leading to a predicted strain power spectral density Sh proportional to f-2." The AI system can effectively search for the stochastic gravitational wave background by analyzing the PSD derived from this ensemble.

5. Distinguishability assessment of waveform corrections by estimating mismatch metrics, as detailed in Section IV D: The difference between two waveforms h1 and h2 is reflected in their mismatch, defined as M = 1 − p. The improved AI system can reliably distinguish between linear and quadratic corrections to the soft waveform based on the estimated SNR.

6. Enhanced parameter estimation accuracy through Fisher Information Matrix (FIM) analysis, as presented in Section V: In the limit of high SNR, the measurement precision can be estimated using the Fisher information matrix (FIM). The AI system can use this FIM to estimate the root-mean-squared error of parameters and determine when a signal is reliably constrained.

7. Improved source localization accuracy by considering angular resolution derived from FIM estimates, as shown in Section V: The angular resolution of the source sky location, given by the solid-angle error corresponding to the 1σ error ellipse, can be estimated as σomega. The AI system can provide more precise sky location estimates for detected memory events.

Gravitational-wave displacement memory is a remarkable, ubiquitous prediction of general relativity that has not yet been detected. Unlike the oscillatory components of gravitational waveforms, displacement memory is associated with soft gravitons and provides the leading low-frequency observable of its parent event. Analogous soft signals may also be associated with velocity and integrated-displacement memory. The simple, universal spectral shapes of soft waveforms provide effective templates for matched filtering and parameter estimation. In this paper, we investigate the detection prospects for such soft memory signals with future space-based laser interferometers. As realistic examples, we examine the infrared spectral features of gravitational waves from moderately relativistic nonspinning compact binary scattering and comparable-mass quasi-circular, nonprecessing black hole mergers. In both cases, the low-frequency spectrum can be described by a soft displacement-memory waveform with finite-frequency corrections. Using the adopted instrumental-noise models, simulated Bayesian parameter-estimation analyses show that a single LISA-like detector can constrain the memory amplitude and arrival time for signals with matched-filter signal-to-noise ratios of order 10. BBO Stage 1 could measure null displacement memory from sufficiently loud stellar-mass compact binary mergers. We also evaluate the detectability of an idealized stochastic background of soft displacement-memory signals. These results establish gravitational-wave bursts with memory as concrete targets for space-based interferometers.