Improving calibration accuracy with torque coupled gravity field calibrator for sub-Hz gravitational wave observation in CHRONOS

Abstract

A fundamental challenge in low-frequency gravitational-wave detectors is the limited signal-to-noise ratio (SNR) of calibration lines, particularly in torsion-bar systems where the response is governed by rotational dynamics. In this work, we resolve this issue by optimizing the geometrical configuration of a torque-coupled gravity field calibrator (GCal), achieving an improvement in calibration-line SNR by more than an order of magnitude compared to conventional layouts. For the Cryogenic sub-Hz cROss torsion-bar detector with quantum NOn-demolition Speed-meter (CHRONOS), the calibration signal appears as a monochromatic line within the $0.1$--$10~\mathrm{Hz}$ band. At $1~\mathrm{Hz}$, the strain-equivalent calibration amplitude reaches $|h_{\rm GCal}| = 1.18 \times 10^{-14}$, corresponding to an SNR density of $|h_{\rm GCal}|/S_h = 4.25 \times 10^{3}$. This demonstrates for the first time that a high-SNR calibration line can be directly injected into the sub-Hz band of a torsion-bar detector. A first-order perturbative error propagation analysis yields a total fractional systematic uncertainty of $δh_{\rm GCal}/h_{\rm GCal} = 0.24\%$, dominated by geometric alignment uncertainties, while contributions from mass uncertainties and the gravitational constant remain subdominant. The corresponding absolute systematic uncertainty is $δh_{\rm GCal} \sim 10^{-17}$ at $1~\mathrm{Hz}$. These results establish torque-coupled gravitational calibration as a practical solution to the longstanding low-SNR problem in sub-Hz torsion-bar detectors and provide a robust pathway toward precision absolute calibration in the low-frequency regime.

Figures (4)

• Figure 1: Conceptual comparison between conventional force-coupled GCal and the torque-coupled configuration proposed in this work. (Left) In conventional implementations used in kilometer-scale laser interferometers, the GCal produces a time-varying gravitational force that induces a small translational motion of the test mass. The calibration signal is therefore coupled through displacement. (Right) In the torque-coupled configuration adopted for CHRONOS, a rotating quadrupole rotor is placed directly beneath the torsion-bar test mass, allowing the gravitational interaction to couple directly to the rotational degree of freedom. The resulting excitation produces a deterministic gravitational torque, which directly drives the fundamental observable of torsion-bar detectors. This direct torque coupling significantly enhances the calibration signal in the sub-Hz frequency band and constitutes the central concept of this work.
• Figure 2: Schematic illustration of the torque-coupled GCal configuration implemented in CHRONOS. A rotating quadrupole rotor is positioned directly beneath the torsion-bar test mass. The time-varying Newtonian gravitational field generated by the rotating masses produces a periodic torque acting on the torsional degree of freedom of the bar. The quadrupole symmetry suppresses lower-order harmonics and generates a dominant calibration signal at twice the rotation frequency ($2f_{\rm rot}$), enabling a narrow-band and spectrally clean excitation of the detector response suitable for precision calibration in the sub-Hz regime.
• Figure 3: Calculated strain sensitivity of the CHRONOS (red curve) together with the predicted GCal response. The colored curves show the expected GCal-induced differential angular motion for calibration rotors made of tungsten (black solid), SUS304 stainless steel (black dashed), and aluminum alloy A5083 (black dotted), assuming identical rotor geometry. Because the gravitational torque scales linearly with the calibration mass, the response amplitude follows the material density. The comparison illustrates the achievable calibration margin for different practical material choices without modifying the mechanical configuration of the detector.
• Figure 4: SNR density of the GCal-induced calibration signal for the CHRONOS. The curves correspond to calibration rotors made of tungsten (solid), SUS304 stainless steel (dashed), and aluminum alloy A5083 (dotted). Because the GCal excitation is monochromatic, the SNR is determined by the noise spectral density at the excitation frequency. The large SNR observed in the sub-Hz band originates from the combination of line excitation and the $1/\Omega^2$ inertial response of the torsion-bar mode.