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The space-based gravitational-wave observatory LISA relies on a form of synthetic interferometry (time-delay interferometry, or TDI) where the otherwise overwhelming laser phase noise is canceled by linear combinations of appropriately delayed phase measurements. These observables grow in length and complexity as the realistic features of the LISA orbits are taken into account. In this paper we outline an implicit formulation of TDI where we write the LISA likelihood directly in terms of the basic phase measurements, and we marginalize over the laser phase noises in the limit of infinite laser-noise variance. Equivalently, we rely on TDI observables that are defined numerically (rather than algebraically) from a discrete-filter representation of the laser propagation delays. Our method generalizes to any time dependence of the armlengths; it simplifies the modeling of gravitational-wave signals; and it allows a straightforward treatment of data gaps and missing measurements.
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Space-based gravitational wave detectors cannot keep rigid structures and precise arm length equality, so the precise equality of detector arms which is required in a ground-based interferometer to cancel the overwhelming laser noise is impossible. The time-delay interferometry method is applied to unequal arm lengths to cancel the laser frequency noise. We give analytical formulas of the averaged response functions for tensor, vector, breathing and longitudinal polarizations in different TDI combinations, and obtain their asymptotic behaviors. At low frequencies, $fll f_*$, the averaged response functions of all TDI combinations increase as $f^2$ for all six polarizations. The one exception is the averaged response functions of $zeta$ for all six polarizations increase as $f^4$ in the equilateral-triangle case. At high frequencies, $fgg f_*$, the averaged response functions of all TDI combinations for the tensor and breathing modes fall off as $1/f^2$, the averaged response functions of all TDI combinations for the vector mode fall off as $ln(f)/f^2$ , and the averaged response functions of all TDI combinations for the longitudinal mode fall as $1/f$. We also give LISA and TianQin sensitivity curves in different TDI combinations for tensor, vector, breathing and longitudinal polarizations.
The future space-based gravitational wave observatory LISA will consist of a constellation of three spacecraft in a triangular constellation, connected by laser interferometers with 2.5 million-kilometer arms. Among other challenges, the success of the mission strongly depends on the quality of the cancellation of laser frequency noise, whose power lies eight orders of magnitude above the gravitational signal. The standard technique to perform noise removal is time-delay interferometry (TDI). TDI constructs linear combinations of delayed phasemeter measurements tailored to cancel laser noise terms. Previous work has demonstrated the relationship between TDI and principal component analysis (PCA). We build on this idea to develop an extension of TDI based on a model likelihood that directly depends on the phasemeter measurements. Assuming stationary Gaussian noise, we decompose the measurement covariance using PCA in the frequency domain. We obtain a comprehensive and compact framework that we call PCI for principal component interferometry, and show that it provides an optimal description of the LISA data analysis problem.
We introduce a generic algorithm to determine the time delays and spacecraft (S/C) positions to compose any time-delay interferometry (TDI) channel in the dynamical case and evaluate its sensitivity by using a full numerical method. We select 11 second-generation TDI channels constructed from four approaches and investigate their gravitational wave responses, noise levels, and averaged sensitivities under a numerical LISA orbit. The sensitivities of selected channels are various especially for frequencies lower than 20 mHz. The optimal channel A$_2$ (or equivalently E$_2$) combined from second-generation Michelson TDI channels (X$_1$, X$_2$, and X$_3$) achieves the best sensitivity among the channels, while the Sagnac $alpha_1$ channel shows the worse sensitivity. Multiple channels show better sensitivities at some characteristic frequencies compared to the fiducial X$_1$ channel. The joint $mathrm{A_2+E_2+T_2}$ observation not only enhances the sensitivity of the X$_1$ channel by a factor of $sqrt{2}$ to 2 but also improves the capacity of sky coverage.
The ongoing development of the space-based laser interferometer missions is aiming at unprecedented gravitational wave detections in the millihertz frequency band. The spaceborne nature of the experimental setups leads to a degree of subtlety regarding the otherwise overwhelming laser frequency noise. The cancellation of the latter is accomplished through the time-delay interferometry technique. Moreover, to eventually achieve the desired noise level, the phase fluctuations of the onboard ultra-stable oscillator must also be suppressed. This can be fulfilled by introducing sideband signals which, in turn, give rise to an improved cancellation scheme accounting for the clock-jitter noise. Nonetheless, for certain Sagnac-type interferometry layouts, it can be shown that resultant residual clock noise found in the literature can be further improved. In this regard, we propose refined cancellation combinations for two specific clock noise patterns. This is achieved by employing the so-called geometric time-delay interferometry interpretation. It is shown that for specific Sagnac combinations, the residual noise diminishes significantly to attain the experimentally acceptable sensitivity level. Moreover, we argue that the derived combination, in addition to the existing ones in the literature, furnishes a general-purpose cancellation scheme that serves for arbitrary time-delay interferometry combinations. The subsequential residual noise will only involve factors proportional to the commutators between the delay operators. Our arguments reside in the form of the clock noise expressed in terms of the coefficients of the generating set of the first module of syzygies, the linear combination of which originally constitutes the very solution for laser noise reduction.
ASTROD-GW (ASTROD [Astrodynamical Space Test of Relativity using Optical Devices] optimized for Gravitation Wave detection) is an optimization of ASTROD to focus on the goal of detection of gravitation waves. The detection sensitivity is shifted 52 times toward larger wavelength compared to that of LISA. The mission orbits of the 3 spacecraft forming a nearly equilateral triangular array are chosen to be near the Sun-Earth Lagrange points L3, L4 and L5. The 3 spacecraft range interferometrically with one another with arm length about 260 million kilometers. In order to attain the requisite sensitivity for ASTROD-GW, laser frequency noise must be suppressed below the secondary noises such as the optical path noise, acceleration noise etc. For suppressing laser frequency noise, we need to use time delay interferometry (TDI) to match the two different optical paths (times of travel). Since planets and other solar-system bodies perturb the orbits of ASTROD-GW spacecraft and affect the (TDI), we simulate the time delay numerically using CGC 2.7 ephemeris framework. To conform to the ASTROD-GW planning, we work out a set of 20-year optimized mission orbits of ASTROD-GW spacecraft starting at June 21, 2028, and calculate the residual optical path differences in the first and second generation TDI for one-detector case. In our optimized mission orbits for 20 years, changes of arm length are less than 0.0003 AU; the relative Doppler velocities are less than 3m/s. All the second generation TDI for one-detector case satisfies the ASTROD-GW requirement.
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