Space-based gravitational wave (GW) detectors are designed for wave sources in the millihertz band with different locations and orientations.
Time-delay interferometry (TDI) technique is an indispensable ingredient in space-borne GW detection that effectively suppresses the laser phase noise.
The abundant TDI solutions derived in the literature also feature distinct angular-dependent sensitivities.
Because a GW source's angular location is unknown prior to the signals' detection, a solid-angle average is often performed when analyzing the sensitivity function of a given TDI combination.
The present study explores the angular dependence of the detector's sensitivity.
This detail is relevant, because once the initial detection is achieved, the source's location can be extracted and used to provide information on a refined TDI combination tailored for the specific GW source.
As the TDI technique is a post-processing algorithm, such a procedure can be implemented in practice.
We evaluate the angular dependence of the detector's response function to the GW signals for different TDI combinations as a function of the orientation angles.
Moreover, we classify the response functions into seven categories at the low-frequency limit, leveraging the characteristics of the underlying geometrical TDI combinations.
By further averaging out the azimuthal angle $ϕ_D$ in the detector's plane, the main features of the resulting response functions and their zenithal dependence with respect to the GW source are scrutinized.
The findings presented in this work provide pertinent insights for ongoing space-borne detector programs.
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…
With the purpose of understanding how time delay interferometry (TDI) combinations can best be used for the characterisation of LISA instrumental noise, we revisit their laser frequency noise cancellation properties. We have developed an algorithm to search for all possible combinations that suppress noise at the same level as the X, Y and Z classical combination. The algorithm calculates delays using symbolic formulas that explicitly include velocities and accelerations of satellites up to the relevant order. In addition, once a combination has been identified, delays are verified by solving …
With a laser interferometric gravitational-wave detector in separate free flying spacecraft, the only way to achieve detection is to mitigate the dominant noise arising from the frequency fluctuations of the lasers via postprocessing. The noise can be effectively filtered out on the ground through a specific technique called time-delay interferometry (TDI), which relies on the measurements of time-delays between spacecraft and careful modeling of how laser noise enters the interferometric data. Recently, this technique has been recast into a matrix-based formalism by several authors, offering …
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 t…
We report on the first demonstration of time-delay interferometry (TDI) for LISA, the Laser Interferometer Space Antenna. TDI was implemented in a laboratory experiment designed to mimic the noise couplings that will occur in LISA. TDI suppressed laser frequency noise by approximately 10^9 and clock phase noise by 6x10^4, recovering the intrinsic displacement noise floor of our laboratory test bed. This removal of laser frequency noise and clock phase noise in post-processing marks the first experimental validation of the LISA measurement scheme.
Time-Delay Interferometry (TDI) is the data processing technique that cancels the large laser phase fluctuations affecting the one-way Doppler measurements made by unequal-arm space-based gravitational wave interferometers. By taking finite linear combinations of properly time-shifted Doppler measurements, laser phase fluctuations are removed at any time t and gravitational wave signals can be studied at a requisite level of sensitivity. In the past, other approaches to this problem have been proposed. Recently, matrix based approaches have been put forward; two such approaches are by Vallisne…
The LISA mission will likely be a signal dominated detector, such that one challenge is the separation of the different astrophysical sources, and to distinguish between them and the instrumental noise. One of the goals of LISA is to probe the early Universe by detecting stochastic GW backgrounds. As correlation with other detectors is not possible for LISA, discrimination of such a GW background from the instrumental noise requires a good estimate of the latter. To this purpose we have revisited Time Delay Interferometry (TDI) to look for new TDI signal combinations that fulfill the laser fre…
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 whi…
Time-delay interferometry (TDI) is a data processing technique for LISA designed to suppress the otherwise overwhelming laser noise by several orders of magnitude. It is widely believed that TDI can only be applied once all phase or frequency measurements from each spacecraft have been synchronized to a common time frame. We demonstrate analytically, using as an example the commonly-used Michelson combination X, that TDI can be computed using the raw, unsynchronized data, thereby avoiding the need for an initial synchronization processing step and significantly simplifying the initial noise re…
The Laser Interferometer Space Antenna is a joint ESA-NASA space-mission to detect and study mHz cosmic gravitational waves. The trajectories followed by its three spacecraft result in unequal- and time-varying arms, requiring use of the Time-Delay Interferometry (TDI) post- processing technique to cancel the laser phase noises affecting the heterodyne one-way Doppler measurements. Although the second-generation formulation of TDI cancels the laser phase noises when the array is both rotating and "flexing", second-generation TDI combinations for which the phase fluctuations of the onboard ultr…
We generalize the combinatorial algebraic approach first proposed by Dhurandhar et al. to construct various classes of modified second-generation time-delay interferometry (TDI) solutions. The main idea behind the algorithm is to enumerate, in a given order, a specific type of commutator between two monomials defined by the products of particular time-displacement operators. On the one hand, the above commutators can be systematically rewritten as the elements of a left ideal, defined by the l.h.s. of the relevant equation for the TDI solution. On the other hand, these commutators are shown to…
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, $f\ll f_*$, the averaged resp…
Time-delay interferometry (TDI) is a processing step essential for the scientific exploitation of LISA, as it reduces the otherwise overwhelming laser noise in the interferometric measurements. The fundamental idea is to define new laser-noise-free observables by combining appropriately time-shifted measurements. First- and second-generation TDI combinations cancel laser noise under the assumption that the LISA armlengths are constant or evolve linearly with time, respectively. We recently extended TDI by solving for the laser-noise-free combinations implicitly, writing the likelihood of the d…
In order to attain the requisite sensitivity for LISA - a joint space mission of the ESA and NASA- the laser frequency noise must be suppressed below the secondary noises such as the optical path noise, acceleration noise etc. By combining six appropriately time-delayed data streams containing fractional Doppler shifts - a technique called time delay interferometry (TDI) - the laser frequency noise may be adequately suppressed. We consider the general model of LISA where the armlengths vary with time, so that second generation TDI are relevant. However, we must envisage the possibility, that n…
Time delay interferometry (TDI) is a post-processing technique used in the Laser Interferometer Space Antenna (LISA) to reduce laser frequency noise by building an equal-arm interferometer via combining time-shifted raw phase measurements. The set of so-called 2nd generation TDI variables which sufficiently suppress laser frequency noise considering realistic LISA orbital dynamics has recently been expanded by a large number of additional solutions. In this paper, we characterize these new TDI channels by relating them to the well-known 1st generation variables $α$, $β$, $γ$, and $ζ$. We compu…
Time-delay interferometry (TDI) is a post-processing technique used to reduce laser noise in heterodyne interferometric measurements with unequal armlengths, a situation characteristic of space gravitational detectors such as Laser Interferometer Space Antenna (LISA). This technique consists in properly time-shifting and linearly combining the interferometric measurements in order to reduce the laser noise by several orders of magnitude and to detect gravitational waves. In this communication, we show that the Doppler shift due to the time evolution of the armlengths leads to an unacceptably l…
Time delay interferometry with minimal null frequencies and shortened time span
Institute of Fundamental Physics and Quantum Technology, Ningbo University（宁波大学基础物理与量子技术研究院）；Department of Physics, School of Physical Science and Technology, Ningbo University（宁波大学物理科学与技术学院物理系）；Shanghai Astronomical Observatory, Chinese Academy of Sciences（中国科学院上海天文台）
Laser frequency noise (LFN) is the dominant source of noise expected in the Laser Interferometer Space Antenna (LISA) mission, at $\sim$7 orders of magnitude greater than the typical signal expected from gravitational waves (GWs). Time-delay interferometry (TDI) suppresses LFN to an acceptable level by linearly combining measurements from individual spacecraft delayed by durations that correspond to their relative separations. Knowledge of the delay durations is crucial for TDI effectiveness. The work reported here extends upon previous studies using data-driven methods for inferring the delay…
eLISA/NGO is a new gravitational wave detection proposal with arm length of 10^6 km and one interferometer down-scaled from LISA. Just like LISA and ASTROD-GW, in order to attain the requisite sensitivity for eLISA/NGO, laser frequency noise must be suppressed to below the secondary noises such as the optical path noise, acceleration noise etc. In previous papers, we have performed the numerical simulation of the time delay interferometry (TDI) for LISA and ASTROD-GW with one arm dysfunctional by using the CGC 2.7 ephemeris. The results are well below their respective limits which the laser fr…
The proposed space-borne laser interferometric gravitational wave (GW) observatory TianQin adopts a geocentric orbit for its nearly equilateral triangular constellation formed by three identical drag-free satellites. The geocentric distance of each satellite is $\approx 1.0 \times 10^{5} ~\mathrm{km}$, which makes the armlengths of the interferometer be $\approx 1.73 \times 10^{5} ~\mathrm{km}$. It is aimed to detect the GWs in $0.1 ~\mathrm{mHz}-1 ~\mathrm{Hz}$. For space-borne detectors, the armlengths are unequal and change continuously which results in that the laser frequency noise is nea…
Time-Delay Interferometry (TDI) is the data processing technique that cancels the large laser phase fluctuations affecting the one-way Doppler measurements made by unequal-arm space-based gravitational wave interferometers. By taking finite linear combinations of properly time-shifted Doppler measurements, laser phase fluctuations are removed at any time $t$ and gravitational wave signals can be studied at a requisite level of precision. In this article we show the delay operators used in TDI can be represented as matrices acting on arrays associated with the laser noises and Doppler measureme…
Heterodyne laser phase measurements in a space-based gravitational wave interferometer are degraded by the phase fluctuations of the onboard clocks, resulting in unacceptable sensitivity performance levels of the interferometric data. In order to calibrate out the clock phase noises it has been previously suggested that additional inter-spacecraft phase measurements must be performed by modulating the laser beams. This technique, however, considerably increases system complexity and probability of subsystem failure. With the advent of self-referenced optical frequency combs, it is possible to …
In order to attain the requisite sensitivity for LISA, laser frequency noise must be suppressed below the secondary noises such as the optical path noise, acceleration noise etc. In a previous paper (Dhurandhar et al., Class. Quantum Grav., 27, 135013, 2010), we have found a large family of second-generation analytic solutions of time delay interferometry with one arm dysfunctional, and we also estimated the laser noise due to residual time-delay semi-analytically from orbit perturbations due to Earth. Since other planets and solar-system bodies also perturb the orbits of LISA spacecraft and a…
The success of LISA Pathfinder in demonstrating the LISA drag-free requirement paved the road of using space missions for detecting low-frequency and middle-frequency GWs. The new LISA GW mission proposes to use arm length of 2.5 Gm (1 Gm = 106 km). The TAIJI GW mission proposes to use arm length of 3 Gm. In order to attain the requisite sensitivity, laser frequency noise must be suppressed to below the secondary noises such as the optical path noise, acceleration noise etc. In previous papers, we have performed the numerical simulation of the time delay interferometry (TDI) for original LISA,…
ASTROD-GW (ASTROD [Astrodynamical Space Test of Relativity using Optical Devices] optimized for Gravitational Wave detection) is a gravitational-wave mission with the aim of detecting gravitational waves from massive black holes, extreme mass ratio inspirals (EMRIs) and galactic compact binaries, together with testing relativistic gravity and probing dark energy and cosmology. Mission orbits of the 3 spacecrafts forming a nearly equilateral triangular array are chosen to be near the Sun-Earth Lagrange points L3, L4 and L5. The 3 spacecrafts range interferometrically with one another with arm l…
LISA will detect gravitational waves (GWs) in the milli-Hz frequency band in space. Time-delay interferometry (TDI) is developed to suppress laser frequency noise beneath the acceleration noise and optical metrology noise. To identify stochastic GW signals, it would be required to characterize these noise components entangled in TDI data streams. In this work, we investigate noises characterization by combining the first-generation TDI channels from Michelson and Relay configurations. The Michelson channels are helpful to characterize acceleration noises in the lower frequency band, and the Re…
Alternative approach to time-delay interferometry with optical frequency comb
University of Maryland, Baltimore County（马里兰大学巴尔的摩县分校）；NASA Goddard Space Flight Center（美国宇航局戈达德太空飞行中心）；Center for Research and Exploration in Space Science and Technology, NASA/GSFC（NASA/戈达德太空飞行中心空间科学与技术研究探索中心）；Massachusetts Institute of Technology（麻省理工学院）；University of Arizona（亚利桑那大学）
TianQin is a proposed geocentric space-based gravitational wave observatory mission, which requires time-delay interferometry (TDI) to cancel laser frequency noise. With high demands for precision, solar-wind plasma environment at $\sim 10^5$ km above the Earth may constitute a non-negligible noise source to laser interferometric measurements between satellites, as charged particles perturb the refractivity along light paths. In this paper, we first assess the plasma noises along single links from space-weather models and numerical orbits, and analyze the time and frequency domain characterist…
The time delay interferometry (TDI) is an algorithm proposed to suppress the laser frequency noise in space-borne gravitational wave detectors. As a post-processing technique, it is implemented by constructing a virtual equal arm interferometer through an appropriate combination of the time-shifted data streams. Such an approach is tailored to the intrinsic feature of the space-based gravitational wave detection, namely, the distances between spacecraft are governed by the orbit dynamics and thus can not be held constant. Among different implementations, the geometric TDI was introduced as a m…
