Cancellation of laser frequency noise in interferometers is crucial for attaining the requisite sensitivity of the triangular 3-spacecraft LISA configuration. Raw laser noise is several orders of magnitude above the other noises and thus it is essential to bring it down to the level of other noises such as shot, acceleration, etc. Since it is impossible to maintain equal distances between spacecrafts, laser noise cancellation must be achieved by appropriately combining the six beams with appropriate time-delays. It has been shown in several recent papers that such combinations are possible. In this paper, we present a rigorous and systematic formalism based on algebraic geometrical methods involving computational commutative algebra, which generates in principle {\it all} the data combinations cancelling the laser frequency noise. The relevant data combinations form the first module of syzygies, as it is called in the literature of algebraic geometry. The module is over a polynomial ring in three variables, the three variables corresponding to the three time-delays around the LISA triangle. Specifically, we list several sets of generators for the module whose linear combinations with polynomial coefficients generate the entire module. We find that this formalism can also be extended in a straight forward way to cancel Doppler shifts due to optical bench motions. The two modules are infact isomorphic. We use our formalism to obtain the transfer functions for the six beams and for the generators. We specifically investigate monochromatic gravitational wave sources in the LISA band and carry out the maximisiation over linear combinations of the generators of the signal-to-noise ratios with the frequency and source direction angles as parameters.Comment: 27 Pages, 6 figure

The joint NASA–ESA mission, LISA, relies crucially on the stability of the three-spacecraft constellation. Each of the spacecraft is in heliocentric orbit forming a stable triangle. In this paper we explicitly show with the help of the Clohessy–Wiltshire equations that any configuration of spacecraft lying in the planes making angles of ±60° with the ecliptic and given suitable initial velocities within the plane, can be made stable in the sense that the inter-spacecraft distances remain constant to first order in the dimensions of the configuration compared with the distance to the Sun. Such analysis would be useful in order to carry out theoretical studies on the optical links, simulators, etc.

The joint ESA–NASA mission LISA relies crucially on the stability of that three spacecraft constellation. All three spacecraft are on heliocentric and weakly eccentric orbits forming a nearly stable triangle. It has been shown that for certain spacecraft orbits, the arms keep constant distances to the first order in eccenticities. However, exact orbitography exhibits the so-called ‘breathing modes’ resulting in slow variations of the armlengths on the timescale of one year. In this paper, we analyse the breathing modes (flexing of the arms) with the help of the geodesic deviation equation up to the octupole order, which is shown to be equivalent to higher order Clohessy–Wiltshire equations. We analytically show that the flexing of the arms can be reduced to a peak-to-peak variation of about 50 000 km, and the corresponding peak-to-peak variation in the Doppler laser frequency shift to about 8 m s−1. This is achieved by slightly changing the well-known tilt of 60°. We further show that it is the minimum within the assumption of equivalent spacecraft orbits, where the orbit of each spacecraft is rotated by 120° from the preceding one.

We present results from the first directed search for nontensorial gravitational waves. While general relativity allows for tensorial (plus and cross) modes only, a generic metric theory may, in principle, predict waves with up to six different polarizations. This analysis is sensitive to continuous signals of scalar, vector, or tensor polarizations, and does not rely on any specific theory of gravity. After searching data from the first observation run of the advanced LIGO detectors for signals at twice the rotational frequency of 200 known pulsars, we find no evidence of gravitational waves of any polarization. We report the first upper limits for scalar and vector strains, finding values comparable in magnitude to previously published limits for tensor strain. Our results may be translated into constraints on specific alternative theories of gravity.

LISA is a joint space mission of the NASA and the ESA for detecting lowfrequency gravitational waves in the band 10 −5 to 1 Hz. In order to attain the requisite sensitivity for LISA, the laser frequency noise must be suppressed below the other secondary noises such as the optical path noise, acceleration noise, etc. This is achieved by the technique called time delay interferometry (TDI) in which the data are combined with appropriate time delays. In this paper we approximately compute the spacecraft orbits in the gravitational field of the Sun and Earth. We have written a numerical code which computes the optical links (time delays) in the general relativistic framework within an accuracy of ∼10 m, which is sufficient for TDI. Our computation of the optical links automatically takes into account the effects such as the Sagnac, Shapiro delay, etc. We show that by optimizing LISA orbits, and using the symmetries inherent in the configuration of LISA and in the physics, the residual laser noise in the modified first-generation TDI can be adequately suppressed. We demonstrate our results for some important TDI observables.

Abstract. It has been shown in several recent papers that the six Doppler data streams obtained from a triangular LISA configuration can be combined by appropriately delaying the data streams for cancelling the laser frequency noise. Raw laser noise is several orders of magnitude above the other noises and thus it is essential to bring it down to the level of other noises such as shot, acceleration, etc. A rigorous and systematic formalism using the powerful techniques of computational commutative algebra was developed which generates in principle all the data combinations cancelling the laser frequency noise. The relevant data combinations form a first module of syzygies.In this paper we use this formalism to advantage for optimising the sensitivity of LISA by analysing the noise and signal covariance matrices. The signal covariance matrix is calculated for binaries whose frequency changes at most adiabatically and the signal is averaged over polarisations and directions. We then present the extremal SNR curves for all the data combinations in the module. They correspond to the eigenvectors of the noise and signal covariance matrices. A LISA 'network' SNR is also computed by combining the outputs of the eigenvectors. We show that substantial gains in sensitivity can be obtained by employing these strategies. The maximum SNR curve can yield an improvement upto 70 % over the Michelson, mainly at high frequencies, while the improvement using the network SNR ranges from 40 % to over 100 %.Finally, we describe a simple toy model, in which LISA rotates in a plane. In this analysis, we estimate the improvement in the LISA sensitivity, if one switches from one data combination to another as it rotates. Here the improvement in sensitivity, if one switches optimally over three cyclic data combinations of the eigenvector is about 55 % on an average over the LISA band-width. The corresponding SNR improvement increases to 60 %, if one maximises over the module.

It was shown in a previous work that the data combinations canceling laser frequency noise constitute a module -the module of syzygies. The cancellation of laser frequency noise is crucial for obtaining the requisite sensitivity for LISA. In this work we show how the sensitivity of LISA can be optimised for a monochromatic source -a compact binary -whose direction is known, by using appropriate data combinations in the module. A stationary source in the barycentric frame appears to move in the LISA frame and our strategy consists of coherently tracking the source by appropriately switching the data combinations so that they remain optimal at all times. Assuming that the polarisation of the source is not known, we average the signal over the polarisations. We find that the best statistic is the 'network' statistic, in which case LISA can be construed of as two independent detectors. We compare our results with the Michelson combination, which has been used for obtaining the standard sensitivity curve for LISA, and with the observable obtained by optimally switching the three Michelson combinations. We find that for sources lying in the ecliptic plane the improvement in SNR increases from 34% at low frequencies to nearly 90% at around 20 mHz. Finally we present the signal-to-noise ratios for some known binaries in our galaxy. We also show that, if at low frequencies SNRs of both polarisations can be measured, the inclination angle of the plane of the orbit of the binary can be estimated.

In order to attain the requisite sensitivity for LISA (Laser Interferometric Space Antenna)—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 not all the optical links of LISA will be operating at all times, and therefore, we here consider the case of LISA operating with two arms only. As shown earlier in the literature, obtaining even approximate solutions of TDI to the general problem is very difficult. Since here only four optical links are relevant, the algebraic problem simplifies considerably. We are then able to exhibit a large number of solutions (from a mathematical point of view an infinite number) and further present an algorithm to generate these solutions.