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    Algebraic approach to time-delay data analysis for LISA
    (2001-12-20) Dhurandhar, Sanjeev; Nayak, K. R.; Vinet, J-Y.
    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 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.
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    Improving the sensitivity of LISA
    (2002-10-04) Nayak, K. R.; Pai, A.; Dhurandhar, Sanjeev
    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
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    Optimising the directional sensitivity of LISA
    (2011-07-05) Nayak, K. R.; Dhurandhar, Sanjeev; Pai, A.
    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 in
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    Fundamentals of the LISA Stable Flight Formation
    (2011-07-06) Dhurandhar, Sanjeev; Nayak, K. R.; Koshti, S.
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    Reducing the flexing of the arms of LISA
    (2005-07-01) Nayak, K. R.; Koshti, S.; Dhurandhar, Sanjeev; et al.
    The joint NASA-ESA mission LISA relies crucially on the stability of the three spacecraft constellation. All three spacecraft are on heliocentric and weakly eccentric orbits forming a stable triangle. It has been shown that for certain spacecraft orbits, the arms keep constant distances to the first order in the eccentricities. However, exact orbitography shows the so-called ‘breathing modes’ of the arms where the arms slowly change their lengths over the time-scale of a year. In this paper we analyse the breathing modes (the flexing of the arms) with the help of the geodesic deviation equations to octupole order which are shown to be equivalent to higher order Clohessy-Wiltshire equations. We show that the flexing of the arms of LISA as given by the ‘exact’ solution of Keplerian orbits, which gives constant armlengths to the first order in eccentricity and whose maximum flexing amplitude is ∼ 115, 000 km, can be improved, by tilting the plane of the LISA triangle slightly from the proposed orientation of 60◦ with the ecliptic to obtain a maximum flexing amplitude of ∼ 48, 000 km, reducing it by a factor of ∼ 2.4. The reduction factor is even larger if we consider the corresponding Doppler shifts, for which the reduction factor reaches almost a factor of 6. We solve the second order equations and obtain the general solution. We then use the general solution to establish the optimality of the solutions that we have found.
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    Physical and algebraical models of LISA
    (2007-11-27) Dhurandhar, Sanjeev
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    On searches for gravitational waves from mini creation events by laser interferometric detectors
    (2006-01-10) Sarmah, B. P.; Banerjee, S.K.; Dhurandhar, Sanjeev
    As an alternative view to the standard big bang cosmology the quasi-steady state cosmol- ogy(QSSC) argues that the universe was not created in a single great explosion; it neither had a beginning nor will it ever come to an end. The creation of new matter in the universe is a regular feature occurring through nite explosive events. Each creation event is called a mini-bang or, a mini creation event(MCE). Gravitational waves are expected to be generated due to any anisotropy present in this process of creation. Mini creation event ejecting matter in two oppositely directed jets is thus a source of gravitational waves which can in principle be detected by laser interferometric detectors. In the present work we consider the gravitational waveforms propagated by linear jets and then estimate the response of laser interferometric detectors like LIGO and LISA.
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    Time-delay interferometry and the relativistic treatment of LISA optical links
    (2008-08-20) Dhurandhar, Sanjeev
    LISA is a joint spae mission of the ESA and NASA for deteting low frequeny gravitational radiation in the band 10−5 −1 Hz. In order to attain the requisite sensitivity for LISA, the laser frequeny noise must be suppressed below the other seondary noises su h asthe optial path noise, a eleration noise et. This is ahieved beause of the redundan y in the data, more spe i ally, by ombining six appropriately time-delayed data streams ontain ingfrational Doppler shifts - time delay interferometry (TDI). The orbits of the spa eraft are omputed in the gravitational eld of the Sun and Earth in the Newtonian framework, while the optial links are treated fully general relativisti ally and thus, e ets suh as the Sagna, Shapiro delay, et. are automatially in orporated. We show that in the model of LISA that we onsider here, there are symmetries inherent in the physis, whih may be used e e tively to suppress the residual laser frequen y noise and simplify the algebrai approa h to TDI.
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    Numerical simulation of time delay interferometry for LISA with one arm dysfunctional
    (2011-02-25) Dhurandhar, Sanjeev; Ni, W.-T.; Wang, G.
    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 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 affect the time delay interferometry (TDI), we simulate the time delay numerically in this paper for all solutions with n ≤ 3. To conform to the actual LISA planning, we have worked out a set of 3-year optimized mission orbits of LISA spacecraft starting at June 21, 2021 using CGC2.7 ephemeris framework. We then use this numerical solution to calculate the residual optical path differences in the second generation solutions of our previous paper, and compare with the semi-analytic error estimate. The accuracy of this calculation is better than 1 cm (or 30 ps). The maximum path length difference, for all configuration calculated, is below 1 m (3 ns). This is well below the limit under which the laser frequency noise is required to be suppressed.