Sanjeev Dhurandhar

Time-Delay Interferometry

Equal-arm detectors of gravitational radiation allow phase measurements many orders of magnitude below the intrinsic phase stability of the laser injecting light into their arms. This is because the noise in the laser light is common to both arms, experiencing exactly the same delay, and thus cancels when it is differenced at the photo detector. In this situation, much lower level secondary noises then set the overall performance. If, however, the two arms have different lengths (as will necessarily be the case with space-borne interferometers), the laser noise experiences different delays in the two arms and will hence not directly cancel at the detector. In order to solve this problem, a technique involving heterodyne interferometry with unequal arm lengths and independent phase-difference readouts has been proposed. It relies on properly time-shifting and linearly combining independent Doppler measurements, and for this reason it has been called time-delay interferometry (TDI).This article provides an overview of the theory, mathematical foundations, and experimental aspects associated with the implementation of TDI. Although emphasis on the application of TDI to the Laser Interferometer Space Antenna (LISA) mission appears throughout this article, TDI can be incorporated into the design of any future space-based mission aiming to search for gravitational waves via interferometric measurements. We have purposely left out all theoretical aspects that data analysts will need to account for when analyzing the TDI data combinations.

A data-analysis strategy for detecting gravitational-wave signals from inspiraling compact binaries with a network of laser-interferometric detectors

A data-analysis strategy based on the maximum-likelihood method (MLM) is presented for the detection of gravitational waves from inspiraling compact binaries with a network of laser-interferometric detectors having arbitrary orientations and arbitrary locations around the globe. For simplicity, we restrict ourselves to the Newtonian inspiral wave form. However, the formalism we develop here is also applicable to a wave form with post-Newtonian (PN) corrections. The Newtonian wave form depends on eight parameters: the distance r to the binary, the phase {delta}{sub c} of the wave form at the time of final coalescence, the polarization-ellipse angle {psi}, the angle of inclination {epsilon} of the binary orbit to the line of sight, the source-direction angles {l_brace}{theta},{phi}{r_brace}, the time of final coalescence t{sub c} at the fiducial detector, and the chirp time {xi}. All these parameters are relevant for a chirp search with multiple detectors, unlike the case of a single detector. The primary construct on which the MLM is based is the network likelihood ratio (LR). We obtain this ratio here. For the Newtonian inspiral wave form, the LR is a function of the eight signal parameters. In the MLM-based detection strategy, the LR must be maximized over all of these parameters.morexa0» Here, we show that it is possible to maximize it analytically with respect to four of the eight parameters, namely, {l_brace}r,{delta}{sub c},{psi},{epsilon}{r_brace}. Maximization over the time of arrival is handled most efficiently by using the fast-Fourier-transform algorithm, as in the case of a single detector. This not only allows us to scan the parameter space continuously over these five parameters but also cuts down substantially on the computational costs. The analytical maximization over the four parameters yields the optimal statistic on which the decision must be based. The value of the statistic also depends on the nature of the noises in the detectors. Here, we model these noises to be mainly Gaussian, stationary, and uncorrelated for every pair of detectors. Instances of non-Gaussianity, as are present in detector outputs, can be accommodated in our formalism by implementing vetoing techniques similar to those applied for single detectors. Our formalism not only allows us to express the likelihood ratio for the network in a very simple and compact form, but also is at the basis of giving an elegant geometric interpretation to the detection problem. Maximization of the LR over the remaining three parameters is handled as follows. Owing to the arbitrary locations of the detectors in a network, the time of arrival of a signal at any detector will, in general, be different from those at the others and, consequently, will result in signal time delays. For a given network, these time delays are determined by the source-direction angles {l_brace}{theta},{phi}{r_brace}. Therefore, to maximize the LR over the parameters {l_brace}{theta},{phi}{r_brace} one needs to scan over the possible time delays allowed by a network. We opt for obtaining a bank of templates for the chirp time and the time delays. This means that we construct a bank of templates over {xi}, {theta}, and {phi}. We first discuss idealized networks with all the detectors having a common noise curve for simplicity. Such an exercise nevertheless yields useful estimates about computational costs, and also tests the formalism developed here. We then consider realistic cases of networks comprising the LIGO and VIRGO detectors: These include two-detector networks, which pair up the two LIGOs or VIRGO with one of the LIGOs, and the three-detector network that includes VIRGO and both the LIGOs. For these networks we present the computational speed requirements, network sensitivities, and source-direction resolutions.«xa0less

Algebraic approach to time-delay data analysis for LISA

Cancellation of laser frequency noise in interferometers is crucial for attaining the requisite sensitivity of the triangular three-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 canceling 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 straightforward way to cancel Doppler shifts due to optical bench motions. The two modules are in fact 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 maximization over linear combinations of the generators of the signal-to-noise ratios with the frequency and source direction angles as parameters.

Fundamentals of the LISA stable flight formation

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.

Gravitational wave radiometry : Mapping a stochastic gravitational wave background

The problem of the detection and mapping of a stochastic gravitational wave background (SGWB), either cosmological or astrophysical, bears a strong semblance to the analysis of the cosmic microwave background (CMB) anisotropy and polarization, which too is a stochastic field, statistically described in terms of its correlation properties. An astrophysical gravitational wave background (AGWB) will likely arise from an incoherent superposition of unmodelled and/or unresolved sources and cosmological gravitational wave backgrounds (CGWB) are also predicted in certain scenarios. The basic statistic we use is the cross correlation between the data from a pair of detectors. In order to “point” the pair of detectors at different locations one must suitably delay the signal by the amount it takes for the gravitational waves (GW) to travel to both detectors corresponding to a source direction. Then the raw (observed) sky map of the SGWB is the signal convolved with a beam response function that varies with location in the sky. We first present a thorough analytic understanding of the structure of the beam response function using an analytic approach employing the stationary phase approximation. The true sky map is obtained by numerically deconvolving the beam function in the integral (convolution) equation. We adopt the maximum likelihood framework to estimate the true sky map using the conjugate gradient method that has been successfully used in the broadly similar, well-studied CMB map-making problem. We numerically implement and demonstrate the method on signal generated by simulated (unpolarized) SGWB for the GW radiometer consisting of the LIGO pair of detectors at Hanford and Livingston. We include “realistic” additive Gaussian noise in each data stream based on the LIGO-I noise power spectral density. The extension of the method to multiple baselines and polarized GWB is outlined. In the near future the network of GW detectors, including the Advanced LIGO and Virgo detectors that will be sensitive to sources within a thousand times larger spatial volume, could provide promising data sets for GW radiometry.

Searching for continuous gravitational wave sources in binary systems

We consider the problem of searching for continuous gravitational wave (cw) sources orbiting a companion object. This issue is of particular interest because the Low mass x-ray binaries (LMXBs), and among them Sco X-1, the brightest x-ray source in the sky, might be marginally detectable with {approximately}2 y coherent observation time by the Earth-based laser interferometers expected to come on line by 2002 and clearly observable by the second generation of detectors. Moreover, several radio pulsars, which could be deemed to be cw sources, are found to orbit a companion star or planet, and the LIGO-VIRGO-GEO600 network plans to continuously monitor such systems. We estimate the computational costs for a search launched over the additional five parameters describing generic elliptical orbits (up to e < {approximately}0.8) using match filtering techniques. These techniques provide the optimal signal-to-noise ratio and also a very clear and transparent theoretical framework. Since matched filtering will be implemented in the final and the most computationally expensive stage of the hierarchical strategies, the theoretical framework provided here can be used to determine the computational costs. In order to disentangle the computational burden involved in the orbital motion of the cw source from the other source parameters (positionmorexa0» in the sky and spin down) and reduce the complexity of the analysis, we assume that the source is monochromatic (there is no intrinsic change in its frequency) and its location in the sky is exactly known. The orbital elements, on the other hand, are either assumed to be completely unknown or only partly known. We provide ready-to-use analytical expressions for the number of templates required to carry out the searches in the astrophysically relevant regions of the parameter space and how the computational cost scales with the ranges of the parameters. We also determine the critical accuracy to which a particular parameter must be known, so that no search is needed for it; we provide rigorous statements, based on the geometrical formulation of data analysis, concerning the size of the parameter space so that a particular neutron star is a one-filter target. This result is formulated in a completely general form, independent of the particular kind of source, and can be applied to any class of signals whose waveform can be accurately predicted. We apply our theoretical analysis to Sco X-1 and the 44 neutron stars with binary companions which are listed in the most updated version of the radio pulsar catalog. For up to {approximately}3 h of coherent integration time, Sco X-1 will need at most a few templates; for 1 week integration time the number of templates rapidly rises to {approx_equal}5{times}10{sup 6}. This is due to the rather poor measurements available today of the projected semi-major axis and the orbital phase of the neutron star. If, however, the same search is to be carried out with only a few filters, then more refined measurements of the orbital parameters are called for{emdash}an improvement of about three orders of magnitude in the accuracy is required. Further, we show that the five NSs (radio pulsars) for which the upper limits on the signal strength are highest require no more than a few templates each and can be targeted very cheaply in terms of CPU time. Blind searches of the parameter space of orbital elements are, in general, completely un-affordable for present or near future dedicated computational resources, when the coherent integration time is of the order of the orbital period or longer. For wide binary systems, when the observation covers only a fraction of one orbit, the computational burden reduces enormously, and becomes affordable for a significant region of the parameter space.«xa0less

On the minimum flexing of LISA's arms

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.

Detecting gravitational waves from inspiraling binaries with a network of detectors: Coherent versus coincident strategies

We compare two strategies of multidetector detection of compact binary inspiral signals, namely, the coincidence and the coherent. For simplicity we consider here two identical detectors having the same power spectral density of noise, that of initial LIGO, located in the same place and having the same orientation. We consider the cases of independent noise as well as that of correlated noise. The coincident strategy involves separately making two candidate event lists, one for each detector, and from these choosing those pairs of events from the two lists which lie within a suitable parameter window, which then are called coincidence detections. The coherent strategy on the other hand involves combining the data phase coherently, so as to obtain a single network statistic which is then compared with a single threshold. Here we attempt to shed light on the question as to which strategy is better. We compare the performances of the two methods by plotting the receiver operating characteristics (ROC) for the two strategies. Several of the results are obtained analytically in order to gain insight. Further we perform numerical simulations in order to determine certain parameters in the analytic formulae and thus obtain the final complete results. We considermorexa0» here several cases from the relatively simple to the astrophysically more relevant in order to establish our results. The bottom line is that the coherent strategy although more computationally expensive in general than the coincidence strategy, is superior to the coincidence strategy--considerably less false dismissal probability for the same false alarm probability in the viable false alarm regime.«xa0less

DETECTION OF GRAVITATIONAL WAVES FROM INSPIRALING, COMPACT BINARIES USING A NETWORK OF INTERFEROMETRIC DETECTORS

We formulate the data analysis problem for the detection of the Newtonian waveform from an inspiraling, compact binary by a network of arbitrarily oriented and arbitrarily located laser interferometric gravitational-wave detectors. We obtain for the first time the relation between the optimal statistic and the magnitude of the network correlation vector, which is constructed from the matched network-filter.

Astrophysical motivation for directed searches for a stochastic gravitational wave background

The nearby Universe is expected to create an anisotropic stochastic gravitational-wave background (SGWB). Different algorithms have been developed and implemented to search for isotropic and anisotropic SGWBs. The aim of this paper is to quantify the advantage of an optimal anisotropic search, specifically comparing a point source with an isotropic background. Clusters of galaxies appear as point sources to a network of ground-based laser-interferometric detectors. The optimal search strategy for these sources is a ``directed radiometer search. We show that the flux of SGWBs created by the millisecond pulsars in the Virgo cluster produces a significantly stronger signal than the nearly isotropic background of unresolved sources of the same kind. We compute their strain power spectra for different cosmologies and the distribution of populations over redshifts. We conclude that a localized source, like the Virgo cluster, can be resolved from the isotropic background with very high significance using the directed-search algorithm. For backgrounds dominated by nearby sources, up to a redshift of about 3, we show that the directed search for a localized source can have a signal-to-noise ratio that is greater than that for the all-sky integrated isotropic search.