On the background estimation by time slides in a network of gravitational wave detectors

Time shifting the outputs of Gravitational Wave detectors operating in coincidence is a convenient way to estimate the background in a search for short duration signals. However this procedure is limited as increasing indefinitely the number of time shifts does not provide better estimates. We show that the false alarm rate estimation error saturates with the number of time shifts. In particular, for detectors with very different trigger rates this error saturates at a large value. Explicit computations are done for 2 detectors, and for 3 detectors where the detection statistic relies on the logical ``OR'' of the coincidences of the 3 couples in the network.Comment: accepted for publication in CQ

Rotating relativistic stars

In this Thesis, we study the dynamics of rotating relativistic stars. We are mainly interested in the dynamics of slowly rotating stars with crusts and rapidly rotating uid stars. We also study the dynamics of non - rotating strongly magnetized stars with crusts. After stressing the importance of our studies in modern Astrophysics (Chapter 1) we describe rotating or / and magnetized relativistic stars in equilibrium (Chapter 2). We rst try to describe them with methods that involve multipole expansions. Later on, we show that two - dimensional methods are preferable. Then, we study small - amplitude pulsations of such stars. We rst derive the relevant general - relativistic equations (Chapter 3). We focus on the dynamics of the uid (we work in the Cowling approximation) and then on that of the spacetime (Inverse Cowling approximation). Subsequently, we solve these equations to obtain numerical results (Chapters 4, 5, 6 and 7). First, we present results for non - rotating neutron stars, then we move on to slowly rotating stars and nally we examine non - rotating magnetized stars with crusts. Neutron stars with crusts are met in many astrophysical cases. For example, phenomena involving rotating neutron stars with crusts and magnetic elds are the radio - emission of pulsars and the X - ray emission of magnetars. In the physical scenaria underlying these phenomena, the magnetic eld lines are frozen into the crust regions and any magnetic - eld perturbations induce elastic perturbations in the stellar crusts. Inversely, elastic perturbations in neutron star crusts shake the magnetic eld lines and modify the lightcurves of such objects. Rotating relativistic stars are also potential sources of gravitational radiation. It was found by Chandrasekhar, Friedman and Schutz that neutron stars could be unstable to the emission of gravitational radiation, if rotating quite rapidly. Their instability could grow high and limit their rotational periods considerably, emitting, at the same time, copious amounts of gravitational radiation. On the other hand, the description of equilibrium con gurations and dynamics of rapidly rotating stars is a di#cult task as it demands two - dimensional numerical techniques. In Chapter 2 we describe how neutron star models are being constructed. We start with the easiest construction, that of a static, spherically symmetric star. We continue by describing the structure equations of a slowly rotating relativistic star. We, rst, keep only linear terms in (the angular velocity of the star) and construct slowly rotating - but spherically symmetric - stars. Then, we derive the structure equations in second order in and we see that the crust now enters into the structure equations and plays an active role in the deformation of the star. An alternative, two - dimensional construction is then presented. Finally, we show how magnetized neutron stars are being constructed. In Chapter 3 we consider small - amplitude pulsations of these stars. We work in the framework of General Relativity but we also adopt the Cowling approximation. We focus on pulsations of slowly rotating relativistic stars with crusts, magnetized relativistic stars with crusts and rapidly rotating uid relativistic stars. In the case of relativistic stars with crusts, we solve eigenvalue problems while, in the case of rapidly rotating uid stars, we work with two - dimensional evolution equations. These equations do not come straightforward; we work on Einstein equations and on the equations of the uid in order to bring them in a good form for their numerical solution. We present our results in Chapters 4, 5, 6 and 7. In the rst one, we study pulsation modes of non - rotating, non - magnetic neutron stars with crusts. We calculate many spheroidal and toroidal modes for polytropic and for realistic neutron star models. In the next two chapters, we evaluate the e ects of rotation and magnetic elds, respectively, on these modes. We compare with previous Newtonian treatments and we quantify the relativistic corrections. In the last one, we attempt to calculate modes with two - dimensional numerical evolutions but we nd ourselves successful only in the case of non - rotating stars. For rotating stars, numerical instabilities take place and do not allow us to extract useful physical conclusions