Topics of LIGO Physics: Template Banks for the Inspiral of Precessing, Compact Binaries, and Design of the Signal-Recycling Cavity for Advanced LIGO

In the next decade, the detection of gravitational-wave signals by ground-based laser interferometric detectors (e.g., the Laser Interferometer Gravitational-Wave Observatory, or LIGO) will provide new information on the structure and dynamics of compact objects such as neutron stars (NS) and black holes (BH), both isolated and in binary systems. Efforts to detect the intrinsically weak gravitational-wave signals involve the development of high-quality detectors, the precise modeling of expected signals, and the development of efficient data analysis techniques. This thesis concerns two topics in these areas: methods to detect signals from the inspiral of precessing NS-BH and BH-BH binaries, and the design of the signal-recycling cavity for Advanced LIGO (the second generation LIGO detector). The detection of signals from the inspiral of precessing binaries using the standard matched filter technique, is complicated by the large number (12 at least) of parameters required to describe the complex orbital-precession dynamics of the binary and the consequent modulations of the gravitational-wave signals. To extract these signals from the noisy detector output requires a discrete bank of a huge number of signal templates that cover the 12-dimensional parameter space; and processing data with all these templates requires computational power far exceeding what is available with current technology. To solve this problem, Buonanno, Chen, and Vallisneri (BCV) proposed the use of detection template families (DTFs) --- phenomenological templates that are capable of mimicking rather accurately the inspiral waveform calculated by the post-Newtonian (PN) approach, while having a simpler functional form to reduce the computational cost. In particular, BCV proposed the so called BCV2 DTF for the precessing-binary inspiral, which has 12 parameters (most of them phenomenological). Of these, 8 are extrinsic parameters that can be searched over analytically, and only four of them are intrinsic parameters that need be searched over in a numerical one-by-one manner. The signal-matching efficiency of the BCV2 DTF has been shown to be satisfactory for signals from comparable mass BH-BH binaries. In Chapter 2 (in collaboration with Alessandra Buonanno, Yanbei Chen, Hideyuki Tagoshi, and Michele Vallisneri), I test the signal-matching efficiency of the BCV2 DTF for signals from a wide sample of precessing BH-BH and NS-BH binaries that covers the parameter range of interest for LIGO and other ground-based gravitational-wave detectors, and I study the mapping between the physical and phenomenological parameters. My colleagues and I calculate the template-match metric, propose the template-placement strategy in the intrinsic parameter space and estimate the number of templates needed (and thus equivalently the computational cost) to cover the parameter space. We also propose a so called BCV2P DTF that replaces the phenomenological parameters in the BCV2 DTF by physical parameters, which can be used to estimate the actual parameters of the binary that emitted any detected signal. In Chapters 3 and 4 (in collaboration with Alessandra Buonanno, Yanbei Chen, and Michele Vallisneri), I investigate a physical template family (PTF) suggested by BCV. This PTF uses the most accurate known waveforms for inspiraling, precessing binaries (the adiabatic PN waveforms), formulated using a new precessing convention such that five parameters become extrinsic. PTF has the obvious advantages over the DTFs of a perfect match with target signals, a lower false-alarm rate at fixed threshold, and an ability to directly estimate the physical parameters of any detected signal. In Chapter 3, we focus on the simpler single-spin binaries in which only four parameters out of nine remain intrinsic. We propose a two-stage scheme to search over the five extrinsic parameters quickly, and investigate the false-alarm statistics in each of the two stages. We define and calculate the metric of the full template space, and the projected metric and average metric of the intrinsic parameter subspace, and use these metrics to develop the method of template placement. Finally, we estimate that the number of templates needed to detect single-spin binary inspirals is within the reach of the current available computational power. In Chapter 4, we generalize the use of the single-spin PTF to double-spin binaries, based on the fact that most double-spin binaries have similar dynamics to the single-spin ones. Since the PTF in this case is, strictly speaking, only quasi-physical, we test and eventually find satisfactory signal-matching performance. We also investigate, both analytically and numerically, the difference between the single-spin and double-spin dynamics, and gain an intuition into where in the parameter space the PTF works well. We estimate the number of templates needed to cover all BH-BH and NS-BH binaries of interest to ground-based detectors, which turns out to be roughly at the limit of currently available computational power. Since the PTF is not exactly physical for double-spin binaries, it introduces systematic errors in parameter estimation. We investigate these, and find that they are either comparable to or overwhelmed by statistical errors, for events with moderate signal-to-noise ratio. BCV and I are currently systematically investigating parameter estimation with the PTF. The second part of this thesis concerns the design of the signal-recycling cavity for Advanced LIGO. In the planned Advanced-LIGO-detector upgrades from the first-generation LIGO, a signal-recycling mirror (SRM) is introduced at the dark output port. This SRM forms a signal-recycling cavity (SRC) with the input test masses. This signal-recycling design offers several advantages and brings new physics to LIGO. However, there is a problem in the current design of the SRC: the SRC is nearly degenerate, i.e., it does not distinguish transverse optical modes; and as a result, mode coupling due to mirror deformation will strongly reduce the optical power in the fundamental mode, and thus reduce the signal strength, which is roughly proportional to it. In Chapter 5, I investigate this problem using a numerical simulation of the propagation of the optical field in an Advanced LIGO interferometer. I find that if the current degenerate design for the SRC is used, there will be a serious and perhaps unattainable constraint on the magnitude of mirror deformations, in order to keep the reduction of signal-to-noise ratio below a few percent. This conclusion is consistent with previous order of magnitude estimates. This constraint poses practical difficulties on the quality of mirror polishing and the control of thermal aberration of the mirrors. Based on my simulation results, for a range of degeneracies of the SRC, I find the optimal level of degeneracy, which minimizes the reduction of signal-to-noise ratio. That optimum is nearly non-degenerate. I also discuss possible modifications to the current design that can achieve this optimal degeneracy.