(PHD, 2023)

Abstract:

This thesis covers topics in gravitational wave physics, including optomechanical measurement theory, novel detection schemes (PT-symmetric interferometer, matter-wave interferometer), and modeling of binary black hole ringdown waveform.

Measurements are accomplished through the interaction between signal and measurement devices. Identifying the nature of couplings is an important step in designing setups for specific applications. In Chapter II, we develop a general framework based on the system Hamiltonian to unambiguously classify optomechanical couplings. We add the new type, ``coherent coupling’’, where the mechanical oscillation couples several non-degenerate optical modes supported in the cavity. We give examples of different couplings, discuss in detail one particular case of the coherent coupling, and demonstrate its benefits in optomechanical experiments. Our general framework allows the design of optomechanical systems in a methodological way, to precisely exploit the strengths of some particular optomechanical couplings.

Conventional resonant detectors are subject to bandwidth-peak sensitivity trade-off, which can be traced back to the quantum Cramer-Rao Bound. Chapters III and IV in this thesis are devoted to the study of PT-symmetric amplifier, which is a stable quantum amplification scheme enabled by two-mode non-degenerate parametric amplification. In Chapter III, we study stability and sensitivity improvements for laser-interferometric gravitational-wave detectors and microwave cavity axion detectors, under Hamiltonian formalism adopting single-mode and resolved-sideband approximations. In Chapter IV, we go beyond these approximations and consider realistic parameters in the optomechanical realization of PT-symmetric interferometer for gravitational detection. We show that the main conclusion concerning stability remains intact using Nyquist analysis and a detailed time-domain simulation.

The detection method of gravitational waves is developed with linear quantum measurement theory. In Chapter V, we extend the usage of this theory to another kind of measurement device — matter-wave interferometers, which have been widely discussed as an important platform for many high-precision measurements. This theory allows us to consider fluctuations from both atoms and light and leads to a detailed analysis of back-action (of light back onto the atoms) and its effect on dynamics and measurement noise in atom interferometry. From this analysis, we obtain a Standard Quantum Limit for matter-wave interferometry. We also give a comparison between the LIGO detector and matter-wave interferometer from the perspective of quantum measurement.

In Chapter VI, we switch focus from measurement to gravitational wave sources. Specifically, we study high-frequency gravitational radiation from the ringdown of a binary black hole merger. We study the high-precision modeling on both temporal and spatial features of ringdown wave to propose a more complete test of General Relativity. We show that spin-weighted spheroidal harmonics, rather than spin-weighted spherical harmonics, better represent ringdown angular patterns. We also study the correlation between progenitor binary properties and the excitation of quasinormal modes, including higher-order angular modes, overtones, prograde and retrograde modes. This chapter seeks to provide an analytical strategy and inspire the future development of ringdown tests using data from real gravitational wave events.

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(PHD, 2022)

Abstract:

In 2015, the detection of gravitational waves (GWs) from merging black holes by the LIGO Scientific Collaboration and the VIRGO Collaboration opened a new era of observational astronomy. This thesis covers a range of topics on how to test the general theory of relativity using current and future GW detectors — both ground- and space-based. Starting from general principles, in Chapter 2, we survey how well the so-called parameterized post-Einstein parameters for binary black hole GWs can be constrained by multi-band GW detection, which employs both ground-based detectors (including Einstein Telescope and Cosmic Explorer) and space-based detectors (including the Laser Interferometer Space Antenna and deci-Hertz detectors).

In Chapter 3, we address the limitations of the Fisher Information Matrix approach in testing relativity. Chapter 4 proposes a novel experimental strategy for multi-band GW observation. More specifically, the detection of a stellar-mass binary from the Laser Interferometer Space Antenna can provide forewarning for ground-based observations, e.g., by third-generation detectors. Adjusting optical configurations of ground-based detectors targeting this particular binary can significantly improving our accuracy in testing the “no-hair theorem” of black holes. In Chapter 5, we establish a systematic framework that describes how the propagation of GWs can differ from predictions of general relativity, incorporating both dispersion and birefringence. In Chapter 6, we focus the specific example of massive gravitons and show how the so-called Vainshtein screening of the graviton’s mass, by the host galaxy of the source, the Milky way galaxy – and galaxies in between – can be extracted from an ensemble of signals.

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(PHD, 2022)

Abstract:

This thesis studies the near-horizon black hole physics in depth from three perspectives.

An important tool for studying perturbations of black hole spacetime is the linearized Einstein equations (LEE). In the Kerr spacetime, the variables in LEE do not separate, which poses a lot of difficulties to obtaining analytical solutions. By taking the near-horizon limit of extremal Kerr black holes, additional symmetries emerge to make the LEE separable. This is achieved by decomposing the metric perturbations using some basis functions adapted to the symmetry. I further show that in two string-inspired low-energy effective theories of gravity, LEE can be directly solved and analytical black hole solutions can be found.

Naively, the near-horizon perturbations of an extremal black hole may destroy the horizon and make the singularity expose itself. This is a direct challenge of the weak cosmic censorship conjecture (WCCC). Based on Wald’s gendanken experiments to destroy black holes, I examine the WCCC for the extremal charged black hole in possible generalizations of Einstein-Maxwell theory due to the higher-order corrections, up to fourth-derivative terms. It turns out that, provided the null energy condition for the falling matter, the WCCC is preserved for all possible generalizations. I further find that for BTZ black holes, i.e. solutions to (2+1)-Einstein gravity with asymptotically *AdS*_{3} boundary, WCCC is always preserved. Through the AdS/CFT correspondence, this establishes the connections between black hole thermodynamics and WCCC.

From considerations of quantum gravity and quantum information, it has been conjectured that space-time geometry near the horizon can be modified, even at scales larger than the Planck scale. The resulting spacetime is commonly referred to as the exotic compact object (ECO). A viable method to look for the near- horizon quantum structures is searching for gravitational wave echoes in the GW signals. After discussing the stability issues associated with the ECOs, I build up the phenomenology for gravitational echoes. I also introduce a new framework to deal with the near-horizon boundaries by considering the tidal response of the ECO as experienced by zero-angular-momentum fiducial observers. It is then straightforward to apply the boundary condition to computing gravitational-wave echoes from exotic compact objects.

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(PHD, 2021)

Abstract:

Gravitational wave observations are opening the door to test general relativity in regimes far less common than the weak gravitational fields that we experience in the solar system. The first part of this thesis addresses the broad issue of how different exotic predictions of general relativity imprint themselves in gravitational waves.

The ringdown portion of a binary black hole merger is dominated by superposition of quasinormal modes, the resonant modes of a perturbed black hole. The quasinormal mode spectrum of a perturbed black hole mostly reflects the spacetime geometry near the photon orbits. Chapter 2 of this thesis develops a new method for calculating quasinormal mode frequencies for weakly charged, rotating black holes. Chapter 3 uses a variety of analytic approximations to calculate the charged, rotating quasinormal mode frequencies in other cases, including nearly extremal black holes.

The event horizon is one of the most unique predictions of general relativity and it unsurprisingly does not imprint itself in gravitational wave emission. However, alternatives to black holes known as exotic compact objects do leave a unique signature in the form of echoes following the initial signal. Chapter 4 develops a formalism to understand and calculate these echoes.

The second part of this thesis focuses on reducing the noise in gravitational wave measurements using neural networks. Chapter 5 demonstrates on mock data how simple neural networks can use auxiliary measurements from the detector to predict unmodeled noise which can be subtracted offline.

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(PHD, 2019)

Abstract:

In this thesis, I will present various advancements in the modeling of binary black holes (BBHs): two black holes (BHs) that are in orbit around each other. The BHs lose energy to gravitational waves, causing them to spiral towards each other until they eventually merge and leave behind a single BH. BBHs are primary sources for ground based detectors such as the Laser Interferometer Gravitational-Wave Observatory (LIGO).

As the BHs are about to merge, they are moving at about half the speed of light and the spacetime is highly dynamical. All analytical methods break down at this stage, and numerical relativity (NR) simulations of the full Einstein’s equations are necessary. These simulations, however, are very expensive, with each simulation taking a month on a supercomputer. For direct data analysis applications with LIGO, we need a model that can be evaluated in a fraction of a second. Therefore, several approximate but fast models that are calibrated to NR simulations have been developed over the years.

Surrogate modeling provides a more powerful alternative: trained directly against the NR simulations without added assumptions, these models can reproduce the simulations as accurately as the simulations themselves, while taking only a fraction of a second to evaluate on a laptop. In short, surrogate models take BBH NR simulations from supercomputers to your laptop, without a loss of accuracy.

In this thesis, I will present several state-of-the-art surrogate models including (i) the first NR based surrogate model to span the full range of frequencies for ground based detectors, (ii) the first surrogate model for the mass, spin, and kick velocity of the final black hole after merger, and (iii) extension of an existing precessing surrogate model to higher mass ratios. In addition, I will present some work in improving the BBH initial data used in NR simulations, as well as in understanding the systematic biases introduced by approximate waveform models in LIGO data analysis.

As we head into the imminent era of high-precision gravitational wave astronomy, accurate yet fast models such as surrogate models will play a crucial role in maximizing the science output of our detectors.

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(PHD, 2019)

Abstract:

As the aLIGO and Virgo detectors continue to improve their sensitivity for observing gravitational waves from merging compact binaries, they will require ever more precise theoretical predictions to extract a detailed understanding of the physics governing these merging systems. This thesis discusses advancements within computing the gravitational waveforms along two avenues of research: the continued development of a spectral Cauchy-Characteristic Extraction (CCE) code and the presentation of a novel method called ‘Tidal Splicing’ for generating waveforms for binary neutron star (BNS), black hole-neutron star (BHNS), and even Beyond GR systems.

Due to the finite extents of typical 3+1 simulations of merging binaries, the waveforms they generate can suffer from near-zone effects and lingering gauge ambiguities. CCE was developed in order evolve radiating gravitational waves as they propagate outward to future null infinity, allowing studies connecting the dynamical spacetime of binary evolutions to effects seen by distant observers, such as superkicks, and angluar and linear momentum fluxes. A recent spectral version of CCE showed promising improvements in accuracy and efficiency over the older finite-differencing code, PittNull. However, lingering issues with the numerics and implementation of the theory prevented it from wide spread use. We detail the developments updated its initial release and demonstrate the enhancement in accuracy they yield beyond the capabilities of PittNull.

The method of Tidal Splicing enhances the inexpensive Post-Newtonian (PN) tidal corrections with BBH waveforms from numerical simulations to generate waveforms corresponding to inpsiraling BNS or BHNS systems. This leverages the accuracy of numerical BBH waveforms to effectively replace the corresponding unknown PN terms. In addition, by picking individual terms in the PN tidal expansions to include, then comparing with existing numerical simulations, we are able to probe the significance of each contribution to the total difference in evolution between BBH and BNS or BHNS inspirals. We also demonstrate how the splicing concepts used for tidal effects can extended in order to model waveforms with corrections according to theories beyond GR using an example case of a resonating ultra-compact object.

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(PHD, 2019)

Abstract:

In general relativity, black hole is the simplest macroscopic object in the universe: any black hole can be completely described by its mass, charge and angular mo- mentum. However, such a simple picture might be changed if the gravitational field equations are modified or quantum effects are taken into consideration. These additional hairs of black hole, if exist, may provide valuable information to reveal the deepest mystery of the universe: quantum theory of gravity.

In this thesis, we try to relate the hypothetical extra hairs of black hole with the ob- servational evidence as gravitational waves – another prediction of general relativity and are recently detected. In Chapter I, we provide a pedagogical introduction to the black hole hairs introduced by modified gravity and quantum mechanics, and lay out a mathematical framework to describe the gravitational wave emission with the existence of near-horizon quantum hair. In Chapter II we show that in scalar-tensor theory of gravity, the formation process of a black hole from gravitational collapse is accompanied with the emission of scalar hair. This mechanism gives rise to a scalar type memory effect of gravitational wave, which does not exist in general relativity. This phenomenon can further be used to study the parameter space of the scalar-tensor theory. In Chapter III, we find the scalar gravitational memory effect from stellar collapses provide the strongest sources for the stochastic gravita- tional wave background with scalar polarization in Brans-Dicke theory. The energy density spectrum for this background is provided and its model dependencies are studied. In Chapter IV, we provide a Green’s function method to study the echoes, which are the gravitational waves reflected by the quantum hair near the event hori- zon of a black hole. In Chapter V, we build phenomenological models to describe the near-horizon quantum hair and predict its implication to the binary black hole stochastic gravitational wave background. Our study indicates that the existence of the quantum hair will significantly increases such a background and pins down the most relevant model parameter to be the area under the effective potential. Further, we also demonstrate that the result is rather robust against the uncertainties about the nature of the near-horizon quantum hair. In the end, a field theory based treatment to the gravitational waves in general relativity is provided as the appendix.

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(PHD, 2019)

Abstract:

Optomechanics has made great strides in theory and experiments over the past decade, which culminated in the first direct detection of gravitational waves in 2015 by LIGO. This thesis explores how optomechanics can be used to test fundamental physics other than the theory of general relativity. Our emphasis will be on falsifiable theories (ultimately, only experiments can decide whether a theory is correct) that address two outstanding issues in quantum mechanics: the measurement problem, and reconciling quantum mechanics with the theory of general relativity. In particular, we show that the space experiment LISA pathfinder places aggressive bounds on two objective collapse models, which are non-linear stochastic modifications of the Schroedinger equation that can resolve the measurement problem. Moreover, we show that state-of-the-art torsion pendulum experiments can test the Schroedinger-Newton theory, which is the non-relativistic limit of a non-linear theory combining quantum mechanics with a fundamentally classical spacetime.

Along the way, we propose how to resolve two major difficulties with determining the predictions of non-linear quantum mechanics in an actual experiment. First, we cannot use the density matrix formalism in non-linear quantum mechanics and so we have to suggest and justify a particular ensemble for the thermal bath. Separating out quantum and classical fluctuations helped us propose a reasonable ensemble. Second, most researchers believe that deterministic non-linear quantum mechanics must violate the no-signaling condition. We show this isn’t necessarily the case because different interpretations of quantum mechanics make different predictions in non-linear quantum mechanics. We propose an interpretation, the causal-conditional prescription, that doesn’t violate causality by noticing that once we fix an initial state, the evolution of a system under many non-linear theories is equivalent to evolution under a linear Hamiltonian with feedback. The mapping allows us to leverage the tools of quantum control, and it tells us that if the non-linear parameters of a non-linear Hamiltonian respond causally (i.e. with an appropriate delay) to measurement results, then the theory can be made causal.

We also contribute to the theory of quantum optomechanics. We introduce two new bases that one can view environment modes with. In linear optomechanics a system interacts with an infinite number of bath modes. We show that the interaction can be reduced to one with finite degrees of freedom. Moreover, at any particular time, the system is correlated with only a finite number of bath modes. We show that if we make the assumption that we can measure any commuting environment modes, then this basis allows us to understand the one-shot quantum Cramer-Rao bound in a simple way, and allows us to sweep large parameter regimes and so find promising optomechanics topologies for quantum state preparation tasks that we can then analyze without the assumption of being able to measure any observable of the environment. We also use this basis to show that when we are interested in the conditional dynamics of a test mass, we can only adiabatically eliminate a lossy cavity when we measure the optomechanical system at a slow enough rate. Finally, we develop an analytic filter for obtaining the state of a generic optomechanical system that interacts linearly with its environment and is driven by Gaussian states, and where the outgoing light is measured with a non-linear photon-counting measurement. We hope that our work will help researchers explore optomechanics topologies that make use of photon counters.

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(PHD, 2015)

Abstract:

We present a complete system for Spectral Cauchy characteristic extraction (Spectral CCE). Implemented in C++ within the Spectral Einstein Code (SpEC), the method employs numerous innovative algorithms to efficiently calculate the Bondi strain, news, and flux.

Spectral CCE was envisioned to ensure physically accurate gravitational wave-forms computed for the Laser Interferometer Gravitational wave Observatory (LIGO) and similar experiments, while working toward a template bank with more than a thousand waveforms to span the binary black hole (BBH) problem’s seven-dimensional parameter space.

The Bondi strain, news, and flux are physical quantities central to efforts to understand and detect astrophysical gravitational wave sources within the Simulations of eXtreme Spacetime (SXS) collaboration, with the ultimate aim of providing the first strong field probe of the Einstein field equation.

In a series of included papers, we demonstrate stability, convergence, and gauge invariance. We also demonstrate agreement between Spectral CCE and the legacy Pitt null code, while achieving a factor of 200 improvement in computational efficiency.

Spectral CCE represents a significant computational advance. It is the foundation upon which further capability will be built, specifically enabling the complete calculation of junk-free, gauge-free, and physically valid waveform data on the fly within SpEC.

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(PHD, 2014)

Abstract:

This thesis covers a range of topics in numerical and analytical relativity, centered around introducing tools and methodologies for the study of dynamical spacetimes. The scope of the studies is limited to classical (as opposed to quantum) vacuum spacetimes described by Einstein’s general theory of relativity. The numerical works presented here are carried out within the Spectral Einstein Code (SpEC) infrastructure, while analytical calculations extensively utilize Wolfram’s Mathematica program.

We begin by examining highly dynamical spacetimes such as binary black hole mergers, which can be investigated using numerical simulations. However, there are difficulties in interpreting the output of such simulations. One difficulty stems from the lack of a canonical coordinate system (henceforth referred to as gauge freedom) and tetrad, against which quantities such as Newman-Penrose Psi_4 (usually interpreted as the gravitational wave part of curvature) should be measured. We tackle this problem in Chapter 2 by introducing a set of geometrically motivated coordinates that are independent of the simulation gauge choice, as well as a quasi-Kinnersley tetrad, also invariant under gauge changes in addition to being optimally suited to the task of gravitational wave extraction.

Another difficulty arises from the need to condense the overwhelming amount of data generated by the numerical simulations. In order to extract physical information in a succinct and transparent manner, one may define a version of gravitational field lines and field strength using spatial projections of the Weyl curvature tensor. Introduction, investigation and utilization of these quantities will constitute the main content in Chapters 3 through 6.

For the last two chapters, we turn to the analytical study of a simpler dynamical spacetime, namely a perturbed Kerr black hole. We will introduce in Chapter 7 a new analytical approximation to the quasi-normal mode (QNM) frequencies, and relate various properties of these modes to wave packets traveling on unstable photon orbits around the black hole. In Chapter 8, we study a bifurcation in the QNM spectrum as the spin of the black hole a approaches extremality.

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(PHD, 2013)

Abstract:

The Laser Interferometric Gravitational-Wave Observatory (LIGO) is designed to detect the Gravitational Waves (GW) predicted by Albert Einstein’s general theory of relativity. The advanced LIGO project is ongoing an upgrade to increase the detection sensitivity by more than a factor of 10, which will make the events detection a routine occurrence. In addition to using higher power lasers, heavier test mass, and better isolation systems, several new designs and techniques are proposed in the long-term upgrade, such as modifying the optics configuration to reduce the quantum noise, active noise cancellation of the Newtonian noise, optimizing the coating structure, and employing non-Guassian laser beams etc.

In the first part of my thesis (Chapters 2 and 3), I apply statistical mechanics and elastostatics to the LIGO coated mirrors, and study the thermal fluctuations that dominate advanced LIGO’s most sensitive frequency band from 40 Hz to 200 Hz.

In particular, in Chapter 2, I study the so-called coating Brownian noise, fluctuations of mirrors coated with multiple layers of dielectrics due to internal friction. Assuming coating materials to be isotropic and homogeneous, I calculate the cross spectra of Brownian fluctuations in the bulk and shear strains of the coating layers, as well as fluctuations in the height of the coating-substrate interface. The additional phase shifting and back-scattering caused by photo elastic effects are also considered for the first time.

In Chapter 3, I study whether it is realistic to adopt higher-order Laguerre-Gauss modes in LIGO, in order to mitigate the effect of mirror thermal noise. We investigate the effect on the detector’s contrast defect caused by the mode degeneracy. With both analytical calculation and numerical simulation, we show that with this approach, the detector’s susceptibility to mirror figure errors is reduced greatly compared to using the nondegenerate modes, therefore making it unacceptable for LIGO requirements.

For the future GW detectors, with much lower noises and higher sensitivity, this might be used to investigate the quantum behaviors of macroscopic mechanical objects. In recent years the linear optomechanical systems with cavity modes coupling to a mechanical oscillator have been studied extensively. In the second part of my thesis (Chapter 4), I study the interaction between a single photon and a high-finesse cavity with a movable mirror, in the so-called strong coupling regime, where the recoil of the photon can cause significant change in the momentum of the mirror. The results are applied to analyze the case with a Fabry-Perot cavity. We also present that with engineering the photon wave function, it is possible to prepare the oscillator into an arbitrary quantum state.

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(PHD, 2013)

Abstract: This thesis presents recent research into analytic topics in the classical theory of General Relativity. It is a thesis in two parts. The first part features investigations into the spectrum of perturbed, rotating black holes. These include the study of near horizon perturbations, leading to a new generic frequency mode for black hole ringdown; an treatment of high frequency waves using WKB methods for Kerr black holes; and the discovery of a bifurcation of the quasinormal mode spectrum of rapidly rotating black holes. These results represent new discoveries in the field of black hole perturbation theory, and rely on additional approximations to the linearized field equations around the background black hole. The second part of this thesis presents a recently developed method for the visualization of curved spacetimes, using field lines called the tendex and vortex lines of the spacetime. The works presented here both introduce these visualization techniques, and explore them in simple situations. These include the visualization of asymptotic gravitational radiation; weak gravity situations with and without radiation; stationary black hole spacetimes; and some preliminary study into numerically simulated black hole mergers. The second part of thesis culminates in the investigation of perturbed black holes using these field line methods, which have uncovered new insights into the dynamics of curved spacetime around black holes.

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(PHD, 2013)

Abstract:

The theories of relativity and quantum mechanics, the two most important physics discoveries of the 20th century, not only revolutionized our understanding of the nature of space-time and the way matter exists and interacts, but also became the building blocks of what we currently know as modern physics. My thesis studies both subjects in great depths — this intersection takes place in gravitational-wave physics.

Gravitational waves are “ripples of space-time”, long predicted by general relativity. Although indirect evidence of gravitational waves has been discovered from observations of binary pulsars, direct detection of these waves is still actively being pursued. An international array of laser interferometer gravitational-wave detectors has been constructed in the past decade, and a first generation of these detectors has taken several years of data without a discovery. At this moment, these detectors are being upgraded into second-generation configurations, which will have ten times better sensitivity. Kilogram-scale test masses of these detectors, highly isolated from the environment, are probed continuously by photons. The sensitivity of such a quantum measurement can often be limited by the Heisenberg Uncertainty Principle, and during such a measurement, the test masses can be viewed as evolving through a sequence of nearly pure quantum states.

The first part of this thesis (Chapter 2) concerns how to minimize the adverse effect of thermal fluctuations on the sensitivity of advanced gravitational detectors, thereby making them closer to being quantum-limited. My colleagues and I present a detailed analysis of coating thermal noise in advanced gravitational-wave detectors, which is the dominant noise source of Advanced LIGO in the middle of the detection frequency band. We identified the two elastic loss angles, clarified the different components of the coating Brownian noise, and obtained their cross spectral densities.

The second part of this thesis (Chapters 3-7) concerns formulating experimental concepts and analyzing experimental results that demonstrate the quantum mechanical behavior of macroscopic objects - as well as developing theoretical tools for analyzing quantum measurement processes. In Chapter 3, we study the open quantum dynamics of optomechanical experiments in which a single photon strongly influences the quantum state of a mechanical object. We also explain how to engineer the mechanical oscillator’s quantum state by modifying the single photon’s wave function.

In Chapters 4-5, we build theoretical tools for analyzing the so-called “non-Markovian” quantum measurement processes. Chapter 4 establishes a mathematical formalism that describes the evolution of a quantum system (the plant), which is coupled to a non-Markovian bath (i.e., one with a memory) while at the same time being under continuous quantum measurement (by the probe field). This aims at providing a general framework for analyzing a large class of non-Markovian measurement processes. Chapter 5 develops a way of characterizing the non-Markovianity of a bath (i.e.,whether and to what extent the bath remembers information about the plant) by perturbing the plant and watching for changes in the its subsequent evolution. Chapter 6 re-analyzes a recent measurement of a mechanical oscillator’s zero-point fluctuations, revealing nontrivial correlation between the measurement device’s sensing noise and the quantum rack-action noise.

Chapter 7 describes a model in which gravity is classical and matter motions are quantized, elaborating how the quantum motions of matter are affected by the fact that gravity is classical. It offers an experimentally plausible way to test this model (hence the nature of gravity) by measuring the center-of-mass motion of a macroscopic object.

The most promising gravitational waves for direct detection are those emitted from highly energetic astrophysical processes, sometimes involving black holes - a type of object predicted by general relativity whose properties depend highly on the strong-field regime of the theory. Although black holes have been inferred to exist at centers of galaxies and in certain so-called X-ray binary objects, detecting gravitational waves emitted by systems containing black holes will offer a much more direct way of observing black holes, providing unprecedented details of space-time geometry in the black-holes’ strong-field region.

The third part of this thesis (Chapters 8-11) studies black-hole physics in connection with gravitational-wave detection.

Chapter 8 applies black hole perturbation theory to model the dynamics of a light compact object orbiting around a massive central Schwarzschild black hole. In this chapter, we present a Hamiltonian formalism in which the low-mass object and the metric perturbations of the background spacetime are jointly evolved. Chapter 9 uses WKB techniques to analyze oscillation modes (quasi-normal modes or QNMs) of spinning black holes. We obtain analytical approximations to the spectrum of the weakly-damped QNMs, with relative error O(1/L^2), and connect these frequencies to geometrical features of spherical photon orbits in Kerr spacetime. Chapter 11 focuses mainly on near-extremal Kerr black holes, we discuss a bifurcation in their QNM spectra for certain ranges of (l,m) (the angular quantum numbers) as a/M → 1. With tools prepared in Chapter 9 and 10, in Chapter 11 we obtain an analytical approximate for the scalar Green function in Kerr spacetime.

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(PHD, 2012)

Abstract:

Numerical-relativity simulations of black-hole binaries and advancements in gravitational-wave detectors now make it possible to learn more about the collisions of compact astrophysical bodies. To be able to infer more about the dynamical behavior of these objects requires a fuller analysis of the connection between the dynamics of pairs of black holes and their emitted gravitational waves. The chapters of this thesis describe three approaches to learn more about the relationship between the dynamics of black-hole binaries and their gravitational waves: modeling momentum flow in binaries with the Landau-Lifshitz formalism, approximating binary dynamics near the time of merger with post-Newtonian and black-hole-perturbation theories, and visualizing spacetime curvature with tidal tendexes and frame-drag vortexes.

In Chapters 2–4, my collaborators and I present a method to quantify the flow of momentum in black-hole binaries using the Landau-Lifshitz formalism. Chapter 2 reviews an intuitive version of the formalism in the first-post-Newtonian approximation that bears a strong resemblance to Maxwell’s theory of electromagnetism. Chapter 3 applies this approximation to relate the simultaneous bobbing motion of rotating black holes in the superkick configuration—equal-mass black holes with their spins anti-aligned and in the orbital plane—to the flow of momentum in the spacetime, prior to the black holes’ merger. Chapter 4 then uses the Landau-Lifshitz formalism to explain the dynamics of a head-on merger of spinning black holes, whose spins are anti-aligned and transverse to the infalling motion. Before they merge, the black holes move with a large, transverse, velocity, which we can explain using the post-Newtonian approximation; as the holes merge and form a single black hole, we can use the Landau-Lifshitz formalism without any approximations to connect the slowing of the final black hole to its absorbing momentum density during the merger.

In Chapters 5–7, we discuss using analytical approximations, such as post-Newtonian and black-hole-perturbation theories, to gain further understanding into how gravitational waves are generated by black-hole binaries. Chapter 5 presents a way of combining post-Newtonian and black-hole-perturbation theories—which we call the hybrid method—for head-on mergers of black holes. It was able to produce gravitational waveforms and gravitational recoils that agreed well with comparable results from numerical-relativity simulations. Chapter 6 discusses a development of the hybrid model to include a radiation-reaction force, which is better suited for studying inspiralling black-hole binaries. The gravitational waveform from the hybrid method for inspiralling mergers agreed qualitatively with that from numerical-relativity simulations; when applied to the superkick configuration, it gave a simplified picture of the formation of the large black-hole kick. Chapter 7 describes an approximate method of calculating the frequencies of the ringdown gravitational waveforms of rotating black holes (quasinormal modes). The method generalizes a geometric interpretation of black-hole quasinormal modes and explains a degeneracy in the spectrum of these modes.

In Chapters 8–11, we describe a new way of visualizing spacetime curvature using tools called tidal tendexes and frame-drag vortexes. This relies upon a time-space split of spacetime, which allows one to break the vacuum Riemann curvature tensor into electric and magnetic parts (symmetric, trace-free tensors that have simple physical interpretations). The regions where the eigenvalues of these tensors are large form the tendexes and vortexes of a spacetime, and the integral curves of their eigenvectors are its tendex and vortex lines, for the electric and magnetic parts, respectively. Chapter 8 provides an overview of these visualization tools and presents initial results from numerical-relativity simulations. Chapter 9 uses topological properties of vortex and tendex lines to classify properties of gravitational waves far from a source. Chapter 10 describes the formalism in more detail, and discusses the vortexes and tendexes of multipolar spacetimes in linearized gravity about flat space. The chapter helps to explain how near-zone vortexes and tendexes become gravitational waves far from a weakly gravitating, time-varying source. Chapter 11 is a detailed investigation of the vortexes and tendexes of stationary and perturbed black holes. It develops insight into how perturbations of (strongly gravitating) black holes extend from near the horizon to become gravitational waves.

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(BS, 2011)

Abstract: Gravitational waves can be produced when a compact object (stellar-mass black hole, neutron star, or white dwarf) spirals towards a massive black hole. For moderate mass ratios of 10^{-2} - 10^{-4}, these events are known as intermediate mass ratio inspirals (IMRis), whereas for mass ratios below 10^{-4} they are called extreme mass ratio inspirals (EMRis). These events will be important sources for the proposed gravitational-wave detector LISA (Laser Interferometer Space Antenna) and will provide a precise test
of general relativity in the unexplored regime of strong gravitational fields. To detect
these gravitational waves and reliably measure the parameters of the binary producing
them, highly accurate models of gravitational waveforms are needed. The spin of the
orbiting compact object (CO) introduces forces that affect the orbit and waveform,
but these effects are often ignored. The goal of this thesis is to determine under what
circumstances compact-object spin will significantly alter the waveform as measured
by LISA. To do this, a post-Newtonian waveform model will be used to explore CO
spin effects. It is concluded that neglecting CO spin does not effect the detectability
of EMRis or IMRis. Parameter measurement errors introduced by neglecting CO
spin are typically small unless the binary system happens to be particularly close.
Constraining the value of compact object spin is unlikely but may be possible for a
nearby IMRI. Within the approximations used in this study, it appears CO spin is
marginally important, but that conclusion may change if a more realistic waveform
model is used in future work.

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(PHD, 2011)

Abstract:

In Part I, we consider numerical simulations of event horizons. Event horizons are the defining physical features of black hole spacetimes, and are of considerable interest in studying black hole dynamics. Here, we reconsider three techniques to find event horizons in numerical spacetimes, and find that straightforward integration of geodesics backward in time is most robust. We apply this method to various systems, from a highly spinning Kerr hole through to an asymmetric binary black hole inspiral. We find that the exponential rate at which outgoing null geodesics diverge from the event horizon of a Kerr black hole is the surface gravity of the hole. In head-on mergers we are able to track quasi-normal ringing of the merged black hole through seven oscillations, covering a dynamic range of about 10^{5}. In the head-on “kick” merger, we find that computing the Landau-Lifshitz velocity of the event horizon is very useful for an improved understanding of the kick behaviour. Finally, in the inspiral simulations, we find that the topological structure of the black holes does not produce an intermediate toroidal phase, though the structure is consistent with a potential re-slicing of the spacetime in order to introduce such a phase. We further discuss the topological structure of non-axisymmetric collisions.

In Part II, we consider parameter estimation of cosmic string burst gravitational waves in Mock LISA data. A network of observable, macroscopic cosmic (super-)strings may well have formed in the early Universe. If so, the cusps that generically develop on cosmic-string loops emit bursts of gravitational radiation that could be detectable by gravitational-wave interferometers, such as the ground-based LIGO/Virgo detectors and the planned, space-based LISA detector. We develop two versions of a LISA-oriented string-burst search pipeline within the context of the Mock LISA Data Challenges, which rely on the publicly available MultiNest and PyMC software packages, respectively. We use the F-statistic to analytically maximize over the signal’s amplitude and polarization, A and ψ, and use the FFT to search quickly over burst arrival times t_{C}. We also demonstrate an approximate, Bayesian version of the F-statistic that incorporates realistic priors on A and ψ. We calculate how accurately LISA can expect to measure the physical parameters of string-burst sources, and compare to results based on the Fisher-matrix approximation. To understand LISA’s angular resolution for string-burst sources, we draw maps of the waveform fitting factor [maximized over (A, ψ, t_{C})] as a function of sky position; these maps dramatically illustrate why (for LISA) inferring
the correct sky location of the emitting string loop will often be practically impossible. In addition, we identify and elucidate several symmetries that are embedded in this search problem, and we derive the distribution of cut-off frequencies f_{max} for observable bursts.

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(PHD, 2009)

Abstract:

This thesis presents a study of various interesting problems in theoretical astrophysics, including gravitational wave astronomy, gamma ray bursts and cosmology. Chapters 2, 3 and 4 explore prospects for detecting gravitational waves from stellar-mass compact objects spiraling into intermediate-mass black holes with ground-based observatories. It is shown in chapter 2 that if the central body is not a BH but its metric is stationary, axisymmetric, reflection symmetric and asymptotically flat, then the waves will likely be triperiodic, as for a BH. Chapters 3 and 4 show that the evolutions of the waves’ three fundamental frequencies and of the complex amplitudes of their spectral components encode (in principle) details of the central body’s metric, the energy and angular momentum exchange between the central body and the orbit, and the time-evolving orbital elements. Chapter 5 studies a local readout method to enhance the low frequency sensitivity of detuned signal-recycling interferometers. We provide both the results of improvement in quantum noise and the implementation details in Advanced LIGO. Chapter 6 applies and generalizes causal Wiener filter to data analysis in macroscopic quantum mechanical experiments. With the causal Wiener filter method, we demonstrate that in theory we can put the test masses in the interferometer to its quantum mechanical ground states. Chapter 7 presents some analytical solutions for expanding fireballs, the common theoretical model for gamma ray bursts and soft gamma ray repeaters. We apply our results to SGR 1806-20 and rediscover the mismatch between the model and the afterglow observations. Chapter 8 discusses the reconstruction of the scalar-field potential of the dark energy. We advocate direct reconstruction of the scalar field potential as a way to minimize prior assumptions on the shape, and thus minimize the introduction of bias in the derived potential. Chapter 9 discusses gravitational lensing modifications to cosmic microwave background anisotropies and polarization, produced by a stochastic background of primordial gravitational waves between us and the last scattering surface. Chapter 10 calculates the non-Gaussian covariance of CMB B-modes of polarization.

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(PHD, 2007)

Abstract:

This thesis is based on three of the five research projects I have worked on during my graduate student years. Each of these projects resulted in a paper a chapter of this thesis is based on. Two other papers I published during my stay at Caltech are not included, as they are not related to the research reported here.

After a brief introduction in the first chapter, in chapter two I discuss a simple way to greatly reduce tilt instability in Advanced LIGO by changing the mirror shape from nearly-flat previously considered mesa beams to nearly a nearly concentric shape that would provide the beam characteristics without the tilt instability. I also propose a family of hyperboloidal beams that join the nearly-flat and nearly-concentric mesa beams in a continuous manner.

In chapter three I report the results of a recent search for the lowest value of thermal noise that can be achieved in LIGO by changing the mirror shape. I discuss in detail the beam properties and point out some of the characteristics of an advanced LIGO that uses these beams. Such an instrument, if built, would have an event rate roughly three times higher than an advanced LIGO using the Mesa beams introduced in chapter 2.

Chapter four, which is also the last chapter of this thesis, is based on a Numerical Relativity project. I compute embeddings of general two-dimensional surfaces with spherical topology in flat 3D space using a minimization algorithm. The minimization problem is similar to the one solved in chapter three, albeit in a somewhat different setting.

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