Gravitational-wave tests of general relativity with ground-based detectors and pulsar-timing arrays

This review is focused on tests of Einstein’s theory of general relativity with gravitational waves that are detectable by ground-based interferometers and pulsar-timing experiments. Einstein’s theory has been greatly constrained in the quasi-linear, quasi-stationary regime, where gravity is weak and velocities are small. Gravitational waves will allow us to probe a complimentary, yet previously unexplored regime: the non-linear and dynamical strong-field regime. Such a regime is, for example, applicable to compact binaries coalescing, where characteristic velocities can reach fifty percent the speed of light and gravitational fields are large and dynamical. This review begins with the theoretical basis and the predicted gravitational-wave observables of modified gravity theories. The review continues with a brief description of the detectors, including both gravitational-wave interferometers and pulsar-timing arrays, leading to a discussion of the data analysis formalism that is applicable for such tests. The review ends with a discussion of gravitational-wave tests for compact binary systems.

<p indent="0mm">The direct detection of gravitational waves from stellar-mass compact binary merger by ground-based laser interferometer gravitational wave detector LIGO/Virgo has verified the prediction of general relativity and opened a new chapter in gravitational wave astronomy. Up to now, a total of 50 gravitational wave events have been detected and published in GWTC-1 and GWTC-2 catalogue. In the near future, the third-generation ground based gravitational wave detector, such as the Einstein Telescope (ET), will be constructed with sensitivity improved by at least a factor of 10. Tens of thousands of gravitational wave signals are expected to be detected per year in the third-generation detector era. These gravitational wave signals will inevitably overlap with foreground massive celestial bodies (such as black hole, galaxy and galaxy cluster), thus leading to lensed gravitational wave signals which will undoubtedly be another important test of general relativity once detected. Furthermore, strongly lensed gravitational wave signals by galaxy from massive binary black hole could possibly be detected by future space detector, e.g., LISA and DECIGO. Since the wavelengths of gravitational waves are comparable with the size of some lens, the lensed gravitational waves play a unique role in studying the phenomena of wave nature, e.g., interference and diffraction. Lensed gravitational wave-electromagnetic wave system will have a wide range of applications in fundamental physics, cosmology and astrophysics when a series of lensed gravitational wave events and their corresponding electromagnetic counterparts have been detected. The most obvious advantage of lensed gravitational wave-electromagnetic wave system lies in that gravitational wave could provide time delay information with high accuracy, and electromagnetic wave could provide Fermat potential difference with high precision because a relatively complete arc of light could be obtained by electromagnetic wave observations and this is the most important step in measuring the Fermat potential. Thus, by combining the information from both approaches, lensed gravitational wave-electromagnetic wave system could be applied to study the speed of gravitational waves, constrain cosmological parameters, explore the substructure of the dark matter halo and investigate the lens model and so on. In this paper, we will review in detail how to use geodesic equation, lens equation, as well as wave equation to tackle the stationary scattering problem of lensed gravitational waves, and introduce how lensed gravitational wave-electromagnetic wave system could be applied to study the tensor properties, interference and diffraction effects of gravitational wave, as well as its applications in gravitational wave velocity, Hubble constant, cosmic curvature, lens mass, substructure and so on.

This thesis investigates the dynamics of binary systems composed of spinning compact objects (such as white dwarfs, neutron stars, and black holes) in the context of general relativity. In particular, we use the method of Lyapunov exponents to determine whether such systems are chaotic. Compact binaries are promising sources of gravitational radiation for both ground- and space-based gravitational-wave detectors, and radiation from chaotic orbits would be difficult to detect and analyze. For chaotic orbits, the number of waveform templates needed to match a given gravitational-wave signal would grow exponentially with increasing detection sensitivity, rendering the preferred matched filter detection method computationally impractical. It is therefore urgent to understand whether the binary dynamics can be chaotic, and, if so, how prevalent this chaos is. We first consider the dynamics of a spinning compact object orbiting a much more massive rotating black hole, as modeled by the Papapetrou equations in Kerr spacetime. We find that many initial conditions lead to positive Lyapunov exponents, indicating chaotic dynamics. The Lyapunov exponents come in positive/negative pairs, a characteristic of Hamiltonian dynamical systems. Despite the formal existence of chaotic solutions, we find that chaos occurs only for physically unrealistic values of the small body's spin. As a result, chaos will not affect theoretical templates in the extreme mass-ratio limit for which the Papapetrou equations are valid. Chaos will therefore not affect the ability of space-based gravitational-wave detectors (such as LISA, the Laser Interferometer Space Antenna) to perform precision tests of general relativity using extreme mass-ratio inspirals. We next consider the dynamics of spinning black-hole binaries, as modeled by the post-Newtonian (PN) equations, which are valid for orbital velocities much smaller than the speed of light. We study thoroughly the special case of quasi-circular orbits with comparable mass ratios, which are particularly relevant from the perspective of gravitational wave generation for LIGO (the Laser Interferometer Gravitational-wave Observatory) and other ground-based interferometers. In this case, unlike the extreme mass-ratio case, we find chaotic solutions for physically realistic values of the spin. On the other hand, our survey shows that chaos occurs in a negligible fraction of possible configurations, and only for such small radii that the PN approximation is likely to be invalid. As a result, at least in the case of comparable mass black-hole binaries, theoretical templates will not be significantly affected by chaos. In a final, self-contained chapter, we discuss various methods for the calculation of Lyapunov exponents in systems of ordinary differential equations. We introduce several new techniques applicable to constrained dynamical systems, developed in the course of studying the dynamics of spinning compact binaries. Considering the Papapetrou and post-Newtonian systems together, our most important general conclusion is that we find no chaos in any relativistic binary system for orbits that clearly satisfy the approximations required for the equations of motion to be physically valid.

Advanced LIGO's first observing run marked the birth of gravitational-wave astronomy through the first detection of gravitational waves from coalescing black holes-GW150914. Advanced LIGO's second and Advanced Virgo's first observing run marked the birth of multimessenger astronomy with first joint observations of gravitational and electromagnetic radiation associated with coalescing neutron stars-GW170817. The electromagnetic observations included detection of a burst of gamma rays produced by the merger, and a kilonova powered by the radioactive decay of r-process nuclei synthesized in the neutron star coalescence ejecta. Gravitational waves from compact binary coalescences carry fingerprints of the sources that generated them. Studying them allows us to test Einstein’s general relativity in the strongest regimes, where it has never been tested before, and study matter at densities beyond reach of the most powerful laboratories on our planet. Moreover, we can gain insight about the evolution of stars, galaxies and even the Universe as a whole by studying the merger rate of compact objects. Joint electromagnetic and gravitational-wave observations help develop our understanding of the physical processes that occur in such systems, and provide a new method of probing cosmological parameters. GW170817 was detected by the GstLAL pipeline in low-latency making the extensive electromagnetic followup possible. The GstLAL pipeline is a matched filtering pipeline that uses compact binary coalescence waveform models to filter the data from gravitational-wave detectors in the time-domain. It can detect gravitational waves from coalescing compact binaries in near real time and provide point estimates for binary parameters. This thesis describes the methods, developments, and the results from the GstLAL pipeline over the course of the first two observing runs of Advanced LIGO, focusing on the contributions made by the author. We also present a study about the prospects of observing a cosmological stochastic background which is expected to be buried under the astrophysical background from the population of coalesceing compact binaries with third-generation gravitational-wave detectors.

Laser interferometer gravitational wave observatory (LIGO) consists of two complex large-scale laser interferometers designed for direct detection of gravitational waves from distant astrophysical sources in the frequency range 10Hz - 5kHz. Direct detection of space-time ripples will support Einstein's general theory of relativity and provide invaluable information and new insight into physics of the Universe. Initial phase of LIGO started in 2002, and since then data was collected during six science runs. Instrument sensitivity was improving from run to run due to the effort of commissioning team. Initial LIGO has reached designed sensitivity during the last science run, which ended in October 2010. In parallel with commissioning and data analysis with the initial detector, LIGO group worked on research and development of the next generation detectors. Major instrument upgrade from initial to advanced LIGO started in 2010 and lasted till 2014. This thesis describes results of commissioning work done at LIGO Livingston site from 2013 until 2015 in parallel with and after the installation of the instrument. This thesis also discusses new techniques and tools developed at the 40m prototype including adaptive filtering, estimation of quantization noise in digital filters and design of isolation kits for ground seismometers. The first part of this thesis is devoted to the description of methods for bringing interferometer to the linear regime when collection of data becomes possible. States of longitudinal and angular controls of interferometer degrees of freedom during lock acquisition process and in low noise configuration are discussed in details. Once interferometer is locked and transitioned to low noise regime, instrument produces astrophysics data that should be calibrated to units of meters or strain. The second part of this thesis describes online calibration technique set up in both observatories to monitor the quality of the collected data in real time. Sensitivity analysis was done to understand and eliminate noise sources of the instrument. Coupling of noise sources to gravitational wave channel can be reduced if robust feedforward and optimal feedback control loops are implemented. The last part of this thesis describes static and adaptive feedforward noise cancellation techniques applied to Advanced LIGO interferometers and tested at the 40m prototype. Applications of optimal time domain feedback control techniques and estimators to aLIGO control loops are also discussed. Commissioning work is still ongoing at the sites. First science run of advanced LIGO is planned for September 2015 and will last for 3-4 months. This run will be followed by a set of small instrument upgrades that will be installed on a time scale of few months. Second science run will start in spring 2016 and last for about 6 months. Since current sensitivity of advanced LIGO is already more than factor of 3 higher compared to initial detectors and keeps improving on a monthly basis, upcoming science runs have a good chance for the first direct detection of gravitational waves.

The Advanced LIGO and Advanced Virgo gravitational wave (GW) detectors will begin operation in the coming years, with compact binary coalescence events a likely source for the first detections. The gravitational waveforms emitted directly encode information about the sources, including the masses and spins of the compact objects. Recovering the physical parameters of the sources from the GW observations is a key analysis task. This work describes the LALInference software library for Bayesian parameter estimation of compact binary signals, which builds on several previous methods to provide a well-tested toolkit which has already been used for several studies. We show that our implementation is able to correctly recover the parameters of compact binary signals from simulated data from the advanced GW detectors. We demonstrate this with a detailed comparison on three compact binary systems: a binary neutron star, a neutron star black hole binary and a binary black hole, where we show a cross-comparison of results obtained using three independent sampling algorithms. These systems were analysed with non-spinning, aligned spin and generic spin configurations respectively, showing that consistent results can be obtained even with the full 15-dimensional parameter space of the generic spin configurations. We also demonstrate statistically that the Bayesian credible intervals we recover correspond to frequentist confidence intervals under correct prior assumptions by analysing a set of 100 signals drawn from the prior. We discuss the computational cost of these algorithms, and describe the general and problem-specific sampling techniques we have used to improve the efficiency of sampling the compact binary coalescence parameter space.

In an earlier work [S. Kastha et al., Phys. Rev. D 98, 124033 (2018)], we developed the parametrized multipolar gravitational wave phasing formula to test general relativity, for the nonspinning compact binaries in quasicircular orbit. In this paper, we extend the method and include the important effect of spins in the inspiral dynamics. Furthermore, we consider parametric scaling of post-Newtonian (PN) coefficients of the conserved energy for the compact binary, resulting in the parametrized phasing formula for nonprecessing spinning compact binaries in quasicircular orbit. We also compute the projected accuracies with which the second and third generation ground-based gravitational wave detector networks as well as the planned space-based detector LISA will be able to measure the multipole deformation parameters and the binding energy parameters. Based on different source configurations, we find that a network of third-generation detectors would have comparable ability to that of LISA in constraining the conservative and dissipative dynamics of the compact binary systems. This parametrized multipolar waveform would be extremely useful not only in deriving the first upper limits on any deviations of the multipole and the binding energy coefficients from general relativity using the gravitational wave detections, but also for science case studies of next generation gravitational wave detectors.

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

Gravitational wave (GW) observations of coalescing compact binaries will be unique probes of strong-field, dynamical aspects of relativistic gravity. We present a short review of various schemes proposed in the literature to test general relativity (GR) and alternative theories of gravity using inspiral waveforms. Broadly these schemes may be classified into two types: model dependent and model independent. In the model dependent category, GW observations are compared against a specific waveform model representative of a particular theory or a class of theories such as scalar-tensor theories, dynamical Chern–Simons theory and massive graviton theories. Model independent tests are attempts to write down a parametrized gravitational waveform where the free parameters take different values for different theories and (at least some of) which can be constrained by GW observations. We revisit some of the proposed bounds in the case of downscaled LISA configuration (eLISA) and compare them with the original LISA configuration. We also compare the expected bounds on alternative theories of gravity from ground-based and space-based detectors and find that space-based GW detectors can test GR and other theories of gravity with unprecedented accuracies. We then focus on a recent proposal to use singular value decomposition of the Fisher information matrix to improve the accuracies with which post-Newtonian theory can be tested. We extend those results to the case of space-based detector eLISA and discuss its implications.

The advent of gravitational wave (GW) astronomy has provided a new window through which to view and understand the Universe. To fully exploit the potential of GW astronomy, an understanding of all the potential electromagnetic counterparts to a gravitational wave detected source will help maximise the science returns. Here I present a study of the electromagnetic emission from relativistic jets that accompany the merger of binary neutron stars or black hole-neutron star systems. These counterparts provide a probe for the structure and dynamics of these relativistic outflows. Binary neutron star, or neutron star-black hole, mergers are thought to be the dominant progenitor of short gamma-ray bursts (GRBs). Here we investigate the possibility that there is a hidden population of low-Lorentz factor jets resulting in failed GRBs, on-axis orphan afterglows, and what kind of counterparts can be expected given a merger-jet population dominated by these failed-GRB jets. I find that for GW detected mergers, ∼ 80% of the population of on-axis events may result in a failed GRB afterglow. The afterglow of a failed GRB is characterised by the lack of any prompt emission; where the γ-rays are emitted within an optically thick region of the low-Lorentz factor (Γ) outflow and significant suppression via pair production and a high opacity results in the photons coupled to the pair plasma. This plasma will undergo adiabatic expansion, and the photons will decouple at the photospheric radius. The energy in the prompt photons, for a sufficiently low-Γ outflow, will have been significantly suppressed. GW detected mergers have a Malmquist bias towards on-axis events (i.e. the rotational axis of the system), where the peak of the probability distribution is an inclination ∼ 30⁰. If the jets from these mergers have an intrinsic structure out to wider angles, then the majority of mergers will be accompanied by electromagnetic counterparts from these various jet structures. By making some simple assumptions about the energetic structure of a jet outside of a bright core region, the various temporal features that result from a given jet structure can be predicted. Where the population of merger jets is dominated by a single structure model, I show the expected fractions of optical counterparts brighter than m_AB = 21. On 17 August 2017, the Light Interferometer Gravitational Wave Observatory (LIGO) in collaboration with Virgo detected the merger of a binary neutron star system. Various electromagnetic counterparts were detected: the GRB 170817A by Fermi/GBM and INTEGRAL; an optical, blue to red, macro/kilo-nova from ∼ 1/2 day post merger to ∼ 5 − 10 days; and a brightening radio, and X-ray counterpart from ∼ 10 days. Optical detection of this counterpart at a magnitude ∼ 26 was made at ∼ 100 days post-merger. Analysis of this counterpart is consistent with the afterglow of a Gaussian structured jet viewed at the system inclination, ∼ 18 ± 8⁰. If all short GRB jets have a similar jet structure, then the rates of orphan afterglows in deep drilling blind surveys e.g. the Large Synoptic Survey Telescope (LSST), will be higher than those expected from a homogeneous, or ‘top-hat’ jet, population. The rates for the various jet structures for orphan afterglows from mergers are discussed, showing that for a population of failed GRBs, or an intrinsic Gaussian structure, an excess in the orphan rate may be apparent. Understanding the dynamics and structure for the jets from black-hole systems born at the merger of a compact binary can help give clues as to the nature of jets from black holes on all scales. As an aside, I show empirically that regardless of black hole mass or system phenomenology, the relativistic jets from such systems share a universal scaling for the jet power and emitted γ-ray luminosity. This scaling could be due to the similar efficiencies of various processes, or alternatively, the scaling may be able to give insights into the emission and physical processes that are responsible for high-energy photons from these outflows. GW astronomy offers a probe of the most extreme relativistic outflows in the Universe, GRBs. The predicted electromagnetic counterparts from these outflows, in association with GW detections, provides a way to probe the Lorentz-factor distribution for merger-jets. Additionally, the phenomenological shape of the afterglows, at various inclinations, gives an indication of the intrinsic structure of these jets. An understanding of these dynamical and structural qualities can be used to constrain the parent population, merger rates, and binary evolution models for compact binary systems.

In this thesis, I explore several avenues for learning about fundamental physics from gravitational-wave (GW) observations. In particular, I focus on the phenomenological study of basic properties of GWs in ways that require minimal assumptions about the underlying nature of gravity. I place a special emphasis on GW polarizations, but also consider their speed and possible dispersion. To constrain possible modifications to general relativity, I develop data-analysis frameworks to measure these properties with both transient and persistent signals detected by ground-based detectors. This includes, among other results, the analysis of two LIGO and Virgo compact-binary detections, GW170814 and GW170817, to produce the first direct observational statements about the local geometry of GW polarizations. I also present constraints on the potential amplitude of nontensorial monochromatic signals from 200 known pulsars in the Milky Way and describe in detail the methods used to obtain them. Because stochastic signals will be a great resource for studying GW properties, I also carefully review the assumptions that go into standard stochastic analyses and explore their applicability beyond general relativity, concluding that those measurements will have to be interpreted carefully to make meaningful statements about corrections to Einstein’s theory. Besides the properties of the waves themselves, I also study the prospect for using GWs as a means to uncover signatures of new ultralight bosons—an exciting possibility that could bring particle physics into the reach of GW astronomy. I explore the potential of current and future detectors to detect these conjectured particles, concluding that third-generation instruments are certain to place theoretically interesting constraints. Because little can be done in the absence of signals, I also propose data-analysis methods to improve LIGO and Virgo’s chances of detecting both transient and continuous signals. The latter have now been used to diagnose real detection candidates on several occasions.

We study the propagation of cosmological gravitational wave (GW) backgrounds from the early radiation era until the present day in modified theories of gravity. Comparing to general relativity (GR), we study the effects that modified gravity parameters, such as the GW friction α M and the tensor speed excess α T, have on the present-day GW spectrum. We use both the WKB estimate, which provides an analytical description but fails at superhorizon scales, and numerical simulations that allow us to go beyond the WKB approximation. We show that a constant α T makes relatively insignificant changes to the GR solution, especially taking into account the constraints on its value from GW observations by the LIGO-Virgo collaboration, while α M can introduce modifications to the spectral slopes of the GW energy spectrum in the low-frequency regime depending on the considered time evolution of α M. The latter effect is additional to the damping or growth occurring equally at all scales that can be predicted by the WKB approximation. In light of the recent observations by pulsar timing array (PTA) collaborations, and the potential observations by future detectors such as SKA, LISA, DECIGO, BBO, or ET, we show that, in most of the cases, constraints cannot be placed on the effects of α M and the initial GW energy density ℰ* GW separately, but only on the combined effects of the two, unless the signal is observed at different frequency ranges. In particular, we provide some constraints on the combined effects from the reported PTA observations.

Pulsars are very stable clocks in space which have many applications to problems in physics and astrophysics. Observations of double-neutron-star binary systems have given the first observational evidence for the existence of gravitational waves (GWs) and shown that Einstein's general theory of relativity is an accurate description of gravitational interactions in the regime of strong gravity. Observations of a large sample of pulsars spread across the celestial sphere forming a 'Pulsar Timing Array' (PTA), can in principle enable a positive detection of the GW background in the Galaxy. The Parkes Pulsar Timing Array (PPTA) is making precise timing measurements of 20 millisecond pulsars at three radio frequencies and is approaching the level of timing precision and data spans which are needed for GW detection. These observations will also allow us to establish a 'Pulsar Timescale' and to detect or limit errors in the Solar System ephemerides used in pulsar timing analyses. Combination of PPTA data with that of other groups to form an International Pulsar Timing Array (IPTA) will enhance the sensitivity to GWs and facilitate reaching other PTA goals. The principal source of GWs at the nanoHertz frequencies to which PTAs are sensitive is believed to be super-massive binary black holes in the cores of distant galaxies. Current results do not signficantly limit models for formation of such black-hole binary systems, but in a few years we expect that PTAs will either detect GWs or seriously constrain current ideas about black-hole formation and galaxy mergers. Future instruments such as the Square Kilometre Array (SKA) should not only detect GWs from astrophysical sources but also enable detailed studies of the sources and the gravitational theories used to account for the GW emission.

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.

Second generation interferometric gravitational wave detectors, such as Advanced LIGO and Advanced Virgo, are expected to begin operation by 2015. Such instruments plan to reach sensitivities that will offer the unique possibility to test General Relativity in the dynamical, strong field regime and investigate departures from its predictions, in particular using the signal from coalescing binary systems. We introduce a statistical framework based on Bayesian model selection in which the Bayes factor between two competing hypotheses measures which theory is favored by the data. Probability density functions of the model parameters are then used to quantify the inference on individual parameters. We also develop a method to combine the information coming from multiple independent observations of gravitational waves, and show how much stronger inference could be. As an introduction and illustration of this framework - and a practical numerical implementation through the Monte Carlo integration technique of nested sampling - we apply it to gravitational waves from the inspiral phase of coalescing binary systems as predicted by General Relativity and a very simple alternative theory in which the graviton has a non-zero mass. This method can trivially (and should) be extended to more realistic and physically motivated theories.