School of Physics - Theses
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ItemStrong gravitational fields and radiation from neutron stars( 2017)Part 1 of this thesis is dedicated to the study of the gravitational field in the strong field regime. In particular, we focus on modified theories of gravity, exploring the implications of high-energy corrections to Einstein's equations in the context of black holes and gravitational waves. Chapter 1 offers a literature review of general relativity and modified theories of gravity. In general relativity, multipole moment constructions allow one to decompose a general metric tensor in order to analyse the physical properties of the solution in detail. For example, the Kerr metric has two free parameters in it, which can be identified as the mass and angular momentum of the black hole through multipole analysis. In Chapter 2 we generalise these multipole moment constructions, with the goal of classifying black holes in modified theories of gravity. One can attempt to design experiments aimed at searching for `non-Einstein' black hole parameters. We explore whether the frequencies of quasi-periodic oscillations from galactic microquasars are consistent with a particular generalised-Kerr black hole solution, after a suitable interpretation of non-Kerr parameters has been made. We use X-ray measurements of the microquasars GRS $1915+105$ and GRO J$1665-40$ to constrain the properties of their black holes. The Ernst formalism of general relativity is a tool used to reduce the tensorial Einstein equations to a single, non-linear differential equation for a complex-valued scalar function. The formalism can be used to generate exact solutions to the Einstein equations. In Chapter 3 we present a generalisation to the $f(R)$ theory of gravity, which allows us to find new exact solutions. We find some exact solutions which represent the gravitational field outside of ellipsoidal, compact objects, and we find some solutions representing exact, solitonic gravitational waves which propagate with arbitrary phase velocity $v \neq c$. We show that neutron stars can be arbitrarily `hairy' in some modified theories of gravity, in the sense that the metric coefficients can depend on an arbitrary number of free parameters. Using the Ernst formalism developed in Chapter 3, we focus on the nature of causality in modified gravity. In Chapter 4, we show that some theories predict the existence of exact, faster-than-light gravitational waves, even when the linearised theory forbids them. The implication is that gravitational information may propagate at a different speed to electromagnetic information, which would have astrophysical consequences for highly relativistic systems such as black holes surrounded by plasma. We show that some theories permitting tachyonic gravitational waves are consistent with data coming from solar system and pulsar timing experiments. In Chapter 5 we investigate black hole structure in modified theories of gravity. We ask the question of what restrictions are placed on a modified theory of gravity if one assumes that event horizons are spherical; a feature of black holes expected from our intuitive understanding of gravitational collapse. We show that spherical horizons are guaranteed if a certain differential inequality is satisfied. This inequality can be used to place constraints on any given theory of gravity. In the context of $f(R)$ gravity, we show that the spherical-horizons condition is consistent with the `no-ghosts' (no negative energy modes) condition. Chapters 2 through 5 are therefore geared towards trying to reduce the pool of possible modified theories of gravity through various theoretical and empirical tests. When considering modified theories of gravity, one faces a degeneracy problem of sorts; non-Einstein gravity with ordinary matter might look like Einstein gravity but with exotic matter, or vice-versa. Part 2 of this thesis is concerned with the other half of the field equations; the matter portion. We focus in particular on the physics of neutron stars with asymmetric mass-density and fluid-velocity distributions, i.e. on continuous gravitational radiation from neutron stars. Our goal is to quantitatively explore the astrophysical properties of neutron stars from gravitational and electromagnetic observations. Chapter 6 serves as a literature review aimed at exploring the physics of neutron stars in various settings. Time-dependent multipole moments lead to gravitational radiation as the star spins. A detection of this radiation would then give us some information, otherwise invisible in the electromagnetic spectrum, about how the star is behaving. In Chapter 7 we consider quantum spin effects of the neutrons, protons, and electrons which comprise the stellar fluid. We find that the resulting quantum force terms may be large $(\chi \gtrsim 1)$ or small $(\chi \lesssim 10^{-3})$ in a neutron star environment depending on the degree of saturation physics, such as the formation of magnetic domains, limiting the value of the magnetic susceptibility $\chi$. We show that the terms are likely to be small except perhaps in magnetar $\left[|\boldsymbol{B}| \gtrsim 10^{13} \left(T_{e}/10^{9} \text{ K}\right) \text{ G}\right]$ environments, where paramagnetic forces can amplify (up to a factor $\sim 10$) the strength of any and all gravitational radiation. In some neutron stars, there is an observational discrepancy between magnetic field strengths inferred from cyclotron lines and spin-down estimates. It has been suggested that strong, non-dipole components may be present in the magnetic field structure, which would account for the apparent inconsistency. In Chapter 8 we investigate a non-ideal magnetohydrodynamic phenomena, known as Hall drift, in the context of neutron star models. In particular, we show that strong, non-dipole components can emerge naturally through Hall drift on timescales of $t \gtrsim 10^{4} \text{ yr}$. The Hall drift necessarily modifies the equilibrium structure of the star, which bolsters the expected gravitational wave luminosity. We find that, if Hall drift is responsible for the magnetic field strength discrepancy, old $(\gtrsim 10^{5} \text{ yr})$ radio pulsars should emit continuous gravitational wave signals which are large enough $(h_{0} \gtrsim 10^{-26})$ to be measurable in the near future with ground-based interferometers such as the Laser Interferometer Gravitational-Wave Observatory (LIGO). In accreting neutron star systems like low-mass X-ray binaries (LMXBs), the time-averaged spin-up torque from accretion is expected to add angular momentum to the neutron star at a rate of $N_{a} \approx \dot{M} \left( G M_{\star} R_{\star} \right)^{1/2}$, where $\dot{M}$ is the accretion rate. It is therefore puzzling that we observe no neutron stars spinning faster than $\nu_{\text{observed}} \lesssim 700 \text{ Hz}$, which is much less than the theoretical maximum set by the break-up limit $\nu_{\text{max}} \gtrsim 1.5 \text{ kHz}$. In Chapter 9 we consider mass-density asymmetries in the form of physical mounds (`magnetic mountains') which build up on neutron star surfaces, when matter is accreted from a main sequence companion. The built-up mountains reduce the global dipole moment of the neutron star by forcing magnetic field lines to buckle underneath infalling matter, and also excite gravitational radiation in the process. The gravitational radiation reaction associated with the magnetic mountain may be responsible for capping the spin frequency of the star. We investigate the characteristics of magnetic mountains when thermal conduction is treated for the first time in self-consistent, numerical simulations. We find that thermal conduction has the effect of softening the equation of state of accreted material on timescales $t \gtrsim 10 \text{ s}$, thereby amplifying the gravitational wave signal (factor $\sim 2$). This strengthens the argument for targeting LMXBs such as Sco X-1 in searches with facilities like LIGO. We show that the clumping of accreted matter leads to the formation of hot regions $(T \gtrsim 10^{8} \text{ K})$ throughout the mountain body, which has implications for type I X-ray burst activity. The excitation of inertial $(r-)$ modes within neutron stars leads to a fluid-velocity asymmetry, which generates a time-dependent current quadrupole moment. Depending on the equation of state, the resulting gravitational radiation may cause $r$-mode amplitudes to grow through an instability known as the Chandrasekhar-Friedman-Schutz (CFS) instability. In particular, it has been suggested that the gravitational radiation reaction resulting from the CFS instability may explain the observation that LMXBs have a narrow range of spin frequencies $0.1 < \nu / \left( \text{ kHz}\right) < 0.7$. However, for barotropic neutron stars, $r$-modes are often described by hyperbolic boundary-value problems (HBVPs). HBVPs are known to admit singularities and other pathological properties in certain circumstances, indicating that some aspects of the model may be unphysical. In Chapter 10 we present models for $r$-modes in nonbarotropic (stratified) neutron stars, and demonstrate that the HBVP nature of the problem cannot arise in certain, well-defined circumstances. We find that nonbarotropic $r$-modes are subject to the CFS instability for magnetic field strengths satisfying $|\boldsymbol{B}| \lesssim 10^{12} \text{ G}$, and spin frequencies in the range $3.0 \times 10^{-2} \leq \nu / \left( \text{ kHz} \right) \leq 1.5$.
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ItemGravitational waves from low-mass X-ray binaries: a search for Scorpius X-1( 2015)Gravitational wave astronomy is an exciting prospect within the grasp of advanced interferometric detectors such as the Laser Interferometer Gravitational-Wave Observatory (LIGO). Ground-based detectors are sensitive to a range of sources and wave forms, including persistent (i.e. continuous-wave) emission from non-axisymmetric rotating neutron stars in accreting binary systems. The notoriously weak signal amplitudes inherent to continuous-wave sources combined with the extensive parameter space volume introduced by a binary orbit when integrating over long observation times can be a formidable data analysis challenge. In this thesis we design, develop, test and implement a semi-coherent data analysis technique, known as the sideband method, to search for persistent gravitational waves from neutron stars in electromagnetically observed low-mass X-ray binaries (LMXBs). X-ray emission resulting from the accretion in these systems can be used as an indicator of the gravitational wave strain. The brightest LMXB Scorpius X-1 (Sco X-1) is one of the most promising potential sources of continuous gravitational wave emission. We describe the practical implementation of the sideband search for periodic gravitational waves from neutron stars in binary systems. The orbital motion of the neutron star causes frequency modulation in the matched filtering F-statistic. The sideband search is based on the incoherent summation of these F-statistic sidebands. For a known sky position, it uses electromagnetic measurements of the orbital period and semi-major axis to construct a sideband template, which is convolved with the F-statistic to produce a new detection statistic called the C-statistic. The analysis was tested on ten simulated low-mass X-ray binary (LMXB) signals which were hardware injected into LIGO data shortly after the end of the S5 run. Hardware injections are incorporated into the data by physical actuation of the test masses, to mimic an actual signal. The results validate the performance of the sideband algorithm on sources with a range of signal strengths. The search returns a clear detection for five of the ten signals with detected strain amplitudes as low as 3.48 × 10^{−24}. The injected amplitudes of the other five signals were below the expected sensitivity of the search. We present results of the sideband search for persistent gravitational wave emission from Sco X-1 in 10 days of LIGO S5 data ranging from 50-550 Hz. Candidates remaining after the removal of known noise lines and spurious noise artefacts were found to be consistent with noise at a 99% confidence level. We present Bayesian 95% confidence upper limits on the gravitational-wave strain amplitude of Sco X-1 using two different prior distributions: a standard one, with no a priori assumptions about the orientation of the spin axis of Sco X-1 relative to the observer; and an angle-restricted one, which uses a prior on the orientation derived from electromagnetic observations. Median strain upper limits of 1.3 × 10^{−24} and 8 × 10^{−25} are reported at 150 Hz for the standard and angle-restricted searches respectively. This analysis improves upon previous upper limits by factors of 1.4 and 2.3 for the standard and angle-restricted searches. We investigate the application of the search to other known LMXBs. After Sco X-1, the next most energetic X-ray source candidates include 4U 1820-30 and Cyg X-2 which, like Sco X-1, exhibit quasi-periodic oscillations. Although dimmer, burst sources such as 4U 1636-536 and XB 1658-298 could also be interesting targets, as the burst oscillations imply the existence of a hard neutron star surface. The computational efficiency of the search means that it can produce prompt results for LMXB searches, when the Advanced Detector Era dawns.
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ItemContinuous and stochastic gravitational wave emission from neutron star interior flows and oscillations( 2014)This thesis investigates continuous and stochastic gravitational wave signals from neutron star interior flows and oscillations. Neutron stars provide a unique laboratory to test physics at extremes of density, gravity, and magnetism. Gravitational waves directly probe the neutron star interior, carrying information about the properties of bulk nuclear matter. Glitches are rotational irregularities occasionally observed in pulsars. We calculate analytically the nonaxisymmetric Ekman spin-up flow following a glitch and its associated gravitational wave signal in the context of an idealised model. A large glitch with $\delta\Omega/\Omega = 10^{-4}$ in a pulsar rotating at $\sim 100$ Hz may be detectable by second- and third-generation interferometers. The signal depends on the inclination angle of the pulsar and the interior viscosity, compressibility, and stratification, which can be inferred gravitational wave data. Superfluid turbulence in neutron stars, driven by crust-core differential rotation, emits stochastic gravitational radiation. We calculate the stochastic background for a Universal neutron star population and two subpopulations: radio-loud pulsars and accreting millisecond pulsars. Non-detection of the stochastic background by LIGO implies an upper limit on the relaxation parameter $\tau_d = \Delta\Omega / \dot{\Omega}$, where $\dot{\Omega}$ is the spin-down rate, of $\tau_d \lesssim 10^5$ yr for radio-loud pulsars and $\tau_d \lesssim 10^7$ yr for accreting millisecond pulsars. Turbulent convection in main-sequence stars also emits gravitational radiation. We calculate the gravitational wave strain power spectral density for an individual star and a Universal stellar population. Due to its proximity, the signal from the Sun dominates the integrated background, but both fall well below the detection threshold of proposed space-based interferometers. Inside the gravitational wave near zone, the signal scales more steeply with distance ($\propto d^{-5}$) and is amplified relative to the far-zone signal ($\propto d^{-1}$). We calculate Rømer and Doppler timing residuals for a pulsar orbiting in the near zone of a high-mass main-sequence star and compare with observed of timing noise in three high-mass systems. The largest predicted root-mean-squared residuals, $\Delta T_{rms} = 2.8$ μs for PSR J0045-7319 at periastron, are a factor $\sim 10^3$ smaller than those observed. We propose a new gravitational wave detection statistic based on a modified form of higher criticism, a statistical method designed to indirectly detect a collection of sources too weak to be detected individually. Using higher criticism to reanalyse \mathcal{C}-statistic values for a simulated search of a low mass X-ray binary, we find higher criticism is sensitive to wave strain $\sim 6\%$ lower than the \mathcal{C}-statistic threshold. Higher criticism makes fewer assumptions about the source frequency and is more robust to error caused by accretion-driven phase wandering or an incorrect orbital period. Finally, we present preliminary results from two projects studying magnetar bursts. We propose a simple model of non-linear crust cracking and shear wave propagation to investigate the transient behaviour observed in the quasiperiodic oscillations detected in magnetar giant flares. The resulting frequency-time spectrograms contain features like frequency drifting, mode splitting and rotational phase dependence of oscillation frequencies. We also extend an existing smooth-particle-magnetohydrodynamics code to build a neutron star model. We validate the code against $f$-mode oscillation frequencies, observe rotational splitting, and present early progress towards implementing a rigid crust. Future applications include simulating crust-core magnetar oscillations as well as long-term spin down and post-glitch circulation.
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ItemCollective superfluid vortex dynamics and pulsar glitches( 2011)Pulsar glitches offer a way of studying the dynamics of cold, ultradense matter in systems of stellar dimensions, under extremes of density, temperature and magnetisation unattainable on Earth. This thesis aims to build a robust model of pulsar glitches, based on the superfluid vortex unpinning paradigm, which relates the physical parameters of the pulsar interior to the observed distribution of glitch sizes and waiting times (power laws and exponentials respectively). Our modelling efforts draw together knowledge about superfluid vortex dynamics and pinning, garnered from condensed matter and nuclear physics, the observational facts gathered by pulsar astronomers, and the theoretical framework of non-equilibrium stochastic systems, such as those exhibiting self-organised criticality. In each case, we emphasise the necessity of collective mechanisms in triggering avalanche-like vortex unpinning events. We begin by studying the dynamics of superfluid vortices from first principles, using numerical solutions of the Gross-Pitaevskii equation (GPE). We solve the GPE in the presence of a lattice of pinning sites, in a container that is decelerated at a constant rate, mimicking the electromagnetic spin-down torque on a pulsar. The superfluid spins down spasmodically, as vortices unpin and hop between pinning sites when the Magnus force, due to the lag between the superfluid and vortex line velocities, exceeds a threshold. Torque feedback between the superfluid and its container regulates the lag between the superfluid and crust, resulting in abrupt increases in the container angular velocity. We study how the statistics of the sizes and waiting times between spin-up events change with the mean and dispersion of pinning strengths, the electromagnetic spin-down torque, the relative number of vortices compared to pinning sites, and the ratio of the crust and superfluid moment of inertia - all parameters of interest in neutron stars. We find that mean glitch size increases with mean pinning strength and the ratio of the moments of inertia. It is independent of the relative number of pinning sites and vortices, suggesting that vortices move a characteristic distance before repinning, rather than repinning at the next available site. The mean waiting time decreases with the number of pinning sites and vortices, the ratio of the moments of inertia and the spin-down torque, and it increases with the width of the pinning strength distribution. In order to explain the broad range of observed glitch sizes using the vortex unpinning paradigm, a collective unpinning mechanism is required. Using numerical solutions of the GPE, we study how the unpinning of one vortex can cause other vortices to unpin. We identify two knock-on triggers: acoustic pulses emitted as a vortex repins, and the increased repulsive force between vortices locally, when an unpinned vortex approaches its nearest neighbours. In the second half of the thesis, we construct a suite of three large-scale stochastic models of glitches. We are inspired to prosecute this program by similarities between the statistics of archetypal self-organised critical systems, such as earthquakes and sand piles, and those of pulsar glitches. The essential features of the vortex dynamics observed in the GPE simulations are abstracted and condensed into a set of iterative rules that form the basis of automata and analytic glitch models. A cellular automaton model, in which vortices interact with nearest neighbours via the Magnus force, reveals that when all pinning sites are of the same strength, large-scale inhomogeneities in the pinned vortex distribution are necessary to produce a broad range of glitch sizes. In this case, glitch sizes and durations are power-law-distributed, and waiting times obey an exponential distribution. We find no evidence of history-dependent glitch sizes or aftershocks. A coherent noise model, based on a similar model developed to study atom hopping in glasses, in which pinning strength varies from site to site, but the pinned vortex distribution is assumed to be spatially homogeneous, exhibits power-law-distributed glitch sizes. Exponential waiting times are put in by hand, by assuming that the stress released in a glitch accumulates over exponentially-distributed time intervals. A wide range of pinning strengths is needed to find agreement with radio timing data. Mean pinning strength is found to decrease with increasing characteristic pulsar age. Finally, we construct a statistical model that tracks the vortex unpinning rate as a function of the stochastically fluctuating global lag between the superfluid and container. Monte-Carlo simulations and a jump-diffusion master equation reveal that a knock-on mechanism that is finely tuned with respect to the pinning strength, is essential to producing a broad range of glitch sizes. Estimates of the power dissipated in acoustic waves during repinning, and of the strength of the proximity effect, do not meet the fine-tuning criteria. We propose to extend this promising model to include nearest-neighbour interactions in the future, in the hope that this may lessen the need for fine tuning. The non-axisymmetric rearrangement of the superfluid velocity field during a vortex-avalanche-driven glitch is a source of gravitational radiation. We calculate the gravitational wave strain using the characteristic vortex motion observed in the GPE simulations. We set an upper bound on the wave strain of h ~ 10-23 for a glitch resulting from an unpinning avalanche of the maximum observed size. We also estimate the contribution to the stochastic gravitational wave background from the superposition of many glitches from a Galactic neutron star population. We place an upper bound on the signal-to-noise ratio of the background of ~ 10-5 for the Advanced LIGO (Laser Interferometer Gravitational-wave Observatory) detector. Detection of a gravitational wave signal from glitches can teach us about the physics of matter at nuclear densities, from the equation of state to transport coefficients like viscosity.