Black Hole Formation and Growth pp 192  Cite as
Black Hole Merging and Gravitational Waves
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Abstract
I was tasked with covering a wide swath of gravitational wave astronomy—including theory, observation, and data analysis—and to describe the detection techniques used to span the gravitational wave spectrum—pulsar timing, ground based interferometers and their future space based counterparts. For good measure, I was also asked to include an introduction to general relativity and black holes. Distilling all this material into nine lectures was quite a challenge. The end result is a highly condensed set of lecture notes that can be consumed in a few hours, but may take weeks to digest.
1 Introduction
In writing up these lecture notes I have mostly followed the order in which the material was presented in Saas Fee, with the exception of the discussion of the detectors, which have been grouped here in a single section. My goal is not to write a textbook on each topic—many excellent texts and review articles on general relativity and gravitational wave astronomy already exist (see e.g. [11, 16, 37, 40, 50]). Rather, I try to highlight the key concepts and techniques that underpin each topic. I also strive to provide a unified picture that emphasizes the similarities between pulsar timing, ground base detectors and space based detectors, and commonalities in how the data is analyzed across the spectrum and across source types.
2 General Relativity
The historical course that lead Einstein to develop the general theory of relativity had many twists and turns, but as Einstein reflected in 1922, one of his primary goals was to understand the equivalence between inertial mass and gravitational mass “It was most unsatisfactory to me that, although the relation between inertia and energy is so beautifully derived [in Special Relativity], there is no relation between inertia and weight. I suspected that this relationship was inexplicable by means of Special Relativity” [43]. Einstein found the resolution to this conundrum by adopting a geometrical picture that generalized Minkowski’s description of special relativity to allow for spacetime curvature.
2.1 Special Relativity
2.2 The Equivalence Principle
Einstein set out to incorporate this insight into a modification of special relativity that could account for gravitational effects. The connection to coordinate transformations suggested a geometrical approach, which caused Einstein to pay more attention to Minkowski’s geometrical formulation of special relativity. Einstein began by showing that the path of light seen by a uniformly accelerated observer could be interpreted in terms of spacetime geometry where the speed of light depends on position.
2.3 Tides and Curvature
2.4 Newtonian Gravity in Geometric Form
2.5 Einstein Equations
2.6 Black Holes
3 Gravitational Wave Theory
Gravitational waves are generated by flows of energymomentum. When you wave to someone you are generating gravitational waves, though with amplitudes that are far to weak to be detected using existing technologies. The waves we can detect come from violent astrophysical events where large concentrations of mass move at close to the speed on light.
3.1 Newtonian Limit Redux
3.2 Waves in Vacuum
3.3 Making Waves
3.4 Energy and Momentum of a Gravitational Wave
Gravitational waves carry energy and momentum away from a source, and through the nonlinearity of Einstein’s equations, become sources that modify the background geometry and even generate waves of their own. The energy and angular momentum carried by gravitational waves causes binary stars to spiral inward and eventually merge. The linear momentum carried by gravitational waves can lead to recoil kicks during black hole mergers that send the merged black hole racing away at thousands of kilometers per hour. The calculation of the energy and momentum carried by gravitational waves raises several subtle issues that deserve a more careful treatment that can be squeezed into these lectures, so here I sketch out the main results.
4 Gravitational Wave Detection
4.1 Photon Timing
The time that it takes a photon to propagate between two points in space will be perturbed by the presence of gravitational waves. Figure 11 illustrates the measurement principle behind pulsar timing, spacecraft doppler tracking and laser interferometers. The pulsar timing approach to gravitational wave detection operates directly on this principle. The highly regular radio pulses from a millisecond pulsar will arrive a little earlier or a little later than they would if no gravitational waves were perturbing the spacetime geometry. Spacecraft doppler tracking measures changes in the frequency of radio signals sent from Earth and transponded back from a satellite. Here the measurement is proportional to the time derivative of the photon propagation time. Laser interferometers measure the phase shifts imparted on a laser signal that is sent down two paths and reflected or transponded back to a common point where the phase of the two beams can be compared. The phase shift is directly proportional to the difference in propagation time long the two paths. To calculate the response of each detector type we only need to calculate the general expression for the change in propagation time caused by gravitational waves for photons propagating between two points in space. The response is then found by combining the effects along the entire photon path, which amounts to a single pass for pulsar timing, two passes for spacecraft doppler tracking and four passes for a Michelson interferometer. More complicated measurement paths, such as a Michelson interferometer with Fabry–Perot cavities, or a space interferometer using time delay interferometry, just require additional single passes to be added together.
5 Gravitational Wave Observatories
Gravitational wave detection efforts began in the 1950s with Joseph Weber’s development of acoustic “bar” detectors. Weber’s announcement of a detection in 1969, while ultimately discredited, spurred further interest, even prompting theorists such as Steven Hawking and his student Gary Gibbons to try their hand at gravitational wave detection! The possibility of using laser interferometry as a detection method was proposed in 1963, and the first experimental studies of this approach occurred in 1971. The following year, Rainer Weiss published a landmark study that laid out the basic design for a practical laser interferometer gravitational wave detector, paying particular attention to the various sources of noise and how they might be mitigated. The idea of launching a laser interferometer into space appeared a few years later in a report by Weiss, Bender, Pound and Misner. At around the same time, Davies, Anderson, Estabrook and Wahlquist were developing the idea of using spacecraft doppler tracking for gravitational wave detection, which gave Detweiler the idea of using pulsar timing to search for gravitational waves. By the end of the 1970s the basic idea behind the three major detection techniques being pursed today were in place: ground and space based interferometers, and pulsar timing (Fig. 16).
5.1 Ground Based Laser Interferometers
Ground based laser interferometers operate in the audio frequency band \(f\sim [10,10^4]\, \mathrm{Hz}\), where the primary targets are stellarremnant mergers of neutron stars and black holes. Other potential sources in the audio band include isolated distorted neutron stars, corecollapse supernovae, low mass Xray binaries, collapsars and cosmic strings.
The overall size and shape of the detectors follow from some simple considerations. As to the shape, a rightangle configuration is chosen since it maximizes the differential armlength change caused by a quadrupole radiation pattern, as is evident from Fig. 10. As to the size, the longer the arms the larger the response, at least until when the armlength becomes comparable to the gravitational wavelength, a condition which defines the transfer frequency, \(f_* = c/(2\pi L)\), where L is the optical path length. For \(f < f_*\) we have \(\varDelta T \sim h L/c\), so to get a large time delay we need a large detector. If the detector is too large we go outside the low frequency limit and the response is diminished. Setting a maximum frequency of 1 kHz defines an optimal size of around 50 km. But building a detector this large would be very costly, so instead resonant cavities are used to fold the light so that the signal gets built up over multiple bounces using much shorter arms—4 km for the LIGO detectors and 3 km for the Virgo detector.
There is a lot more that could be said about the operation of the LIGO and Virgo detectors that goes well beyond the brief description given here. Topics of particular importance that are not covered here are how the control system keeps the interferometers in resonance, and how the output from this control loop is used to calculate the calibrated strain. To learn more, see Abbott et al. [2], Izumi and Sigg [31].
5.2 Space Based Laser Interferometers
To detect signals below \(f\sim 1\) Hz we need to get away from gravity gradient and seismic noise. This can be achieved by launching a detector into deep space. The frequency of a gravitational wave signal scales inversely with the size of the source, so space based detectors can detect much larger systems, such as massive black hole mergers and stellar binaries on wide orbits.
The basic idea behind the LISA mission is to use laser interferometry to precisely track the distance between widely separated free flying proof masses. It is necessary to house the proof masses inside a spacecraft to protect them from nongravitational disturbances such as solar radiation pressure and the solar wind. The spacecraft also provide the platform to house the lasers, optical benches and telescopes for the interferometry system. Figure 22 shows a schematic of the LISA design. The LISA design employs two optical benches on each spacecraft, comprised of a free floating gold/platinum cube and a \(\sim \)25 cm diameter telescope to transmit and receive the laser signal along each arm. Laser signals are transmitted from each optical bench, producing a total of six laser links. The gold/platinum proof masses are housed in metal cages, and capacitive sensing is used to track the distance between the cubes and the sides of the cage. MicroNewton trusters gently maneuver the spacecraft to maintain separation between the proof masses and the cages.
5.3 Pulsar Timing Arrays
Nature has provided us with extremely accurate galactic clocks in the form of pulsars—rapidly rotating Neutron stars which emit beams of radio waves that sweep past the Earth as the Neutron star rotates. The first pulsar, PSR B1919\(+\)21, was discovered by Jocelyn Bell and Antony Hewish in 1967. The number of pulsars known today is approaching 2,000. Most of these are socalled “classical” pulsars, with spin periods of 0.1–10 s. In 1982 the first millisecond pulsar was discovered [5]. Millisecond pulsars have rotational period in the range of about 1–10 ms, and are thought to be classical pulsars that have been spunup by accreting material from a binary companion. The accretion is also thought to bury the magnetic field, which reduces the pulsar winds and results in a slower spin down rate. Millisecond pulsars typically have more consistent pulse profiles and more regular pulse periods than classical pulsars, making them much better clocks to use for gravitational wave detection. There are currently 300 known millisecond pulsars, and several new ones are discovered each year.
Using pulsars to detect gravitational waves sounds simple enough—gravitational waves should perturb the arrival time of the pulses and lead to the distinct correlation pattern predicted by Hellings and Downs—but in practice there are many challenges to overcome. First, both the pulsars and the radio receivers are in constant relative motion, so we have to account for a multitude of effects that contribute to changes in the light propagation time. Another issue is that the individual radio pulses have different shapes, and many thousands of pulses have to be stacked together to arrive at a consistent pulse profile that can be used for the timing. Telescope availability and constraints on observing schedules mean that the timing measurements are irregularly spaced with gaps of one two weeks between observations. The noise levels in each measurement can also vary widely.
6 Gravitational Waves from Binary Systems
Binary systems are a prime target for gravitational wave detectors. Binaries made up of ordinary stars are not so promising since they glom together before reaching significant orbital velocities. Compact stellar remnants, such as white dwarfs, Neutron stars and black holes are much better candidates, as are the supermassive black holes found near the centers of galaxies. Binary black hole system are able to reach orbital velocities close to the speed of light before the individual black holes merge to form a larger black hole. The binary dynamics, for both black holes and Neutron stars, is usually divided into three regions: inspiral, merger and ringdown. During the early inspiral the stars follow almost Keplerian orbits, but as the orbit shrinks due to the emission of gravitational waves the orbits become increasingly nonKeplerian, exhibiting interesting relativistic effects such as periastron precession and orbital plane precession. The merger is highly relativistic, and sources strong, dynamical gravitational fields. Neutron star mergers also involve high density material colliding at high velocities. Two black holes merge to form a single distorted black hole, which then sheds the distortions during the ringdown phase. The end state of a Neutron star merger is more complicated, and may involve the formation of a single massive Neutron star that later collapses to form a black hole. Both the massive Neutron star and the final black hole produce ringdown radiation.
In these lectures I will give a brief introduction to the PN expansion, and I will skip the description of the self force program, EOB and numerical relativity. Excellent reviews of these other approaches, along with a far more thorough treatment of the the PN approach, can be found in Poisson [44], Van de Meent [53], Poisson and Will [45], Buonanno and Sathyaprakash [10], Brügmann [8].
6.1 PostNewtonian Expansion
6.2 Circular Newtonian Binary
6.3 Stationary Phase Approximation
6.4 Eccentric Newtonian Binary
Going beyond the leading order Newtonian description of the orbital motion introduces qualitatively new effects, the most important being periapse precession, which enters at 1PN order.
6.5 Spinning Binaries
Figure 36 shows the plus and cross waveforms for a spinning binary computed at 2.5PN order. The system shown had \(m_1= 20 M_\odot \), \(m_2= 15 M_\odot \), \(e=0\) and spin magnitudes \(\chi _1 = \mathbf{S}_1/m_1^2=0.7\) and \(\chi _2 = \mathbf{S}_2/m_2^2=0.5\). The spins were misaligned with the orbital angular momentum such that configuration at \(t=0\) had \(\mathrm{acos}(\hat{L} \cdot \hat{S}_1) = 85^\circ \), \(\mathrm{acos}(\hat{L} \cdot \hat{S}_2) = 82^\circ \) and \(\mathrm{acos}(\hat{S}_1 \cdot \hat{S}_2) = 110^\circ \).
7 Science Data Analysis
The output from a gravitational wave detector is a time series d(t). Usually we have multiple detectors and hence multiple times series. The data can be aggregated into a vector \(\mathbf{d}\), with components that are labeled by detector name and a time stamp. Fundamentally, gravitational wave data analysis is timeseries analysis, and many of the standard tools of time series analysis get applied, such as bandpass filters, windows, FFTs, spectral estimators, wavelet transforms etc. In these lectures I will gloss over these low level (yet essential) data processing steps and focus on the higher level aspects of the analysis.
For binary mergers the waveform templates \(\mathbf{h}(\varvec{\theta })\) are built from the \(h_+,h_\times \) polarizations states computed using the techniques discussed in Sect. 6. The source frame waveforms have to be convolved with the instrument response as described in Sect. 4. For a fully general binary black hole system the parameter vector \(\varvec{\theta }\) will have 17 components. Seven of the parameters are time invariant: the two masses \(m_1,m_2\); the dimensionless spin magnitudes \(\chi _1 = \mathbf{S}_1/m_1^2\), \(\chi _2 = \mathbf{S}_2/m_2^2\), the sky location \((\theta ,\phi )\) and the luminosity distance to the source \(D_L\). The other ten parameters have to be referenced to some particular orbital separation, which defines a reference time \(t_*\). The parameters defined at \(t_*\) are the four spin components \(\hat{S}_1,\hat{S}_2\); the overall phase \(\phi _*\); the eccentricity e and periapse angle \(\phi _e\); and two angles that define the orientation of the total angular momentum vector \(\mathbf{J } = \mathbf{L} + \mathbf{S}_1 + \mathbf{S}_2\). There are many alternative ways to parameterize the signals. For example, the spin/orbit parameters can also be described in terms of the angles \((\theta _L, \phi _L)\), \((\theta _1,\phi _1)\) and \((\theta _2,\phi _2)\) between \(\mathbf{L}, \mathbf{S}_1,\mathbf{S}_2\) and \(\mathbf{J }\) at the reference time. The merger time \(t_c\) is often used to set the time reference, and the chirp mass \(\mathcal{M}\) and total mass M are often used in place of the individual masses \(m_1,m_2\). Priors can be placed on these parameters using information from past astronomical observations and from theoretical considerations. For example, we might assume that binary systems follow the distribution of galaxies on the sky, which at large distances goes over to a uniform distribution. The range of spin magnitudes can be limited to the region [0, 1] so as to avoid naked singularities.
7.1 Posterior Distributions, Bayesian Learning and Model Evidence
7.2 Maximum Likelihood and the Fisher Information Matrix
Before continuing the discussion of signal detection and parameter estimation, it is instructive to digress a little and consider maximum likelihood parameter estimation and error forecasting using the Fisher Information Matrix. These topics are usually discussed in the classical, or frequentist, approach to gravitational wave detection, but they are also closely related to Taylor series expansion of Bayesian posterior distributions.
7.3 Frequentist Detection Statistics
Figure 41 shows the theoretical probability distribution for the \(\varLambda \) detection statistic for pure noise, and for data containing a \(\mathrm{SNR}=5\) signal. A false alarm occurs when we incorrectly conclude there is a signal present. Setting a false alarm probability of <1% corresponds to requiring that \(\varLambda > 13.28\). A false dismissal occurs when we conclude there is no signal when one is indeed present. Using a \(1\%\) false alarm threshold implies that there is a \(4.38\%\) chance that we will falsely dismiss a \(\mathrm{SNR}=5\) signal. For the first detections of gravitational waves the LIGO and Virgo collaborations were very conservative, and demanded that the false alarm rate, i.e. the number of random events mistaken for gravitational wave signals per unit time, should be very small. Setting a false alarm rate (FAR) of one per 100,000 years over a one year stretch of observation corresponds to a false alarm probability of \(p_\mathrm{FA} = T_\mathrm{obs} \times \mathrm{FAR} = 10^{5}\).
An important caveat to the preceding discussion is that the distribution shown for the \(\varLambda (\varvec{\kappa })\) pertains for fixed values of the parameters \(\varvec{\kappa }\). In an actual search the correct parameter values are not known a priori, and they must be searched over to find the values that maximize the log likelihood. The probability distribution for the search statistic maximized over all parameters, \(\varLambda _\mathrm{max}\), does not follow a Rayleigh distribution, and in all but the simplest cases must be computed numerically.
7.4 Searches for Gravitational Waves
The approach to detecting gravitational waves vary between and within collaborations. For example, the Parkes Pulsar Timing Array (PPTA) collaboration have traditionally used frequentist techniques, as have groups within the LIGO and Virgo collaborations that search for compact binary mergers. In contrast, the North American NanoHertz Gravitational Observatory (NANOGrav) collaboration takes a predominantly Bayesian approach, as does the LIGOVirgo BayesWave group. For strong signals the likelihood is highly peaked around the maximum value, and the Bayesian and frequentist approaches yield very similar results. To keep the discussion focused, I will restrict attention to the template based searches for compact binary mergers of the kind performed by the LIGO and Virgo collaborations. As we saw in the previous section, the Bayesian evidence for a signal being present in stationary, Gaussian data can be approximated by the maximum likelihood statistic \(\varLambda \). In practice the LIGO and Virgo data are not perfectly stationary and Gaussian, and slightly different search statistics are used, and the probability distribution for these statistics under the noise hypothesis are derived empirically from the data. Since it is not known a priori if a given stretch of data contains a signal, the noise properties are determined by first scrambling the data to remove the possibility of detecting a signal. This is done by introducing relative time shifts to the data that are greater than the light travel time between the detectors, ensuring that any signals present appear as noise fluctuations in the shifted data. Just a few weeks of data can be used to simulate millions of years of signalfree observation, making it possible to estimate the probability distribution for the noise down to very small false alarm probabilities (or false alarm rates).
7.5 Bayesian Parameter Estimation
We have seen that Bayesian inference can be used to compute posterior distributions for the gravitational waveforms \(\mathbf{h}( \varvec{\theta })\) and the parameters \(\varvec{\theta }\) that describe the signal model, and additionally the model evidence. Bayes’ theorem tells us that once the signal model and likelihood are defined and the prior distributions specified, the calculation of the posterior distributions and evidence comes down to computing a challenging multidimensional integral. It is only in the last two decades that efficient computational techniques, coupled with a increase in microprocessor speed, have made it possible to carry out the necessary computations for realworld applications. Bayesian inference is now rapidly supplanting classical (frequentist) statistics in many branches of science, including gravitational wave astronomy. There are two main approaches used to carry out the Bayesian computation. The first is the Markov Chain Monte Carlo (MCMC) approach [7, 24], which has its roots in statistical mechanics, and the second is Nested Sampling [52], which uses a stochastic Lebesgue integration technique. The MCMC approach produces samples from the posterior distribution without directly evaluating the evidence integral, while Nested Sampling computes the evidence without directly sampling the posterior distribution. With a little extra work the MCMC approach can be used to compute the evidence, and the posterior distributions can be recovered as a byproduct of the Nested Sampling approach, so both methods provide a comprehensive framework in which to carry out Bayesian inference. In my own research I exclusively use the MCMC approach as I find it to be better suited to the kinds of models I work with, which are generally of the transdimensional variety. Transdimensional modeling expands the usual sampling of model parameters to sampling across models in a large model space. Here I will focus on the MCMC approach as it is the one I am most familiar with.

propose a new state \(\mathbf{y} \sim q(\mathbf{y}  \mathbf{x}_i)\)

evaluate the MH ratio \(H(\mathbf{y}  \mathbf{x}_i)\)

draw a random deviate \(\alpha \sim U(0,1)\)

if \(H(\mathbf{y}  \mathbf{x}_i) > \alpha \) accept the new state, \(\mathbf{x}_{i+1}=\mathbf{y}\), otherwise \(\mathbf{x}_{i+1}=\mathbf{x}_i\)

increment \(i\rightarrow i+1\) and repeat
Assuming the process has converged, the collection of samples \(\{ \mathbf{x}_1, \mathbf{x}_2, \ldots \}\) generated by this algorithm represent fairs draws from the posterior distribution \(p(\mathbf{x}  \mathbf{d}, M)\). The posterior samples can be used to estimate confidence intervals etc. It is often necessary to discard some number of samples from the beginning of the chain since it can take many iterations before the chain locks onto the region of high posterior density (known as the burnin phase). With efficient proposal distributions the burnin phase can be kept very short. The number of iterations needed depends on several factors. One factor is the degree of correlation between successive samples. The MH procedure generates correlated samples, and the degree of correlation can be measured by computing, for example, the autocorrelation length for each parameter. The number of independent samples can be estimated by dividing the total number of samples by the autocorrelation length of the most highly correlated parameter. But then there is the question of how many independent samples are needed, to which the answer depends on what you want to compute, and to what accuracy. For example, it takes many more samples to estimate a \(95\%\) credible region to \(1\%\) relative error than it does to estimate a \(90\%\) credible region to \(10\%\) relative error. The cost also increases with dimension, for example computing credible regions for the 2d sky position of a source takes many more samples than computing the equivalent credible region for just the azimuthal angle.
The most important ingredient in a MCMC implementation is the proposal distribution. From (233) we see that the ideal proposal distribution would be the target distribution, \(q(\mathbf{x}  \mathbf{y}) = \pi (\mathbf{x})\), since then \(H(\mathbf{y}  \mathbf{x})= 1\) and every proposed jump would be accepted, and each sample would be independent. But if we knew the target distribution (in our case the posterior distribution), and how to draw from it, there would be no need to perform the MCMC! In lieu of using the posterior distribution, we can instead compute approximations to the posterior distribution and use those as proposal distributions. For example, we can approximate the posterior distribution in the neighborhood of a local maximum using multivariate normal distributions with correlation matrices given by the inverse of the Fisher information matrix, as was done in Eqs. (208) and (209). To do this we need to locate maxima of the likelihood, which can be done using the algebraically maximized log likelihood employed in the searches (see Sect. 7.4), and either a grid search, or more efficient maximization schemes such as random restart hill climbers, particle swarms, or genetic algorithms. Finding maxima of the likelihood surface can be computationally challenging, especially when the model dimension its high and/or the likelihood is expensive to compute, making it necessary to reduce the parameter dimension by ignoring less important parameters, and by using approximations to the likelihood function that are less expensive to compute. These approximate maps of the likelihood surface make for good global proposal distributions that can help the MCMC explore all the modes of a multimodal posterior distribution.
The spacing of the temperature ladder and the temperate range covered must be carefully chosen. A good rule of thumb is that the hottest chain should have \(\beta \approx 1/\mathrm{SNR}^2\). The reasoning is that \(\beta \) factor rescales the noise weighted inner product such that the effective signaltonoise is \(\mathrm{SNR}^2_\beta = \beta \, \mathrm{SNR}^2\), and we want \(\mathrm{SNR}^2_\beta \approx 1\) for the hottest chain, rending the annealed likelihood sufficiently flat that the hot chain explores the full prior volume. The spacing of the chains has to be chosen such that chain exchanges are often accepted. If the chains are too widely spaced the interchain exchange probability gets very small, and the chains stop communicating. Conversely, if the chains are spaced too closely it takes a prohibitively large number of chains to cover the necessary temperature range. If the likelihood is well approximated by a multivariate normal distribution it can be shown that the optimal spacing in geometric: \(\beta _{i+1} = c \, \beta _i\) for some constant c. For the more complicated likelihood surfaces encounter in realworld analyses it is often necessary to use adaptive schemes to find the optimal placement of the temperature ladder. For simple examples a geometric spacing with \(c=0.8\) is usually a good choice, which for a \(\mathrm{SNR}=20\) signal requires \(N_c = 2 \log (\mathrm{SNR})/\log c = 27\) chains for full coverage.
7.6 Worked Example—Sinusoidal Signal
Signal Search
The probability distribution for the search statistic, \(\varLambda _\mathrm{max}\), maximized over \(A_0,\phi _0\) and \(f_0\), was determined empirically by repeating the search using \(10^4\) simulated noise realizations. The distributions are displayed in Fig. 46 for pure noise, and for noise coadded to a signal with \(\mathrm{SNR}=7\). Setting a \(1\%\) false alarm rate yields a detection threshold of \(\varLambda _\mathrm{max} = 22\), and a false dismissal probability of \(0.5\%\) for signals with \(\mathrm{SNR} = 7\).
The output of the grid search over \(f_0\) for the simulated data set is shown in Fig. 47. The maximum likelihood template had \(f_0=0.999\), \(A_0=0.898\) and \(\phi _0=2.75\). The match between this template and the injected signal was \(\mathrm{M}=0.98\). The false alarm probability is too low to be reliably estimated from the empirically derived probability distribution for the noise hypothesis shown in Fig. 46, but the significance is greater than \(3\sigma \) (Gaussian equivalent, \(p_\mathrm{FA} < 0.3\%\)).
Parameter Estimation
The posterior distributions for this idealized example were monomodal and well approximated by a multivariate normal distribution, and the full MCMC machinery we employed was not needed. We can however make the problem a little more challenging by widening the prior range on the amplitude to include negative values, \(A_0 \in [10,10]\) which results in a multimodal likelihood surface since solutions with parameters \((A_0, \phi _0+\pi )\) produce identical likelihoods to those with parameters \((A_0,\phi _0)\). Figure 50 shows the posterior distributions for the noise and signal parameters when the amplitude is allowed to take negative values. The combination of the global proposal and parallel tempering allow the MCMC algorithm to fully explore both modes. Without parallel tempering or the global proposal the chain remains stuck on a single mode of the posterior.
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