Numerical Waveforms Research Articles

Towards a Fourier domain waveform for non-spinning binaries with arbitrary eccentricity

Although the gravitational waves observed by advanced LIGO and Virgo are consistent with compact binaries in a quasi-circular inspiral prior to coalescence, eccentric inspirals are also expected to occur in nature. Due to their complexity, we currently lack ready-to-use, analytic waveforms in the Fourier domain valid for sufficiently high eccentricity, and such models are crucial to coherently extract weak signals from the noise. We here take the first steps to derive and properly validate an analytic waveform model in the Fourier domain that is valid for inspirals of arbitrary orbital eccentricity. As a proof-of-concept, we build this model to leading post-Newtonian order by combining the stationary phase approximation, a truncated sum of harmonics, and an analytic representation of hypergeometric functions. Through comparisons with numerical post-Newtonian waveforms, we determine how many harmonics are required for a faithful (matches above 99%) representation of the signal up to orbital eccentricities as large as 0.9. As a first byproduct of this analysis, we present a novel technique to maximize the match of eccentric signals over time of coalescence and phase at coalescence. As a second byproduct, we determine which of the different approximations we employ leads to the largest loss in match, which could be used to systematically improve the model because of our analytic control. The future extension of this model to higher post-Newtonian order will allow for an accurate and fast phenomenological hybrid that can account for arbitrary eccentricity inspirals and mergers.

Tracking Black Hole Kicks from Gravitational-Wave Observations.

Coalescing binary black holes emit anisotropic gravitational radiation. This causes a net emission of linear momentum that produces a gradual acceleration of the source. As a result, the final remnant black hole acquires a characteristic velocity known as recoil velocity or gravitational kick. The symmetries of gravitational wave emission are reflected in the interactions of the gravitational wave modes emitted by the binary. In this Letter, we make use of the rich information encoded in the higher-order modes of the gravitational wave emission to infer the component of the kick along the line of sight (or radial kick). We do this by performing parameter inference on simulated signals given by numerical relativity waveforms for nonspinning binaries using numerical relativity templates of aligned-spin (nonprecessing) binary black holes. We find that for suitable sources, namely those with mass ratio q≥2 and total mass M∼100 M_{⊙}, and for modest radial kicks of 120 km/s, the 90% credible intervals of our posterior probability distributions can exclude a zero kick at a signal-to-noise ratio of 15, using a single Advanced LIGO detector working at its early sensitivity. The measurement of a nonzero radial kick component would provide the first observational signature of net transport of linear momentum by gravitational waves away from their source.

Extensions to the Groningen ground-motion model for seismic risk calculations: component-to-component variability and spatial correlation

A bespoke ground-motion model has been developed for the prediction of response spectral accelerations, peak ground velocity and significant duration due to induced earthquakes in the Groningen gas field in the Netherlands. For applications to the calculation of risk to the exposed building stock, extensions to the model are required. The use of the geometric mean horizontal component in the ground-motion predictions and the arbitrary horizontal component for the building fragility functions requires the addition of component-to-component variability. A model for this variability has been developed that both reflects the strong horizontal polarisation of motions observed in many Groningen records obtained at short distances and the fact that the strong polarisation is unlikely to persist at larger magnitudes. The other extension of the model is the spatial correlation of ground motions for the calculation of aggregated risk, which can be approximated through simple rules for sampling the variance within site response zones. Making use of ground-motion recordings from several networks in the field and the results of finite difference waveform simulations, a Groningen-specific spatial correlation model has been developed. The new model also combines results from traditional variogram fitting approaches with a new method to infer spatial correlation lengths from observed variance reduction. The development of the new spatial correlation model relaxes the need to approximate spatial correlation through the sampling of site response, although the results obtained herein suggest that similar results could be obtained using either approach. The preliminary consideration of the numerical waveform modelling results in this study paves the way for significant extensions to be made for the modelling of spatial correlations and the decomposition of apparent spatial variability into systematic and random components within a fully non-ergodic framework .

Numerical investigation of a feed-forward linewidth reduction scheme using a mode-locked laser model of reduced complexity

We provide numerical verification of a feed-forward, heterodyne-based phase noise reduction scheme using single-sideband modulation that obviates the need for optical filtering at the output. The main benefit of a feed-forward heterodyne linewidth reduction scheme is the simultaneous reduction of the linewidth of all modes of a mode-locked laser (MLL) to that of a narrow-linewidth single-wavelength laser. At the heart of our simulator is an MLL model of reduced complexity. Importantly, the main issue being treated is the jitter of MLLs and we show how to create numerical waveforms that mimic the random-walk nature of timing jitter of pulses from MLLs. Thus, the model does not need to solve stochastic differential equations that describe the MLL dynamics, and the model calculates self-consistently the line-broadening of the modes of the MLL and shows good agreement with both the optical linewidth and jitter. The linewidth broadening of the MLL modes are calculated after the phase noise reduction scheme and we confirm that the phase noise contribution from the timing jitter still remains. Finally, we use the MLL model and phase noise reduction simulator within an optical communications system simulator and show that the phase noise reduction technique could enable MLLs as optical carriers for higher-order modulation formats, such as 16-state and 64-state quadrature amplitude modulation.

Observations of experimental and numerical waveforms of piezoelectric signals generated in bovine cancellous bone by ultrasound waves

Experimental and numerical waveforms of piezoelectric signals generated in the bovine cancellous bone by ultrasound waves at 1.0 MHz were observed. The experimental observations were performed using a “piezoelectric cell (PE-cell)”, in which an air-saturated cancellous bone specimen was electrically shielded. The PE-cell was used to receive burst ultrasound waves. The numerical observations were performed using a piezoelectric finite-difference time-domain (PE-FDTD) method, which was an elastic FDTD method with piezoelectric constitutive equations. The cancellous bone model was reconstructed from the three-dimensional X-ray microcomputed tomographic image of the specimen used in the experiments. Both experimental and numerical results showed that the repetitive piezoelectric signals could be generated by the multireflected ultrasound waves within the cancellous bone specimen. Moreover, it was shown that the output piezoelectric signal in the PE-cell could be the overlap of the local signals in the trabecular elements at various depths (or thicknesses) in the cancellous bone specimen.

Hangup effect in unequal mass binary black hole mergers and further studies of their gravitational radiation and remnant properties

We present the results of 74 new simulations of nonprecessing spinning black hole binaries with mass ratios $q=m_1/m_2$ in the range $1/7\leq q\leq1$ and individual spins covering the parameter space $-0.95\leq\alpha_{1,2}\leq0.95$ with one runs with spins of $\pm0.95$. We supplement those runs with 107 previous simulations to study the hangup effect in black hole mergers, i.e. the delay or prompt merger of spinning holes with respect to non spinning binaries. We perform the numerical evolution for typically the last ten orbits before the merger and down to the formation of the final remnant black hole. This allows us to study the hangup effect for unequal mass binaries leading us to identify the spin variable that controls the number of orbits before merger as $\vec{S}_{hu}\cdot{\hat{L}},$ where $\vec{S}_{hu}=(1+\frac12\frac{m_2}{m_1})\vec{S}_1+(1+\frac12\frac{m_1}{m_2})\vec{S}_2$. We also combine the total results of those 181 simulations to obtain improved fitting formulae for the remnant final black hole mass, spin and recoil velocity as well as for the peak luminosity and peak frequency of the gravitational strain, and find new correlations among them. This accurate new set of simulations enhances the number of available numerical relativity waveforms available for parameter estimation of gravitational wave observations.

Black hole spectroscopy: Systematic errors and ringdown energy estimates

The relaxation of a distorted black hole to its final state provides important tests of general relativity within the reach of current and upcoming gravitational wave facilities. In black hole perturbation theory, this phase consists of a simple linear superposition of exponentially damped sinusoids (the quasinormal modes) and of a power-law tail. How many quasinormal modes are necessary to describe waveforms with a prescribed precision? What error do we incur by only including quasinormal modes, and not tails? What other systematic effects are present in current state-of-the-art numerical waveforms? These issues, which are basic to testing fundamental physics with distorted black holes, have hardly been addressed in the literature. We use numerical relativity waveforms and accurate evolutions within black hole perturbation theory to provide some answers. We show that (i) a determination of the fundamental $l=m=2$ quasinormal mode to within $1\%$ or better requires the inclusion of at least the first overtone, and preferably of the first two or three overtones; (ii) a determination of the black hole mass and spin with precision better than $1\%$ requires the inclusion of at least two quasinormal modes for any given angular harmonic mode $(\ell,\,m)$. We also improve on previous estimates and fits for the ringdown energy radiated in the various multipoles. These results are important to quantify theoretical (as opposed to instrumental) limits in parameter estimation accuracy and tests of general relativity allowed by ringdown measurements with high signal-to-noise ratio gravitational wave detectors.

Application of infrared spectroscopy for the identification of squamous cell carcinoma (lung cancer). Preliminary study

ObjectivesThe aim of the study was to compare the spectra of squamous cell carcinoma (SCC) with normal pulmonary tissue (collected from the area outside of the cancer infiltration) and control samples (normal tissue collected from a patient without lung cancer) using FTIR spectroscopy. Material and methodsTumor tissue fragments obtained from 10 patients (males/females 7/3, G1-two pts, G2-five pts, G3-three pts) with diagnosed SCC, according to WHO 2015 classification were analyzed. FTIR spectroscopy spectrum (range of 1100–3500 cm−1) was obtained and analyzed using the Bruker Vertex 70 spectrometer. Statistical principal component analysis (PCA) was used for analysis of the obtained spectra. ResultsThe analysis of the obtained FTIR spectra revealed the existence of characteristic differences in cancer tissue compared to normal and control tissue. Comparison of individual spectra showed changes in the intensity of the absorbance and transfer of major absorption bands present in all of the tested range, suggesting a potential difference in metabolism in the analyzed SCC samples. The numerous waveform shifts in the range of proteins observed in these spectra, probably are the result of changes in their conformation and composition. The PCA analysis performed in the protein region did not show significant differences between normal lung tissue in patients with SCC and control group. Analysis of tumor tissue spectrum has confirmed its significant difference compared to the spectrum of tissue without tumor. ConclusionsAnalysis of changes in spectra in SCC well differentiates tumor tissue from normal, so FTIR spectroscopy can be helpful in SCC histopathological diagnosis and in the determination of microscopic radical tumor resection. Further studies in larger patient groups will determine the suitability of this method in routine SCC histopathological diagnostics.

Eccentric, nonspinning, inspiral, Gaussian-process merger approximant for the detection and characterization of eccentric binary black hole mergers

We present $\texttt{ENIGMA}$, a time domain, inspiral-merger-ringdown waveform model that describes non-spinning binary black holes systems that evolve on moderately eccentric orbits. The inspiral evolution is described using a consistent combination of post-Newtonian theory, self-force and black hole perturbation theory. Assuming eccentric binaries that circularize prior to coalescence, we smoothly match the eccentric inspiral with a stand-alone, quasi-circular merger, which is constructed using machine learning algorithms that are trained with quasi-circular numerical relativity waveforms. We show that $\texttt{ENIGMA}$ reproduces with excellent accuracy the dynamics of quasi-circular compact binaries. We validate $\texttt{ENIGMA}$ using a set of $\texttt{Einstein Toolkit}$ eccentric numerical relativity waveforms, which describe eccentric binary black hole mergers with mass-ratios between $1 \leq q \leq 5.5$, and eccentricities $e_0 \lesssim 0.2$ ten orbits before merger. We use this model to explore in detail the physics that can be extracted with moderately eccentric, non-spinning binary black hole mergers. We use $\texttt{ENIGMA}$ to show that GW150914, GW151226, GW170104, GW170814 and GW170608 can be effectively recovered with spinning, quasi-circular templates if the eccentricity of these events at a gravitational wave frequency of 10Hz satisfies $e_0\leq \{0.175,\, 0.125,\,0.175,\,0.175,\, 0.125\}$, respectively. We show that if these systems have eccentricities $e_0\sim 0.1$ at a gravitational wave frequency of 10Hz, they can be misclassified as quasi-circular binaries due to parameter space degeneracies between eccentricity and spin corrections. Using our catalog of eccentric numerical relativity simulations, we discuss the importance of including higher-order waveform multipoles in gravitational wave searches of eccentric binary black hole mergers.

Python Open source Waveform ExtractoR (POWER): an open source, Python package to monitor and post-process numerical relativity simulations

Numerical simulations of Einstein’s field equations provide unique insights into the physics of compact objects moving at relativistic speeds, and which are driven by strong gravitational interactions. Numerical relativity has played a key role to firmly establish gravitational wave astrophysics as a new field of research, and it is now paving the way to establish whether gravitational wave radiation emitted from compact binary mergers is accompanied by electromagnetic and astro-particle counterparts. As numerical relativity continues to blend in with routine gravitational wave data analyses to validate the discovery of gravitational wave events, it is essential to develop open source tools to streamline these studies. Motivated by our own experience as users and developers of the open source, community software, the Einstein Toolkit, we present an open source, Python package that is ideally suited to monitor and post-process the data products of numerical relativity simulations, and compute the gravitational wave strain at future null infinity in high performance environments. We showcase the application of this new package to post-process a large numerical relativity catalog and extract higher-order waveform modes from numerical relativity simulations of eccentric binary black hole mergers and neutron star mergers. This new software fills a critical void in the arsenal of tools provided by the Einstein Toolkit consortium to the numerical relativity community.

Numerical relativity waveform surrogate model for generically precessing binary black hole mergers

A generic, non-eccentric binary black hole (BBH) system emits gravitational waves (GWs) that are completely described by 7 intrinsic parameters: the black hole spin vectors and the ratio of their masses. Simulating a BBH coalescence by solving Einstein's equations numerically is computationally expensive, requiring days to months of computing resources for a single set of parameter values. Since theoretical predictions of the GWs are often needed for many different source parameters, a fast and accurate model is essential. We present the first surrogate model for GWs from the coalescence of BBHs including all $7$ dimensions of the intrinsic non-eccentric parameter space. The surrogate model, which we call NRSur7dq2, is built from the results of $744$ numerical relativity simulations. NRSur7dq2 covers spin magnitudes up to $0.8$ and mass ratios up to $2$, includes all $\ell \leq 4$ modes, begins about $20$ orbits before merger, and can be evaluated in $\sim~50\,\mathrm{ms}$. We find the largest NRSur7dq2 errors to be comparable to the largest errors in the numerical relativity simulations, and more than an order of magnitude smaller than the errors of other waveform models. Our model, and more broadly the methods developed here, will enable studies that would otherwise require millions of numerical relativity waveforms, such as parameter inference and tests of general relativity with GW observations.

An architecture for efficient gravitational wave parameter estimation with multimodal linear surrogate models

The recent direct observation of gravitational waves has further emphasized the desire for fast, low-cost, and accurate methods to infer the parameters of gravitational wave sources. Due to expense in waveform generation and data handling, the cost of evaluating the likelihood function limits the computational performance of these calculations. Building on recently developed surrogate models and a novel parameter estimation pipeline, we show how to quickly generate the likelihood function as an analytic, closed-form expression. Using a straightforward variant of a production-scale parameter estimation code, we demonstrate our method using surrogate models of effective-one-body and numerical relativity waveforms. Our study is the first time these models have been used for parameter estimation and one of the first ever parameter estimation calculations with multi-modal numerical relativity waveforms, which include all modes. Our grid-free method enables rapid parameter estimation for any waveform with a suitable reduced-order model. The methods described in this paper may also find use in other data analysis studies, such as vetting coincident events or the computation of the coalescing-compact-binary detection statistic.

A Surrogate model of gravitational waveforms from numerical relativity simulations of precessing binary black hole mergers

We present the first surrogate model for gravitational waveforms from the coalescence of precessing binary black holes. We call this surrogate model NRSur4d2s. Our methodology significantly extends recently introduced reduced-order and surrogate modeling techniques, and is capable of directly modeling numerical relativity waveforms without introducing phenomenological assumptions or approximations to general relativity. Motivated by GW150914, LIGO's first detection of gravitational waves from merging black holes, the model is built from a set of $276$ numerical relativity (NR) simulations with mass ratios $q \leq 2$, dimensionless spin magnitudes up to $0.8$, and the restriction that the initial spin of the smaller black hole lies along the axis of orbital angular momentum. It produces waveforms which begin $\sim 30$ gravitational wave cycles before merger and continue through ringdown, and which contain the effects of precession as well as all $\ell \in \{2, 3\}$ spin-weighted spherical-harmonic modes. We perform cross-validation studies to compare the model to NR waveforms \emph{not} used to build the model, and find a better agreement within the parameter range of the model than other, state-of-the-art precessing waveform models, with typical mismatches of $10^{-3}$. We also construct a frequency domain surrogate model (called NRSur4d2s_FDROM) which can be evaluated in $50\, \mathrm{ms}$ and is suitable for performing parameter estimation studies on gravitational wave detections similar to GW150914.

A numerical wave-maker for internal solitary waves with timely updated mass source/sink terms

On the basis of the two-dimensional Navier–Stokes equations, a new numerical method is proposed to generate internal solitary waves (ISWs) of expected parameters by adding a source term above the interface and a sink term below the interface into the continuity equation. Fluxes between the source and the sink are balanced to assure mass conservation, and the source/sink regions (the spatial windows over which the sink/source terms are placed) are adjusted as functions of time with the interface motion. Thus, the nonphysical trailing waves can be eliminated, which makes it easy to assure the prescribed ISW parameters. Moreover, a new layout is presented to avoid the difficulty of sizing and positioning the source/sink region, which has been proved to be an intrinsic drawback of the traditional mass source method. Numerical experiments are performed to validate the proposed method by analyzing the wave displacements and vertical profiles of velocity fields. It is shown that the numerical waveform remains stable with much less trailing waves than previous methods, and the numerical results are in good agreement with theoretical and experimental results. In addition, through sensitivity analysis, a reasonable method to determine the width of source/sink region is recommended.

Crustal velocity structure of central Gansu Province from regional seismic waveform inversion using firework algorithm

The firework algorithm (FWA) is a novel swarm intelligence-based method recently proposed for the optimization of multi-parameter, nonlinear functions. Numerical waveform inversion experiments using a synthetic model show that the FWA performs well in both solution quality and efficiency. We apply the FWA in this study to crustal velocity structure inversion using regional seismic waveform data of central Gansu on the northeastern margin of the Qinghai-Tibet plateau. Seismograms recorded from the moment magnitude (MW) 5.4 Minxian earthquake enable obtaining an average crustal velocity model for this region. We initially carried out a series of FWA robustness tests in regional waveform inversion at the same earthquake and station positions across the study region, inverting two velocity structure models, with and without a low-velocity crustal layer; the accuracy of our average inversion results and their standard deviations reveal the advantages of the FWA for the inversion of regional seismic waveforms. We applied the FWA across our study area using three component waveform data recorded by nine broadband permanent seismic stations with epicentral distances ranging between 146 and 437 km. These inversion results show that the average thickness of the crust in this region is 46.75 km, while thicknesses of the sedimentary layer, and the upper, middle, and lower crust are 3.15, 15.69, 13.08, and 14.83 km, respectively. Results also show that the P-wave velocities of these layers and the upper mantle are 4.47, 6.07, 6.12, 6.87, and 8.18 km/s, respectively.

A nearly analytic discrete method for solving the acoustic-wave equations in the frequency domain

We have developed a nearly analytic discrete (NAD) method to discretize frequency-domain acoustic-wave equations with an absorbing boundary condition. We evaluate in detail the discrete process of wave equations to derive a linear system. The sparse structure and eigenproperties of its coefficient matrix (also called the impedance matrix) were analyzed to reveal the intrinsic difficulty in solving the linear system efficiently. To accelerate the forward-modeling process in the frequency domain, we introduce a class of inexact rotated block triangular preconditioners incorporated with Krylov subspace methods to solve this linear system and test their numerical behaviors by comparing with other two commonly used preconditioners. To this end, we perform wavefield simulation by the NAD method and another two conventional numerical schemes in various media. Numerical dispersion analysis and waveform comparison are also implemented for these numerical schemes. Our results show the superiority of our proposed methods.

Duty‐cycle detection for noisy arbitrary waveforms using artificial neural networks

A novel technique for detecting duty cycles of noisy arbitrary waveforms by employing artificial neural networks trained with empirical moments of asynchronously sampled waveforms’ amplitudes is presented. The proposed technique is software-based and hence can be applied flexibly to numerous waveform types without requiring any hardware changes. Furthermore, in contrast to existing duty-cycle detection methods, this technique is capable of detecting duty cycles of noisy waveforms. Results obtained for square and sawtooth waveforms demonstrate wide duty-cycle detection range of 10–95% with mean absolute percentage errors < 0.4%.

Complexity and Hopf Bifurcation Analysis on a Kind of Fractional-Order IS-LM Macroeconomic System

On the basis of our previous research, we deepen and complete a kind of macroeconomics IS-LM model with fractional-order calculus theory, which is a good reflection on the memory characteristics of economic variables, we also focus on the influence of the variables on the real system, and improve the analysis capabilities of the traditional economic models to suit the actual macroeconomic environment. The conditions of Hopf bifurcation in fractional-order system models are briefly demonstrated, and the fractional order when Hopf bifurcation occurs is calculated, showing the inherent complex dynamic characteristics of the system. With numerical simulation, bifurcation, strange attractor, limit cycle, waveform and other complex dynamic characteristics are given; and the order condition is obtained with respect to time. We find that the system order has an important influence on the running state of the system. The system has a periodic motion when the order meets the conditions of Hopf bifurcation; the fractional-order system gradually stabilizes with the change of the order and parameters while the corresponding integer-order system diverges. This study has certain significance to policy-making about macroeconomic regulation and control.

A simple estimate of gravitational wave memory in binary black hole systems

A simple estimate is given of gravitational wave memory for the inspiral and merger of a binary black hole system. Here the memory is proportional to the total energy radiated and has a simple angular dependence. Estimates of this sort might be helpful as a consistency check for numerical relativity memory waveforms.

Binary neutron star merger simulations with different initial orbital frequency and equation of state

We present results from three-dimensional general relativistic simulations of binary neutron star coalescences and mergers using public codes. We considered equal mass models where the baryon mass of the two neutron stars is , described by four different equations of state (EOS) for the cold nuclear matter (APR4, SLy, H4, and MS1; all parametrized as piecewise polytropes). We started the simulations from four different initial interbinary distances (, and 60 km), including up to the last 16 orbits before merger. That allows us to show the effects on the gravitational wave (GW) phase evolution, radiated energy and angular momentum due to: the use of different EOS, the orbital eccentricity present in the initial data and the initial separation (in the simulation) between the two stars. Our results show that eccentricity has a major role in the discrepancy between numerical and analytical waveforms until the very last few orbits, where ‘tidal’ effects and missing high-order post-Newtonian coefficients also play a significant role. We test different methods for extrapolating the GW signal extracted at finite radii to null infinity. We show that an effective procedure for integrating the Newman–Penrose signal to obtain the GW strain h is to apply a simple high-pass digital filter to h after a time domain integration, where only the two physical motivated integration constants are introduced. That should be preferred to the more common procedures of introducing additional integration constants, integrating in the frequency domain or filtering before integration.