A Simple Family of Analytical Trumpet Slices of the Schwarzschild Spacetime

We describe a simple family of analytical coordinate systems for the Schwarzschild spacetime. The coordinates penetrate the horizon smoothly and are spatially isotropic. Spatial slices of constant coordinate time $t$ feature a trumpet geometry with an asymptotically cylindrical end inside the horizon at a prescribed areal radius $R_0$ (with $0<R_{0}\leq M$) that serves as the free parameter for the family. The slices also have an asymptotically flat end at spatial infinity. In the limit $R_{0}=0$ the spatial slices lose their trumpet geometry and become flat -- in this limit, our coordinates reduce to Painlev\'e-Gullstrand coordinates.Comment: 7 pages, 3 figure

Equilibrium initial data for moving puncture simulations: The stationary 1+log slicing

We propose and explore a "stationary 1+log" slicing condition for the construction of solutions to Einstein's constraint equations. For stationary spacetimes, these initial data will give a stationary foliation when evolved with "moving puncture" gauge conditions that are often used in black hole evolutions. The resulting slicing is time-independent and agrees with the slicing generated by being dragged along a time-like Killing vector of the spacetime. When these initial data are evolved with moving puncture gauge conditions, numerical errors arising from coordinate evolution are minimized. In the construction of initial data for binary black holes it is often assumed that there exists an approximate helical Killing vector that generates the binary's orbit. We show that, unfortunately, 1+log slices that are stationary with respect to such a helical Killing vector cannot be asymptotically flat, unless the spacetime possesses an additional axial Killing vector.Comment: 20 pages, 3 figures, published versio

Method to estimate ISCO and ring-down frequencies in binary systems and consequences for gravitational wave data analysis

Recent advances in the description of compact binary systems have produced gravitational waveforms that include inspiral, merger and ring-down phases. Comparing results from numerical simulations with those of post-Newtonian (PN), and related, expansions has provided motivation for employing PN waveforms in near merger epochs when searching for gravitational waves and has encouraged the development of analytic fits to full numerical waveforms. The models and simulations do not yet cover the full binary coalescence parameter space. For these yet un-simulated regions, data analysts can still conduct separate inspiral, merger and ring-down searches. Improved knowledge about the end of the inspiral phase, the beginning of the merger, and the ring-down frequencies could increase the efficiency of both coherent inspiral-merger-ring-down (IMR) searches and searches over each phase separately. Insight can be gained for all three cases through a recently presented theoretical calculation, which, corroborated by the numerical results, provides an implicit formula for the final spin of the merged black holes, accurate to within 10% over a large parameter space. Knowledge of the final spin allows one to predict the end of the inspiral phase and the quasinormal mode ring-down frequencies, and in turn provides information about the bandwidth and duration of the merger. In this work we will discuss a few of the implications of this calculation for data analysis.Comment: Added references to section 3 14 pages 5 figures. Submitted to Classical and Quantum Gravit

Distributional sources for black hole initial data

Black hole initial data is usually produced using Bowen-York type puncture initial data or by applying an excision boundary condition. The benefits of the Bowen-York initial data are the ability to specify the spin and momentum of the system as parameters of the initial data. In an attempt to extend these benefits to other formulations of the Einstein constraints, the puncture method is reformulated using distributions as source terms. It is shown how the Bowen-York puncture black hole initial data and the trumpet variation is generated by distributional sources. A heuristic argument is presented to argue that these sources are the general sources of spin and momentum. In order to clarify the meaning of other distributional sources, an exact family of initial data with generalized sources to the Hamiltonian constraint are studied; spinning trumpet black hole initial data and black hole initial data with higher order momentum sources are also studied.Comment: Code available at https://github.com/SwampWalker/LeapingMonke

Complete phenomenological gravitational waveforms from spinning coalescing binaries

The quest for gravitational waves from coalescing binaries is customarily performed by the LIGO-Virgo collaboration via matched filtering, which requires a detailed knowledge of the signal. Complete analytical coalescence waveforms are currently available only for the non-precessing binary systems. In this paper we introduce complete phenomenological waveforms for the dominant quadrupolar mode of generically spinning systems. These waveforms are constructed by bridging the gap between the analytically known inspiral phase, described by spin Taylor (T4) approximants in the restricted waveform approximation, and the ring-down phase through a phenomenological intermediate phase, calibrated by comparison with specific, numerically generated waveforms, describing equal mass systems with dimension-less spin magnitudes equal to 0.6. The overlap integral between numerical and phenomenological waveforms ranges between 0.95 and 0.99.Comment: Proceeding for the GWDAW-14 conference. Added reference in v

Conformally flat black hole initial data, with one cylindrical end

We give a complete analytical proof of existence and uniqueness of extreme-like black hole initial data for Einstein equations, which possess a cilindrical end, analogous to extreme Kerr, extreme Reissner Nordstrom, and extreme Bowen-York's initial data. This extends and refines a previous result \cite{dain-gabach09} to a general case of conformally flat, maximal initial data with angular momentum, linear momentum and matter.Comment: Minor changes and formula (21) revised according to the published version in Class. Quantum Grav. (2010). Results unchange

LISA as a dark energy probe

Recently it was shown that the inclusion of higher signal harmonics in the inspiral signals of binary supermassive black holes (SMBH) leads to dramatic improvements in parameter estimation with the Laser Interferometer Space Antenna (LISA). In particular, the angular resolution becomes good enough to identify the host galaxy or galaxy cluster, in which case the redshift can be determined by electromagnetic means. The gravitational wave signal also provides the luminosity distance with high accuracy, and the relationship between this and the redshift depends sensitively on the cosmological parameters, such as the equation-of-state parameter $w=p_{\rm DE}/\rho_{\rm DE}$ of dark energy. With a single binary SMBH event at $z < 1$ having appropriate masses and orientation, one would be able to constrain $w$ to within a few percent. We show that, if the measured sky location is folded into the error analysis, the uncertainty on $w$ goes down by an additional factor of 2-3, leaving weak lensing as the only limiting factor in using LISA as a dark energy probe.Comment: 11pages, 1 Table, minor changes in text, accepted for publication in Classical and Quantum Gravity (special issue for proceedings of 7th LISA symposium

Connecting Numerical Relativity and Data Analysis of Gravitational Wave Detectors

Gravitational waves deliver information in exquisite detail about astrophysical phenomena, among them the collision of two black holes, a system completely invisible to the eyes of electromagnetic telescopes. Models that predict gravitational wave signals from likely sources are crucial for the success of this endeavor. Modeling binary black hole sources of gravitational radiation requires solving the Eintein equations of General Relativity using powerful computer hardware and sophisticated numerical algorithms. This proceeding presents where we are in understanding ground-based gravitational waves resulting from the merger of black holes and the implications of these sources for the advent of gravitational-wave astronomy.Comment: Appeared in the Proceedings of 2014 Sant Cugat Forum on Astrophysics. Astrophysics and Space Science Proceedings, ed. C.Sopuerta (Berlin: Springer-Verlag

Can a combination of the conformal thin-sandwich and puncture methods yield binary black hole solutions in quasi-equilibrium?

We consider combining two important methods for constructing quasi-equilibrium initial data for binary black holes: the conformal thin-sandwich formalism and the puncture method. The former seeks to enforce stationarity in the conformal three-metric and the latter attempts to avoid internal boundaries, like minimal surfaces or apparent horizons. We show that these two methods make partially conflicting requirements on the boundary conditions that determine the time slices. In particular, it does not seem possible to construct slices that are quasi-stationary and avoid physical singularities and simultaneously are connected by an everywhere positive lapse function, a condition which must obtain if internal boundaries are to be avoided. Some relaxation of these conflicting requirements may yield a soluble system, but some of the advantages that were sought in combining these approaches will be lost.Comment: 8 pages, LaTeX2e, 2 postscript figure

Are moving punctures equivalent to moving black holes?

When simulating the inspiral and coalescence of a binary black-hole system, special care needs to be taken in handling the singularities. Two main techniques are used in numerical-relativity simulations: A first and more traditional one ``excises'' a spatial neighbourhood of the singularity from the numerical grid on each spacelike hypersurface. A second and more recent one, instead, begins with a ``puncture'' solution and then evolves the full 3-metric, including the singular point. In the continuum limit, excision is justified by the light-cone structure of the Einstein equations and, in practice, can give accurate numerical solutions when suitable discretizations are used. However, because the field variables are non-differentiable at the puncture, there is no proof that the moving-punctures technique is correct, particularly in the discrete case. To investigate this question we use both techniques to evolve a binary system of equal-mass non-spinning black holes. We compare the evolution of two curvature 4-scalars with proper time along the invariantly-defined worldline midway between the two black holes, using Richardson extrapolation to reduce the influence of finite-difference truncation errors. We find that the excision and moving-punctures evolutions produce the same invariants along that worldline, and thus the same spacetimes throughout that worldline's causal past. This provides convincing evidence that moving-punctures are indeed equivalent to moving black holes.Comment: 4 pages, 3 eps color figures; v2 = major revisions to introduction & conclusions based on referee comments, but no change in analysis or result