Constraint Damping in First-Order Evolution Systems for Numerical Relativity

A new constraint suppressing formulation of the Einstein evolution equations is presented, generalizing the five-parameter first-order system due to Kidder, Scheel and Teukolsky (KST). The auxiliary fields, introduced to make the KST system first-order, are given modified evolution equations designed to drive constraint violations toward zero. The algebraic structure of the new system is investigated, showing that the modifications preserve the hyperbolicity of the fundamental and constraint evolution equations. The evolution of the constraints for pertubations of flat spacetime is completely analyzed, and all finite-wavelength constraint modes are shown to decay exponentially when certain adjustable parameters satisfy appropriate inequalities. Numerical simulations of a single Schwarzschild black hole are presented, demonstrating the effectiveness of the new constraint-damping modifications.Comment: 11 pages, 5 figure

Evolution of Magnetic Fields in Freely Decaying Magnetohydrodynamic Turbulence

We study the evolution of magnetic fields in freely decaying magnetohydrodynamic turbulence. By quasi-linearizing the Navier-Stokes equation, we solve analytically the induction equation in quasi-normal approximation. We find that, if the magnetic field is not helical, the magnetic energy and correlation length evolve in time respectively as E_B \propto t^{-2(1+p)/(3+p)} and \xi_B \propto t^{2/(3+p)}, where p is the index of initial power-law spectrum. In the helical case, the magnetic helicity is an almost conserved quantity and forces the magnetic energy and correlation length to scale as E_B \propto (log t)^{1/3} t^{-2/3} and \xi_B \propto (log t)^{-1/3} t^{2/3}.Comment: 4 pages, 2 figures; accepted for publication in PR

Making use of geometrical invariants in black hole collisions

We consider curvature invariants in the context of black hole collision simulations. In particular, we propose a simple and elegant combination of the Weyl invariants I and J, the {\sl speciality index} ${\cal S}$. In the context of black hole perturbations $\cal S$ provides a measure of the size of the distortions from an ideal Kerr black hole spacetime. Explicit calculations in well-known examples of axisymmetric black hole collisions demonstrate that this quantity may serve as a useful tool for predicting in which cases perturbative dynamics provide an accurate estimate of the radiation waveform and energy. This makes ${\cal S}$ particularly suited to studying the transition from nonlinear to linear dynamics and for invariant interpretation of numerical results.Comment: 4 pages, 3 eps figures, Revte

On the energy and baseline optimization to study effects related to the δ-phase (CP-/T-violation) in neutrino oscillations at a neutrino factory

In this paper we discuss the detection of CP- and T-violation effects in the framework of a neutrino factory. We introduce three quantities, which are good discriminants for a non-vanishing complex phase (δ) in the 3 × 3 neutrino mixing matrix: Δδ, ΔCP and ΔT. We find that these three discriminants (in vacuum) all scale with L/Ev, where L is the baseline and Ev the neutrino energy. Matter effects modify the scaling, but these effects are large enough to spoil the sensitivity only for baselines larger than 5000 km. So, in the hypothesis of constant neutrino factory power (i.e., number of muons inversely proportional to muon energy), the sensitivity on the δ-phase is independent of the baseline chosen. Specially interesting is the direct measurement of T-violation from the "wrong-sign" electron channel (i.e., the ΔT discriminant), which involves a comparison of the ve → vμ and vμ → ve oscillation rates. However, the vμ → ve measurement requires magnetic discrimination of the electron charge, experimentally very challenging in a neutrino detector. Since the direction of the electron curvature has to be estimated before the start of the electromagnetic shower, low-energy neutrino beams and hence short baselines, are preferred. In this paper we show, as an example, the exclusion regions in the Δm212-δ plane using the ΔCP and ΔT discriminants for two concrete cases keeping the same L/Ev ratio (730 km/7.5 GeV and 2900 km/30 GeV). We obtain a similar excluded region provided that the electron detection efficiency is ∼20% and the charge confusion 0.1%. The Δm212 compatible with the LMA solar data can be tested with a flux of 5 × 1021 muons. We compare these results with the fit of the visible energy distributions. © 2002 Elsevier Science B.V. All rights reserved

The Lazarus project: A pragmatic approach to binary black hole evolutions

We present a detailed description of techniques developed to combine 3D numerical simulations and, subsequently, a single black hole close-limit approximation. This method has made it possible to compute the first complete waveforms covering the post-orbital dynamics of a binary black hole system with the numerical simulation covering the essential non-linear interaction before the close limit becomes applicable for the late time dynamics. To determine when close-limit perturbation theory is applicable we apply a combination of invariant a priori estimates and a posteriori consistency checks of the robustness of our results against exchange of linear and non-linear treatments near the interface. Once the numerically modeled binary system reaches a regime that can be treated as perturbations of the Kerr spacetime, we must approximately relate the numerical coordinates to the perturbative background coordinates. We also perform a rotation of a numerically defined tetrad to asymptotically reproduce the tetrad required in the perturbative treatment. We can then produce numerical Cauchy data for the close-limit evolution in the form of the Weyl scalar $\psi_4$ and its time derivative $\partial_t\psi_4$ with both objects being first order coordinate and tetrad invariant. The Teukolsky equation in Boyer-Lindquist coordinates is adopted to further continue the evolution. To illustrate the application of these techniques we evolve a single Kerr hole and compute the spurious radiation as a measure of the error of the whole procedure. We also briefly discuss the extension of the project to make use of improved full numerical evolutions and outline the approach to a full understanding of astrophysical black hole binary systems which we can now pursue.Comment: New typos found in the version appeared in PRD. (Mostly found and collected by Bernard Kelly

The close limit from a null point of view: the advanced solution

We present a characteristic algorithm for computing the perturbation of a Schwarzschild spacetime by means of solving the Teukolsky equation. We implement the algorithm as a characteristic evolution code and apply it to compute the advanced solution to a black hole collision in the close approximation. The code successfully tracks the initial burst and quasinormal decay of a black hole perturbation through 10 orders of magnitude and tracks the final power law decay through an additional 6 orders of magnitude. Determination of the advanced solution, in which ingoing radiation is absorbed by the black hole but no outgoing radiation is emitted, is the first stage of a two stage approach to determining the retarded solution, which provides the close approximation waveform with the physically appropriate boundary condition of no ingoing radiation.Comment: Revised version, published in Phys. Rev. D, 34 pages, 13 figures, RevTe

The importance of precession in modelling the direction of the final spin from a black-hole merger

The prediction of the spin of the black hole resulting from the merger of a generic black-hole binary system is of great importance to study the cosmological evolution of supermassive black holes. Several attempts have been recently made to model the spin via simple expressions exploiting the results of numerical-relativity simulations. Here, I first review the derivation of a formula, proposed in Barausse & Rezzolla, Apj 704 L40, which accurately predicts the final spin magnitude and direction when applied to binaries with separations of hundred or thousands of gravitational radii. This makes my formula particularly suitable for cosmological merger-trees and N-body simulations, which provide the spins and angular momentum of the two black holes when their separation is of thousands of gravitational radii. More importantly, I investigate the physical reason behind the good agreement between my formula and numerical relativity simulations, and nail it down to the fact that my formula takes into account the post-Newtonian precession of the spins and angular momentum in a consistent manner.Comment: 6 pages, 2 figures. Panel added to fig 2, discussion extended to comply with referee's comments. Version accepted for publication as proceeding of the 8th Amaldi International Conference on Gravitational Waves, NYC, 21-26 June 200

Plunge waveforms from inspiralling binary black holes

We study the coalescence of non-spinning binary black holes from near the innermost stable circular orbit down to the final single rotating black hole. We use a technique that combines the full numerical approach to solve Einstein equations, applied in the truly non-linear regime, and linearized perturbation theory around the final distorted single black hole at later times. We compute the plunge waveforms which present a non negligible signal lasting for $t\sim 100M$ showing early non-linear ringing, and we obtain estimates for the total gravitational energy and angular momentum radiated.Comment: Corrected typos in the radiated ang momentum and frequenc

A perturbative solution for gravitational waves in quadratic gravity

We find a gravitational wave solution to the linearized version of quadratic gravity by adding successive perturbations to the Einstein's linearized field equations. We show that only the Ricci squared quadratic invariant contributes to give a different solution of those found in Einstein's general relativity. The perturbative solution is written as a power series in the $\beta$ parameter, the coefficient of the Ricci squared term in the quadratic gravitational action. We also show that, for monochromatic waves of a given angular frequency $\omega$, the perturbative solution can be summed out to give an exact solution to linearized version of quadratic gravity, for $0<\omega<c/\mid\beta\mid^{1/2}$. This result may lead to implications to the predictions for gravitational wave backgrounds of cosmological origin.Comment: 9 pages, to appear in CQ

Testing the Isotropy of the Universe with Type Ia Supernovae

We analyze the magnitude-redshift data of type Ia supernovae included in the Union and Union2 compilations in the framework of an anisotropic Bianchi type I cosmological model and in the presence of a dark energy fluid with anisotropic equation of state. We find that the amount of deviation from isotropy of the equation of state of dark energy, the skewness \delta, and the present level of anisotropy of the large-scale geometry of the Universe, the actual shear \Sigma_0, are constrained in the ranges -0.16 < \delta < 0.12 and -0.012 < \Sigma_0 < 0.012 (1\sigma C.L.) by Union2 data. Supernova data are then compatible with a standard isotropic universe (\delta = \Sigma_0 = 0), but a large level of anisotropy, both in the geometry of the Universe and in the equation of state of dark energy, is allowed.Comment: 12 pages, 7 figures, 2 tables. Union2 analysis added. New references added. To appear in Phys. Rev.