Numerical approach for high precision 3-D relativistic star models

A multi-domain spectral method for computing very high precision 3-D stellar models is presented. The boundary of each domain is chosen in order to coincide with a physical discontinuity (e.g. the star's surface). In addition, a regularization procedure is introduced to deal with the infinite derivatives on the boundary that may appear in the density field when stiff equations of state are used. Consequently all the physical fields are smooth functions on each domain and the spectral method is absolutely free of any Gibbs phenomenon, which yields to a very high precision. The power of this method is demonstrated by direct comparison with analytical solutions such as MacLaurin spheroids and Roche ellipsoids. The relative numerical error reveals to be of the order of $10^{-10}$. This approach has been developed for the study of relativistic inspiralling binaries. It may be applied to a wider class of astrophysical problems such as the study of relativistic rotating stars too.Comment: Minor changes, Phys. Rev. D in pres

The generalized F-statistic: multiple detectors and multiple GW pulsars

The F-statistic, derived by Jaranowski, Krolak & Schutz (1998), is the optimal (frequentist) statistic for the detection of nearly periodic gravitational waves from known neutron stars, in the presence of stationary, Gaussian detector noise. The F-statistic was originally derived for the case of a single detector, whose noise spectral density was assumed constant in time, and for a single known neutron star. Here we show how the F-statistic can be straightforwardly generalized to the cases of 1) a network of detectors with time-varying noise curves, and 2) a population of known sources. Fortunately, all the important ingredients that go into our generalized F-statistics are already calculated in the single-source/single-detector searches that are currently implemented, e.g., in the LIGO Software Library, so implementation of optimal multi-detector, multi-source searches should require negligible additional cost in computational power or software development.Comment: 6 pages, 0 figures, submitted to PRD; section IV substantially enlarged and revised, and a few typos correcte

Constraining crystalline color superconducting quark matter with gravitational-wave data

We estimate the maximum equatorial ellipticity sustainable by compact stars composed of crystalline color-superconducting quark matter. For the theoretically allowed range of the gap parameter $\Delta$, the maximum ellipticity could be as large as $10^{-2}$, which is about 4 orders of magnitude larger than the tightest upper limit obtained by the recent science runs of the LIGO and GEO600 gravitational wave detectors based on the data from 78 radio pulsars. We point out that the current gravitational-wave strain upper limit already has some implications for the gap parameter. In particular, the upper limit for the Crab pulsar implies that $\Delta$ is less than O(20) MeV for a range of quark chemical potential accessible in compact stars, assuming that the pulsar has a mass $1.4 M_{\odot}$, radius 10 km, breaking strain $10^{-3}$, and that it has the maximum quadrupole deformation it can sustain without fracturing.Comment: Minor changes to match the published versio

Numerical models of irrotational binary neutron stars in general relativity

We report on general relativistic calculations of quasiequilibrium configurations of binary neutron stars in circular orbits with zero vorticity. These configurations are expected to represent realistic situations as opposed to corotating configurations. The Einstein equations are solved under the assumption of a conformally flat spatial 3-metric (Wilson-Mathews approximation). The velocity field inside the stars is computed by solving an elliptical equation for the velocity scalar potential. Results are presented for sequences of constant baryon number (evolutionary sequences). Although the central density decreases much less with the binary separation than in the corotating case, it still decreases. Thus, no tendency is found for the stars to individually collapse to black hole prior to merger.Comment: Minor corrections, improved figure, 5 pages, REVTeX, Phys. Rev. Lett. in pres

Equilibrium sequences of irrotational binary polytropic stars : The case of double polytropic stars

Solutions to equilibrium sequences of irrotational binary polytropic stars in Newtonian gravity are expanded in a power of $\epsilon=a_0/R$, where R and $a_0$ are the orbital separation of the binary system and the radius of each star for $R=\infty$. For each order of $\epsilon$, we should solve ordinary differential equations for arbitrary polytropic indices n. We show solutions for polytropic indices n= 0.5, 1, 1.5 and 2 up to $\epsilon^6$ orders. Our semi-analytic solutions can be used to check the validity of numerical solutions.Comment: 59 pages including 15 tables and 13 figures, revtex, accepted to Phys. Rev.

On the Maximum Mass of Differentially Rotating Neutron Stars

We construct relativistic equilibrium models of differentially rotating neutron stars and show that they can support significantly more mass than their nonrotating or uniformly rotating counterparts. We dynamically evolve such ``hypermassive'' models in full general relativity and show that there do exist configurations which are dynamically stable against radial collapse and bar formation. Our results suggest that the remnant of binary neutron star coalescence may be temporarily stabilized by differential rotation, leading to delayed collapse and a delayed gravitational wave burst.Comment: 4 pages, 2 figures, uses emulateapj.sty; to appear in ApJ Letter

A Fluid-Dynamical Subgrid Scale Model for Highly Compressible Astrophysical Turbulence

We formulate and implement the Euler equations with SGS dynamics and provide numerical tests of an SGS turbulence energy model that predicts the turbulent pressure of unresolved velocity fluctuations and the rate of dissipation for highly compressible turbulence. We test closures for the turbulence energy cascade by filtering data from high-resolution simulations of forced isothermal and adiabatic turbulence. Optimal properties and an excellent correlation are found for a linear combination of the eddy-viscosity closure that is employed in LES of weakly compressible turbulence and a term that is non-linear in the Jacobian matrix of the velocity. Using this mixed closure, the SGS turbulence energy model is validated in LES of turbulence with stochastic forcing. It is found that the SGS model satisfies several important requirements: 1. The mean SGS turbulence energy follows a power law for varying grid scale. 2. The root mean square (RMS) Mach number of the unresolved velocity fluctuations is proportional to the RMS Mach number of the resolved turbulence, independent of the forcing. 3. The rate of dissipation and the turbulence energy flux are constant. Moreover, we discuss difficulties with direct estimates of the turbulent pressure and the dissipation rate on the basis of resolved flow quantities that have recently been proposed. In combination with the energy injection by stellar feedback and other unresolved processes, the proposed SGS model is applicable to a variety of problems in computational astrophysics. Computing the SGS turbulence energy, the treatment of star formation and stellar feedback in galaxy simulations can be improved. Further, we expect that the turbulent pressure on the grid scale affects the stability of gas against gravitational collapse.Comment: 19 pages, 16 figures, submitted to A&

Binary black holes in circular orbits. II. Numerical methods and first results

We present the first results from a new method for computing spacetimes representing corotating binary black holes in circular orbits. The method is based on the assumption of exact equilibrium. It uses the standard 3+1 decomposition of Einstein equations and conformal flatness approximation for the 3-metric. Contrary to previous numerical approaches to this problem, we do not solve only the constraint equations but rather a set of five equations for the lapse function, the conformal factor and the shift vector. The orbital velocity is unambiguously determined by imposing that, at infinity, the metric behaves like the Schwarzschild one, a requirement which is equivalent to the virial theorem. The numerical scheme has been implemented using multi-domain spectral methods and passed numerous tests. A sequence of corotating black holes of equal mass is calculated. Defining the sequence by requiring that the ADM mass decrease is equal to the angular momentum decrease multiplied by the orbital angular velocity, it is found that the area of the apparent horizons is constant along the sequence. We also find a turning point in the ADM mass and angular momentum curves, which may be interpreted as an innermost stable circular orbit (ISCO). The values of the global quantities at the ISCO, especially the orbital velocity, are in much better agreement with those from third post-Newtonian calculations than with those resulting from previous numerical approaches.Comment: 27 pages, 20 PostScript figures, improved presentation of the regularization procedure for the shift vector, new section devoted to the check of the momentum constraint, references added + minor corrections, accepted for publication in Phys. Rev.

Perturbative approach to the structure of rapidly rotating neutron stars

We construct models of rotating stars using the perturbative approach introduced by J. Hartle in 1967, and a set of equations of state proposed to model hadronic interactions in the inner core of neutron stars. We integrate the equations of stellar structure to third order in the angular velocity and show, comparing our results to those obtained with fully non linear codes, to what extent third order corrections are needed to accurately reproduce the moment of inertia of a star which rotates at rates comparable to that of the fastest isolated pulsars.Comment: 17 pages, 5 figures, minor changes to match version accepted by Phys. Rev.