Instabilities of rotating compact stars: a brief overview

Direct observations of gravitational waves will open in the near future new windows on the Universe. Among the expected sources, instabilities of rotating compact astrophysical objects are waited to be detected with some impatience as this will sign the birth of ``gravitational waves asteroseismology'', a crucial way to improve our knowledge of matter equation of state in conditions that cannot be reproduced in a lab. However, the theoretical work needed to really get informations from to-be-detected signals is still quite large, numerical simulations having become a necessary key ingredient. This article tries to provide a short overview of the main physical topics involved in this field (general relativity, gravitational waves, instabilities of rotating fluids, {\it etc.}), concluding with a brief description of the work that was done in Paris-Meudon Observatory by Silvano Bonazzola and collaborators.Comment: 19 pages, Proceeding of Cargese School "Astrophysical fluid dynamics" (May 2005) organized by B. Dubrulle and M. Rieutord in honour of J.-P. Zahn and S. Bonazzola. Slightly upgraded version: references added, summary on compact stars birth clarifie

Merger states and final states of black hole coalescences: a numerical-relativity-assisted effective-one-body approach

We study to what extent the effective-one-body description of the dynamical state of a nonspinning, coalescing binary black hole (considered either at merger, or after ringdown) agrees with numerical relativity results. This comparison uses estimates of the integrated losses of energy and angular momentum during ringdown, inferred from recent numerical-relativity data. We find that the values, predicted by the effective-one-body formalism, of the energy and angular momentum of the system agree at the per mil level with their numerical-relativity counterparts, both at merger and in the final state. This gives a new confirmation of the ability of effective-one-body theory to accurately describe the dynamics of binary black holes even in the strong-gravitational-field regime. Our work also provides predictions (and analytical fits) for the final mass and the final spin of coalescing black holes for all mass ratiosComment: 10 pages, 4 figures. Submitted to Phys. Rev.

Inertial modes in stratified rotating neutron stars : An evolutionary description

With (non-barotropic) equations of state valid even when the neutron, proton and electron content of neutron star cores is not in beta equilibrium, we study inertial and composition gravity modes of relativistic rotating neutron stars. We solve the relativistic Euler equations in the time domain with a three dimensional numerical code based on spectral methods, in the slow rotation, relativistic Cowling and anelastic approximations. Principally, after a short description of the gravity modes due to smooth composition gradients, we focus our analysis on the question of how the inertial modes are affected by non-barotropicity of the nuclear matter. In our study, the deviation with respect to barotropicity results from the frozen composition of non-superfluid matter composed of neutrons, protons and electrons, when beta equilibrium is broken by millisecond oscillations. We show that already for moderatly fast rotating stars the increasing coupling between polar and axial modes makes those two cases less different than for very slowly rotating stars. In addition, as we directly solve the Euler equations, without coupling only a few number of spherical harmonics, we always found, for the models that we use, a discrete spectrum for the $l = m = 2$ inertial mode. Finally, we find that, for non-barotropic stars, the frequency of this mode, which is our main focus, decreases in a non-negligible way, whereas the time dependence of the energy transfer between polar and axial modes is substantially different due to the existence of low-frequencies gravity modes.Comment: 34 pages, 24 figures, published versio

MHD of rotating compact stars with spectral methods: description of the algorithm and tests

A flexible spectral code for the study of general relativistic magnetohydrodynamics is presented. Aiming at investigating the physics of slowly rotating magnetized compact stars, this new code makes use of various physically motivated approximations. Among them, the relativistic anelastic approximation is a key ingredient of the current version of the code. In this article, we mainly outline the method, putting emphasis on algorithmic techniques that enable to benefit as much as possible of the non-dissipative character of spectral methods, showing also a potential astrophysical application and providing a few illustrative tests.Comment: 15 pages, 4 figures (new figure added, misprints corrected) Article accepted for publication in a special issue of Classical and Quantum Gravity "New Frontiers in Numerical Relativity

1D Cahn-Hilliard dynamics : coarsening and interrupted coarsening

Many systems exhibit a phase where the order parameter is spatially modulated. These patterns can be the result of a frustration caused by the competition between interaction forces with opposite effects. In all models with local interactions, these ordered phases disappear in the strong segregation regime (low temperature). It is expected however that these phases should persist in the case of long range interactions, which can't be correctly described by a Ginzburg-Landau type model with only a finite number of spatial derivatives of the order parameter. An alternative approach is to study the dynamics of the phase transition or pattern formation. While, in the usual process of Ostwald ripening, succession of doubling of the domain size leads to a total segregation, or macro-segregation, C. Misbah and P. Politi have shown that long-range interactions could cause an interruption of this coalescence process, stabilizing a pattern which then remains in a micro-structured state or super-crystal. We show that this is the case for a modified Cahn-Hilliard dynamics due to Oono which includes a non local term and which is particularly well suited to describe systems with a modulated phase

Phase transition of the three-dimensional chiral Ginzburg-Landau model -- search for the chiral phase

Nature of the phase transition of regularly frustrated vector spin systems in three dimensions is investigated based on a Ginzburg-Landau-type effective Hamiltonian. On the basis of the variational analysis of this model, Onoda et al recently suggested the possible occurrence of a chiral phase, where the vector chirality exhibits a long-range order without the long-range order of the spin [Phys. Rev. Lett. 99, 027206 (2007)]. In the present paper, we elaborate their analysis by considering the possibility of a first-order transition which was not taken into account in their analysis. We find that the first-order transition indeed occurs within the variational approximation, which significantly reduces the stability range of the chiral phase, while the chiral phase still persists in a restricted parameter range. Then, we perform an extensive Monte Carlo simulation focusing on such a parameter range. Contrary to the variational result, however, we do not find any evidence of the chiral phase. The range of the chiral phase, if any, is estimated to be less than 0.1% in the temperature width.Comment: 19 pages, 17 figure

Tails of Localized Density of States of Two-dimensional Dirac Fermions

The density of states of Dirac fermions with a random mass on a two-dimensional lattice is considered. We give the explicit asymptotic form of the single-electron density of states as a function of both energy and (average) Dirac mass, in the regime where all states are localized. We make use of a weak-disorder expansion in the parameter g/m^2, where g is the strength of disorder and m the average Dirac mass for the case in which the evaluation of the (supersymmetric) integrals corresponds to non-uniform solutions of the saddle point equation. The resulting density of states has tails which deviate from the typical pure Gaussian form by an analytic prefactor.Comment: 8 pages, REVTeX, 1 eps figure; to appear in Annalen der Physi

Inertial modes in slowly rotating stars : an evolutionary description

We present a new hydro code based on spectral methods using spherical coordinates. The first version of this code aims at studying time evolution of inertial modes in slowly rotating neutron stars. In this article, we introduce the anelastic approximation, developed in atmospheric physics, using the mass conservation equation to discard acoustic waves. We describe our algorithms and some tests of the linear version of the code, and also some preliminary linear results. We show, in the Newtonian framework with differentially rotating background, as in the relativistic case with the strong Cowling approximation, that the main part of the velocity quickly concentrates near the equator of the star. Thus, our time evolution approach gives results analogous to those obtained by Karino {\it et al.} \cite{karino01} within a calculation of eigenvectors. Furthermore, in agreement with the work of Lockitch {\it et al.} \cite{lockandf01}, we found that the velocity seems to always get a non-vanishing polar part.Comment: 36 pages, 27 figures, accepted for publication in Phys. Rev. D (discussion added in the introduction