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

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

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

Effects of semiclassical spiral fluctuations on hole dynamics

We investigate the dynamics of a single hole coupled to the spiral fluctuations related to the magnetic ground states of the antiferromagnetic J_1-J_2-J_3 Heisenberg model on a square lattice. Using exact diagonalization on finite size clusters and the self consistent Born approximation in the thermodynamic limit we find, as a general feature, a strong reduction of the quasiparticle weight along the spiral phases of the magnetic phase diagram. For an important region of the Brillouin Zone the hole spectral functions are completely incoherent, whereas at low energies the spectral weight is redistributed on several irregular peaks. We find a characteristic value of the spiral pitch, Q=(0.7,0.7)\pi, for which the available phase space for hole scattering is maximum. We argue that this behavior is due to the non trivial interference of the magnon assisted and the free hopping mechanism for hole motion, characteristic of a hole coupled to semiclassical spiral fluctuations.Comment: 6 pages, 5 figure

Order by disorder and gauge-like degeneracy in quantum pyrochlore antiferromagnet

The (three-dimensional) pyrochlore lattice antiferromagnet with Heisenberg spins of large spin length $S$ is a highly frustrated model with an macroscopic degeneracy of classical ground states. The zero-point energy of (harmonic order) spin wave fluctuations distinguishes a subset of these states. I derive an approximate but illuminating {\it effective Hamiltonian}, acting within the subspace of Ising spin configurations representing the {\it collinear} ground states. It consists of products of Ising spins around loops, i.e has the form of a $Z_2$ lattice gauge theory. The remaining ground state entropy is still infinite but not extensive, being $O(L)$ for system size $O(L^3)$. All these ground states have unit cells bigger than those considered previously.Comment: 4pp, one figur

Dipolar spin correlations in classical pyrochlore magnets

We study spin correlations for the highly frustrated classical pyrochlore lattice antiferromagnets with O(N) symmetry in the limit T->0. We conjecture that a local constraint obeyed by the extensively degenerate ground states dictates a dipolar form for the asymptotic spin correlations, at all N $\ne$ 2 for which the system is paramagnetic down to T=0. We verify this conjecture in the cases N=1 and N=3 by simulations and to all orders in the 1/N expansion about the solvable N=infinity limit. Remarkably, the N=infinity formulae are an excellent fit, at all distances, to the correlators at N=3 and even at N=1. Thus we obtain a simple analytical expression also for the correlations of the equivalent models of spin ice and cubic water ice, I_h.Comment: 4 pages revtex

Ordering in a frustrated pyrochlore antiferromagnet proximate to a spin liquid

We perform a general study of spin ordering on the pyrochlore lattice with a 3:1 proportionality of two spin polarizations. Equivalently, this describes valence bond solid conformations of a quantum dimer model on the diamond lattice. We determine the set of likely low temperature ordered phases, on the assumption that the ordering is weak, i.e the system is close to a ``U(1)'' quantum spin liquid in which the 3:1 proportionality is maintained but the spins are strongly fluctuating. The nature of the 9 ordered states we find is determined by a ``projective symmetry'' analysis. All the phases exhibit translational and rotational symmetry breaking, with an enlarged unit cell containing 4 to 64 primitive cells of the underlying pyrochlore. The simplest of the 9 phases is the same ``R'' state found earlier in a theoretical study of the ordering on the magnetization plateau in the $S=3/2$ materials \cdaf and \hgaf. We suggest that the spin/dimer model proposed therein undergoes a direct transition from the spin liquid to the R state, and describe a field theory for the universal properties of this critical point, at zero and non-zero temperatures

Non-equilibrium beta processes in superfluid neutron star cores

The influence of nucleons superfluidity on the beta relaxation time of degenerate neutron star cores, composed of neutrons, protons and electrons, is investigated. We numerically calculate the implied reduction factors for both direct and modified Urca reactions, with isotropic pairing of protons or anisotropic pairing of neutrons. We find that due to the non-zero value of the temperature and/or to the vanishing of anisotropic gaps in some directions of the phase-space, superfluidity does not always completely inhibit beta relaxation, allowing for some reactions if the superfluid gap amplitude is not too large in respect to both the typical thermal energy and the chemical potential mismatch. We even observe that if the ratio between the critical temperature and the actual temperature is very small, a suprathermal regime is reached for which superfluidity is almost irrelevant. On the contrary, if the gap is large enough, the composition of the nuclear matter can stay frozen for very long durations, unless the departure from beta equilibrium is at least as important as the gap amplitude. These results are crucial for precise estimation of the superfluidity effect on the cooling/slowing-down of pulsars and we provide online subroutines to be implemented in codes for simulating such evolutions.Comment: 11 pages, 6 Figs., published, minor changes, subroutines can be found on line at http://luth2.obspm.fr/~etu/villain/Micro/Resolution.htm

Quantum and Classical Spins on the Spatially Distorted Kagome Lattice: Applications to Volborthite

In Volborthite, spin-1/2 moments form a distorted Kagom\'e lattice, of corner sharing isosceles triangles with exchange constants $J$ on two bonds and $J'$ on the third bond. We study the properties of such spin systems, and show that despite the distortion, the lattice retains a great deal of frustration. Although sub-extensive, the classical ground state degeneracy remains very large, growing exponentially with the system perimeter. We consider degeneracy lifting by thermal and quantum fluctuations. To linear (spin wave) order, the degeneracy is found to stay intact. Two complementary approaches are therefore introduced, appropriate to low and high temperatures, which point to the same ordered pattern. In the low temperature limit, an effective chirality Hamiltonian is derived from non-linear spin waves which predicts a transition on increasing $J'/J$, from $\sqrt 3\times \sqrt 3$ type order to a new ferrimagnetic {\em striped chirality} order with a doubled unit cell. This is confirmed by a large-N approximation on the O($n$) model on this lattice. While the saddle point solution produces a line degeneracy, $O(1/n)$ corrections select the non-trivial wavevector of the striped chirality state. The quantum limit of spin 1/2 on this lattice is studied via exact small system diagonalization and compare well with experimental results at intermediate temperatures. We suggest that the very low temperature spin frozen state seen in NMR experiments may be related to the disconnected nature of classical ground states on this lattice, which leads to a prediction for NMR line shapes.Comment: revised, section V about exact diagonalization is extensively rewritten, 17 pages, 11 figures, RevTex 4, accepted by Phys. Rev.