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Register a new userConstraint damping of the conformal and covariant formulation of the Z4 system in simulations of binary neutron stars
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Following previous work in vacuum spacetimes, we investigate the constraint-damping properties in the presence of matter of the recently developed traceless, conformal and covariant Z4 (CCZ4) formulation of the Einstein equations. First, we evolve an isolated neutron star with an ideal gas equation of state and subject to a constraint-violating perturbation. We compare the evolution of the constraints using the CCZ4 and Baumgarte-Shibata-Shapiro-Nakamura-Oohara-Kojima (BSSNOK) systems. Second, we study the collapse of an unstable spherical star to a black hole. Finally, we evolve binary neutron star systems over several orbits until the merger, the formation of a black hole, and up to the ringdown. We show that the CCZ4 formulation is stable in the presence of matter and that the constraint violations are one or more orders of magnitude smaller than for the BSSNOK formulation. Furthermore, by comparing the CCZ4 and the BSSNOK formulations also for neutron star binaries with large initial constraint violations, we investigate their influence on the errors on physical quantities. We also give a new, simple and robust prescription for the damping parameter that removes the instabilities found when using the fully covariant version of CCZ4 in the evolution of black holes. Overall, we find that at essentially the same computational costs the CCZ4 formulation provides solutions that are stable and with a considerably smaller violation of the Hamiltonian constraint than the BSSNOK formulation. We also find that the performance of the CCZ4 formulation is very similar to another conformal and traceless, but noncovariant formulation of the Z4 system, i.e. the Z4c formulation.
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We present a new formulation of the Einstein equations based on a conformal and traceless decomposition of the covariant form of the Z4 system. This formulation combines the advantages of a conformal decomposition, such as the one used in the BSSNOK formulation (i.e. well-tested hyperbolic gauges, no need for excision, robustness to imperfect boundary conditions) with the advantages of a constraint-damped formulation, such as the generalized harmonic one (i.e. exponential decay of constraint violations when these are produced). We validate the new set of equations through standard tests and by evolving binary black hole systems. Overall, the new conformal formulation leads to a better behavior of the constraint equations and a rapid suppression of the violations when they occur. The changes necessary to implement the new conformal formulation in standard BSSNOK codes are very small as are the additional computational costs.
We adopt a reference-metric approach to generalize a covariant and conformal version of the Z4 system of the Einstein equations. We refer to the resulting system as ``fully covariant and conformal, or fCCZ4 for short, since it is well suited for curvilinear as well as Cartesian coordinates. We implement this fCCZ4 formalism in spherical polar coordinates under the assumption of spherical symmetry using a partially-implicit Runge-Kutta (PIRK) method and show that our code can evolve both vacuum and non-vacuum spacetimes without encountering instabilities. Our method does not require regularization of the equations to handle coordinate singularities, nor does it depend on constraint-preserving outer boundary conditions, nor does it need any modifications of the equations for evolutions of black holes. We perform several tests and compare the performance of the fCCZ4 system, for different choices of certain free parameters, with that of BSSN. Confirming earlier results we find that, for an optimal choice of these parameters, and for neutron-star spacetimes, the violations of the Hamiltonian constraint can be between 1 and 3 orders of magnitude smaller in the fCCZ4 system than in the BSSN formulation. For black-hole spacetimes, on the other hand, any advantages of fCCZ4 over BSSN are less evident.
We present a strongly hyperbolic first-order formulation of the Einstein equations based on the conformal and covariant Z4 system (CCZ4) with constraint-violation damping, which we refer to as FO-CCZ4. As CCZ4, this formulation combines the advantages of a conformal and traceless formulation, with the suppression of constraint violations given by the damping terms, but being first order in time and space, it is particularly suited for a discontinuous Galerkin (DG) implementation. The strongly hyperbolic first-order formulation has been obtained by making careful use of first and second-order ordering constraints. A proof of strong hyperbolicity is given for a selected choice of standard gauges via an analytical computation of the entire eigenstructure of the FO-CCZ4 system. The resulting governing partial differential equations system is written in non-conservative form and requires the evolution of 58 unknowns. A key feature of our formulation is that the first-order CCZ4 system decouples into a set of pure ordinary differential equations and a reduced hyperbolic system of partial differential equations that contains only linearly degenerate fields. We implement FO-CCZ4 in a high-order path-conservative arbitrary-high-order-method-using-derivatives (ADER)-DG scheme with adaptive mesh refinement and local time-stepping, supplemented with a third-order ADER-WENO subcell finite-volume limiter in order to deal with singularities arising with black holes. We validate the correctness of the formulation through a series of standard tests in vacuum, performed in one, two and three spatial dimensions, and also present preliminary results on the evolution of binary black-hole systems. To the best of our knowledge, these are the first successful three-dimensional simulations of moving punctures carried out with high-order DG schemes using a first-order formulation of the Einstein equations.
We have studied the dynamics of an equal-mass magnetized neutron-star binary within a resistive magnetohydrodynamic (RMHD) approach in which the highly conducting stellar interior is matched to an electrovacuum exterior. Because our analysis is aimed at assessing the modifications introduced by resistive effects on the dynamics of the binary after the merger and through to collapse, we have carried out a close comparison with an equivalent simulation performed within the traditional ideal magnetohydrodynamic approximation. We have found that there are many similarities between the two evolutions but also one important difference: the survival time of the hyper massive neutron star increases in a RMHD simulation. This difference is due to a less efficient magnetic-braking mechanism in the resistive regime, in which matter can move across magnetic-field lines, thus reducing the outward transport of angular momentum. Both the RMHD and the ideal magnetohydrodynamic simulations carried here have been performed at higher resolutions and with a different grid structure than those in previous work of ours [L. Rezzolla, B. Giacomazzo, L. Baiotti, J. Granot, C. Kouveliotou, and M. A. Aloy, Astrophys. J. Letters 732, L6 (2011)], but confirm the formation of a low-density funnel with an ordered magnetic field produced by the black hole--torus system. In both regimes the magnetic field is predominantly toroidal in the highly conducting torus and predominantly poloidal in the nearly evacuated funnel. Reconnection processes or neutrino annihilation occurring in the funnel, none of which we model, could potentially increase the internal energy in the funnel and launch a relativistic outflow, which, however, is not produced in these simulations.
Using our recent proposal for defining gauge invariant averages we give a general-covariant formulation of the so-called cosmological backreaction. Our effective covariant equations allow us to describe in explicitly gauge invariant form the way classical or quantum inhomogeneities affect the average evolution of our Universe.
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