Neutron star (NS) mergers where both stars have negligible spins are commonly considered as the most likely ‘standard’ case. In globular clusters, however, the majority of NSs have been spun up to millisecond (ms) periods and, based on observed systems, we estimate that a non-negligible fraction of all double NS mergers (⁠∼4±2 per cent⁠) contains one component with a spin of a (few) ms. We use the Lagrangian numerical relativity code SPHINCS_BSSN to simulate mergers where one star has no spin and the other has a dimensionless spin parameter of χ = 0.5. Such mergers exhibit several distinct signatures compared to irrotational cases. They form only one, very pronounced spiral arm and they dynamically eject an order of magnitude more mass of unshocked material at the original, very low electron fraction. One can therefore expect particularly bright, red kilonovae. Overall, the spinning case collisions are substantially less violent and they eject smaller amounts of shock-generated semirelativistic material. Therefore, the ejecta produce a weaker blue/ultraviolet kilonova precursor signal, but – since the total amount is larger – brighter kilonova afterglows months after the merger. The spinning cases also have significantly more fallback accretion and thus could power late-time X-ray flares. Since the post-merger remnant loses energy and angular momentum significantly less efficiently to gravitational waves, such systems can delay a potential collapse to a black hole and are therefore candidates for merger-triggered gamma-ray bursts with longer emission time-scales.

We present version 1.0 of our Lagrangian numerical relativity code SPHINCS_BSSN. This code evolves the full set of Einstein equations, but contrary to other numerical relativity codes, it evolves the matter fluid via Lagrangian particles in the framework of a high-accuracy version of smooth particle hydrodynamics (SPH). The major new elements introduced here are: (i) a new method to map the stress–energy tensor (known at the particles) to the spacetime mesh, based on a local regression estimate; (ii) additional measures that ensure the robust evolution of a neutron star through its collapse to a black hole; and (iii) further refinements in how we place the SPH particles for our initial data. The latter are implemented in our code SPHINCS_ID which now, in addition to LORENE, can also couple to initial data produced by the initial data library FUKA. We discuss several simulations of neutron star mergers performed with SPHINCS_BSSN_v1.0, including irrotational cases with and without prompt collapse and a system where only one of the stars has a large spin (χ = 0.5).

We present the first neutron star merger simulations performed with the newly developed Numerical Relativity code SPHINCS_BSSN. This code evolves the spacetime on a mesh using the BSSN formulation, but matter is evolved via Lagrangian particles according to a high-accuracy version of general-relativistic Smooth Particle Hydrodynamics (SPH). Our code contains a number of new methodological elements compared to other Numerical Relativity codes. The main focus here is on the new elements that were introduced to model neutron star mergers. These include (a) a refinement (fixed in time) of the spacetime-mesh, (b) corresponding changes in the particle–mesh mapping algorithm and (c) a novel way to construct SPH initial data for binary systems via the recently developed “Artificial Pressure Method.” This latter method makes use of the spectral initial data produced by the library LORENE, and is implemented in a new code called SPHINCS_ID. While our main focus is on introducing these new methodological elements and documenting the current status of SPHINCS_BSSN, we also show as a first application a set of neutron star merger simulations employing “soft” (Γ=2.00Γ=2.00) and “stiff” (Γ=2.75Γ=2.75) polytropic equations of state.

The observation of gravitational waves from compact objects has now become an active part of observational astronomy. For a sound interpretation, one needs to compare such observations against detailed Numerical Relativity simulations, which are essential tools to explore the dynamics and physics of compact binary mergers. To date, essentially all simulation codes that solve the full set of Einstein’s equations are performed in the framework of Eulerian hydrodynamics. The exception is our recently developed Numerical Relativity code SPHINCS_BSSN which solves the commonly used BSSN formulation of the Einstein equations on a structured mesh and the matter equations via Lagrangian particles. We show here, for the first time, SPHINCS_BSSN neutron star merger simulations with piecewise polytropic approximations to four nuclear matter equations of state. In this set of neutron star merger simulations, we focus on perfectly symmetric binary systems that are irrotational and have 1.3 M⊙ masses. We introduce some further methodological refinements (a new way of steering dissipation, an improved particle–mesh mapping), and we explore the impact of the exponent that enters in the calculation of the thermal pressure contribution. We find that it leaves a noticeable imprint on the gravitational wave amplitude (calculated via both quadrupole approximation and the Ψ4 formalism) and has a noticeable impact on the amount of dynamic ejecta. Consistent with earlier findings, we only find a few times 10−3M⊙ as dynamic ejecta in the studied equal mass binary systems, with softer equations of state (which are more prone to shock formation) ejecting larger amounts of matter. In all of the cases, we see a credible high-velocity (∼0.5…0.7c) ejecta component of ∼10−4M⊙ that is launched at contact from the interface between the two neutron stars. Such a high-velocity component has been suggested to produce an early, blue precursor to the main kilonova emission, and it could also potentially cause a kilonova afterglow. 

We present bimEX, a Mathematica package for exact computations in 3 + I bimetric relativity. It is based on the xAct bundle, which can handle computations involving both abstract tensors and their components. In this communication, we refer to the latter case as concrete computations. The package consists of two main parts. The first part involves the abstract tensors, and focuses on how to deal with multiple metrics in xAct. The second part takes an ansatz for the primary variables in a chart as the input, and returns the covariant BSSN bimetric equations in components in that chart. Several functions are implemented to make this process as fast and user-friendly as possible. The package has been used and tested extensively in spherical symmetry and was the workhorse in obtaining the bimetric covariant BSSN equations and reproducing the bimetric 3+1 equations in the spherical polar chart. Program summary Program Title: bimEX Program Files doi: http://dx.doi.org/10.17632/2s5d7csc9w.1 Licensing provisions: GNU General Public License 3.0 Programming language: Mathematica Supplementary material: 1. README file, containing instructions about how to use the working example. 2. Working example, constituted by the notebooks: (a) bimEX_Working_Example.nb (b) bimEX_Decomposition_Lists_Loader.nb (c) bimEX_Decomposition_xAct_Loader.nb Nature of problem: Writing the bimetric covariant BSSN equations in any desired ansatz and chart. Solution method: Definition of functions within the Mathematica package xAct, which computes all the components of the defined abstract tensors and reduce the abstract tensors to their representation in components. Additional comments: GitHub repository at https://github.com/nubirel/bimEX

Numerical integration of the field equations in bimetric relativity is necessary to obtain solutions describing realistic systems. Thus, it is crucial to recast the equations  as a well-posed problem. In general relativity, under certain assumptions, the covariant BSSN formulation is a strongly hyperbolic formulation of the Einstein equations, hence its Cauchy problem is well-posed. In this paper, we establish the covariant BSSN formulation of the bimetric field equations. It shares many features with the corresponding formulation in general relativity, but there are a few fundamental differences between them. Some of these differences depend on the gauge choice and alter the hyperbolic structure of the system of partial differential equations compared to general relativity. Accordingly, the strong hyperbolicity of the system cannot be claimed yet, under the same assumptions as in general relativity. In the paper, we stress the differences compared with general relativity and state the main issues that should be tackled next, to draw a roadmap towards numerical bimetric relativity.

In general relativity, the endpoint of spherically symmetric gravitational collapse is a Schwarzschild-[(A)dS] black hole. In bimetric gravity, it has been speculated that a static end state must also be Schwarzschild-[(A)dS]. To this end, we present a set of exact solutions, including collapsing massless dust particles. For these, the speculation is confirmed.

We present a method for solving the constraint equations in the Hassan-Rosen bimetric theory to determine the initial data for the gravitational collapse of spherically symmetric dust. The setup leads to equations similar to those for a polytropic fluid in general relativity, here called Lane-Emden-like equations. Using a numerical code which solves the evolution equations in the standard 3 + 1 form, we also obtain a short-term development of the initial data for these bimetric spherical clouds. The evolution highlights some important features of the bimetric theory such as the interwoven and oscillating null cones representing the essential nonbidiagonality in the dynamics of the two metrics. The simulations are in the strong-field regime and show that, at least at an early stage, if the bimetric initial data are close to those for general relativity, the bimetric evolution stays close to the evolution in general relativity as well, and with no instabilities, albeit with small oscillations in the metric fields. In addition, we determine initial data and first evolution for vacuum bimetric spherically symmetric nonstationary solutions, providing generic counterexamples to a statement analog to Jebsen-Birkhoff theorem in bimetric relativity.

Bimetric gravity can reproduce the accelerated expansion of the Universe, without a cosmological constant. However, the stability of these solutions to linear perturbations has been questioned, suggesting exponential growth of structure in this approximation. We present a simple model of structure formation, for which an analytical solution is derived. The solution is well-behaved, showing that there is no physical instability with respect to these perturbations. The model can yield a growth of structure exhibiting measurable differences from ACDM.

General relativity (GR) is the standard physical theory describing gravitational interactions. All astrophysical and cosmological observations are compatible with its predictions, provided that unknown matter and energy components are included. These are called dark matter and dark energy.

In addition, GR describes the nonlinear self-interaction of a massless spin-2 field. In particle physics, there are both massless and massive fields having spin 0, 1 and 1/2. It is then well-justified to ask whether a mathematically consistent nonlinear theory describing a massive spin-2 field exists.

The Hassan–Rosen bimetric relativity (BR) is a mathematically consistent theory describing the nonlinear interaction between a massless and a massive spin-2 field. These fields are described by two metrics, out of which only one can be directly coupled to us and determines the geometry we probe.

Since it includes GR, BR is an extension of it and provides us with new astrophysical and cosmological solutions. These solutions, which may give hints about the nature of dark matter and dark energy, need to be tested against observations in order to support or falsify the theory. This requires predictions for realistic physical systems. One such system is the spherically symmetric gravitational collapse of a dust cloud, and its study is the overarching motivation behind the thesis.

Studying realistic physical systems in BR requires the solving of the nonlinear equations of motion of the theory. This can be done in two ways: (i) looking for methods that simplify the equations in order to solve them exactly, and (ii) solving the equations numerically.

The studies reviewed in the thesis provide results for both alternatives. In the first case, the results concern spacetime symmetries (e.g., spherical symmetry) and how they affect particular solutions in BR, especially those describing gravitational collapse. In the second case, inspired by the success of numerical relativity, the results initiate the field of numerical bimetric relativity. The simulations provide us with the first hints about how gravitational collapse works in BR.