ADAPTACIÓN CURRICULAR:

The initial data for GR simulations has to satisfy two types of constraints : the mathematical constraints required for the initial data to be compatible with Ein- stein’s equations, and the astrophysical constraints used to make the initial data correspond to a realistic situation.

The choice of initial data is typically made using the 3+1 decomposi- tion, in which spacetime is foliated by 3-dimensional spatial hypersurfaces parametrized by a time coordinate t. The metric is then

ds2 = −α2dt2+ φ4γ˜ij dxi+ βidt

dxj+ βjdt
, (1.2) where α is the lapse, φ the conformal factor, ˜γij the conformal 3-metric and βi

the shift. The mathematical constraints on the metric components on a surface of constant t can then be written as a set of elliptical equations of the type

∇2_{φ = ...}

(1.3) ∇2_{(αφ) = ...}

(1.4)

Mijk∇i∇jβk = ... (1.5)

where the right-hand sides are functions of the stress-energy tensor and the met- ric variables and their derivatives (see Section 2.2.1 for a more detailed discus- sion of the constraints). In this formalism, the free parameters for the initial data are the conformal metric ˜gij, the trace of the extrinsic curvature Kij (∼ the

derivative of the metric along the normal to the spatial slice, see eq. [2.9]), and their derivatives along the ‘time’ coordinate t. These freely specifiable func- tions, combined with a choice of boundary conditions for the variables α, φ and βi, will determine the physical properties of the initial configuration.

In the case of a binary system, it is fairly natural to require that the time derivatives vanish in the coordinate frame comoving with the compact objects (quasi-equilibrium condition). This assumption would be exact for a binary in circular orbit in the absence of gravitational radiation, and remains very accu- rate except during the plunge. Let us note, however, that this condition fixes the time derivatives of the free variables but introduce a new freedom: the choice of the comoving frame. We generally choose a frame rotating at constant angu- lar velocity Ω and with a radial velocity vr = ar. Ω and a then determine the

eccentricity and orbital phase of the binary.

Choosing ˜gij and K is more difficult. We use either flat space (’conformally

flat’ initial data), or some superposition of analytical solutions for isolated ob- jects. However, neither of these choices is exact for a binary with finite orbital separation. In practice, this means that in numerical simulations the binary will relax from the chosen initial data to a more physical quasi-stationary orbit — usually through quasi-normal ringing of the black hole(s), the emission of ‘un- physical’ gravitational waves, and/or oscillations in the star(s). Reducing the importance of these unphysical effects and choosing the initial parameters so that the final configuration corresponds to a binary with the desired physical parameters (mass and spin of the compact objects, orbital parameters) is one of the main challenges in our choice of initial data.

For BHNS and NS-NS binaries, an additional complication comes from the necessity to determine the initial value of the matter fields (density ρ, tempera- ture T , velocity ui). The assumption of quasi-equilibrium allows us to determine

the density : the requirement that the matter fields are constant in the comov- ing frame turns out to give an algebraic relation between the metric and matter

fields, up to a constant fixing the mass of the star. The temperature can quite naturally be set to T = 0, as the temperature in the star is expected to be neg- ligible compared to the Fermi energy of the degenerate neutrons. The velocity, however, has to be solved for: the requirement that the fluid in the star is in an irrotational configuration gives us an elliptic equation for the velocity potential Ψ, where ∇iΨ ∼ ui.

Finally, the equations for the metric and the matter fields are coupled: ρ and Ψ are source terms in the elliptic equations for α, φ and β, while knowledge of the metric is required to solve for Ψ and ρ. Solving the whole system thus requires the following ingredients:

• An elliptic solver to find the constrained metric variables α, φ and β as well as the velocity potential Ψ, when the source terms of the various elliptic equations are known.

• A prescription for the free functions ˜g, K and their time derivatives, so that the initial data is as close as possible to the desired physical configuration. • An iterative procedure allowing us to solve the coupled problems of satis- fying the constraints (5 elliptic equations), determining the fluid velocity (1 elliptic equation), and finding the matter density within the star (1 alge- braic equation with a free parameter chosen iteratively so that the star has the desired mass). The procedure must converge towards a self-consistent solution with the desired physical properties.

Chapter 2 details the numerical methods used to choose initial data for our simulations, and presents tests of the accuracy of the algorithm. We verified that at large orbital separations our results agree with analytical approximations to

Einstein’s equations, as well as with previous numerical results by Taniguchi et al. [131]. Through the use of spectral methods we obtained exponential con- vergence for the solution, at least in the case of simple equations of state. We also adapted methods used for binary black holes to obtain low-eccentricity or- bits [88] and high spin [69] to black hole-neutron star systems. The resulting al- gorithm is thus capable of efficiently generating accurate initial data for a wide range of black hole-neutron star systems, with orbital parameters reproducing the most likely astrophysical configurations.

Increased accuracy requirements as well as the need to generate initial data for misaligned black hole spins led to regular improvements to the initial data solver after the publication of Chapter 2. These recent modifications are dis- cussed in Appendix A.