Capabilities

GRFolres uses a lot of the features and follows a similar design philosophy to GRChombo. The key difference is that it includes the contributions to the Einstein Equations coming from certain modified gravity theories, in particular, it currently implements the Four-Derivative Scalar-Tensor theory of gravity ($4\partial ST$) - the most general parity invariant EFT of gravity with terms up to derivatives to the 4th power (but still with second order equations of motion), for which the action is given in Variable conventions.

For this class of modified gravity the standard $3+1$ CCZ4 formulation used in GRChombo is not well-posed, hence the code also implements a modified puncture gauge - a version of the modified Generalised Harmonic Gauge suggested by Kovacs and Reall adapted for singularity avoiding coordinates.

The design of the code is sufficiently flexible that we can add alternative modifications in future with minimal changes, such as the general expression of the Horndeski action.

Current key features in the public code

• Modified puncture gauge - The equations are written in a formalism that differs from the standard CCZ4 approach by adding extra spatially dependent functions $a(x)$ and $b(x)$, which modify the speed of propagation of the gauge modes and ensure well-posedness when adding the modified gravity contributions, as in this paper.

• Matrix inversion - For $4\partial ST$ (and many other modified gravity theories) there are terms in the time derivatives (aka, the right hand side or RHS) of the $3+1$ variables that depend (linearly) on the time derivatives of other variables. Therefore, in order to solve the full non-linear equations one needs to express the evolution of those variables at each spatial point as a linear system. The function that computes the matrix on the LHS and solves the linear system is included in the class FourDerivScalarTensor over which the ModifiedCCZ4 class is templated.

• Coupling and potential - The FourDerivScalarTensor class is templated over the coupling to the modified gravity terms and the potential of the scalar field, therefore the user needs to create a class to compute the following quantities:

  • dfdphi: the derivative of the coupling to the Gauss-Bonnet curvature $f(\phi)$ with respect to $\phi$.
  • d2fdphi2: the second derivative of the coupling $f(\phi)$ with respect to $\phi$.
  • g2: the coupling to the square of the kinetic term $X^2$ of the scalar field $g_2$.
  • V_of_phi: the potential of the scalar field $\phi$.
  • dVdphi: the derivative of the potential with respect to the field $\phi$.

Several examples of such classes are provided, for shift symmetric and exponential couplings, which should be easy to adapt to other cases.

• Excision - In the puncture approach, numerical excision of the region inside the apparent horizon is not needed (in contrast to Generalised Harmonic Coordinates). However, the equations can in principle be changed in this region without affecting the rest of the spacetime, and this is usually necessary since close to the singularity EsGB type theories develop non hyperbolic regions. Therefore the code allows the user to modify the coupling $f(\phi)$ so that it vanishes in the singularity and then smoothly transitions to its chosen value before reaching the apparent horizon. This has been implemented in two ways,
  • using the radial coordinate $r$ that measures the distance to the black hole: $f(\phi)\to f(\phi)/(1+e^{-\tau(r-r_0)})$, or
  • using the conformal factor $\chi$: $f(\phi)\to f(\phi)/(1+e^{-\tau(\chi-\chi_0)})$, where $\tau$, $r_0$ and $\chi_0$ have to be chosen so that the function is smooth enough and only affects the region inside the apparent horizon.

Forthcoming features

The following are things for which we know code exists in some form with some member(s) of the collaboration. Our goal is always to add code to the public repository after a reasonable period of testing (usually after a research article using the code has been accepted for publication). However, there is sometimes a lag in this happening since we have higher standards for public code that often necessitate a rewrite of even working and well-tested code. If you plan to develop one of these features we suggest you Contact us to see if we already have code that we can share with you (at your own risk!).

• EsGB initial conditions - The CTTK approach has been adapted to treat the additional EsGB terms for general initial configurations of the scalar field. (GRChombo responsible members: S Brady/K Clough)

• Higher dimensional Gauss-Bonnet gravity - The equations for Einstein-Gauss-Bonnet theory in 4+1 dimensions have been implemented as in GRFolres using the Cartoon formalism. (GRChombo responsible members: S Yao/P Figueras)