Variable Conventions

Modified puncture gauge

For the modified puncture approach used we perform the following replacement in the equations of motions:

$$R^{\mu\nu}-\tfrac{1}{2}R g^{\mu\nu}\to R^{\mu\nu}-\tfrac{1}{2}R g^{\mu\nu}+2\big(\delta_{\alpha}^{(\mu}\hat{g}^{\nu)\beta}-\tfrac{1}{2}\delta_{\alpha}^{\beta}\hat{g}^{\mu\nu}\big)\nabla_{\beta}Z^{\alpha}-\kappa_1\big[2n^{(\mu}Z^{\nu)}+\kappa_2n^{\alpha}Z_{\alpha}g^{\mu\nu} \big],$$ where $Z^{\mu}$ are the constraints (as in the standard puncture gauge) and $\hat{g}^{\mu\nu}$ and $\tilde{g}^{\mu\nu}$ are two auxiliary Lorentzian metrics that ensure that gauge modes propagate at distinct speeds from the physical modes,

$$\tilde{g}^{\mu\nu}=g^{\mu\nu}-a(x)n^{\mu}n^{\nu},\qquad \hat{g}^{\mu\nu}=g^{\mu\nu}-b(x)n^{\mu}n^{\nu},$$

with $a(x)$ and $b(x)$ being arbitrary functions such that $0\lt a(x)\lt b(x)$ (in practise setting them to be spatially constant has been sufficient to obtain a stable evolution in most cases) and $n^{\mu}$ is the unit timelike vector normal to the t=const. hypersurfaces. Note that $a(x)=b(x)=0$ leads to the standard puncture gauge formulation , including the damping terms, whose coefficients $\kappa_1>0$ and $\kappa_2>-\tfrac{2}{2+b(x)}$ guarantee that constraint violating modes are exponentially suppressed. See this work for further information.

Four-Derivative Scalar-Tensor theory

The action of the Four-Derivative Scalar-Tensor theory ($4\partial ST$) yields

$$S=\int d^4\sqrt{-g}\left(\frac{R}{16\pi G}+X - V(\phi) + g_2(\phi)X^2 + f(\phi)\mathcal{R}^{\text{GB}} \right),$$

where $X=-\frac{1}{2}(\partial_{\mu}\phi)^2$, $\mathcal{R}^{\text{GB}}=R^2-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$ and $f(\phi)$ and $g_2(\phi)$ are the couplings to the modified gravity contributions. Note that some proportionality factors in the code differ from the ones described in the literature (the goal in the code is that the example should reproduce exactly the minimally coupled scalar field class when the additional modified gravity terms are turned off). We can easily match the other conventions by suitably rescaling $\phi$, $g_2(\phi)$ and $f(\phi)$.

Regarding the expressions for the couplings $g_2(\phi)$ and $f(\phi)$, the first one is generally set to a constant value while the second one has to depend on $\phi$ (otherwise the Gauss-Bonnet contribution becomes trivial in 4 spacetime dimensions). Its possible forms can be separated into two categories:

• Type I coupling: those such that $f'(0)\neq 0$, for which all black holes have a non-trivial scalar field. The main example of this is:
  • Shift-symmetric: $f(\phi)\propto\phi$.
• Type II coupling: those such that $f'(0)=0$, for which black holes can either be hairy or non-hairy. Some examples are:
  • Quadratic: $f(\phi)\propto\phi^2$.
  • Exponential: $f(\phi)\propto 1-e^{-\beta\phi^2(1+\kappa\phi^2)}$, where $\beta$ and $\kappa$ are constants.

We provide examples for both types of coupling in the code.