Gauge conditions for Generalized Harmonic Evolution that reduce the deformations of the apparent horizon.

Good gauge conditions are essential for stable numerical relativity simulations. The choice of gauge also affects the coordinate shape of apparent horizons. For codes using spectral methods, like SpEC and SpECTRE, the apparent horizons are represented using spherical harmonics. Having apparent horizons close to spherical is desirable because they can then be accurately represented using low-order spherical harmonics, which will in turn make the code more efficient. In the generalized harmonic formalism, the stability of the system restricts the gauge conditions that we can use. We instead use gauge driver extension of the generalized harmonic formalism, which allows us to set any gauge by giving a target for the contracted Christoffel symbol $Gamma_a$ . We set the target $Gamma_a$ around each apparent horizon to be that of a Kerr black hole of the correct mass and spin. Using this gauge, we can keep the shape of both apparent horizons close to a sphere during the evolution, substantially reducing the number of spherical harmonics required to represent them. The next step is to try setting the gauge directly without using the gauge driver, which would make the simulations even more efficient.