A well-posed BSSN-type formulation for scalar-tensor theories of gravity with second-order field equations

TL;DR

The paper develops a well-posed, BSSN-type formulation (mBSSN) for scalar-tensor and related beyond-GR theories by exploiting a modified gauge framework that decouples physical, pure-gauge, and gauge-condition-violating modes. It derives the mBSSN system from the modified CCZ4 (mCCZ4) formalism through a chain of symmetry-breaking steps (Z4 to Z3 to BSSN) and assesses well-posedness in GR via a strong-hyperbolicity analysis, yielding real eigenvalues grouped into physical, gauge, and pure-gauge sectors. The beyond-GR extension relies on weak coupling to theories known to be well-posed in mCCZ4 (e.g., Einstein-Gauss-Bonnet and 4-derivative scalar-tensor theories), arguing that hyperbolicity carries over when the modified gravity corrections are treated as an effective stress-energy tensor with a decoupled principal part. Numerically, the authors implement mBSSN in GRFolres and compare against mCCZ4 in EsGB gravity through two BH benchmarks (an isolated spinning BH and head-on mergers), finding strong agreement in scalar and gravitational-wave signals and comparable constraint behavior, thereby enabling broader adoption of beyond-GR simulations in puncture-gauge NR codes.

Abstract

Recent developments in the modified harmonic and modified puncture gauges have opened new possibilities for performing stable numerical evolutions beyond General Relativity. In this work, we utilise techniques developed in the aforementioned formalisms to derive a BSSN-type formalism compatible with certain classes of modified gravity theories. As an intermediate step, we also derived modified versions of the Z4 and Z3 formalisms, thereby completing the connection between these formalisms beyond General Relativity. We then test the robustness of the new modified BSSN formalism by simulating the dynamics of black hole systems and benchmarking the results against the modified CCZ4 formulation. These developments enable the exploration of theories beyond General Relativity in many well-known Numerical Relativity codes that use different versions of the puncture gauge approach.

Figures (8)

• Figure 1: Scalar waves of an isolated spinning BH. The real part of the $(0,0)$ mode of the scalar field $\phi$ extracted at $r=100M$. The scalar waves from both mBSSN and mCCZ4 are compared, showing a strong agreement between the formalisms.
• Figure 2: Apparent horizon mass of an isolated spinning BH. The AH mass of the BH as a function of time. We compare the mass of the BH as measured at the apparent horizon for both formalisms. We find that the mass experiences a slight increase at late times for the mCCZ4 formalism, while the mBSSN formalism exhibits a slight decrease.
• Figure 3: Apparent horizon spin of an isolated spinning BH. The AH spin of the BH as a function of time. We compare the spin of the BH as measured at the apparent horizon for both formalisms. We find that mBSSN agrees better with mCCZ4 at late times, though one can still observe a slight increase in mBSSN and decrease in mCCZ4.
• Figure 4: Hamiltonian constraint violation of an isolated spinning BH.$L^{2}$ norm of the Hamiltonian, $\mathcal{H}$, across the numerical grid as a function of time. The mBSSN formulation shows slightly larger Hamiltonian constraint violation than mCCZ4, although the latter remains approximately constant at late times. This trend is consistent with previous comparisons between the standard BSSN and CCZ4 formulations Sanchis-Gual:2014nha.
• Figure 5: Gravitational waves of a head-on merger. The real part of the main $(2,2)$ mode of the Newman-Penrose scalar $\Psi_{4},$ as a function of time, extracted at $r=100M$, for GR and EsGB simulations, using the mCCZ4 and mBSSN formalisms, respectively. We find here a very good agreement between both formalisms and especially in the observed dephasing between GR and non-GR.