# Initial Configurations

### Gaseous Disk

We follow the same equilibrium disk models as used in predecoupling case (Gold et. all. PRD 89, 064060, 2014). The gaseous disk obeys a polytropic equation of state, $P=K\rho^\Gamma_0$, at $t=0$. It is evolved according to the ideal gas law $P=(\Gamma-1)\rho_0\epsilon$, where P is the pressure, K is the polytropic gas constant, ε is the internal specific energy, Γ is the adiabatic index, and ρ0 the rest-mass density. We chose $\Gamma = 4/3$ appropriate for radiation pressure-dominated, thermal disks, which are typically optically thick. The initial disk satisfies the density profile of a disk that would be in equilibrium about a single black hole with the same mass as the binary. We seed a small, purely poloidal B-field in the initial disk and then evolve it with the magnetized fluid using GRMHD simulations. We allow the binary-disk system to relax to quasiequilibrium by evolving the pre-decoupling phase for $10^4M$. Here we use the conformal thin-sandwich (CTS) metric, which remains fixed in the rotating frame.

### Black Hole Binary

The initial binary is in a quasiequilibrium circular orbital with initial frequency of MΩ ~ 0.028. The matter and gravitational field variables are determined by using the conformal-thin-sandwich (CTS) formalism. BH equilibrium boundary conditions are imposed on the BH horizon. The mass ratio of the binary varies from 1:1 to 1:4.

The CTS initial data we adopt possess a helical Killing vector, which implies that the gravitational fields are stationary in a frame corotating with the binary. As a result, we can perform the metric evolution in the center-of-mass frame of the binary by simply rotating the initial gravitational fields. This technique simplifies our computations substantially for pre-decouplings. Following decoupling, we track the metric evolution by solving the BSSN equations and the fluid and magnetic field by solving the equations of relativistic MHD. These integrations are performed by employing the Illinois GRMHD code.

### Cooling Law

In both cases, we adopted "rapid" radiative cooling by setting the cooling time scale equal to 10% of the local, Keplerian time scale $\tau_{cool}(r)/M = 0.1P_{Kep}(r)/M=0.1 \cdot 2\pi(r/M)^{3/2}$, where r is the cylindrical radial coordinate measured from the center of mass of the binary.