Redshift factor and the small mass-ratio limit in binary black hole simulations

We present a calculation of the Detweiler redshift factor in binary black hole simulations based on its relation to the surface gravity. The redshift factor has far-reaching applications in analytic approximations, gravitational self-force calculations, and conservative two-body dynamics. By specializing to non-spinning, quasi-circular binaries with mass ratios ranging from $m_A/m_B = 1$ to $m_A/m_B = 9.5$ we are able to recover the leading small-mass-ratio (SMR) prediction with relative differences of order $10^{-5}$ from simulations alone. The next-to-leading order term that we extract agrees with the SMR prediction arising from self-force calculations, with differences of a few percent. These deviations from the first-order conservative prediction are consistent with non-adiabatic effects that can be accommodated in an SMR expansion. This fact is also supported by a comparison to the conservative post-Newtonian prediction of the redshifts. For the individual redshifts, a re-expansion in terms of the symmetric mass ratio $\nu$ does not improve the convergence of the series. However we find that when looking at the sum of the redshift factors of both back holes, $z_A + z_B$, which is symmetric under the exchange of the masses, a re-expansion in $\nu$ accelerates its convergence. Our work provides further evidence of the surprising effectiveness of SMR approximations in modeling even comparable mass binary black holes.