Effective-one-body Hamiltonian with next-to-leading order spin-spin coupling for two nonprecessing black holes with aligned spins

The canonical Arnowitt-Deser-Misner (ADM) Hamiltonian with next-to-leading order spin-spin coupling [J. Steinhoff, S. Hergt, and G. Sch\"afer] is converted into the EOB formalism of T. Damour, P. Jaranowski, and G. Sch\"afer for the special case of spinning black hole binaries whose spins are aligned with the angular momentum. In particular, we propose to include the new terms by adding a dynamical term of next-to-leading order to the Kerr parameter squared entering the effective metric. The modified EOB Hamiltonian consistently reduces to the Kerr Hamiltonian as the mass ratio tends to zero; moreover, it predicts the existence of an Innermost Stable Circular Orbit. We also derive, for the general case of arbitrarily oriented spins but in the vanishing mass-ratio limit, a coordinate transformation that maps the next-to-leading order spin-spin contribution of the ADM Hamiltonian to the EOB Hamiltonian.