Reduced Hamiltonian for next-to-leading order Spin-Squared Dynamics of General Compact Binaries

TL;DR

This paper provides the first fully reduced (SSC-eliminated) NLO spin-squared Hamiltonian for general compact binaries in the post-Newtonian regime, incorporating spin-induced quadrupole deformations via C_Q. It uses the ADM canonical formalism and Newton-Wigner SSC to obtain an explicit Hamiltonian H^{ADM can}_{NLO S1^2}, along with corresponding spin, position, and momentum equations in canonical form. The authors also establish consistency with EFT-based nonreduced potentials by transforming Porto-Rothstein Routhians to a reduced Hamiltonian and showing agreement through a canonical transformation, thereby validating the result. The derived Hamiltonian can improve GW templates for binaries with spinning components (BHs/NSs) and aid in constraining the equation of state through the C_Q parameter.

Abstract

Within the post Newtonian framework the fully reduced Hamiltonian (i.e., with eliminated spin supplementary condition) for the next-to-leading order spin-squared dynamics of general compact binaries is presented. The Hamiltonian is applicable to the spin dynamics of all kinds of binaries with self-gravitating components like black holes and/or neutron stars taking into account spin-induced quadrupolar deformation effects in second post-Newtonian order perturbation theory of Einstein's field equations. The corresponding equations of motion for spin, position and momentum variables are given in terms of canonical Poisson brackets. Comparison with a nonreduced potential calculated within the Effective Field Theory approach is made.