An "inside-out" approach to modeling supermassive black hole binary inspiral

The inspiral and merger of two supermassive black holes (SMBHs) releases immense energy in low-frequency gravitational waves (GWs). Recent pulsar timing array (PTA) observations of the stochastic nHz GW background (GWB) are consistent with a SMBH binary origin. GW data from PTAs and from the upcoming Laser Interferometer Space Antenna (LISA) can probe late-stage binary evolution where electromagnetic constraints are scarce. However, the complexity of the relevant astrophysics necessitates a well optimized approach. I argue that a key physical quantity to constrain with PTAs is the orbital semi-major axis at which binary inspiral transitions from the astrophysical to the GW-dominated regime (a GW). This quantity should be treated as a free parameter in analysis of the GWB. Using this premise, I present a simple analytic framework for modeling SMBH binary inspiral in an "inside-out" fashion. At orbital separations slightly larger than a GW, a power-law scaling for the astrophysical inspiral timescale is assumed, while the outermost phase (prior to the PTA regime) is simply modeled via a delay time. I show that the GWB spectral shape and amplitude are most sensitive to a GW (normalized to 10 9 M, equal-mass binaries and expressed in gravitational units), with weaker dependence on the inner power-law index and the outer delay time. The GWB is largely insensitive to the mass and mass-ratio scaling of a GW and to the boundary between the inner and outer astrophysical inspiral regimes. I compare with models for gas- and stellar-driven binary inspiral and discuss implications for LISA and for GW source parameter inference.