Accelerating the time-domain LISA response model with central finite differences and hybridization techniques

Accurate and efficient modeling of the Laser Interferometer Space Antenna (LISA) response is crucial for gravitational-wave (GW) data analysis. A key computational challenge lies in evaluating time-delay interferometry (TDI) variables, which require projecting GW polarizations onto the LISA arms at different retarded times. Without approximations, the full LISA response is computationally expensive, and traditional approaches, such as the long-wavelength approximation, accelerate the response calculation at the cost of reducing accuracy at high frequencies. In this work, we introduce a novel hybrid time-domain response for LISA that balances computational efficiency and accuracy across the binary's evolution. Our method is applicable to massive black hole binaries and implements a fast low-frequency approximation during the early inspiral$\unicode{x2013}$where most of these binaries spend most of the time in the sensitive frequency band of LISA$\unicode{x2013}$while reserving the computationally intensive full-response calculations for the late inspiral, merger, and ringdown phases. The low-frequency approximation (LFA) is based on Taylor expanding the response quantities around a chosen evaluation time such that time delays correspond to central finite differences. Our hybrid approach supports CPU and GPU implementations, TDI generations 1.5 and 2.0, and flexible time-delay complexity, and has the potential to accelerate parts of the global fit and reduce energy consumption. We also test our LFA and hybrid responses on eccentric binaries, and we perform parameter estimation for a "golden" binary. Additionally, we assess the efficacy of our low-frequency response for "deep alerts" by performing inspiral-only Bayesian inference.