The Laser Interferometer Space Antenna (LISA) is due to launch in the mid-2030s. A key challenge for LISA data analysis is efficient Bayesian inference with parametrised gravitational-wave models, particularly for early inspirals of low- and intermediate-mass black-hole binaries, where time series can contain ∼ 10^8 – 10^9 samples and naive likelihood evaluations become prohibitively expensive. We present a time-domain likelihood-approximation scheme for such signals. The method retains a small subset of samples and defines a modified noise-weighted inner product on this subset that closely reproduces the original inner product on the waveform manifold. Because this alters the effective noise model, the scheme is intended for analysis of simulated, rather than real LISA data. In our examples, the resulting posteriors closely agree with those obtained using much denser sampling, while retaining only 10^3 – 10^4 samples. The computational cost scales linearly with the number of retained samples N_s , so the speed-up is roughly N_f/N_s , where N_f is the original data length; for realistic LISA-like datasets with N_f∼ 10^8 – 10^9 this would correspond to gains of order 10^4 – 10^6 over straightforward frequency-domain likelihood evaluations of the full dataset, although a small number of additional runs is required to verify convergence. The time-domain formulation is particularly convenient for modelling effects naturally expressed through modified time evolution, such as those induced by non-trivial astrophysical environments or additional dynamical fields in beyond-GR theories. This paper provides the theoretical basis for the software package Dolfen.
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