Fast likelihood approximations for simulated LISA data by downsampling

Abstract 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 $$\sim 10^8$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>∼</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>8</mml:mn> </mml:msup> </mml:mrow> </mml:math> – $$10^9$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>9</mml:mn> </mml:msup> </mml:math> 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 $$10^3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>3</mml:mn> </mml:msup> </mml:math> – $$10^4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>4</mml:mn> </mml:msup> </mml:math> samples. The computational cost scales linearly with the number of retained samples $$N_\textrm{s}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>N</mml:mi> <mml:mtext>s</mml:mtext> </mml:msub> </mml:math> , so the speed-up is roughly $$N_\textrm{f}/N_\textrm{s}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>N</mml:mi> <mml:mtext>f</mml:mtext> </mml:msub> <mml:mo>/</mml:mo> <mml:msub> <mml:mi>N</mml:mi> <mml:mtext>s</mml:mtext> </mml:msub> </mml:mrow> </mml:math> , where $$N_\textrm{f}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>N</mml:mi> <mml:mtext>f</mml:mtext> </mml:msub> </mml:math> is the original data length; for realistic LISA-like datasets with $$N_\textrm{f}\sim 10^8$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>N</mml:mi> <mml:mtext>f</mml:mtext> </mml:msub> <mml:mo>∼</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>8</mml:mn> </mml:msup> </mml:mrow> </mml:math> – $$10^9$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>9</mml:mn> </mml:msup> </mml:math> this would correspond to gains of order $$10^4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>4</mml:mn> </mml:msup> </mml:math> – $$10^6$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>6</mml:mn> </mml:msup> </mml:math> 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 .