Constraining the evolution of Newton's constant with slow inspirals observed from spaceborne gravitational-wave detectors

Spaceborne gravitational-wave (GW) detectors observing at milli-Hz and deci-Hz frequencies are expected to detect large numbers of quasi-monochromatic signals. The first and second time-derivative of the GW frequency (\(\dot f_0\) and \(\ddot f_0\)) can be measured for the most favourable sources and used to look for negative post-Newtonian corrections, which can be induced by the source's environment or modifications of general relativity. We present an analytical, Fisher-matrix-based approach to estimate how precisely such corrections can be constrained. We use this method to estimate the bounds attainable on the time evolution of the gravitational constant \(G(t)\) with different classes of quasi-monochromatic sources observable with LISA and DECIGO, two representative spaceborne detectors for milli-Hz and deci-Hz GW frequencies. We find that the most constraining source among a simulated population of LISA galactic binaries could yield \(\dot G/G_0 \lesssim 10^{-6}\text{yr}^{-1}\), while the best currently known verification binary will reach \(\dot G/G_0 \lesssim 10^{-4}\text{yr}^{-1}\). We also perform Monte-Carlo simulations using quasi-monochromatic waveforms to check the validity of our Fisher-matrix approach, as well as inspiralling waveforms to analyse binaries that do not satisfy the quasi-monochromatic assumption. We find that our analytical Fisher matrix produces good order-of-magnitude constraints even for sources well beyond its regime of validity. Monte-Carlo investigations also show that chirping stellar-mass compact binaries detected by DECIGO-like detectors at cosmological distances of tens of Mpc can yield constraints as tight as \(\dot G/G_0 \lesssim 10^{-11}\text{yr}^{-1}\).