Modulator noise suppression in the LISA Time-Delay Interferometric combinations

We previously showed how the measurements of some eighteen time series of relative frequency or phase shifts could be combined (1) to cancel the phase noise of the lasers, (2) to cancel the Doppler fluctuations due to non-inertial motions of the six optical benches, and (3) to remove the phase noise of the onboard reference oscillators required to track the photodetector fringes, all the while preserving signals from passinggravitational waves. Here we analyze the effect of the additional noise due to the optical modulators used for removing the phase fluctuations of the onboard reference oscillators. We use a recently measured noise spectrum of an individual modulator to quantify the contribution of modulator noise to the first and second-generation Time-Delay Interferometric (TDI) combinations as a function of the modulation frequency. We show that modulator noise can be made smaller than the expected proof-mass acceleration and optical-path noises if the modulation frequencies are larger than $approx 682$ MHz in the case of the unequal-arm Michelson TDI combination $X_1$, $approx 1.08$ GHz for the Sagnac TDI combination $alpha_1$, and $approx 706$ MHz for the symmetrical Sagnac TDI combination $zeta_1$. These modulation frequencies are substantially smaller than previously estimated and may lead to less stringent requirements on the LISAs oscillator noise calibration subsystem.