Sensitivity functions of space-borne gravitational wave detectors for arbitrary time-delay interferometry combinations regarding nontensorial polarizations

Direct observation of gravitational waves offers us numerous novel possibilities to further explore the Universe. In practice, the efficiency of the experimental facility for a given gravitational wave source can be measured in terms of the sensitivity function of the instrument. The latter essentially indicates the least incident amplitude required in order to achieve the desired level of the signal-to-noise ratio. Among others, the resultant sensitivity function depends on the specific polarization state of the incident gravitational wave as well as the spatial layout of the detector and its orientation to the wave source. Theoretically, Einstein's general relativity predicts two tensor polarization modes, from a total of six possible modes arising from an arbitrary perturbation of the spacetime metric. Therefore, in this context, the feasibility of measuring nontensorial polarization states provides access to an alternative theory of gravity. In the present study, we analytically evaluate the response functions for arbitrary time-delay interferometry combinations while enumerating all possible polarization modes. The derivation is accomplished by separating the average on the all-sky solid angle from the remaining expression, which, in turn, gives rise to a few factors independent of the time-delay interferometry combination in question. Moreover, we applied the obtained results to the LISA and TianQin missions, and the asymptotic behavior of the resultant sensitivity functions is analyzed and discussed. Among others, it is observed that, for Sagnac combinations, the averaged response function of the breathing mode attains zero at specific discrete frequencies. Such a frequency value corresponds to a multiple of the reciprocal of the one-way light propagation time along the detector arm, irrelevant to the orientation of the wave source. For all six polarization modes, the present findings can be readily applied to an arbitrary time-delay interferometry combination with improved efficiency and accuracy.