Frequency response of time-delay interferometry for space-based gravitational wave antenna

Space-based gravitational wave detectors cannot keep rigid structures and precise arm length equality, so the precise equality of detector arms which is required in a ground-based interferometer to cancel the overwhelming laser noise is impossible. The time-delay interferometry method is applied to unequal arm lengths to cancel the laser frequency noise. We give analytical formulas of the averaged response functions for tensor, vector, breathing, and longitudinal polarizations in different TDI combinations, and obtain their asymptotic behaviors. At low frequencies, f≪f*, the averaged response functions of all TDI combinations increase as f2 for all six polarizations. The one exception is that the averaged response functions of ζ for all six polarizations increase as f4 in the equilateral-triangle case. At high frequencies, f≫f*, the averaged response functions of all TDI combinations for the tensor and breathing modes fall off as 1/f2, the averaged response functions of all TDI combinations for the vector mode fall off as ln(f)/f2, and the averaged response functions of all TDI combinations for the longitudinal mode fall as 1/f. We also give LISA and TianQin sensitivity curves in different TDI combinations for tensor, vector, breathing, and longitudinal polarizations.