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

## TL;DR

This paper derives analytical formulas for the frequency response of various TDI combinations in space-based gravitational wave detectors, analyzing their behavior across frequencies and polarization modes, and provides sensitivity curves for LISA and TianQin.

## Contribution

It offers the first comprehensive analytical formulas for the averaged response functions of different polarizations in TDI combinations, including their asymptotic behaviors.

## Key findings

- Response functions increase as $f^2$ at low frequencies for all polarizations.
- Response functions for tensor and breathing modes fall off as $1/f^2$ at high frequencies.
- Sensitivity curves for LISA and TianQin are provided for multiple polarizations.

## Abstract

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, $f\ll f_*$, the averaged response functions of all TDI combinations increase as $f^2$ for all six polarizations. The one exception is the averaged response functions of $\zeta$ for all six polarizations increase as $f^4$ in the equilateral-triangle case. At high frequencies, $f\gg f_*$, the averaged response functions of all TDI combinations for the tensor and breathing modes fall off as $1/f^2$, the averaged response functions of all TDI combinations for the vector mode fall off as $\ln(f)/f^2$ , 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.