Unlike ground-based gravitational wave detectors, space-based gravitational wave detectors can detect the ringdown signals from massive black hole binary mergers with large signal-to-noise ratio, and help to extract the source parameters and localize
the source. To reduce the computation time in Fisher information matrix, we derive the analytical formulas of frequency-domain ringdown signals for heliocentric detectors and geocentric detectors by considering the effect of the harmonic phases, the rotation period of the geocentric detector, and the detector arm length. We explore the median errors of parameter estimation and source localization with ringdown singals from binaries with different masses and different redshifts. Using a binary source with the total mass $M=10^7 M_odot$ at the redshift $z=1$, we analyze the dependence of these errors on the sky position. We find that the network of space-based gravitational wave detectors can significantly improve the source localization at the ringdown stage.
The discovery of gravitational waves, which are ripples of space-time itself, opened a new window to test general relativity, because it predicts that there are only plus and cross polarizations for gravitational waves. For alternative theories of gr
avity, there may be up to six polarizations. The measurement of the polarization is one of the major scientific goals for future gravitational wave detectors. To evaluate the capability of the detector, we need to use the frequency dependent response functions averaged over the source direction and polarization angle. We derive the full analytical formulas of the averaged response functions for all six possible polarizations and present their asymptotic behaviors based on these analytical formulas. Compared with the numerical simulation, the full analytical formulas are more efficient and valid for any equal-arm interferometric gravitational wave detector without optical cavities in the arms and for a time-delay-interferometry Michelson combination.
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. T
he 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, $fll f_*$, 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, $fgg f_*$, 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.
