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On angular dependent response to gravitational-wave signals for time-delay interferometry combinations

Pan-Pan Wang, Hao-Kang Chen, Wei-Liang Qian, Rui Luo, Jing Zhou, Wei-Sheng Huang, Yu-Jie Tan, Cheng-Gang Shao

Abstract

Space-based gravitational wave (GW) detectors are designed for wave sources in the millihertz band with different locations and orientations. Time-delay interferometry (TDI) technique is an indispensable ingredient in space-borne GW detection that effectively suppresses the laser phase noise. The abundant TDI solutions derived in the literature also feature distinct angular-dependent sensitivities. Because a GW source's angular location is unknown prior to the signals' detection, a solid-angle average is often performed when analyzing the sensitivity function of a given TDI combination. The present study explores the angular dependence of the detector's sensitivity. This detail is relevant, because once the initial detection is achieved, the source's location can be extracted and used to provide information on a refined TDI combination tailored for the specific GW source. As the TDI technique is a post-processing algorithm, such a procedure can be implemented in practice. We evaluate the angular dependence of the detector's response function to the GW signals for different TDI combinations as a function of the orientation angles. Moreover, we classify the response functions into seven categories at the low-frequency limit, leveraging the characteristics of the underlying geometrical TDI combinations. By further averaging out the azimuthal angle $φ_D$ in the detector's plane, the main features of the resulting response functions and their zenithal dependence with respect to the GW source are scrutinized. The findings presented in this work provide pertinent insights for ongoing space-borne detector programs.

On angular dependent response to gravitational-wave signals for time-delay interferometry combinations

Abstract

Space-based gravitational wave (GW) detectors are designed for wave sources in the millihertz band with different locations and orientations. Time-delay interferometry (TDI) technique is an indispensable ingredient in space-borne GW detection that effectively suppresses the laser phase noise. The abundant TDI solutions derived in the literature also feature distinct angular-dependent sensitivities. Because a GW source's angular location is unknown prior to the signals' detection, a solid-angle average is often performed when analyzing the sensitivity function of a given TDI combination. The present study explores the angular dependence of the detector's sensitivity. This detail is relevant, because once the initial detection is achieved, the source's location can be extracted and used to provide information on a refined TDI combination tailored for the specific GW source. As the TDI technique is a post-processing algorithm, such a procedure can be implemented in practice. We evaluate the angular dependence of the detector's response function to the GW signals for different TDI combinations as a function of the orientation angles. Moreover, we classify the response functions into seven categories at the low-frequency limit, leveraging the characteristics of the underlying geometrical TDI combinations. By further averaging out the azimuthal angle in the detector's plane, the main features of the resulting response functions and their zenithal dependence with respect to the GW source are scrutinized. The findings presented in this work provide pertinent insights for ongoing space-borne detector programs.

Paper Structure

This paper contains 22 sections, 80 equations, 15 figures, 6 tables.

Table of Contents

  1. Introduction
  2. The space-based GW detector and TDI algorithm
  3. The space-based GW detector and data streams
  4. The TDI technique
  5. The relationship between the barycentric and detector frames
  6. The coordinate transform between the barycentric and the LISA detector frames
  7. The coordinate transform between the barycentric and the TianQin detector frames
  8. Angular dependent response functions to GW signals for an arbitrary TDI combination
  9. The angular dependence of the response functions
  10. Angular dependent response function for an arbitrary TDI combination
  11. The noise PSD
  12. Angular dependent response functions for the forty-five geometric TDI combinations
  13. Specific TDI combinations at the low-frequency limit
  14. Angular dependent response functions
  15. Angular dependent response functions for TianQin detector
  16. ...and 7 more sections

Figures (15)

  • Figure 1: The spatial layout of the three ongoing space-based GW detectors.
  • Figure 2: An illustration of a space-based detector's various arm lengths tdi-03.
  • Figure 3: An illustration of the Michelson TDI combination from a geometric TDI perspective tdi-geometric-2022. Left: The virtual laser beams' trajectories that furnish the Michelson combination. Right: The equivalent space-time diagram of the combination. While the colors of the trajectories remain unchanged, one uses dashed lines to represent propagations in the negative direction of time.
  • Figure 4: The source trajectory in the LISA reference frame when $(\theta_B,\phi_B)$ are set to $\left( {\frac{\pi }{2},0} \right),\left( {\frac{\pi }{2},\frac{\pi }{3}} \right), \left( {0,\frac{\pi }{2}} \right),\left( {0,\frac{{3\pi }}{2}} \right)$.
  • Figure 5: The source trajectory in the LISA reference frame when $\left( {0,0} \right),\left( {\frac{\pi }{3},0} \right), \left( {\frac{\pi }{2},0} \right),\left( {\frac{{3\pi }}{2},0} \right)$.
  • ...and 10 more figures