Comparison of various methods to extract ringdown frequency from gravitational wave data

The ringdown part of gravitational waves in the final stage of the merger of compact objects tells us the nature of strong gravity and hence can be used for testing theories of gravity. The ringdown waveform, however, fades out in a very short time with a few cycles, and hence it is challenging to extract the ringdown frequency and its damping timescale. We here propose to build up a suite of mock data of gravitational waves to compare the performance of various approaches developed to detect the dominant quasinormal mode from an excited black hole after merger. In this paper, we present our initial results of comparisons of the following five methods: (1) plain matched filtering with ringdown part method, (2) matched filtering with both merger and ringdown parts method, (3) Hilbert-Huang transformation method, (4) autoregressive modeling method, and (5) neural network method. After comparing the performances of these methods, we discuss our future projects.

Figure 1

Examples of sets A and B. (Inset) The ringdown part. Here, set A [(red) thick] with SXS:BBH:0174 and set B [(blue) thin] with SXS:BBH:0002 are shown. The solid lines denote the modified amplitude $A_{22}(t)$, and the dashed lines are the GW frequency ${\omega }_{22}(t)/(2\pi )$. The total mass is $M=60M_{\odot }$, and the real and imaginary parts of the ringdown frequency are 300 and 40 Hz, respectively. The real frequency is obtained by multiplying by 538.609 Hz, and the real amplitude of set A is derived by dividing by 1.37903. The large difference in the inspiral phase is due to the difference of the binary parameters.

Figure 2

Plots of the base 10 logarithm of the error in the estimate for all test data [(a) and (b) for the real part for sets A and B, respectively, and (c) and (d) for the imaginary part for sets A and B, respectively]. The data number is sorted for each method in ascending order of the magnitude of the error.

Figure 3

The root-mean-square error in percent of each method for three levels of SNR [(a) for both sets A and B, (b) for only set A, and (c) for only set B]. The solid and dashed lines represent the real and imaginary parts, respectively.

Figure 4

The averaged error in percent of each method for three levels of SNR [(a) for both sets A and B, (b) for only set A, and (c) for only set B]. The solid and dashed lines represent the real and imaginary parts, respectively.