MatBYIB: A MATLAB-Based Toolkit for Parameter Estimation of Eccentric Gravitational Waves from EMRIs
Abstract
1. Introduction
2. Theroy
2.1. Waveform Generation
2.2. Response Function
2.3. Fisher Information Matrix
2.4. Markov Chain Monte Carlo
Algorithm 1 Pseudo-code of Metropolis–Hastings |
|
2.5. Convergence Diagnostics
3. Software Architecture
- Common_CF.m: This module defines the values of physical constants and the function to read input files (Readinput( )). Users can fill in the corresponding parameters as needed for their models.
- eqs( ): Defines a set of ordinary differential equations (ODEs) (Equation (1)) and returns the values of the orbital parameters as they evolve. Note that there may be slight differences in the ODE solvers due to the different versions of MATLAB.
- evolution( ): Calculates the solutions of the kludge ODE equations defined in eqs( ). To solve the ODE system, we utilize the ODE45 function in MATLAB. This choice facilitates the switch between different native solvers available in the library.
- get_Aplus_A_cross( ): This calls evolution( ) and calculates the intensity of the different-order harmonic GW in the source coordinate system according to the Equation (3).
- waveform_td( ): This calls eqs( ), evolution( ), and get_Aplus_A_cross( ), to compute the time-domain GW including the detector response function from the detector module and then performs its Fourier transform via the Fourier_tran( ) function in Common_CF.m. This process is integrated into the functions Fisher_Matrix( ) and MCMC_run( ) detailed below.
- Detector.m: This module defines the sensitivity curves of various GW detectors, including LISA [57], Taiji [6], and Tianqin [7], as well as their response functions. Users can modify the detector sensitivity curve directly in the main.m file. For example, they can call the Sen_curve_LISA( ) function from the Detector.m module to obtain the LISA sensitivity curve.
- get_noise( ): This defines the detector noise function, and the noise satisfies the Gaussian distribution.
- Fisher_Matrix.m: This module defines functions related to the calculation of the FIM, including:
- Diff_param( ): This function defines the partial derivatives of the GW signal with respect to different parameters. It generates the frequency domain function by calling the waveform_fd( ) and uses the central difference to obtain the partial derivatives of the GW signal with respect to different parameters.
- Matrix( ): This calls diff_param( ) and inv( ) functions, with the latter being MATLAB’s built-in function for calculating the inverse of a matrix.
- MCMC.m: This module consists of functions related to MCMC. The functions mainly include:
- lprior( ), lpost( ): These functions define the prior distribution and the posterior probability, respectively. They can be manually set to specific ranges.
- llike( ): This function computes the likelihood according to Equation (7). It is used in MCMC_run( ) to compute the likelihood at each step.
- accept_reject( ): This function corresponds to Equation (12).
- MCMC_run( ): This calls waveform_fd( ) to generate the GW signal, calculates the posterior distribution at each step according to Equation (7), and calls accept_reject( ) to determine whether to retain the current particle’s posterior distribution value, Equation (12), and then continues to iterate.
- converg( ): The function uses Equations (13)–(15) to assess the convergence of sampling. If , the chain is considered non-convergent, and we will increase the sampling points. Typically, in our program, the number of points is increased to 1.25 times the original total number, after which MCMC_contin( ) is invoked to resume sampling.
- MCMC_contin( ): Its usage is identical to that of MCMC_run( ).
4. Test and Numerical Examples
4.1. GW Waveform
4.2. Fisher Information Matrix
4.3. MCMC
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
MBH Mass | |||||
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References
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Line Number | Values | Units | Parameters | Physical Quantity |
---|---|---|---|---|
1 | 1 × 106 | the mass of central BH | ||
2 | 10 | the mass of rotating object | ||
3 | magnitude of spin angular momentum of MBH | |||
4 | where is the last stable circular orbital eccentricity | |||
5 | z | red shift | ||
6 | 60 | ° | ||
7 | 0 | where is the last stable circular orbital mean anomaly | ||
8 | 60 | ° | where is the angle (in orbital plane) between and pericenter | |
9 | 60 | ° | where is the azimuthal direction of in the orbital plane | |
10 | 60 | ° | the source direction’s polar angle | |
11 | 60 | ° | azimuthal direction to source | |
12 | 60 | ° | the polar angle of MBH’s spin | |
13 | 60 | ° | azimuthal direction of MBH’s spin | |
14 | 0 | ° | the mass of rotating object | |
15 | 3.14 × 106 | s | is time where orbit is last stable circular orbit |
Iteration | ||||
---|---|---|---|---|
FIM | ||||
mcmcstat | ||||
2000 | ||||
5000 | ||||
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Li, G.; Zhao, S.; Guo, H.; Su, J.; Lin, Z. MatBYIB: A MATLAB-Based Toolkit for Parameter Estimation of Eccentric Gravitational Waves from EMRIs. Universe 2025, 11, 259. https://doi.org/10.3390/universe11080259
Li G, Zhao S, Guo H, Su J, Lin Z. MatBYIB: A MATLAB-Based Toolkit for Parameter Estimation of Eccentric Gravitational Waves from EMRIs. Universe. 2025; 11(8):259. https://doi.org/10.3390/universe11080259
Chicago/Turabian StyleLi, Genliang, Shujie Zhao, Huaike Guo, Jingyu Su, and Zhenheng Lin. 2025. "MatBYIB: A MATLAB-Based Toolkit for Parameter Estimation of Eccentric Gravitational Waves from EMRIs" Universe 11, no. 8: 259. https://doi.org/10.3390/universe11080259
APA StyleLi, G., Zhao, S., Guo, H., Su, J., & Lin, Z. (2025). MatBYIB: A MATLAB-Based Toolkit for Parameter Estimation of Eccentric Gravitational Waves from EMRIs. Universe, 11(8), 259. https://doi.org/10.3390/universe11080259