# Parameter Inference for Coalescing Massive Black Hole Binaries Using Deep Learning

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## Abstract

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## 1. Introduction

## 2. Model

## 3. Datasets

## 4. Results

## 5. Summary and Discussions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Framework of the neural network used in this work. The small black dashed box represents the CNN used to extract key features from the strain data s. The large black dashed box depicts the block of the NF, and the number of the blocks is set to 22. The initial variable z is transformed to $\theta $ after passing through the whole neural network.

**Figure 2.**P-P plot for redshifted total mass M (blue), mass ratio q (green), coalescence time ${t}_{c}$ (orange), and luminosity distance ${d}_{L}$ (red) of MBHB. We generate a test dataset composed of 1000 simulated strain data to conduct the KS test for our model, and the p-values are listed in the upper left corner. The model produces 20,000 samples for each test data. The colored lines denote the empirical CDF provided by the model, and the black dashed line denotes the true CDF. The grey regions represent the $1\sigma $, $2\sigma $ and $3\sigma $ confidence bounds.

**Figure 3.**Reduction in parameter space volume as a function of SNR. The value of reduction is calculated based on a 90% credible region obtained from the posterior samples produced by our model in the KS test. The blue line denotes the average value calculated for each bin of SNR.

**Figure 4.**Posterior distributions of the redshifted total mass M, mass ratio q, coalescence time ${t}_{c}$, and luminosity distance ${d}_{L}$ produced by our model. The orange line denotes the result for a single MBHB signal and the blue line denotes the result for the same signal overlapped with the inspiral phases of other 14 MBHB signals. The model produces 50,000 posterior samples for each case.

Parameter | Prior |
---|---|

M | $\mathrm{LogUniform}{[10}^{6}{M}_{\odot},{10}^{7}{M}_{\odot}]$ |

q | $\mathrm{Uniform}[1,5]$ |

${t}_{c}$ | $\mathrm{Uniform}[3\phantom{\rule{3.33333pt}{0ex}}\mathrm{d},365\phantom{\rule{3.33333pt}{0ex}}\mathrm{d}]$ |

${d}_{L}$ | $\mathrm{Uniform}{[2910\phantom{\rule{3.33333pt}{0ex}}\mathrm{Mpc},47,312\phantom{\rule{3.33333pt}{0ex}}\mathrm{Mpc}]}^{3}$ |

$({s}_{1z},{s}_{2z})$ | $\mathrm{Uniform}[-1,1]$ |

$cos\iota $ | $\mathrm{Uniform}[-1,1]$ |

$sin\beta $ | $\mathrm{Uniform}[-1,1]$ |

$\lambda $ | $\mathrm{Uniform}[0,2\pi ]$ |

${\varphi}_{c}$ | $\mathrm{Uniform}[0,2\pi ]$ |

$\psi $ | $\mathrm{Uniform}[0,\pi ]$ |

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**MDPI and ACS Style**

Ruan, W.; Wang, H.; Liu, C.; Guo, Z.
Parameter Inference for Coalescing Massive Black Hole Binaries Using Deep Learning. *Universe* **2023**, *9*, 407.
https://doi.org/10.3390/universe9090407

**AMA Style**

Ruan W, Wang H, Liu C, Guo Z.
Parameter Inference for Coalescing Massive Black Hole Binaries Using Deep Learning. *Universe*. 2023; 9(9):407.
https://doi.org/10.3390/universe9090407

**Chicago/Turabian Style**

Ruan, Wenhong, He Wang, Chang Liu, and Zongkuan Guo.
2023. "Parameter Inference for Coalescing Massive Black Hole Binaries Using Deep Learning" *Universe* 9, no. 9: 407.
https://doi.org/10.3390/universe9090407