# Preliminary study on parameter estimation accuracy of supermassive black hole binary inspirals for TianQin

## TL;DR

This study evaluates how accurately TianQin can estimate parameters of supermassive black hole binaries using Fisher matrix analysis, considering spin, eccentricity, and post-Newtonian effects, with simulations showing promising precision levels.

## Contribution

It introduces a comprehensive Fisher matrix analysis including spin, eccentricity, and higher-order post-Newtonian effects for TianQin's parameter estimation of SMBH binaries.

## Key findings

- Mass parameters can be estimated with sub-percent accuracy.
- Luminosity distance can be measured within 1-3%.
- Angular resolution is better than 12 square degrees.

## Abstract

We use the Fisher information matrix method to calculate the parameter estimation accuracy of inspiraling supermassive black holes binaries for TianQin, a space-borne laser interferometric detector aimed at detecting gravitational waves in the millihertz frequency band. The `restricted' post-Newtonian waveform in which third order post-Newtonian (3PN) phase including spin effects (spin-orbit $\beta$ and spin-spin $\sigma$) and first-order eccentricity contribution is employed. Monte Carlo simulations using $10^3$ binaries for mass pairs with component masses in the range of $({10^5},{10^7}){M_ \odot }$ and cosmological redshift $z=0.5$ show that the medians of the root-mean-square error distributions for the chirp mass $M_c$ and symmetric mass ratio $\eta$ are in the range of $\sim 0.02\% - 0.7\% $ and $\sim 4\% - 8\% $, respectively. The luminosity distance $D_L$ can be determined to be $\sim 1\% - 3\% $, and the angular resolution of source $\Delta \Omega $ is better than 12 deg$^2$. The corresponding results for $z=1.0$ and $2.0$, which are deteriorated with the decreasing of the signal-to-noise ratio, have also been given. We show that adding spin parameters degrades measurement accuracy of the mass parameters (${M_c}$, $\eta$), and the time and the orbital phase of coalescence ($t_c$, $\phi _c$); the inclusion of the first-order eccentricity correction to the phase worsens the estimation accuracy comparing with the circular cases. We also show the effects of post-Newtonian order on parameter estimation accuracy by comparing the results based on second order and third order post-Newtonian phases. Moreover, we calculate the horizon distance of supermassive black hole binaries for TianQin.