Cramér-Rao lower bounds on parameter estimation for a circular binary of supermassive black holes in gravitational wave observations using pulsar timing array

Pulsar timing arrays (PTAs) are effective in detecting low-frequency gravitational waves (GWs), especially in circular binary systems of supermassive black holes. To evaluate the effectiveness and accuracy of PTAs in searching for GW parameter estimation, we use the Cramer-Rao lower bound to estimate the parameters of the Virgo source under various search conditions, including both evolving and non-evolving scenarios, as well as Earth and pulsar term searches, and only Earth term searches. The results show that the estimation accuracy of the inclination angle and GW strain is lower in the evolving condition than in the non-evolving condition because the two parameters participate in the evolving process. In addition, we find that the parameters estimable relates to signal-to-noise ratio, which is beneficial for reducing the dimensionality in the search process. In summary, our method serves as a reference for exploring lower parameter limits under different conditions and assists in assessing the optimization of parameter estimation in GW detection algorithms. This helps lay the theoretical foundation for advances in the field of GWs and the optimization of PTAs.