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

Abstract

Abstract Pulsar timing arrays, as an effective low frequency gravitational source identification tool, show excellent applicability in circular supermassive binary black hole systems. In order to evaluate the efficiency and accuracy of the array, we introduce the Cramer-Rao lower bound to estimate the detection efficiency of the pulsar timing array. We use CRLB to estimate the parameters of Virgo source under both evolutionary and non-evolving search conditions, as well as full term and Earth term search conditions. The results show that the estimation accuracy of the two parameters is lower in the evolutionary condition than in the non evolving condition because the inclination Angle and the gravitational wave strain participate in the evolving process. Under the condition of Full term and Earth term search, because the Earth term search does not involve pulsar distance, the estimation accuracy of the whole parameter is higher than that of full term search. This results in a significant reduction in the uncertainty of pulsar parameters, enabling more accurate gravitational wave detection. In addition, in the case of relatively dense pulsars, the detection accuracy of gravitational wave 
direction will also be significantly reduced. In general, our method provides a reference for searching the lower limit of parameters under different objective conditions, and helps to evaluate whether the parameter estimation of the gravitational wave detection algorithm 
reaches the optimal level. This provides a theoretical basis for the progress in the field of gravitational waves and the optimization of pulsar timing arrays.