Multifractal Analysis of Pulsar Timing Residuals: Assessment of Gravitational Wave Detection
We introduce a pipeline including multifractal detrended cross-correlation analysis (MF-DXA) modified by either singular value decomposition or the adaptive method to examine the statistical properties of the pulsar timing residual ($PTR$) induced by a gravitational wave (GW) signal. We propose a new algorithm, the so-called irregular-MF-DXA, to deal with irregular data sampling. Inspired by the quadrupolar nature of the spatial cross-correlation function of a gravitational wave background, a new cross-correlation function, $\barσ_{\times}$, derived from irregular-MF-DXA will be introduced. We show that, this measure reveals the quadrupolar signature in the $PTRs$ induced by stochastic GWB. We propose four strategies based on the $y$-intercept of fluctuation functions, the generalized Hurst exponent, and the width of the singularity spectrum to determine the dimensionless amplitude and power-law exponent of the characteristic strain spectrum as $\mathcal{H}_c(f)\sim\mathcal{A}_{yr}(f/f_{yr})^ζ$ for stochastic GWB. Using the value of Hurst exponent, one can clarify the type of GWs. We apply our pipeline to explore 20 millisecond pulsars observed by Parkes Pulsar Timing Array. The computed scaling exponents confirm that all data are classified into a nonstationary class implying the universality feature. The value of the Hurst exponent is in the range $H\in [0.56,0.87]$. The $q$-dependency of the generalized Hurst exponent demonstrates that the observed $PTRs$ have multifractal behavior, and the source of this multifractality is mainly attributed to the correlation of data which is another universality of the observed datasets. Multifractal analysis of available $PTRs$ datasets reveals an upper bound on the dimensionless amplitude of the GWB, $\mathcal{A}_{yr}< 2.0\times 10^{-15}$.