Accurate pulsar timing array residual variances and correlation of the stochastic gravitational wave background

Abstract

Pulsar timing arrays have reported a compelling evidence of a nanohertz stochastic gravitational wave background. However, the origin of the signal remains undetermined, largely because its spectrum is bluer for an astrophysical source and can be explained by cosmological models. In this letter, we revisit the frequency- and Fourier-domain analysis of the signal by deriving theoretically accurate expressions for the Fourier bin variances and correlation of pulsar timing residuals, and demonstrate their outstanding agreement with point source astrophysical simulations. In contrast, we show that a common power law (or a diagonal covariance approximation) traditionally used to interpret a stochastic gravitational wave background signal is generally faced with systematic errors, one of which is the illusion of a bluer signal. This hints at a conservative solution, supportive of an astrophysical source, to the observed correlated common spectrum process in pulsar timing arrays.

Figures (5)

• Figure 1: Low-pass and high-pass transfer functions \ref{['eq:aa_filter']} and \ref{['eq:bb_filter']} in the 1st and 10th frequency bins with $T=15$ yr; $f_1 \sim 2$ nHz, $f_{10}=10 f_1\sim 20$ nHz.
• Figure 2: Sky positions of the Meerkat PTA pulsars (red stars) Miles:2022lkg with a smoothed projection/map of a SGWB.
• Figure 3: Constraints on the power spectrum models $I_{{\rm M}_0}(f)=I_{\rm ref} \overline{f}^{-7/3}$ and $I_{{\rm M}_1}(f)=I_{\rm ref} \overline{f}^{2-\gamma}$ with accurate bin variances (\ref{['eq:xy_correlation']}-\ref{['eq:bb_filter']}) using 15-yr Meerkat PTA mock data with input GWs at frequencies $f\in (1, 100)$ nHz from ${\cal O}(10^3)$ SMBHBs, $A_{\rm gw}=2.4 \times 10^{-15}$, and $\gamma_{\rm gw}=13/3$.
• Figure 4: Simulated bin variances in 15-yr Meerkat PTA mock data (red and blue points/error bars) with input GWs at frequencies $f\in (1, 100)$ nHz from ${\cal O}(10^3)$ SMBHBs, $A_{\rm gw}=2.4 \times 10^{-15}$, and $\gamma_{\rm gw}=13/3$. Black stars and green pentagons are best fits from models ${\rm M}_1^*(I_{\rm ref}^*, \eta^*)$ and ${\rm M}_0^*(I_{\rm ref}^*)$, respectively, and (\ref{['eq:xy_correlation']}-\ref{['eq:bb_filter']}). Violet '$\times$' and green '$+$' are bin variances corresponding to a common power law model saturated to fit both $\alpha$- and $\beta$-bin variances of the mock data. Shaded regions below thresholds $\sim10^{-3}$$\mu$s$^2$ and $\sim10^{-2}$$\mu$s$^2$ depict typical one-point noise levels.
• Figure 5: Correlation in the 1st frequency bin, $f_1\sim 2$ nHz, in 15-yr Meerkat PTA mock data (red points/error bars) with input GWs at frequencies $f\in (1, 100)$ nHz from ${\cal O}(10^3)$ SMBHBs, $A_{\rm gw}=2.4 \times 10^{-15}$, and $\gamma_{\rm gw}=13/3$. Black-solid and blue-dashed lines show the best fits ${\rm M}_1^*(I_{\rm ref}^*, \eta^*)$ and ${\rm M}_0^*(I_{\rm ref}^*)$, respectively.