Can we hear beats with pulsar timing arrays?

TL;DR

The paper investigates whether the characteristic cross-correlation pattern of a gravitational-wave background, as captured by pulsar timing arrays, can be distinguished from or perturbed by the presence of a secondary SMBHB whose GWs have near-identical amplitudes but different frequencies. It develops a two-source, monochromatic framework in which cross-terms factorize into a geometry-dependent component $\Pi(\gamma)$ and a beat-dependent component $B(t_i,t_f)$, with $\omega_B=2\pi|f_2-f_1|$ and beat phase $\Phi$, and shows how four equal time-domain partitions yield an inverse problem whose solution provides $\cos\eta$ with $\eta=\omega_B T_{obs}/4$, enabling inference of the beat frequency $f_{beat}$. The authors illustrate the method with a numerical example and discuss the validity range of the monochromatic approximation, extensions to scenarios with stochastic backgrounds, and conditions under which beat modulations could be observed in PTA data. Overall, the work offers a principled approach to detecting beat-driven modulations in PTA angular correlations, potentially revealing multiple dominant nano-HHz GW sources and improving interpretation of PTA signals.

Abstract

An isolated supermassive black hole binary (SMBHB) produces an identical cross-correlation pattern of pulsar timings as an isotropic stochastic background gravitational waves (GWs) generated possibly by inflation. Can there remain the identical cross-correlation pattern in the presence of a secondary SMBHB? To address this issue, the present paper focuses on GWs with similar amplitudes but slightly different frequencies $f_1$ and $f_2$ coming from two different directions. Beats between the two GWs can modify angular correlation patterns. The beat-induced correlation patterns are not stationary but modulated with a beat frequency $f_{beat} \equiv |f_1 - f_2|$. We obtain an analytic solution that allows us to infer $f_{beat}$ from the modulated angular correlations.

Figures (4)

• Figure 1: Beat between two sinusoidal waves traveling along the same line. $h_1$ and $f_1$ are the amplitude and frequency of the primary wave, while $h_2$ and $f_2$ are those of the secondary one. Here, $h_2/ h_1 = 0.5$ and $f_2/f_1 = 0.8$ are chosen. The blue, red and black (in color) curves denote the primary, secondary, and sum of sinusoidal waves and respectively, where a green curve means an envelope wave of frequency $(f_1 - f_2)/2$.
• Figure 2: Geometrical factor $\Pi(\gamma)$ for $\beta = 0, \pi/4, \pi/2, 3\pi/4, \pi$, where $h_{1+} = h_{1\times}=1$ and $h_2 \equiv h_{2+} = h_{2\times}=0.5$ are chosen for its simplicity. See e.g. CornishAllen2024Sasaki for a set of coordinates convenient for numerical calculations of $\int_{S^2} d\Omega_{pa}$ and $\int_{S^2} d\Omega_{pb}$.
• Figure 3: Angular correlations for GWs with $\beta = \pi/2$. For the curves to be recognized by eye, $h_2/ h_1 = 0.5$, $f_2/f_1 = 0.8$ and $\Phi = \pi/5$ are chosen as exaggerations. For the convenience in numerical plots, $h_1 \equiv h_{1+} = h_{1\times}$ and $h_2 \equiv h_{2+} = h_{2\times}$ are assumed, which affect only the magnitude of $\Pi(\gamma)$ but do not influence our discussions especially including Figure \ref{['fig-B']}.
• Figure 4: Cross-term correlation $\Pi(\gamma) B_K$ by Eq. (\ref{['crossterm']}) dependent on $D_K$. The parameters correspond to those in Figure \ref{['fig-correlation']}.