Forecasting Sensitivity to Modified Dispersion Effects in Pulsar Timing Arrays

Abstract

The pulsar timing array systems have reported a detection of a nanohertz-band stochastic gravitational wave background in our galaxy. It is of interest to use this observation to probe modified gravity and to forecast the sensitivity with which certain deviations can be tested in the coming years. In this paper, we focus on the modified dispersion relation of the tensor modes and its effect on the overlap reduction function of the timing residual cross-correlations. We perform a comprehensive forecast of the phase velocity uncertainty, $σ_v$, using a Fisher analysis validated by a mock-data study to account for potential non-Gaussian behavior. We also take into account the sample variance effect and provide an observational timeline for future PTA sensitivity: detecting a $10\%$ or $-1\%$ deviation from the speed of light at the $3σ$ level requires $\mathcal{O}(30)$ years of observations.

Figures (10)

• Figure 1: The sample variance in modified gravity vs. GR. The right panels show the overlap reduction function $\Gamma(\xi)$, and the left panels show the coefficients of the Legendre polynomials. The blue lines and shaded areas represent the ORF for GR and its variance, the red lines correspond to the massive-gravity–type modified dispersion with $v_p > 1$, and the green lines correspond to the case with $v_p < 1$.
• Figure 2: Left panel: Forecast $\sigma_v$ (log-scaled) from the Fisher formalism as a function of $v_p^{fid}$, using the angular separations $\xi_{ij}$ and uncertainties $\sigma_{ij}$ from the NANOGrav's $15$ yrs dataset maximum-likelihood noise parameters optimal statistic Agazie_2023the_nanograv_collaboration_2025_16051178 (green curve), and a random angular distribution following Eq. \ref{['pdf_xi']} and constant $\bar{\sigma}$ (blue and red dashed curve). We use the same number of pulsars ($N_{psr}=67$) for all curves and the same positions for the blue and dashed red curves. For the blue curve, we choose $\bar{\sigma}=\sqrt{\frac{N_{pairs}}{\sum_{\langle i,j\rangle}1/\sigma_{ij}^2}}\approx1.3$ where $\sigma_{ij}$ are the uncertainties from the NANOGrav's optimal statistic. For the red dashed curve, we choose $\bar{\sigma}=0.9$ such that the red and green curves overlap for $v_p>1$. Right panel: Distribution of the angular separations associated with the blue (uniformly distributed) and green curves (NANOGrav's data) in the left-hand side plot.
• Figure 3: Whisker plot of the forecast confidence levels for different $v_p^{fid}$ and uncertainties $\tilde{\sigma}$. The $1\sigma$ (solid lines) and $2\sigma$ (dashed lines) are shown for comparison to the GR case ($v_p=1$, dashed black line). For large $v_p^{fid}$ and $\tilde{\sigma}$, the errors bars exceed the window width. Positive deviations ($v_p^{fid}>1.1$) only start to be distinguishable at $2\sigma$ level for $\tilde{\sigma}=0.2$.
• Figure 4: Total Fisher information in Eq. \ref{['def_fisher_w_SV']} as a function of the average uncertainty $\bar{\sigma}$ for $67$ uniformly distributed pulsars and three phase velocities, $v_p = 1.1, 1, 0.9$. Solid lines show the full Fisher information; dashed lines show the result in Eq. \ref{['def_fisher_w_SV']} obtained by approximating $\sigma_{\rm tot} \approx \bar{\sigma}$; and dotted lines show the maximum value in the limit $\sigma_{\rm SV} \gg \bar{\sigma}$. The black dotted lines separate the different approximation regimes, and the breakpoints indicate the threshold where sample variance cannot be neglected. We see that the $\bar{\sigma}^{-4}$ scaling is never reached.
• Figure 5: Whisker plot of the forecast confidence levels for different $v_p^{fid}$ and uncertainties $\tilde{\sigma}$, including the sample variance effect described in Sec. \ref{['1.D']}. The $1\sigma$ (solid lines) and $2\sigma$ (dashed lines) are shown for comparison to the GR case ($v_p=1$, dashed black line). For large $v_p^{fid}$ and $\tilde{\sigma}$, the error bars exceed the window width. The results are slightly modified compared to Fig. \ref{['whisk_fcst']} for $v_p\geq1.1$ and not even visible for $v_p<1.1$.