A novel probe of graviton dispersion relations at nano-Hertz frequencies

TL;DR

This work generalizes Phinney's practical theorem to accommodate modified graviton dispersion relations with a frequency-dependent GW speed $c_T(f)$. A tanh-like transition model $c_T(f,\sigma,f_*)$ induces a localized distortion, or bump, in the SGWB spectrum $\Omega_{\rm GW}(f)$ near the transition frequency $f_*$, governed by $\Delta = 1 - c_T(f_d)/c_T(f_s)$ and the source redshift distribution. The authors derive a modified energy-density relation and provide a Padé-fit template $\tilde{\Omega}_{\rm GW}$ to quantify deviations from the canonical $f^{2/3}$ scaling. They perform Fisher forecasts for Pulsar Timing Arrays, using Hellings-Downs correlations and a realistic pulsar-noise model, showing that with large numbers of pulsars and plausible SMBH populations, constraints on $1-c_0$ can reach the $\sim 10^{-2}$ to $10^{-3}$ level, while also constraining the redshift distribution of SGWB sources. The results highlight nano-Hertz GW probes as a testbed for modified gravity and suggest a potential role as a cosmic distance ladder, motivating extensions to other detector bands and more sophisticated noise and population models.

Abstract

We generalise Phinney's 'practical theorem' to account for modified graviton dispersion relations motivated by certain cosmological scenarios. Focusing on specific examples, we show how such modifications can induce characteristic localised distortions, bumps, in the frequency profile of the stochastic gravitational wave background emitted from distant binary sources. We concentrate on gravitational waves at nano-Hertz frequencies probed by pulsar timing arrays, and we forecast the capabilities of future experiments to accurately probe parameters controlling modified dispersion relations. Our predictions are based on properties of gravitational waves emitted in the first inspiral phase of the binary process, and do not rely on assumptions of non-linear effects occurring during the binary merging phase

Figures (5)

• Figure 1: The quantity $c_T$ as a function of $f/f_*$ for $\sigma = 1,\,2,\,5$, demonstrating the behaviour of the ansatz \ref{['ansatz1']} for $c_0 = 0.9$.
• Figure 2: Plot of the quantity ${\Omega}_{\rm GW}(x,\,c_0,\,z_0)$ (with $x = f/f_*$) for $z_0 \,=\, 1,3$ with $c_0 = 0.6, 0.4$ (respectively left and right panels). We also examine the the quality of fit of the model in eq \ref{['omegafit']}.
• Figure 3: Left panel: The relative error in $c_0$ as a function of $N_{psr}$ for $1 - c_0 = 10^{-1}$ to $1 - c_0 = 10^{-2}$ marginalized over redshift. Right panel: The relative error ellipses for $\log_{10}(1-c_0)$ against $z$ for $N_{psr} = 500, 2000$ and $1-c_0 = 5 \times 10^{-2}$ with all sources located at the benchmark value $z = 1$. The quoted errors refer to the the $N_{psr} = 500$ case.
• Figure 4: The SMBH binary population models (1)-(3) of Table \ref{['table:1']}, as a function of $z$ for binaries with chirp mass $\mathcal{M} = 3.2 \times 10^7\mathcal{M}_{\odot}$ emitting at the frequency $f_*$.
• Figure 5: The relative error bounds for $\log_{10}(1-c_0) = -1, -2, -3$ benchmark values, integrating over redshift eq \ref{['popmodel']} to $z = 1$. First row: population model $(1)$. Second row: population model $(3)$. We select $N_{psr}\,=\,200$ and $N_{psr}\,=\,500$. In all plots, the darker bands represent a 1-$\sigma$ deviation, and the lighter bands represent a 2-$\sigma$ deviation. The quoted errors refer to the $N_{psr} = 200$ case with all errors tabulated in Table \ref{['table:2']}.