Strong scale-dependence does not enhance the kinematic boosting of gravitational wave backgrounds

TL;DR

The paper tackles whether Doppler boosting from our Solar System motion can amplify stochastic gravitational wave backgrounds (SGWBs) with strong scale dependence. It derives the exact Lorentz boost for the SGWB phase-space energy density, showing the boosted spectrum is a blue/red shift of the isotropic frame without introducing extra amplification or new spectral features, in contrast to claims based on a truncated Taylor expansion in the boost parameter $\beta$. By analyzing a toy spectral model and a scalar-induced GW scenario, the authors demonstrate that the perceived enhancement in previous work is a artefact of the Taylor-series approach, which breaks down for steep spectral gradients. The results emphasize that the exact boost expression is straightforward to implement in data analyses and pipelines, ensuring reliable interpretation of SGWB anisotropies and preserving the smallness of the kinematic dipole under realistic velocities. This work clarifies the regime of validity for kinematic effects and provides a robust tool for analyzing scale-dependent SGWB scenarios.

Abstract

Existing expressions in the literature appear to indicate that Doppler boosting, due to our proper motion with respect to the isotropic frame of the universe, can amplify stochastic gravitational wave backgrounds whose energy spectra exhibit strong scale dependence, for example, those generated by large scalar perturbations in models of primordial black holes or by astrophysical populations with broken power-law behaviour. It has been suggested that this enhancement could increase the signal-to-noise ratio of such backgrounds in pulsar timing measurements, as well as in ground- and space-based observatories. We show that the reported enhancement is an artefact of a Taylor expansion of the boosted signal, typically performed in the literature under the assumption of a small boosting parameter. This approximation fails to reproduce the correct result for signals with strong scale dependence. When Doppler boosting is treated exactly, the apparent amplification disappears. Using representative spectra, we demonstrate that Doppler motion induces only blue- and red-shifting by the expected amount; it does not lead to additional amplification or introduce new spectral features. The exact expression for the kinematic boost can and should be easily applied in analysing such backgrounds.

Figures (2)

• Figure 1: Toy model GW energy density from the power spectrum of Eq. \ref{['eq:sqrt_H']} in the isotropic frame of the sources (written in terms of $f$ - solid blue curve), the observed spectrum in the direction $u=\hat{v}\cdot\hat{n}=1$ (doshed orange) and its first order Taylor expansion in the kinematic velocity $\beta$ (dotted green). We take the boost factor $\beta =0.00123$, corresponding to the Solar System speed in the CMB rest frame.
• Figure 2: Scalar induced power spectrum of Eq. \ref{['eq:PBH_Omega']} in the reference frame of the sources (solid blue), the observed spectrum in the direction $u \equiv \hat{v}\cdot\hat{n}=1$ (dashed orange) and its first order Taylor expansion in the kinematic velocity $\beta$ (dotted green). A log-normal mass function with parameter $\Delta=0.1$ is considered. The plot of the left panel assumes a large value for the kinematic velocity $\beta=0.5$ to emphasise the discrepancy between the curves, while the right-hand panel shows a detail of the curves around the cusp of the GW spectrum with kinematic boost factor $\beta=0.00123$ given by the velocity of the Solar System in the CMB reference frame.