Comment on "Hearing the signal of dark sectors with gravitational wave detectors" [Phys. Rev. D 94, no. 10, 103519 (2016)]

Abstract

We revisit the calculation of the gravitational wave spectra generated in a classically scale-invariant $SU(2)$ gauge sector with a scalar field in the adjoint representation, as discussed by J.~Jaeckel, et al. The finite-temperature potential at 1-loop level can induce a strong first-order phase transition, during which gravitational waves can be generated. With the accurate numerical computation of the on-shell Euclidean actions of the nucleation bubbles, we find that the triangle approximation employed by J.~Jaeckel, et al. strongly distorts the actual potential near its maximum and thus greatly underestimates the action values. As a result, the gravitational wave spectra predicted by J.~Jaeckel, et al deviate significantly from the exact ones in peak frequencies and shapes.