 Solution

Applying $ϵ=b_i(t)x^i$ to $A_{\mu }=0$ givesIts field strength vanishes, so it is a large gauge transformation of the background. To arise as the zero-momentum limit of a physical transverse perturbation, $b_i(t)$ must obey the zero-momentum vector equation of motion. In the variables used in the question this iswith a constant growing solution and a decaying solution proportional to ${\int }^{t}d{t}^{\prime }/a^3(t^{\prime })$. This condition makes the large gauge mode an adiabatic mode that can be continued to small nonzero momentum.