= Solution
Applying $\epsilon=b_i(t)x^i$ to $A_\mu=0$ gives
$$
A_0=\dot b_i(t)x^i,
\qquad
A_i=b_i(t).
$$
Its 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 is
$$
\boxed{\partial_t(a^3\dot b_i)=0},
$$
with a constant growing solution and a decaying solution proportional to $\int^t dt'/a^3(t')$. This condition makes the large gauge mode an <adiabatic mode> that can be continued to small nonzero momentum.
Back to article page
