  In this note, we obtain various formulas for the curvature of the $L^2$
metric on the direct image of the relative canonical bundle twisted by a
holomorphic line bundle endowed with a positively curved metric with analytic
singularities, generalizing some of Berndtsson's seminal results in the smooth
case. When the twist is assumed to be relatively big, we further provide a very
explicit lower bound for the curvature of the $L^2$ metric.
