Density-gradient-vorticity relation in perfect-fluid Robertson-Walker perturbations

In a previous paper, a second-order propagation equation was derived for covariant and gauge-invariant vector fields characterizing density inhomogeneities in an almost-Friedmann-Lema\^{}tre-Robertson-Walker (-FLRW) perfect-fluid universe. However, an error there led to omission of a term representing an effect of vorticity on spatial density gradients at linear level. Here we determine this interaction (leading to an extra term in the second-order propagation equation for the spatial density gradient), and examine its geometrical and physical meaning. We define a new local decomposition of the observed density gradient and we show that the scalar variable defined in the decomposition naturally describes density clumping, and satisfies the standard Bardeen second-order equation. The physical meaning of the other variables defined in the decomposition is discussed, and their propagation equations are presented. Finally, the vorticity-induced time growth of the density gradient is derived in the long-wavelength limit.