Cosmological Vorticity in a Gravity with Quadratic Order Curvature Couplings

where Lm is the matter part Lagrangian. The gravitational field equation is presented in Eq. (2) of [1]. In contrast with the situation with generalized f(φ,R) gravity in Eq. (27), the gravity with Ricci-curvature square term in the action does not have the conformal symmetry to Einstein gravity [2]. In [1] we analysed the gravitational wave and derived a fourth order differential equation. However, for the background equation we have a symmetry between the R and RRab terms; the effects of the two terms appear in a common manner in the background equation, see Eq. (22). In this paper, we will analyse the vector type perturbation. In contrast with the gravitational wave studied in [1] we will find that the Ricci-curvature square term does not add any new feature to the evolution of the vector type perturbation: the rotational perturbation is described by the angular momentum conservation independently of the generalized nature of the gravity theory. The main result is Eq. (26) which is the same equation valid in Einstein gravity with A = 0 = B. Since the analysis for RRab term is new, we present the necessary details for the derivation in the Appendices. We take the same convention as [1].