On FLRW reductions of N=1 supergravity

An FLRW reduction of N=1 supergravity is troublesome since spatial isotropy prohibits a non-vanishing Rarita-Schwinger field. In view of this, we consider the quadratic form ${\psi }_m^{\alpha }{\psi }_{n\alpha }$ arising from a particular superspace generalization of the metric tensor. Imposing the FLRW form on this object reduces ${\psi }_m^{\alpha }$ to a single 2-component spinor ${\phi }^{\alpha }$, which becomes the superpartner of both the scale factor $a$ and lapse $N$, thus inducing the conformal time gauge, $N=a$. Time-reparametrization invariance is restored if we allow for deviations of FLRW form that are proportional to the gauge fields $N,{\psi }_0^{\alpha }$. We determine the allowed supergravity transformations that preserve the resulting FLRW supermultiplets.