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.