The Volkov–Akulov–Starobinsky supergravity

We construct a supergravity model whose scalar degrees of freedom arise from a chiral superfield and are solely a scalaron and an axion that is very heavy during the inflationary phase. The model includes a second chiral superfield X  , which is subject however to the constraint X2=0X2=0 so that it describes only a Volkov–Akulov goldstino and an auxiliary field. We also construct the dual higher-derivative model, which rests on a chiral scalar curvature superfield RR subject to the constraint R2=0R2=0, where the goldstino dual arises from the gauge-invariant gravitino field strength as γmnDmψnγmnDmψn. The final bosonic action is an R+R2R+R2 theory involving an axial vector AmAm that only propagates a physical pseudoscalar mode.