Tachyon potentials from a supersymmetric FRW model

Considering that the effective theory of closed string tachyons can have world sheet supersymmetry, as shown by Vafa, we study a worldline supersymmetric action in a Friedmann-Robertson-Walker background, for which the superpotential originates a tachyon scalar potential. There are such potentials with spontaneously broken supersymmetry at the instability and supersymmetry after tachyon condensation. Furthermore, given a tachyonic potential, the superpotential can be computed by a power series ansatz and has a free parameter that can be chosen such that complex solutions become real.