Moduli of suspension spectra

For a -connected spectrum , we study the moduli space of suspension spectra which come equipped with a weak equivalence to . We construct a spectral sequence converging to the homotopy of the moduli space in positive degrees. In the metastable range, we get a complete homotopical classification of the path components of the moduli space. Our main tool is Goodwillie's calculus of homotopy functors.