On the E¹-term of the gravity spectral sequence

The author constructed a spectral sequence strongly converging to h_*(Omega^n Sigma^n X) for any homology theory in [Topology 33 (1994) 631-662]. In this note, we prove that the E^1-term of the spectral sequence is isomorphic to the cobar construction, and hence the spectral sequence is isomorphic to the classical cobar-type Eilenberg-Moore spectral sequence based on the geometric cobar construction from the E^1-term. Similar arguments can be also applied to its variants constructed in [Contemp Math 293 (2002) 299-329].