Exact analytic solution for f(H) yielding positive Einstein‑frame de Sitter expansion

Construct an explicit analytic function f(H), defined for H>0 in the O(d,d)-invariant isotropic cosmology, that satisfies the differential condition for an Einstein-frame de Sitter vacuum with time-dependent dilaton, namely (d+1)/(d−1)·s·H·sqrt(H f′(H) − f(H))·f″(H) + (1/(d−1))·(H f′(H) − f(H))·f″(H) + (d/(d−1))·H^2·f″(H) + s·sqrt(H f′(H) − f(H))·f′(H) + (1/2)·H·f′(H)·f″(H) = 0, such that the associated quantity F(H) = (d/(1−d))·[H + s·sqrt(H f′(H) − f(H))] is nonzero and the resulting Einstein-frame Hubble parameter H_E is a strictly positive constant H_* > 0 for a vacuum solution.

Background

In the isotropic setup of O(d,d)-invariant cosmology to all orders in α′, the authors derive a necessary condition for the existence of an Einstein-frame de Sitter solution with a time-dependent dilaton. This condition can be written compactly as (1/F)' + 1/G = 0, which yields the explicit differential equation for f(H) shown above. Here, F(H) = d/(1−d)·[H + s·sqrt(H f′ − f)] and G(H) = s·sqrt(H f′ − f]·f″(H), with s = ±1 determined by the sign of the dilaton’s time derivative.

They present one analytic solution, f̄(H) = (H + H_m)^2 − H_m^2 (for H>0 and s=−1), but this yields F(H)=0 and thus H_* = 0 in Einstein frame. They further report that while numerical solutions exist with H_E>0, they were unable to obtain an exact closed-form f(H) producing nonvanishing F(H) and a positive constant H_E. Establishing such an explicit analytic solution would demonstrate a fully analytic realization of Einstein-frame de Sitter expansion in this non-perturbative framework.