Warped branches of flux compactifications

We consider Freund-Rubin-type compactifications which are described by (p + q)-dimensional Einstein gravity with a positive cosmological constant and a q-form flux. Using perturbative expansions of Kinoshita's ansatz for warped dS(p) x S-q and AdS(p) x S-q spacetimes, we obtain analytical solutions describing the warped branches and their respective phase spaces. These equations are given by inhomogeneous Gegenbauer differential equations which can be solved by the Green's function method. The requirement that the Green's functions are regular provides constraints which determine the structure of the phase space of the warped branches. We apply the perturbation results to calculate the thermodynamic variables for the warped dS(p) x S-q branch. In particular, the first law of thermodynamics can be reproduced using this method.