Instability of brane cosmological solutions with flux compactifications
  We discuss the stability of the higher-dimensional de Sitter (dS) brane solutions with two-dimensional internal space in the Einstein-Maxwel theory. We show that an instability appears in the scalar-type perturbations with respect to the dS spacetime. We derive a differential relation which has the very similar structure to the ordinary laws of thermodynamics as an extension of the work for the six-dimensional model [20]. In this relation, the area of dS horizon (integrated over the two internal dimensions) exactly behaves as the thermodynamical entropy. The dynamically unstable solutions are in the thermodynamically unstable branch. An unstable dS compactification either evolves toward a stable configuration or two-dimensional internal space is decompactified. These dS brane solutions are equivalent to the accelerating cosmological solutions in the six-dimensional Einstein-Maxwell-dilaton theory via dimensional reduction. Thus, if the seed higher-dimensional solution is unstable, the corresponding six-dimensional solution is also unstable. From the effective four-dimensional point of view, a cosmological evolution from an unstable cosmological solution in higher dimensions may be seen as a process of the transition from the initial cosmological inflation to the current dark energy dominated Universe. 
Masato Minamitsuji