Extra-dimensional curvature suppression of the effective four-dimensional vacuum energy density

Considering a very large number of extra dimensions, N → , we show that in the effective four-dimensional picture, to leading order in N, both the cosmological constant in N + 4 dimensions and the curvature of the extra dimensions (curved as spheres) give the same type of contributions. Furthermore, in this limit, the extra-dimensional curvature naturally suppresses the effect of a positive cosmological constant, so that the resulting effective potential governing the vacuum energy in the effective 4 D picture has a leading 1/N dependence (i.e. vanishing in the large N limit). We can understand qualitatively this effect in a heuristic picture by thinking that both visible and extra dimensions have an equal sharing of the curvature caused by Λ. In this case when increasing the overall number of dimensions by adding N extra dimensions, then if N is large, the visible dimensions do not have to curve too much; hence, a small four-dimensional vacuum energy follows. In the large N picture, the potential can also be stabilized by a small (i.e. vanishing at large N) expectation value of a four-index field strength.