Effective potential for the O(N)-symmetric model in static homogeneous spacetimes

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Abstract
The effective potential for a scalar theory with O(N) symmetry is calculated in the large-N limit for the Einstein and static hyperbolic universes. For the Einstein universe it is found that spontaneous symmetry breaking via a second-order phase transition cannot occur regardless of the masses or strengths of the couplings of the fields to the scalar curvature. For the static hyperbolic universe spontaneous symmetry breaking can occur and is enhanced if the coupling to the scalar curvature is positive. For both spacetimes the curvature changes the detailed behavior of the effective potential but, with the exception of spontaneous symmetry breaking, its basic properties are the same as in flat space.