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We analyze the large-order behavior of the perturbative weak-field expansionof the effective Lagrangian density of a massive scalar in de Sitter and antide Sitter space, and show that this perturbative information is not sufficientto describe the non-perturbative behavior of these theories, in contrast to theanalogous situation for the Euler-Heisenberg effective Lagrangian density forcharged scalars in constant electric and magnetic background fields. Forexample, in even dimensional de Sitter space there is particle production, butthe effective Lagrangian density is nevertheless real, even though itsweak-field expansion is a divergent non-alternating series whose formalimaginary part corresponds to the correct particle production rate. Thisapparent puzzle is resolved by considering the full non-perturbative structureof the relevant Feynman propagators, and cannot be resolved solely from theperturbative expansion.Comment: 18 page

Large-order perturbation theory and de Sitter/anti–de Sitter effective actions