Inflationary Constraints on String Theory

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It is an important task to embed inflation in a fundamental microphysical theory such as string theory. Since string theory possesses a vast landscape of 4-dimensional theories, we would like to know which portions contain inflation and which do not. I prove a no-go theorem that inflation and de Sitter vacua are forbidden in an exponentially large number of infinite families of simple and well understood compactifications of type IIA string theory. I also mention more complicated and less well understood compactifications, which may have the ingredients for our cosmology.