Generalizing the Swampland: Embedding $P(X, \varphi)$ Inflationary Theories in a Curved Multi-field Space

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We study the general embedding of a $ P(X, \varphi) $ inflationary theory into a two-field theory with curved field space metric, which was proposed as a possible way to examine the relation between de Sitter Swampland conjecture and \textit{k}-inflation.

• We show that this embedding method fits into the special type of two-field model in which the heavy field can be integrated out at the full action level.
• However, this embedding is not exact due to the upper bound of the effective mass of the heavy field.
• We quantify the deviation between the speed of sound calculated via the $ P(X, \varphi) $ theory and the embedding two-field picture to next leading order terms.