We study the general embedding of a $P(X,\phi )$ inflationary theory into a two-field theory with a curved field space metric, which was proposed as a possible way to examine the relation between de Sitter swampland conjecture and $k$-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 $P(X,\phi )$ theory and the embedded two-field picture to next leading order terms. We especially focus on the first potential slow-roll parameter defined in the two-field picture and obtain an upper bound on it.

• Received 28 January 2021
• Accepted 26 February 2021

DOI:https://doi.org/10.1103/PhysRevD.103.063533

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Published by the American Physical Society