Generalizing the Swampland: Embedding $P(X, \varphi)$ Inflationary
  Theories in a Curved Multi-field Space
  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. We especially focus on the first
potential slow roll parameter defined in the two-field picture and obtain an
upper bound on it.
