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

TL;DR

This paper explores embedding $P(X, )$ inflation models into curved multi-field spaces, analyzing the approximation's accuracy and deriving bounds on slow-roll parameters to understand their relation to the Swampland conjecture.

Contribution

It demonstrates how $P(X, )$ theories can be embedded into two-field models with curved space, quantifies deviations, and establishes bounds on slow-roll parameters.

Findings

Embedding is approximate due to heavy field mass bounds.

Deviation in sound speed is quantified at next-to-leading order.

Upper bound on the first potential slow roll parameter is derived.

Abstract

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.