Recovering P(X) from a canonical complex field

TL;DR

This paper explores the relationship between complex scalar field models and P(X)-theories in cosmology, analyzing their background dynamics and perturbation behaviors, including stability and dispersion relations.

Contribution

It establishes a correspondence between complex scalar fields and P(X)-theories, clarifying their perturbation spectra and stability conditions across different regimes.

Findings

Low momentum dispersion matches P(X)-theory sound speed.

Superluminal case leads to tachyonic instability.

Complex fields impose a cutoff on unstable modes.

Abstract

We study the correspondence between models of a self-interacting canonical complex scalar field and P(X)-theories/shift-symmetric k-essence. Both describe the same background cosmological dynamics, provided that the amplitude of the complex scalar is frozen modulo the Hubble drag. We compare perturbations in these two theories on top of a fixed cosmological background. The dispersion relation for the complex scalar has two branches. In the small momentum limit, one of these branches coincides with the dispersion relation of the P(X)-theory. Hence, the low momentum phase velocity agrees with the sound speed in the corresponding P(X)-theory. The behavior of high frequency modes associated with the second branch of the dispersion relation depends on the value of the sound speed. In the subluminal case, the second branch has a mass gap. On the contrary, in the superluminal case, this branch is vulnerable to a tachyonic instability. We also discuss the special case of the P(X)-theories with an imaginary sound speed leading to the catastrophic gradient instability. The complex field models provide with a cutoff on the momenta involved in the instability.