Quintessence and phantom emerging from the split-complex field and the split-quaternion field

TL;DR

This paper introduces split-complex and split-quaternion scalar fields inspired by hyperbolic number theory, exploring their cosmic evolution and showing they can naturally produce quintessence and phantom fields in a flat universe.

Contribution

It defines new scalar fields based on split-complex and split-quaternion numbers and analyzes their cosmological evolution, linking mathematical structures to dark energy models.

Findings

Both quintessence and phantom fields can emerge from these scalar fields.

The theories are a subclass of multi-field inflationary models.

Cosmic evolution of these fields aligns with observed universe expansion.

Abstract

Motivated by the mathematic theory of split-complex numbers (or hyperbolic numbers, also perplex numbers) and the split-quaternion numbers (or coquaternion numbers), we define the notion of split-complex scalar field and the split-quaternion scalar field. Then we explore the cosmic evolution of these scalar fields in the background of spatially flat Friedmann-Robertson-Walker Universe. We find that both the quintessence field and the phantom field could naturally emerge in these scalar fields. Introducing the metric of field space, these theories fall into a subclass of the multi-field theories which have been extensively studied in inflationary cosmology.