VCDM and Cuscuton

TL;DR

This paper compares two minimally modified gravity theories, VCDM and Cuscuton, analyzing their solutions, relations to General Relativity, and cosmological mappings, revealing conditions under which they align or differ.

Contribution

It provides a detailed comparison of VCDM and Cuscuton theories, identifying solution compatibilities, conditions for GR solutions, and establishing a cosmological mapping between the two.

Findings

All Cuscuton solutions are solutions of VCDM, but not vice versa.

VCDM admits exact GR solutions with or without matter and cosmological constant.

A mapping between VCDM and Cuscuton exists for quadratic potentials, affecting the effective Planck mass.

Abstract

We investigate two Type-IIa Minimally Modified Gravity theories, namely VCDM and Cuscuton theories. We confirm that all acceptable Cuscuton solutions are always solutions for VCDM theory. However, the inverse does not hold. We find that VCDM allows for the existence of exact General Relativity (GR) solutions with or without the presence of matter fields and a cosmological constant. We determine the conditions of existence for such GR-VCDM solutions in terms of the trace of the extrinsic curvature and on the fields which define the VCDM theory. On the other hand, for the Cuscuton theory, we find that the same set of exact GR solutions (such as Schwarzschild and Kerr spacetimes) is not compatible with timelike configurations of the Cuscuton field and therefore cannot be considered as acceptable solutions. Nonetheless, in Cuscuton theory, there could exist solutions which are not the same but close enough to GR solutions. We also show the conditions to determine intrinsic-VCDM solutions, i.e. solutions which differ from GR and do not belong to the Cuscuton model. We finally show that in cosmology a mapping between VCDM and the Cuscuton is possible, for a generic form of the VCDM potential. In particular, we find that for a quadratic potential in VCDM theory, this mapping is well defined giving an effective redefinition of the Planck mass for the cosmological background solutions of both theories.