Geometric formulation of $k$-essence and late-time acceleration

TL;DR

This paper introduces a geometric framework with nonmetricity linked to a vector field, showing it can replicate late-time cosmic acceleration and fit observational data, offering a viable alternative to the standard $\\Lambda$CDM model.

Contribution

The work formulates a geometric model with nonmetricity that is equivalent to $k$-essence, analyzes its stability, and constrains its parameters using observational data, demonstrating its viability as a late-time acceleration model.

Findings

Model reproduces $\\Lambda$CDM phenomenology.

Stable parameter regions identified through physical viability criteria.

Model fits late-time observational data and is statistically indistinguishable from $\\Lambda$CDM.

Abstract

We study a class of geometries in which nonmetricity is fully determined by a vectorial degree of freedom and three independent coefficients. Formulating the simplest linear action in this geometry, implemented through Lagrange multipliers, naturally leads to an equivalence with the purely kinetic $k$-essence models with quadratic kinetic terms. A detailed dynamical systems analysis reveals that the $\Lambda$CDM phenomenology is embedded within the model. Crucially, we find that if stability conditions such as a positive sound speed squared and non-negative energy density are not enforced, the model generically exhibits instabilities and divergent behaviour in the phase space. These physical viability criteria allow us to isolate stable regions of the parameter space and derive well-motivated priors for parameter inference. Using Markov Chain Monte Carlo methods and late-time observational data, including cosmic chronometers, Pantheon$^{+}$ Type Ia supernovae, and DESI baryon acoustic oscillations, we constrain the degrees of freedom associated with nonmetricity and demonstrate the viability of the model. Remarkably, the model is found to be statistically indistinguishable from $\Lambda$CDM at late times. We discuss the implications of these results in light of the recent cosmic tensions, and give a possible explanation as to why the equivalent $k$-essence models have been missed as serious competitors to $\Lambda$CDM in the past. Finally, we review the geometric foundations of the theory and show that the integrable Weyl, Schr\"{o}dinger and completely symmetric geometries are embedded within our framework as special cases.