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

Lehel Csillag, Erik Jensko

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 $Λ$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 $Λ$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 $Λ$CDM in the past. Finally, we review the geometric foundations of the theory and show that the integrable Weyl, Schrödinger and completely symmetric geometries are embedded within our framework as special cases.

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

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 -essence models with quadratic kinetic terms. A detailed dynamical systems analysis reveals that the 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 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 -essence models have been missed as serious competitors to CDM in the past. Finally, we review the geometric foundations of the theory and show that the integrable Weyl, Schrödinger and completely symmetric geometries are embedded within our framework as special cases.

Paper Structure

This paper contains 18 sections, 7 theorems, 121 equations, 9 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Geometric foundations of nonmetricity
  3. Action principles for integrable nonmetricity
  4. Field equations and variations
  5. Scalar-tensor representation
  6. Cosmological dynamics of geometric k-essence
  7. Dynamical systems formulation
  8. Fixed point analysis and stability constraints
  9. Late-time observational constraints
  10. Redshift representation
  11. Methodology and datasets
  12. Results
  13. Discussion and final remarks
  14. Coordinate-free treatment of vectorial nonmetricity
  15. Derivation of the field equations
  16. ...and 3 more sections

Key Result

Proposition A.4

Let $\nabla$ be a connection with vectorial nonmetricity. Then, for all vector fields, which satisfy $\nabla_{X} X=0$, the following are equivalent:

Figures (9)

  • Figure 1: Illustration of vectorial nonmetricity and its special cases.
  • Figure 2: Phase portraits with pressureless matter equation of state $w=0$. The grey area (above the black line) is excluded due to a negative matter density $\Omega_m <0$. Points A, B and C lie on this line with $\Omega_m =0$. The triangle formed from the vertices O, A, C represents the physically relevant phase space, and orbits within this region stay in this region for all time.
  • Figure 3: Evolution of the matter density parameter $\Omega_m$, the $k$-essence density parameter $\Omega_\phi$, and the deceleration parameter $q$ in the geometric $k$-essence model, compared to the standard $\Lambda$CDM scenario.
  • Figure 4: Phase plot showing the physical phase space (black border) and accelerating region (green) used to define the parameter space, see Eq. (\ref{['param_region']}).
  • Figure 5: Corner plots showing the posterior constraints on the model parameters for the geometric $k$-essence model (left panel) and the $\Lambda$CDM model (right panel). Both the $1\sigma$ and $2\sigma$ confidence regions are displayed.
  • ...and 4 more figures

Theorems & Definitions (24)

  • Definition A.1
  • Remark A.2
  • proof
  • Remark A.3
  • Proposition A.4
  • proof
  • Remark A.5
  • Proposition A.6
  • proof
  • Remark A.7
  • ...and 14 more