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Disformal transformations in a Palatini extension of Horndeski's gravity

Aleksander Kozak

Abstract

In this paper, we extend Horndeski's theory into the Palatini approach, assuming that the metric tensor and the (symmetric) connection are a priori independent objects. We introduce an additional transformation of the connection and write down the action functional being form-invariant under both the disformal transformation of the metric and the new transformation of the connection. We show that such a theory reduces on-shell to a metric subclass of Horndeski's gravity called kinetic gravity braiding. We also introduce an invariant metric and connection, and demonstrate that quantities defined in such a way lead to a metric theory. In the second part of the paper, we consider a simple cosmological model within the theory and explore its potential links with $ k$-essence-type theories, with a non-trivial coupling between the scalar field and the matter part of the action in the Einstein frame. We show that there exists a model that reproduces late-time cosmic acceleration, approaching asymptotically the de Sitter phase, motivating further study of the theories.

Disformal transformations in a Palatini extension of Horndeski's gravity

Abstract

In this paper, we extend Horndeski's theory into the Palatini approach, assuming that the metric tensor and the (symmetric) connection are a priori independent objects. We introduce an additional transformation of the connection and write down the action functional being form-invariant under both the disformal transformation of the metric and the new transformation of the connection. We show that such a theory reduces on-shell to a metric subclass of Horndeski's gravity called kinetic gravity braiding. We also introduce an invariant metric and connection, and demonstrate that quantities defined in such a way lead to a metric theory. In the second part of the paper, we consider a simple cosmological model within the theory and explore its potential links with -essence-type theories, with a non-trivial coupling between the scalar field and the matter part of the action in the Einstein frame. We show that there exists a model that reproduces late-time cosmic acceleration, approaching asymptotically the de Sitter phase, motivating further study of the theories.
Paper Structure (14 sections, 67 equations, 3 figures, 2 tables)

This paper contains 14 sections, 67 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Disformal transformations and the transformation of the connection
  3. Action functional and transformation formulae
  4. Related MA theories
  5. Invariant metric and connection
  6. Equations of motion and dynamically equivalent metric theory
  7. Field equation for the scalar field
  8. Cosmology in the disformally invariant Palatini theory
  9. Vanishing energy density
  10. Non-vanishing energy density
  11. Case I: $q = 2$
  12. Case II: $q = 4$
  13. Discussion and conclusions
  14. Scalar field functions in the dynamically-equivalent metric representation

Figures (3)

  • Figure 1: Phase space of the system \ref{['eq:system1']} with $\zeta_1 = \zeta_2$. The red dashed line represents those values of $x_1, x_2$ for which the system becomes singular. The part of the figure in light blue corresponds to the unphysical region within the phase space.
  • Figure 2: Phase space of the system \ref{['eq:system2']}. The red dashed line represents those values of $x_1, x_2$ for which the system becomes singular. The green point $P$ represents the possible current state of the universe. The part of the figure in light blue corresponds to the unphysical region within the phase space.
  • Figure 3: Evolution of the deceleration parameter $q$ and the dark energy dimensionless density fraction $\Omega_\phi$ for a trajectory that passes through the point $P$ in Fig \ref{['fig:both']} at present, i.e. for $\hat{N} = 0$.