[Paper Review] Energy exchange in Weyl geometry

John Miritzis|arXiv (Cornell University)|Jan 23, 2013

Cosmology and Gravitation Theories3 citations

TL;DR

This paper investigates homogeneous and isotropic cosmologies in Weyl geometry, showing that the Weyl vector field acts as a stiff fluid, and when coupled to ordinary matter via energy exchange, the Weyl fluid dominates only at early times. Crucially, in integrable Weyl geometry, the scalar field associated with the Weyl connection behaves as a phantom field for certain parameter choices, providing a geometric mechanism for late-time accelerated expansion without postulating dark energy.

ABSTRACT

We study homogeneous and isotropic cosmologies in a Weyl spacetime. We show that the field equations can be reduced to the Einstein equations with a two-fluid source and analyze the qualitative, asymptotic behavior of the models. Assuming an interaction of the two fluids we impose conditions so that the solutions of the corresponding dynamical system remain in the physically acceptable phase space. We show that in Weyl integrable spacetime, the corresponding scalar field acts as a phantom field and therefore, it may give rise to a late accelerated expansion of the Universe.

Motivation & Objective

• To analyze homogeneous and isotropic cosmological models in Weyl spacetime, extending beyond standard general relativity.
• To model the Weyl vector field as a fluid and study its dynamical interaction with ordinary matter via energy exchange.
• To investigate whether the geometric scalar field in integrable Weyl geometry can generate late-time accelerated expansion.
• To determine the asymptotic behavior of cosmological models under interacting fluid dynamics in Weyl geometry.

Proposed method

• The field equations are derived from a constrained variational principle applied to the Ricci scalar Lagrangian, yielding modified Einstein equations with a two-fluid source.
• The Weyl vector field is expressed as $ Q_{ u} = q u_{ u} $, and its energy-momentum tensor is identified as a stiff fluid with $ \rho_1 = p_1 = \frac{3}{4}q^2 $.
• An energy-exchange model is introduced via the equations $ \dot{\rho}_1 = -3\gamma_1 H \rho_1 - \beta H \rho_1 + \alpha H \rho_2 $ and $ \dot{\rho}_2 = -3\gamma_2 H \rho_2 + \beta H \rho_1 - \alpha H \rho_2 $, ensuring total energy conservation.
• The dynamical system is analyzed in terms of dimensionless variables $ \omega $ and $ \chi $, with phase space restricted to $ D = [-\pi/2, \pi/2] \times [-\chi_\alpha, \chi_\alpha] $.
• A modified Lagrangian $ L = R + \xi \nabla^\mu Q_\mu + L_m $ is used to derive field equations where the scalar field $ \phi $, with $ Q_\mu = \partial_\mu \phi $, couples geometrically to gravity.
• The effective coupling parameter $ \lambda = \frac{4\xi - 3}{2} $ determines whether the scalar field behaves as a phantom field ($ \lambda > 0 $), enabling late-time acceleration.

Experimental results

Research questions

• RQ1 Can Weyl geometry provide a geometric mechanism for late-time cosmic acceleration without introducing dark energy?
• RQ2 How does energy exchange between the Weyl fluid and ordinary matter affect the asymptotic behavior of cosmological models?
• RQ3 What is the role of the Weyl scalar field in determining the late-time dynamics of the universe?
• RQ4 Under what conditions does the scalar field in integrable Weyl geometry behave as a phantom field?

Key findings

• The Weyl fluid acts as a stiff fluid with $ \rho_1 = p_1 = \frac{3}{4}q^2 $, and its contribution to the energy budget is significant only at early times in expanding models.
• In models with energy exchange between fluids, the 'real' fluid (ordinary matter) dominates at late times, while the Weyl fluid dominates near the big bang and in recollapsing models.
• The phase space is restricted to a closed rectangle $ D = [-\pi/2, \pi/2] \times [-\chi_\alpha, \chi_\alpha] $, with $ \chi_\alpha \leq 1 $, indicating reduced phase space compared to non-interacting models.
• The equilibrium point $ F_1^+ $ is a past attractor for all expanding models with $ \Omega > 0 $, indicating that early-time evolution is approximated by a flat FRW model dominated by the Weyl fluid.
• For $ \lambda > 0 $, the scalar field $ \phi $ behaves as a phantom field, which may drive late-time accelerated expansion through purely geometric means.
• The model shows that the Weyl field can induce late-time acceleration without postulating dark energy, provided the coupling parameter satisfies $ \lambda > 0 $.