The Cosmon Model for an Asymptotically Vanishing Time-dependent Cosmological “constant”

We investigate the coupled system of gravity and a scalar with exponential potential. The energy momentum tensor of the scalar field induces a time-dependent cosmological “constant”. This adjusts itself dynamically to become in the “late” universe (including today) proportional to the energy density of matter and radiation. Possible consequences for the present cosmology are shortly discussed. We also address the question of naturalness of the cosmon model. Whenever cosmology encounters potential difficulties in the description of the present universe cosmologists revive the discussion about the cosmological constant [1]. The discrepancy between the critical energy density expected from inflationary cosmology and lower dynamical estimates of this density has been attributed to the cosmological constant [2]. The discussion also pertains to the age of the universe [2] and the formation of structure [3]. In fact, a cosmological constant λ of the order of today’s critical energy density in the universe (λ ≈ (2 · 10eV )) strongly affects the present universe without altering the successful predictions of the hot big bang model at early stages of the evolution of the universe. Despite many attempts [4] we have at present no satisfactory understanding why λ should be much smaller than typical energy scales of the standard model or even the Planck mass Mp. For a time-independent cosmological constant it seems even harder to explain why it should be of the order of the present energy density. The latter depends on the age of the universe rather than on fundamental constants. It looks then not very natural that a constant λ should have a value which equals the energy density just at a time within the present cosmological epoch. In this work we consider a model where the cosmological “constant” varies with time such that the asymptotic solution for late times is characterized by a constant ratio λ(t)/ρ(t)[5], [6]. We discuss consequences for present cosmology and various alternatives how “early cosmology” could have made a transition to this type of “late cosmology”. We also briefly address the question of naturalness of an asymptotically vanishing cosmological “constant”. We start from the field equations for a scalar field φ coupled to gravity in a homogenous and isotropic universe (with k = 0 and H the Hubble parameter) φ̈+ 3Hφ̇+ ∂V ∂φ = q (1) ρ̇+ 3H(ρ+ p) + qφ̇ = 0 (2) H = 1 6M2 (