On curvature coupling and quintessence fine-tuning

– We discuss the phenomenological model in which the potential energy of the quintessence field depends linearly on the energy density of the spatial curvature. We find that the pressure of the scalar field takes a different form when the potential of the scalar field also depends on the scale factor and the energy momentum tensor of the scalar field can be expressed as the form of a perfect fluid. A general coupling is proposed to explain the current accelerating expansion of the Universe and solve the fine-tuning problem. The astronomical observations tell us that the Universe is currently expanding in an acceleration phase, while it was in a deceleration phase in the near past [1,2]. The accelerating expansion of the Universe suggests that 70% of the total matter energy density comes from dark energy (DE) whose pressure is negative. The simplest candidate of DE is the cosmological constant. Due to the unusually small value of the cosmological constant given by observations, other DE models are explored, such as the quintessence models [3] and the holographic DE models [4, 5]. For a review of DE models, please see Ref. [6] and references therein. The quintessence models employ a scalar field that slowly rolls down its potential well to explain the accelerating expansion. In the quintessence model, the potential energy dominates the energy component, so the current value of the potential energy must be of the order of the current critical energy density 3mpH 2 0 = 8.1h 2×10−47 GeV, where the reduced Planck mass mp = (8πG) −1/2 = 1.221× 10 GeV, the current Hubble parameter H0 = 100h km/s/Mpc and h = 0.72±0.08 [7]. It is not natural to get the small potential energy for the quintessential scalar field, this is the fine-tuning problem. Recently, França proposed a phenomenological model in which the quintessential potential depends linearly on the curvature energy density ρk = 3m 2 pk/a 2 to solve the fine-tuning problem of DE model [8]. The existence of a small curvature is consistent with the observation [1]. In this letter, we assume that the quintessential potential depends linearly on k/a and start from the equation of motion of the scalar field to explain the accelerating expansion and solve the fine-tuning problem. (∗) E-mail: [email protected]