Crossing of the Cosmological Constant Boundary - an Equation of State Description

The phenomenon of the dark energy transition between the quintessence regime (w > −1) and the phantom regime (w < −1), also known as the cosmological constant boundary crossing, is analyzed in terms of the dark energy equation of state. It is found that the dark energy equation of state in the dark energy models which exhibit the transition is implicitly defined. The generalizations of the the models explicitly constructed to exhibit the transition are studied to gain insight into the mechanism of the transition. It is found that the cancellation of the terms corresponding to the cosmological constant boundary makes the transition possible. ‡ On leave of absence from the Theoretical Physics Division, Rudjer Bošković Institute, Zagreb, Croatia Crossing of the Cosmological Constant... 2 Among many important cosmological problems, the phenomenon of the present, late-time accelerated expansion of the universe has come to the forefront of the observational and theoretical efforts in several last years. Apart from the exciting series of cosmological observational results confirming the accelerated character of the expansion of the universe [1, 2, 3], we are witnessing many theoretical endeavours aimed at explaining the features of the present expansion of the universe, as well as the revival of some longstanding problems in cosmology and high energy physics, such as the cosmological constant problem [4]. From the theoretical viewpoint there is still no decisive insight into the nature of the accelerating mechanism. However, many promising models have been proposed to explain the acceleration in the universe’s expansion. Some of the interesting approaches include the braneworld models and the modifications of gravity at cosmological scales. The most studied accelerating mechanism is the existence of a cosmic component with negative pressure, a so called dark energy component. Dark energy is a very useful concept since all our ignorance on the acceleration phenomenon is encoded into a single cosmic component. It can also be very useful as an effective description of other approaches to the explanation of the acceleration of the universe. Many models of dark energy have been constructed so far, assigning to dark energy different properties. A very general classification of these models is possible with respect to the parameter w of the dark energy equation of state (EOS), pd = wρd, where pd and ρd refer to the dark energy pressure and energy density, respectively § . The benchmark value for the parameter of the dark energy EOS is w = −1 which is characteristic of the cosmological constant (CC). A problem associated to the CC value predicted by high energy physics, i.e. its discrepancy of many orders of magnitude with the value inferred from observations, is notoriously difficult. Such a situation has stimulated the development of dynamical dark energy models. Some prominent dynamical models of dark energy such as quintessence [5], k-essence [6] or Chaplygin gas [7] are characterized by w > −1. On the other side of the CC boundary are located models of phantom energy [8], with the property w < −1. These models are characterized by some tension between a certain favour from the observational side and certain disfavour from the theoretical side. Many recent analyses of observational data [9], using ingenious parametrizations for the redshift dependence w(z), show that the best fit values imply the transition of the dark energy parameter of EOS from w > −1 to w < −1 at a small redshift. This phenomenon has been referred to in literature as the crossing of the CC boundary, crossing of the phantom divide or the transition between the quintessence and phantom regimes. It is important to stress that currently some other options, like the one of the ΛCDM cosmology, are also consistent with the observational data. Should the future observations confirm the present indications of the crossing, the aspects of the theoretical description of the crossing might provide a useful means of distinguishing and discriminating various dark energy models and other frameworks designed to explain § Since dark energy is the only component discussed in this paper, the subscripts d will be suppressed furtheron. Crossing of the Cosmological Constant... 3 -4 -2 0 2 4 ln(a/a 0 ) -1,4 -1,2 -1 -0,8 -0,6 w γ=−0.6, η=−1.4 γ=−0.7, η=−1.3 γ=−0.8, η=−1.2 Figure 1. The dark energy parameter of EOS w given by (3) as a function of the scale factor a for the present value w0 = −1.1 and three sets of parameters γ and η. the present cosmic acceleration. Therefore, the crossing of the CC boundary is to some extent observationally favoured and its description is a theoretical challenge. A number of approaches have been adopted so far to describe the phenomenon of the CC boundary crossing [10]. In our considerations of the phenomenon of the CC boundary crossing [11], we assume that that dark energy is a single, noninteracting cosmic component. We focus on the question whether the CC boundary crossing can be described using the dark energy EOS and if the answer is yes, which form the dark energy EOS needs to have to make the crossing possible. The equation of state is most frequently formulated as p given as an analytic expression of the energy density ρ. In the considerations given below we use a much broader definition of EOS. We define the equation of state parametrically, i.e. as a pair of quantities depending on the cosmic time (ρ(t), p(t)), or equivalently on the scale factor a in the expanding universe (ρ(a), p(a)). This definition easily comprises broad classes of dark energy models considered in the literature. Let us start by considering a specific dark energy model which describes the CC boundary crossing. The dependence of the dark energy density on the scale factor in this model is given by ρ = C1 ( a a0 ) −3(1+γ) + C2 ( a a0 ) −3(1+η) . (1) where γ > −1 and η < −1. The scaling of this energy density resembles the sum of two independent cosmic components. However, we consider it to be the energy density of a single cosmic component and study its properties. Using the energy-momentum tensor conservation the expression for the dark energy pressure is obtained: p = γC1 ( a a0 ) −3(1+γ) + ηC2 ( a a0 ) −3(1+η) . (2) Combining (1) and (2) the expression for the parameter of the dark energy EOS acquires Crossing of the Cosmological Constant... 4 the following form w = γ + η γ−w0 w0−η (