In the big rip, the scale factor and density diverge in a singularity at a finite future time. In theΛCDM model, there is no such divergence and no disintegration be- cause the dark energy density remains constant. The little rip interpolates between these two cases; mathematically it can be represented as an infinite limit sequence which has the big rip and theΛCDM model as its boundaries. Such models can be represented generically by a density varying with scale factor as in equation (4.19). Physically, in the little rip, the scale factor and the density are never infinite at a finite time. Nevertheless, such models generically lead to structure disintegration at a finite time. For models consistent with current supernova observations, such disintegration can occur either earlier or later in a little rip model than in a big rip model, depending on the parameters chosen for the models. However, for a given present-day value of w, the big rip model with constant wwill necessarily lead to an earlier disintegration than the little rip model with the same present-day value ofw. This results from the fact thatwincreases monotonically in the little rip models, resulting in a smaller value forρat any givenathan in the corresponding constant-wbig rip model, and therefore, a lower expansion rate. Thus, supernova bounds on the epoch of disintegration for constant-w big rip models also apply to little rip models; one cannot simultaneously satisfy supernova constraints and hasten the onset of disintegration to an arbitrarily early time simply by iterating exponentials in the expansion law.
Furthermore, supernova data force both big rip and little rip models into a region of parameter space in which both models resemble ΛCDM. In this limit, big rip and little rip models produce essentially the same expansion law up to the present, despite having very different future evolution. Thus, current data already make it essentially impossible to determine whether or not the universe will end in a future singularity.
Finally, we remark that since the novel and speculative cyclic cosmology pro- posed in Ref. [77] requires only disintegration and not a singularity, such cyclicity would seem to be possible within a little rip model instead of the big rip considered in [77]. This is one potentially fruitful direction for future research.
Chapter 5
Models for Little Rip Dark Energy
1
5.1
Introduction
The current acceleration of the universe is often attributed to dark energy, an un- known fluid with effective equation of state (EoS) parameter wclose to −1. The observational data [13, 12, 75, 79] favorΛCDM with w = −1. However, phantom (w < −1) or quintessence (−1/3 > w > −1) dark energy models are not excluded by observational data [80]. In both cases, it is known that the universe may evolve to a finite-time future singularity. Phantom dark energy models can lead to a sin- gularity in which the scale factor and density become infinite at a finite time; such a singularity is called a big rip [59, 29], or Type I singularity [62]. For quintessence dark energy, one can have a singularity for which the pressure goes to infinity at a fixed time, but the scale factor and density remain finite; this is called a sudden singularity [69, 70], or a Type II singularity [62]). Alternately, the density and pres- sure can both become infinite with a finite scale factor at a finite time (a Type III singularity), or higher derivatives of the Hubble parameterH can diverge (a Type IV singularity) [62]. The occurrence of a singularity at a finite time in the future
may lead to some inconsistencies. Several scenarios to avoid a future singularity have been proposed so far: coupling with dark matter [81], inclusion of quantum effects [82], additional changes in the equation of state [83], or special forms of modified gravity [83].
Recently, a new scenario to avoid a future singularity has been proposed in Ref. [55]. In this scenario, w is less than −1, so that the dark energy density in- creases with time, but w approaches −1 asymptotically and sufficiently rapidly that a singularity is avoided. This proposed non-singular cosmology was called a “little rip” because it leads to a dissolution of bound structures at some point in the future (similar to the effect of a big rip singularity). It can be realized in terms of a general fluid with a complicated EoS [62, 63, 84]. The evolution of the little rip cosmology is close to that ofΛCDM up to the present, and is similarly consistent with the observational data.
The present article is devoted to further study of the properties of the little rip cosmology. In the next section, the inertial force interpretation of the little rip is developed, and it becomes clear why a dissolution of bound structures occurs. Coupling of the little rip fluid with dark matter is considered in Section III. It is shown that as the result of such a coupling an asymptotically de Sitter universe can eventually evolve to have a little or big rip. In Section IV, the little rip cosmology is reconstructed in terms of scalar field models. Our results are summarized in Section V.