Advances in High Energy Physics

Volume 2018, Article ID 6980486, 7 pages

https://doi.org/10.1155/2018/6980486

## From de Sitter to de Sitter: Decaying Vacuum Models as a Possible Solution to the Main Cosmological Problems

^{1}Universidade Federal do ABC (UFABC), Santo André, 09210-580 São Paulo, Brazil^{2}Departamento de Física, Universidade Federal de São Paulo (UNIFESP), 09972-270 Diadema, SP, Brazil^{3}Departamento de Astronomia, Universidade de São Paulo (IAGUSP), Rua do Matão 1226, 05508-900 São Paulo, Brazil

Correspondence should be addressed to J. A. S. Lima; rb.psu.gai@amil.saj

Received 11 April 2018; Accepted 19 June 2018; Published 16 August 2018

Academic Editor: Marek Szydlowski

Copyright © 2018 G. J. M. Zilioti et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

Decaying vacuum cosmological models evolving smoothly between two extreme (very early and late time) de Sitter phases are able to solve or at least to alleviate some cosmological puzzles; among them we have (i) the singularity, (ii) horizon, (iii) graceful-exit from inflation, and (iv) the baryogenesis problem. Our basic aim here is to discuss how the coincidence problem based on a large class of running vacuum cosmologies evolving from de Sitter to de Sitter can also be mollified. It is also argued that even the cosmological constant problem becomes less severe provided that the characteristic scales of the two limiting de Sitter manifolds are predicted from first principles.

#### 1. Introduction

The present astronomical observations are being successfully explained by the so-called cosmic concordance model or CDM cosmology [1]. However, such a scenario can hardly provide by itself a definite explanation for the complete cosmic evolution involving two unconnected accelerating inflationary regimes separated by many aeons. Unsolved mysteries include the predicted existence of a space-time singularity in the very beginning of the Universe, the “graceful-exit” from primordial inflation, the baryogenesis problem, that is, the matter-antimatter asymmetry, and the cosmic coincidence problem. Last but not least, the scenario is also plagued with the so-called cosmological constant problem [2].

One possibility for solving such evolutionary puzzles is to incorporate energy transfer among the cosmic components, as what happens in decaying or running vacuum models or, more generally, in the interacting dark energy cosmologies. Here we are interested in the first class of models because the idea of a time-varying vacuum energy density or -models () in the expanding Universe is physically more plausible than the current view of a strict constant [3–13].

The cosmic concordance model suggests strongly that we live in a flat, accelerating Universe composed of of matter (baryons + dark matter) and of a constant vacuum energy density. The current accelerating period () started at a redshift or equivalently when . Thus, it is remarkable that the constant vacuum and the time-varying matter-energy density are of the same order of magnitude just by now thereby suggesting that we live in a very special moment of the cosmic history. This puzzle (“why now”?) has been dubbed by the cosmic coincidence problem (CCP) because of the present ratio , but it was almost infinite at early times [14, 15]. There are many attempts in the literature to solve such a mystery, some of them closely related to interacting dark energy models [16–18].

Recently, a large class of flat nonsingular FRW type cosmologies, where the vacuum energy density evolves like a truncated power-series in the Hubble parameter , has been discussed in the literature [19–22] (its dominant term behaves like ). Such models has some interesting features; among them, there are (i) a new mechanism for inflation with no “graceful-exit” problem, (ii) the late time expansion history which is very close to the cosmic concordance model, and (iii) a smooth link between the initial and final de Sitter stages through the radiation and matter dominated phases.

In this article we will show in detail how the coincidence problem is also alleviated in the context of this class of decaying vacuum models. In addition, partially based on previous works, we also advocate here that a generic running vacuum cosmology providing a complete cosmic history evolving between two extreme de Sitter phases is potentially able to mitigate several cosmological problems.

#### 2. The Model: Basic Equations

The Einstein equations, [], for an interacting vacuum-matter mixture in the FRW geometry read [19, 20]where and are the total energy density and pressure of the material medium formed by nonrelativistic matter and radiation. Note that the bare appearing in the geometric side was absorbed on the matter-energy side in order to describe the effective vacuum with energy density . Naturally, the time dependence of is provoked by the vacuum energy transfer to the fluid component. In this context, the total energy conservation law, , assumes the following form: *What about the behavior of **?* Assuming that the created particles have zero chemical potential and that the vacuum fluid behaves like a condensate carrying no entropy, as what happens in the Landau-Tisza two-fluid description employed in helium superfluid dynamics[23], it has been shown that as a consequence of the second law of thermodynamics [10], that is, the vacuum energy density diminishes in the course of the evolution. Therefore, in what follows we consider that the coupled vacuum is continuously transferring energy to the dominant component (radiation or nonrelativistic matter components). Such a property defines precisely the physical meaning of decaying or running vacuum cosmologies in this work.

Now, by combining the above field equation it is readily checked thatwhere the equation of state () was used. The above equations are solvable only if we know the functional form of .

The decaying vacuum law adopted here was first proposed based on phenomenological grounds [7–9, 11] and later on suggested by the renormalization group approach techniques applied to quantum field theories in curved space-time [24]. It is given bywhere is an arbitrary time scale describing the primordial de Sitter era (the upper limit of the Hubble parameter), and are dimensionless constants, and is a constant with dimension of .

In a point of fact, the constant above does not represent a new degree of freedom. It can be determined with the proviso that, for large values of , the model starts from a de Sitter phase with and . In this case, from (5) one finds because the first two terms there are negligible in this limit [see Eq. in [9] for the case and [11] for a general ]. The constant can be fixed by the time scale of the final de Sitter phase. For we also see from (4) that , where characterizes the final de Sitter stage (see (6) and (8)). Hence, the phenomenological law (5) assumes the final form: This is an interesting 3-parameter phenomenological expression. It depends on the arbitrary dimensionless constant and also the two extreme Hubble parameters (, ) describing the primordial and late time inflationary phases, respectively. Current observations imply that the value of is very small, [25, 26]. More interestingly, the analytical results discussed below remain valid even for . In this case, we obtain a sort of minimal model defined only by a pair of physical time scales, and , determining the entire evolution of the Universe. As we shall see, the possible existence of these two extreme de Sitter regimes suggests a different perspective to the cosmological constant problem.

By inserting the above expression into (3) we obtain the equation of motion:In principle, all possible de Sitter phases here are simply characterized by a constant Hubble parameter () satisfying the conditions and . For all physically relevant values of and in the present context, we see that the condition is satisfied whether the possible values of are constrained by the algebraic equation involving the arbitrary (initial and final) de Sitter vacuum scales and :In particular, for , the value preferred from the covariance of the action, the exact solution is given by and since we see that the two extreme scaling solutions for are and . However, we also see directly from (8) that the condition , also guarantees that such solutions are valid regardless of the values of . In certain sense, since is only the present day expansion rate, characterizing a quite casual stage of the recent evolving Universe, probably, it is not the interesting scale to be a priori predicted. In what follows we consider that the pair of extreme de Sitter scales () are the physically relevant quantities. This occurs because different from , the expanding de Sitter rates are associated with very specific limiting manifolds. For instance, it is widely known that de Sitter spaces are static when written in a suitable coordinate system. Besides the discussion on the coincidence problem (see next section), a new idea to be advocated here is that the prediction of such scales, at least in principle, should be an interesting theoretical target. Their first principles prediction would open a new and interesting route to investigate the cosmological constant problem.

The solutions for the Hubble parameter describing analytically the transitions vacuum-radiation () and matter-vacuum () can be expressed in terms of the scale factor, the couple of scales (), and free parameters ():We remark that the transition radiation-matter is like that in the standard cosmic concordance model. The only difference is due to the small parameter that can be fixed to be zero (minimal model). Indeed, if one fixes , the matter-vacuum transition is exactly the same one appearing in the flat CDM model. As we shall see below, the final scale can be expressed as a simple function of , , and . Naturally, the existence of such an expression is needed in order to compare with the present observations. However, it cannot be used to hide the special meaning played by in a possible solution of the cosmological constant problem.

#### 3. Alleviating the Coincidence Problem

The so-called coincidence problem is very well known. It comes from the fact that the matter-energy density of the nonrelativistic components (baryons + dark matter) decreases as the Universe expands while the vacuum energy density () is always constant in the cosmic concordance model (CDM). This happens also because the energy densities of the radiation (CMB photons) and neutrinos () are negligible today. Thus, in a broader perspective, one may also say that the ratio , where , was almost infinite at early times, but it is nearly of the order unity today.

The current fine-tuning behind the coincidence problem can also be readily defined in terms of the corresponding density parameters, since ( and ), so that the ratio is of the order unity some 14 billion years later.

In Figure 1 we display the standard view of the coincidence problem in terms of the corresponding density parameters: (baryons + cold dark matter) and (CMB photons + relic neutrinos). As one may conclude from the figure, the ratio was practically infinite at very high redshifts, that is, at the early Universe (say, roughly at the Planck time). However, both densities are nearly coincident at present. The ratio ) is at low redshifts. Note also that, in the far future, that is, very deep in the de Sitter stage, the ratio approaches zero or equivalently the inverse ratio is almost infinite because the vacuum component becomes fully dominant.