Cosmological singularities and analytical solutions in varying vacuum cosmologies

Eur. Phys. J. C (2018) 78:684 https://doi.org/10.1140/epjc/s10052-018-6139-8 Regular Article - Theoretical Physics Cosmological singularities and analytical solutions in varying vacuum cosmologies Spyros Basilakos1,a , Andronikos Paliathanasis2,3,4,b , John D. Barrow5,c , G. Papagiannopoulos6,d 1 Academy of Athens, Research Center for Astronomy and Applied Mathematics, Soranou Efesiou 4, 11527 Athens, Greece 2 Instituto de Ciencias Físicas y Matemáticas, Universidad Austral de Chile, Valdivia, Chile 3 Department of Mathematics and Natural Sciences,Core Curriculum Program, Prince Mohammad Bin Fahd University, AlKhobar 31952, Kingdom of Saudi Arabia 4 Institute of Systems Science, Durban University of Technology, PO Box 1334, Durban 4000, Republic of South Africa 5 DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Rd., Cambridge CB3 0WA, UK 6 Faculty of Physics, Department of Astronomy-Astrophysics-Mechanics, University of Athens,, Panepistemiopolis 157 83 Athens, Greece Received: 20 May 2018 / Accepted: 4 August 2018 / Published online: 27 August 2018 © The Author(s) 2018 Abstract We investigate the dynamical features of a large family of running vacuum cosmologies for which  evolves as a polynomial in the Hubble parameter. Specifically, using the critical point analysis we study the existence and the stability of singular solutions which describe de-Sitter, radiation and matter dominated eras. We find several classes of (H ) cosmologies for which new analytical solutions are given in terms of Laurent expansions. Finally, we show that the Milne universe and the Rh = ct model can be seen as perturbations around a specific (H ) model, but this model is unstable. 1 Introduction Over the last two decades, most studies in cosmology strongly indicate that the universe is spatially flat and it consists of ∼ 4% baryonic matter, ∼ 26% dark matter and ∼ 70% of dark energy (hereafter DE), thought to be driving the phenomenon of cosmic acceleration [1–6]. Although there is a general consensus regarding the main properties of DE, namely it has a negative pressure, the origin of this unexpected component of the universe has yet to be identified. This has given rise to a plethora of alternative cosmological scenarios which mainly generalize the nominal Einstein–Hilbert action of general relativity using either a new field in nature [7–10], or a non-standard gravity theory that increases the number of degrees of freedom [11–16]. a e-mail: b e-mail: c e-mail: d e-mail: The introduction of a cosmological constant, , is probably the simplest modification of the Einstein–Hilbert action which can be considered. In the framework of the so called CDM model, the cosmological constant coexists with cold dark matter (CDM) and ordinary baryonic matter (see [17] for review). Although the CDM model fits accurately, the current cosmological data suffers from two problems [18– 21]. The first is the ’old’ cosmological problem, namely the expected (Planck natural unit) vacuum energy density is ∼ 120 orders of magnitude larger than the presently observed value of . The second problem is the coincidence problem: that the density of dark energy is so similar to the matter density today (the two were equal when the universe had expanded to about 75% of it present expansion scale). An alternative approach to resolving these two problems is to consider the so called (t)CDM models, wherein  is allowed to vary with cosmic time (see [22–29] and references therein). This class of models [30–49] is based on a dynamical vacuum energy density that evolves as a power series of the Hubble rate (for review see [50], [51–53]). Powered by a decaying vacuum energy density, the spacetime can emerges from a pure non-singular initial de Sitter vacuum stage, “gracefully” exit from inflation to a radiation era followed by dark-matter and vacuum-dominated phases, before finally, evolving to a late-time de Sitter phase [26,28,29,54]. Recently, Sola et al. [55] tested the performance of the running vacuum models against the latest cosmological data and they found that the (H ) models are favored than the usual CDM model at ∼ 4σ statistical level (see also [56]). These developments have led to growing interest in (H ) cosmological models. There was a great effort to explore the (H ) models both analytically and observationally but a dynamical analysis 123 684 Page 2 of 13 Eur. Phys. J. C (2018) 78:684 based on critical points is still missing. The purpose of our work is to bridge this gap, by determining the de Sitter phases of a general family of (H ) models to search for singular solutions of the form a (t) ∝ t p which secure the existence of radiation ( p = 1/3) and matter ( p = 2/3) dominated eras, respectively. We will investigate the stability of the critical points in order to understand the dynamical and cosmological properties of the (H ) models. Here, the main mathematical tool that we use is that of the singularity analysis of differential equations, and more specifically we apply the ARS (Ablowitz, Ramani and Segur) algorithm [57–59]. This algorithm provides a method to construct the analytical solution of a differential equation which is expressed as a Laurent expansion around the singular leading-order term (for some applications on gravitational theories see [60–65] and references therein). Information regarding the stability of the trajectories close to the singularity can be extracted directly from the ARS algorithm. The structure of the manuscript is as follows. In Sect. 2 we briefly introduce the concept of the running (H ) cosmologies. In Sects. 3 and 4 we present the main results of our work, namely we study the critical points and their stability as well as we provide the corresponding analytical solutions. Finally, in Sect. 5 we draw our conclusions. 2 -Varying cosmology In this section we briefly present the main points of the running vacuum cosmology. If we model the expanding universe as a perfect fluid with density ρ, and corresponding pressure1 p = wρ, then the energy–momentum tensor is given by Tμν = − p gμν + (ρ + p)Uμ Uν . In this context, the term  gμν can be absorbed by the total energy momentum tensor T̃μν ≡ Tμν + gμν ρ , where ρ = /(8π G) is the vacuum energy density which is related to , while the corresponding equation of state is p = −ρ . Combining the above expressions we arrive at T̃μν = (ρ − p) gμν + (ρm + p)Uμ Uν , (1) and thus the Einstein’s field equations become 1 Rμν − gμν R = 8π G T̃μν . 2 (5) In order to find more general solutions for this scenario, ones in which there is evolution of the form of the expansion rate over time, we need to explore a more general functional form for (H ). The case of viable running vacuum  can be placed in the general framework of quantum field theory in curved spacetime [50,66,67]. Specifically, in ref. [26] the following expression (for recent review see [68]) was proposed fo (...truncated)