Dynamical analysis and cosmological viability of varying G and $$\Lambda $$ cosmology

Eur. Phys. J. C (2018) 78:681 https://doi.org/10.1140/epjc/s10052-018-6165-6 Regular Article - Theoretical Physics Dynamical analysis and cosmological viability of varying G and  cosmology Andronikos Paliathanasis1,2,3,a 1 Instituto de Ciencias Físicas y Matemáticas, Universidad Austral de Chile, Valdivia, Chile Department of Mathematics and Natural Sciences, Core Curriculum Program, Prince Mohammad Bin Fahd University, Al Khobar 31952, Kingdom of Saudi Arabia 3 Institute of Systems Science, Durban University of Technology, PO Box 1334, Durban 4000, Republic of South Africa 2 Received: 13 June 2018 / Accepted: 17 August 2018 / Published online: 27 August 2018 © The Author(s) 2018 Abstract The cosmological viability of varying G (t) and  (t) cosmology is discussed by determining the cosmological eras provided by the theory. Such a study is performed with the determination of the critical points while stability analysis is performed. The application of renormalization group in the ADM formalism of general relativity provides a modified second-order theory of gravity where varying G (t) plays the role of a minimally coupled field, different from that of scalar–tensor theories, while  (t) =  (G (t)) is a potential term. We find that the theory provides two de Sitter phases and a tracking solution. In the presence of matter source, two new critical points are introduced, where the matter source contributes to the universe. One of those points describes the CDM cosmology and in order for the solution at the point to be cosmologically viable, it has to be unstable. Moreover, the second point, where matter exists, describes a universe where the dark energy parameter for the equation of state has a different value from that of the cosmological constant. 1 Introduction The detailed analysis of the cosmological data over the last years supports the assumptions that the universe is spatially flat, it has been through an inflation phase in the past prior to the radiation dominated era, and that, currently. the universe is in a second acceleration epoch [1–5]. The acceleration phase of the universe has been attributed to a matter source in the gravitational field equations which has an equation of state parameter with a negative value. The nature of this exotic matter source has led to the dark energy problem. In the literature one can find various proposals/models to solve the dark energy problem. These proposals can be catea e-mail: gorized in two different families; more specifically, in these where in the context of Einstein’s general relativity, an energy momentum tensor is introduced to explain the acceleration phases [6–16], and these in which the Einstein–Hilbert action is modified, leading to the so-called modified/alternative theories of gravity, such that the origin of the acceleration to correspond to the gravitational theory, for instance, see [17–29] and references therein. A common feature for some of the modified theories of gravity is that Newton’s constant G, is varying; and it is a varying parameter. For instance, in Brans–Dicke theory and in f (R)-gravity someone can define the effective parameters G e f f = Gφ −1 and G e f f = G f  (R) respectively [17,18,30]. Dealing with fundamental constants in physics as parameters is the main concept of the renormalization group [31–33]. Reuter and Weyer in [34] inspired by the property that Brans–Dicke action modify G, reconstructed the Brans– Dicke action with the use of the renormalization group in general relativity, by assuming that G and  (the cosmological constant) are varying. Various alternative gravitational theories [35–50] have been modified by the renormalization group in cosmological systems as also in strong gravitational systems [51–56]. A study which provides important analytical information about the existence of cosmological epochs (such as matter dominated era, acceleration phase and others) and the stability of those epochs is the analysis of critical points of the gravitational filed equations [57,58]. In the dark energy models, the analysis of the critical point provides results for the evolution of the universe [59] and the viability of each model being studied [60]. For some extended applications of the critical point analysis in modified theories of gravity, we refer the reader to [61–68] and references therein. We are interested in the dynamical analysis of the gravitational field equations which follows from the renormaliza- 123 681 Page 2 of 11 tion group in the ADM Lagrangian of general relativity as described in [69]. Specifically, in [69] the authors assumed that G and  are varying parameters such that new degrees of freedom are introduced. The theory, remains of secondorder and the variable G can be seen as a scalar field coupled to gravity, but different from that of Brans–Dicke or from the scalar–tensor theory. The reason for the latter lies in the starting point for the application of the renormalization group. This is the ADM Lagrangian and not the Einstein– Hilbert action as in [34]. Some exact solutions for that specific modified gravitational theory can be found in [70,71]. Cosmological constraints and comparison with the CDM model are given in [72] where it was found that for this specific variable G,  cosmology is compatible with some of the observational data and can explain the late acceleration phase of the universe. More specifically, in this work, we study the existence of critical points in varying G, cosmology [69] in order to explore the possible cosmological eras provided by the theory. We define new dimensionless variables and in terms of the H -normalization [59] we study the critical points of the cosmological model. Because the resulting field equations of [69] have similarities with scalar–tensor theories, our analysis can be compared with the analysis performed for the Brans–Dicke theory in [65]. However, as we shall see, there are essential differences with the scalar–tensor theories. The plan of the paper follows. In Sect. 2 we present the model of our consideration which belongs to the family of varying G and  cosmology. Section 3 includes the main material of our analysis where the analysis of the critical points for dimensionless variables and in the H -normalization is discussed. Our discussion of the results is given in Sect. 4, where we also draw our conclusions. 2 Field equations in varying G and  cosmology In the ADM formalism of general relativity, Bonanno et al. [69] after the application of the renormalization group, proposed the following modification for the ADM Lagrangian of general relativity, √  1 N h (K i j K i j − K 2 + R ∗ − 2 (G))d 3 x 16π G √   μ Ni N h + N −2 (G ,0 )2 − 2 2 G ,0 G ,i 16π G N   iNj  N − hi j − (1) G ,i G , j d 3 x, N2 S = Sm + where Sm describes the action integral of the matter source, and G,  (G) are varying. 123 Eur. Phys. J. C (2018) 78:681 Furthermore, the line element of the backgro (...truncated)