Eur. Phys. J. C (2020) 80:894 https://doi.org/10.1140/epjc/s10052-020-08476-9 Regular Article - Theoretical Physics Dynamical description of a quintom cosmological model nonminimally coupled with gravity Mihai Marciua Faculty of Physics, University of Bucharest, 405 Atomi¸stilor, POB MG-11, 077125 Bucharest-M˘agurele, Romania Received: 4 August 2020 / Accepted: 13 September 2020 / Published online: 26 September 2020 © The Author(s) 2020 Abstract In this work we have studied a cosmological dark energy sector. In order to explain the dynamical evolu- model based on a quintom dark energy model non-minimally tion of the dark energy equation of state various theoretical coupled with gravity, endowed with a specific potential directions have been proposed [15–21] in the form of single energy of the exponential squared type. For this specific type or multiple scalar fields, minimally or non-minimally cou- of potential energy and non-minimal coupling, the dynamical pled with gravity or other possible invariants [22]. properties are analyzed and the corresponding cosmological The behavior of the dark energy equation of state [23–25] effects are discussed. Considering the linear stability method, represents an important aspect when constructing a viable we have investigated the dynamical properties of the phase scalar tensor theory of gravitation [26,27] which can explain space structure, determining the physically acceptable solu- various physical quantities associated to the known Universe. tions. The analysis showed that in this model we can have var- In this case the strange issue related to the crossing over the ious cosmological epochs, corresponding to radiation, matter phantom divide line (the cosmological constant barrier) [25, domination, and de Sitter eras. Each solution is investigated 28] by the dark energy equation of state has been explained by from a physical and cosmological point of view, obtaining adopting a possible extension to the Einstein–Hilbert action possible constraints of the model’s parameters. In principle which adds two scalar fields [29,30], an addition which the present cosmological setup represent a possible viable includes a canonical scalar field and a phantom field, respec- scalar tensor theory which can explain various transitional tively, a composition which violates the null energy condition effects related to the behavior of the dark energy equation of [31]. In scalar tensor theories the quintessence dark energy state and the evolution of the Universe at large scales. models [32,33] represent a possible configuration for the dark energy sector, a canonical direction which can explain vari- ous astrophysical observations. A more exotic configuration 1 Introduction which includes the addition of a negative kinetic energy in the specific action has been suggested [34–36], leading to the In the present days the cosmological context reached a golden formation of phantom dark energy models, a particular theo- age epoch by overturning the fundamental concepts related retical direction which is viable from an observational point to the evolution and the major constituents of the Universe, of view [37–39]. However, such theoretical constructions fracturing our understanding of time and space. The accel- lead to the violation of the null energy condition [36,40,41] erated expansion of the Universe [1] represent an enigma to and can exhibit Big Rip ending scenes. Since the nature of theorists and cosmologists, with deep ramifications in vari- the dark energy section is currently unknown, various exotic ous branches of physics. The basic evidence of the acceler- models have been proposed [22,42–44], adding new intrigu- ated expansion has been probed through various astrophys- ing directions to the cosmic landscape. ical studies [2,3] which included observation from type Ia The development of the scalar tensor theories lead to the supernovae [4,5], baryon acoustic oscillations [6–10] and formation of quintom cosmological models [31], an exotic cosmic microwave background radiation [11–13]. The sim- configuration which might explain some of the dynamical plest scenario of dark energy is represented by the cosmo- aspects associated to the dark energy equation of state. In the logical constant [14] added to the Einstein field equation, a first quintom scenario the two quintom scalar fields were min- proposal which lead to a constant equation of state for the imally coupled in the corresponding action [29,30], explain- ing the astrophysical observations related to the specific a e-mail: [email protected] (corresponding author) 123 894 Page 2 of 13 Eur. Phys. J. C (2020) 80 :894 crossing [45] of the cosmological constant boundary. In the The paper is organized as follows: in Sect. 2 we present recent years the quintom paradigm [31] has been continu- the basic equations which express the evolution relations for ously developed in various studies [46–60] which includes the quintom model non-minimally coupled to scalar curva- the additions of various non-minimal couplings in different ture, endowed with a specific potential energy of exponential scalar tensor theories [61–67]. In spite of the fact that the squared type. Then, in Sect. 3 we propose the auxiliary vari- quintom paradigm implies the violation of the null energy ables and write the autonomous system of equations, deter- condition, embedding a pathological phantom field in the mining the critical points and the dynamical features which corresponding action, it remains as an admissible modified are associated. In the last section Sect. 4, we present the sum- gravity construction which can justify various astrophysical mary of the analytical investigation and the final concluding observations [31]. Although a quintom action based on two remarks. scalar fields include the addition of a phantom field which lead to specific instabilities when possible quantum features are considered, it is consistent with astrophysical observa- 2 The field equations and modified Friedmann relations tions, showing the specific effect related to the crossing of the phantom divide line by the dark energy equation of state, In what follows we shall study a quintom model for the dark a dynamical effect [31] which cannot be explained in single energy component non-minimally coupled with scalar cur- scalar field models with minimal coupling [68]. In an ear- vature, which includes an action corresponding to the mat- lier paper [62], a quintom dark energy extension has been ter component Sm, assuming the following form of the total proposed, where the scalar fields were non-minimally cou- action [62]: pled with scalar curvature, the physical features of the model were analyzed by adopting a numerical approach. In scalar √ √ = + 4 − R + 1 4 − − μν ∂ φ∂ φ tensor theories of gravitation the addition of non-minimal Stot Sm d x g d x g g μ ν 2 2 couplings with gravity represent a viable direction supported μν 2 2 +g ∂μσ∂ν σ − ξ Rφ + ξ Rσ − 2V (φ) − 2V (σ) , by different hypothetical models [69–74]. The effects of the 1 2 1 2 non-minimal couplings with gravity have been investigated (1) in single scalar field theories [75–78], by considering the lin- ear stability theory, showing the viability of the correspond- where φ(t) represents the canonical scalar field (quintessence), ing models [79–86]. Furthermore, in scalar tensor theories and σ(t) the non-canonical (negative kinetic) field with a based on teleparallel gravity the models non-minimally cou- phantom pathological behavior; R denotes the scalar curva- 2 2 pled with gravity are constructed by using the corresponding ture which for the metric descriptor (−1, +a (t), +a (t), + 2( )) = a¨ + ( a˙ )2 analogous invariant scalars, the torsion [87] and boundary a t is equal to R 6 a a . Here we shall assume coupling [88] parameters. From an observational point of the fields φ and σ to be time dependent, and use dots to denote view the non-minimal couplings with curvature have been derivatives with respect to the cosmic time. Also, a(t) is the investigated in different specific models [75,81,89,90]. The usual cosmic scale factor and H =˙a/a the associated Hub- extension of the quintom paradigm towards non-minimal cur- ble parameter. vature couplings represents a particular attempt of correct- The modified Friedmann relations for this specific action ing two scalar field models, a specific model which might are the following [62]: explain the dynamical crossing [28,31] of the cosmologi- 2 3H = ρφ + ρσ + ρ , (2) cal constant boundary in the recent past by the dark energy m ˙ 1 equation of state, a phenomenon presented by recent astro- H =− (ρφ + ρσ + ρm + pφ + pσ + pm), (3) physical observations. In principle the quintom paradigm can 2 also be regarded as a specific multi-fluid scenario [91,92]by with the corresponding energy densities and pressures [83]: embedding the superposition of two scalar fields with kinetic energy of opposite sign. 1 ˙2 2 2 ˙ ρφ = φ + V1(φ) + 3ξ1 H φ + 6ξ1 Hφφ, (4) In this paper we shall further analyze the dynamical 2 features of a specific scalar tensor cosmological scenario 1 2 2 2 ρσ =− σ˙ + V2(σ) − 3ξ2 H σ − 6ξ2 Hσ σ,˙ (5) [62], observing the physical consequences of the couplings 2 1 ˙2 2 2 ˙ between the quintom scalar fields and the curvature in the pφ = φ − V1(φ) − ξ1(φ (3H + 2H) phase space, for a different potential energy, considering the 2 ¨ ˙2 ˙ linear stability method. The potential energy type considered +2φφ + 2φ + 4Hφφ), (6) in the present paper belongs to the exponential squared class, 1 2 2 2 ˙ pσ =− σ˙ − V2(σ) + ξ2(σ (3H + 2H) which have been previously studied [83] in scalar tensor the- 2 ories of gravitation. +2σ σ¨ + 2σ˙ 2 + 4Hσ σ).˙ (7) 123 Eur. Phys. J. C (2020) 80 :894 Page 3 of 13 894 Furthermore, we can define the pressure, the specific cosmological scenario, we choose the following auxiliary variables [83]: pφσ = pφ + pσ , (8) φ˙ x1 = √ , (18) 6H the energy density for the dark energy component, √ V1(φ) y1 = √ , (19) ρ = ρ + ρ , 3H φσ φ σ (9) φ z1 = √ , (20) 6 the dark energy equation of state √ 1 dV1(φ) λ1 =− 6 (21) pφ + pσ V1(φ) dφ wφσ = , (10) σ˙ ρφ + ρσ x2 = √ , (22) √6H and the effective (total) equation of state: V2(σ) y2 = √ , (23) 3H pm + pφσ pm + pφ + pσ σ w = = .