Dynamics of interacting quintessence

Eur. Phys. J. C (2015) 75:395 DOI 10.1140/epjc/s10052-015-3608-1 Regular Article - Theoretical Physics Dynamics of interacting quintessence M. Shahalam1,a , S. D. Pathak2,b , M. M. Verma2,c , M. Yu. Khlopov3,4,d , R. Myrzakulov5,e 1 Center for Theoretical Physics, Jamia Millia Islamia, New Delhi 110025, India 2 Department of Physics, University of Lucknow, Lucknow 226 007, India 3 National Research Nuclear University “MEPHI” (Moscow Engineering Physics Institute), 115409 Moscow, Russia 4 APC Laboratory, 10 rue Alice Domon et Léonie Duquet, 75205 Paris Cedex 13, France 5 Department of General and Theoretical Physics, Eurasian National University, Astana, Kazakhstan Received: 8 May 2015 / Accepted: 6 August 2015 / Published online: 27 August 2015 © The Author(s) 2015. This article is published with open access at Springerlink.com Abstract In this paper, we investigate coupled quintessence with scaling potential assuming specific forms of the coupling as A namely, α ρ̇m , β ρ̇φ and σ (ρ̇m + ρ̇φ ), and present phase space analysis for three different interacting models. We focus on the attractor solutions that can give rise to late time acceleration with DE / DM of order unity in order to alleviate the coincidence problem. 1 Introduction A large number of cosmological observations [1–5] reveal that our Universe is experiencing an accelerated expansion at present and the transition from deceleration phase to acceleration phase took place in the recent past [3]. In the standard Einstein gravity, the late time cosmic acceleration is driven by an exotic energy component with huge negative pressure filling the Universe, known as ‘dark energy’ [6–9]. One of the simplest candidate of dark energy (DE) is the cosmological constant (CC) . However, it is plagued with difficult theoretical issues such as fine tuning and cosmic coincidence problem [10–13]. This is important to explore whether dark energy is cosmological constant or it has dynamics. To this effect, a variety of dynamical dark energy models have been explored in the references [14–48]. The models of unstable dark matter with a non zero cosmological constant can also mimic such dynamics [49–53]. Alternatively, large scale modification of gravity has been used to obtain late time cosmic acceleration. At present late time cosmic acceleration is treated as an established phenomenon however its underlya e-mail: b e-mail: c e-mail: d e-mail: e e-mail: ing cause is still unknown. Within the framework of Einstein gravity and modified theories of gravity, numerous models can explain the said phenomenon. Although CDM model is consistent with present observations, yet there is no satisfactory argument for coincidence problem. Interaction of dark energy with dark matter is one novel approach that might address the mentioned problem. The interacting dark energy models have been recently proposed by several authors [54–59]. The interaction between dark energy and dark matter may enhance the dark matter, and also affect structure formation. The investigation of phase space analysis is the one conclusive test for dark energy models. Specifically, the attractor solutions are independent for a wide range of initial conditions. If the dark energy models have DE /DM of the order 1 and an accelerated scaling attractor solution, then the coincidence problem can be alleviated. The non-interacting quintessence [60– 62] and quintom [63–66] models show late time accelerated attractors, and possess DE as 1, therefore, they do not provide an adequate solution for coincidence problem. In the literature, two forms of interactions have been discussed namely, local and non-local. Local forms of interactions are directly proportional to energy density whereas non-local forms are directly proportional to Hubble parameter H and energy density ρ. In this paper we consider local forms of interactions proportional to energy density. Some of the local forms have been discussed in references [67–71]. Note that some of the choices of interacting terms appeared implicitly in the literature [72]. There is also approach to discuss the interacting term without the assumption of a specific form of interacting term [73]. The plan of the work is organised as follows: In Sect. 2 we establish the interacting quintessence cosmological framework and construct an autonomous dynamical system which is worthy for phase space investigation. In Sect. 3 we discuss phase space analysis and find stationary points and their stability for three 123 395 Page 2 of 9 Eur. Phys. J. C (2015) 75:395 interacting quintessence cosmological models. Our results are presented in Sect. 4. 2 Quintessence cosmology We consider two components first one is canonical scalar field (quintessence) as a source of dark energy in spatially flat Universe, and second one is matter (Baryonic+DM). The total energy density of the Universe is conserved, and the individual components of energy density may not be conserved. Thus, we are considering following conservation equations of energy density as: ρ̇tot + 3H (1 + wtot )ρtot = 0, ρ̇φ + 3H (1 + wφ )ρφ = −A, (1) where ρtot = ρφ + ρm , 1 + wφ = φ̇ 2 /ρφ , wm = 0 is the equation of state of matter, A is the interaction strength and H is the Hubble parameter which is given as 8π G ρtot 3 (2) The sign of A gives information about the direction of flow of energy between two components. There are three cases: Case I: If A > 0, in this case transfer of energy occurs from quintessence to dark matter. Consequently, quintessence losses self strength and gives dark matter. Case II: If A < 0, under this condition dark matter losses its strength and there is energy transfer from dark matter to quintessence. Case III: If A = 0, under this condition quintessence do not interact with dark matter, and no energy transfer at all between two components considered in the literature. Therefore, we are not considering this case. Since, there is no fundamental theory of dark energy and dark matter interaction (interaction in dark sector) at present, therefore it is not possible to construct the functional form of interaction strength A from first principle. Different forms of interaction strength (linear and non-linear) have been considered by several authors [74–83]. Motivated from the left hand side of the energy conservation equation (1) it is natural that A should be the function of Hubble parameter and energy density that is A = A(H, ρm , ρφ ) (3) Here we consider three specific forms of interaction strength heuristically as: A = α ρ̇m , (4) A = β ρ̇φ , (5) A = σ (ρ̇m + ρ̇φ ) (6) 123 κ2 (ρm + ρφ ) 3 κ2 2H Ḣ = (ρ̇m + ρ̇φ ) 3 H2 = (7) where κ 2 = 8π G, ρφ = 21 φ̇ 2 + V (φ) and pφ = 21 φ̇ 2 − V (φ). We introduce following dimensionless parameters κ 2 φ̇ 2 κ2V V 2 (8) ; Y = ; λ = − 6H 2 3H 2 κV to form an autonomous system of evolution equations (1) and (7) as:  3 2 Ḣ dX = −3X + λY − X 2 dN 2 H (9)  Ḣ dY 3 =− λX Y − Y 2 dN 2 H where N = ln (...truncated)