Cosmology of a higher derivative scalar theory with non-minimal Maxwell coupling

Eur. Phys. J. C (2019) 79:448 https://doi.org/10.1140/epjc/s10052-019-6960-8 Regular Article - Theoretical Physics Cosmology of a higher derivative scalar theory with non-minimal Maxwell coupling Shahab Shahidia School of Physics, Damghan University, Damghan 41167-36716, Iran Received: 7 January 2019 / Accepted: 17 May 2019 © The Author(s) 2019 Abstract Higher derivative scalar field theory in curved space-time belongs to the GLPV theory coupled nonminimally to the Maxwell field is considered. We will show that the theory admits two independent exact de Sitter solutions in the FRW background, one driven by the cosmological constant and the other by the GLPV scalar field. The dynamical system analysis of the theory shows that these two exact solutions are stable fixed points. Also, cosmological perturbations over these solutions shows that the cosmological constant based solution is healthy at linear level but the GLPV based solution suffers from a gradient instability in the scalar sector. This proves that the cosmological constant is needed in the GLPV-Maxwell system in order to have a healthy de Sitter solution. Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . 2 The action . . . . . . . . . . . . . . . . . . . . . . . 3 Background cosmology . . . . . . . . . . . . . . . . 3.1 Dynamical system analysis . . . . . . . . . . . . 3.2 de Sitter fixed points . . . . . . . . . . . . . . . 3.3 Matter dominated fixed points . . . . . . . . . . 3.4 General solutions . . . . . . . . . . . . . . . . . 4 Perturbations . . . . . . . . . . . . . . . . . . . . . . 4.1 Tensor perturbation . . . . . . . . . . . . . . . . 4.2 Vector perturbation . . . . . . . . . . . . . . . . 4.3 Scalar perturbation . . . . . . . . . . . . . . . . 4.3.1 -based solution . . . . . . . . . . . . . . 4.3.2 GLPV-based solution . . . . . . . . . . . . 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . a e-mail: 0123456789().: V,-vol 1 Introduction Modifying Einstein’s general relativity has a long history. Perhaps the first modification, can be attributed to the addition of the cosmological constant to the gravitational field equation by Einstein himself [1]. From then, infinite number of modifications have come out, concentrating on both ultraviolet/infrared limits of the Einstein’s field equations [2–4]. Cosmology however, suffers from many problems, one of the most important is the accelerated expansion of the universe at late times. This can be explained by introducing some light degree of freedom (dof) to the Einstein’s field equations, which can be responsible for the IR modification of gravity. Many proposals have been suggested so far in the literature, including the addition of some extra field to the Einstein’s theory, which can be a scalar/vector/tensor field [5], or enriching the gravitational action itself like higher order derivative theories [6–8], Weyl-Cartan theories [9–11] or massive gravity theories [12]. Also one can assume some non-trivial matter-geometry coupling to explain the accelerated expansion of the universe. Among all, addition of a scalar field may be the minimal modification of the theory. This adds one additional dof to the Einstein’s theory (with two dof) if the Lagrangian for the scalar field is healthy. In order for the scalar interactions to becomes healthy, the scalar field should not have more than two time derivatives at the level of equations of motion, and the interaction terms should have a form which avoid gradient/tachyonic instabilities. The scalar field theories is then divided into two major classes; those which produce accelerated expansion from the kinetic interactions [13], and those which do that from non-trivial potential terms [5]. One the most interesting scalar field theories for the above goal, is the so-called Galileon theory [14]. Galileons are scalar fields which has more than second order time derivatives in the action but due to the special form of the interactions, it has at most second order time derivatives in the 123 448 Page 2 of 10 Eur. Phys. J. C equations of motion. This makes the theory free from Ostrogradski instability. Galileon terms has an internal symmetry under which the interaction terms remain invariant if one shift the scalars as φ → φ + bμ x μ + c, where φ is the Galileon scalar and bμ and c are constants. Many works has been done in the literature, considering cosmological [15–22], balck holes [23–25], quantum nature [26–31] and some generalizations of the Galileon scalars [32–35]. However, one of most interesting facts about the Galileons is that they can be interpreted as a position of the 4D brane world embedded in the 5D flat space [36]. This suggests that the Galileon interaction terms can not have an arbitrary form and as a result we have a finite number of Galileon interaction terms in any dimension [14]. Upon generalizing the Galileon interactions to curved space time, one immediately find out that higher order time derivatives come back to the equations of motion [37]. This is due to the fact that in curved space time, partial derivatives do not commute. This problem can be solved by adding to the action some higher order derivative terms which compensate the higher order time derivatives in the equations of motion. However, these terms breaks the Galileon invariance [37]. The most general scalar-tensor interactions in curved space time which has the property that the equations of motion are healthy is called the Horndeski theory [38]. Among all the Horndeski terms, four terms bring more attention in the sense that any combination of these terms have a consistent self-tuning mechanism on FRW background. These terms are well-known as the Fab-four [39] an can be written as √ L john = −gV john (φ)G μν ∇μ φ∇ν φ, √ Lgeorge = −gVgeorge (φ)R, √ L paul = −gV paul (φ)P μναβ ∇μ φ∇α φ∇ν ∇β φ, √ Lringo = −gVringo (φ)G, (1.1) where P μναβ is the double dual of the Riemann tensor and G is the Gauss-Bonnet invariant. Also it is proposed that the Horndeski theory can be generalized further to contain terms proportional to the Levi-Civita tensor [40] β L 4 ⊇ μγ αβ νδρ ∇ μ φ∇ ν φ∇ γ ∇ δ φ∇ ρ ∇ α φ, (1.2) where L 4 is the fourth Horndeski Lagrangian (there is also a similar term for the fifth Horndeski Lagrangian [40]). These term will produce third-order derivative terms in the equations of motion but it can be shown that the extra ghost dof does not appear in this case. The Fab-four terms can be further generalized in the sense that the potentials for John and Paul terms can depend on φ and also on X = ∂μ φ∂ μ φ. The 123 (2019) 79:448 resulting theory is called beyond Fab-four [41]. This new theory however is a subclass of the GLPV theory, as will be reviewed in the next section. In this paper, we will investigate cosmological consequences of a scalar field theory coupled to a Maxwell field. The procedure of defining the ac (...truncated)