Holographic dark energy models and higher order generalizations in dynamical Chern–Simons modified gravity
Dark energy models are here investigated and studied in the framework of the Chern–Simons modified gravity model. We bring into focus the holographic dark energy model with Granda–Oliveros cut-off, the modified holographic Ricci dark energy model and a model with higher derivatives of the Hubble parameter. The relevant expressions of the scale factor \(a(t)\) for a Friedmann–Robertson–Walker Universe are derived and studied, and, in this context, the evolution of the scale factor is shown to be similar to the one displayed by the modified Chaplygin gas in two of the above models.
Cosmological data obtained from different independent observations of SNeIa, CMB radiation anisotropies, X-ray experiments and Large Scale Structures are well known to point toward the accelerated phase of the expansion of the Universe [1, 2, 3, 4, 5, 6, 7].
The cosmological constant \(\varLambda \) model, dark energy (DE) models, and theories of modified gravity, among other attempts, have been considered to provide an explanation for the accelerated expansion of the Universe [8, 9, 10, 11, 12]. The cosmological constant \(\varLambda \) stands for the most straightforward candidate suggested to explain the observational evidence for it. The fine-tuning and the cosmic coincidence problems are questions still underlying the cosmological constant model [13, 14].
A model for DE, motivated by the holographic principle, was proposed  and it has been further studied in [16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. The holographic model of DE (HDE) has been comprehensively investigated [26, 27, 28, 29, 30, 31, 32, 33, 34]. The HDE model has been further employed to drive inflation of the Universe , and considered in [32, 36, 37, 38, 39, 40] with different IR cut-offs, for example the future event and the particle horizons and the Hubble horizon as well. Moreover, correspondences between the HDE model and other scalar field models have been recently suggested . The HDE model fits the cosmological data by CMB radiation anisotropies and SNeIa [42, 43].
Recently, the cosmic acceleration has also been well studied by a promising modified gravity model that has recently come forth: the modified gravity Chern–Simons model . The low-energy limit of string theory comprises an anomaly-canceling correction to the Einstein–Hilbert action, deriving the Chern–Simons modified gravity as an effective theory. Gravitational parity violation was first investigated using this formalism , appearing both in 4D compactifications of perturbative string theory and further in loop quantum gravity as well, when the Barbero–Immirzi parameter emulates a scalar field coupled to the Nieh–Yan invariant [45, 46, 47, 48]. In the Chern–Simons modified gravity, the Pontryagin topological invariant is well known not to affect the field equations, and thus the so-called Chern–Simons correction consists of the product of the Pontryagin density by a scalar field that can be regarded as either a non-dynamical background field or as a dynamical evolving field. In the latter case, the dynamical Chern–Simons modified gravity (DCSMG) is therefore considered [49, 50]. Some efforts have recently provided bounds to the Chern–Simons parameter .
In this paper, we study the many faces of DE models in the context of the dynamic formulation of Chern–Simons gravity, where the coupling constant is promoted to a scalar field. Recent applications include for instance the neutron star binary . We shall investigate three different DE models: the HDE model with Granda–Oliveros (GO) cut-off, the modified holographic Ricci dark energy (MHRDE) model, and a recently proposed model with higher derivatives of the Hubble parameter \(H\)  in the framework of Chern–Simons gravity, in order to obtain the expressions of the scale factor for each model. We prove that both the HDE model with GO cut-off and the model with higher derivatives of the Hubble parameter \(H\) in the framework gravity Chern–Simons are related to the modified Chaplygin gas models [57, 58, 59] that further represent the well known models of dark energy as the Chaplygin gas [54, 55, 56].
The paper is organized as follows. In Sect. 2, we briefly revisit dynamical Chern–Simons modified gravity model. In Sect. 3, we describe the three different DE models considered in this work in the framework of the Chern–Simons modified gravity models and we derive the relevant expressions of the scale factor \(a(t)\). Finally, in Sect. 4, we state the conclusions of this work.
2 Chern–Simons gravity
In what follows we study Eq. (10) in the context of the modified Chern–Simons modified gravity for three different DE models, namely, the HDE model with Granda–Oliveros cut-off, the MHRDE model and a recent DE model which involves the Hubble parameter squared and the first and the second time derivatives of the Hubble parameter. We shall derive an expression of the scale factor for each one of these models.
3 Dark energy models in Chern–Simons gravity
In this section, our aim is to give a brief description of the DE models dealt with and to study their behavior in the framework of Chern–Simons modified gravity model, in order to find the expressions of the scale factor \(a(t)\). In the first subsection, we will consider the HDE model with Granda–Oliveros cut-off, in the second subsection the MHRDE model, while in the third one we treat the model with higher derivatives of the Hubble parameter \(H\).
3.1 The HDE model with GO cut-off
3.2 The MHRDE model
3.3 Model with higher derivatives of the Hubble parameter \(H\)
In this work we studied the behavior of three different DE models: the HDE model with Granda–Oliveros cut-off, the MHRDE model, and the model with higher derivatives of the Hubble parameter \(H\), in the framework of the Chern–Simons modified gravity model. For each of these models, we derived the respective scale factors \(a\left( t \right) \).
For the HDE model with GO cut-off, the scale factor \(a\left( t \right) \) is an hyperbolic sine function of cosmic time. Nevertheless, in the MHRDE model paradigm the scale factor is a power law of the time, and, finally, according to the values of the parameters involved for the model with higher derivatives of the Hubble parameter, we have either a power law solution or \(a\left( t \right) \) proportional to a hyperbolic sine function.
The scale factor obtained in Eq. (28) for the HDE with GO cut-off and in Eqs. (48) and (49) for the model with higher derivatives of \(H\) are similar to those obtained in [62, 64, 65]. For this reason, we conclude that, for suitable choices of the parameters involved, the HDE model with GO cut-off and the model with higher derivatives of the Hubble parameter \(H\) in the framework of Chern–Simons modified gravity have the same results obtained from the modified Chaplygin gas [57, 58, 59]. Namely, the results clearly indicate that there is agreement between both the HDE model with GO cut-off and the model with higher derivatives of the Hubble parameter \(H\) in the framework gravity Chern–Simons, and the modified Chaplygin gas. It is worthwhile to emphasize that as the Ricci dark energy in Chern–Simons modified gravity is related to the Ricci dark energy with a minimally coupled scalar when choosing the FRW metric, the above-mentioned similarity between them is limited to the de Sitter phase derived by the cosmological constant in the future .
Financial support under Grant No. SR/FTP/PS-167/2011 from DST, Govt of India is thankfully acknowledged by the second author. RdR is grateful to CNPq Grants 473326/2013-2 and 303027/2012-6. RdR is also Bolsista da CAPES Proc. 10942/13-0.
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