1 Introduction

Present observational data (Riess et al. 1998; Perlmutter et al. 1999) substantiates the dominance of dark energy (DE) in the composition of our universe that causes the cosmic expansion speedy day by day. There have been many proposals to resolve the mysterious picture of DE, however, none of them is regarded to be much successful (Caldwell et al. 1998; Caldwell 2002; Bertolami and Sen 2002). Among DE candidates, modified gravity theories are the most prominent one and comparatively less ambiguous (Flanagan 2004; Lobo 2008). Brans-Dicke (BD) theory as an alternative gravitational framework obtained by the inclusion of scalar field into the tensor field geometry, tackles enormous cosmological issues successfully (Bertolami and Martins 2000; Banerjee and Pavon 2001). This framework with dynamical gravitational constant \((G=\frac{1}{\phi})\) incorporates Mach’s principle (Brans and Dicke 1961), Dirac’s large number hypothesis (Weinberg 1972) and weak equivalence principle. The compatibility with local gravity tests restricts the BD parameter ω to very large values, i.e., ω≥40,000 (Bertotti et al. 2003; Felice et al. 2006).

The holographic DE (HDE) is an interesting dynamical candidate of DE that originates from the holographic principle of quantum gravity which is defined as in a physical system, there should be finite number of degrees of freedom which scale within the bounding area of that system instead of its volume (Horava and Minic 2000). In cosmology, the application of this principle yields the well-known HDE model that connects DE density to cosmic horizon, a global facet of the universe and the spacetime foam Fischler and Susskind (1998). By taking the assumption that HDE and matter cannot be conserved separately, it can yield a possible solution to the cosmic coincidence problem. Li (2004) argued that the energy density should be constrained by the inequality in terms of infrared (IR) cut-off radius L and reduced Planck mass M p given by \(\rho_{D}\leq3c^{2}M_{p}^{2}/L^{2}\), where c 2 is a dimensionless constant. Since the entropy-area relationship implies the Friedmann equation as well as the well-known expression of HDE, therefore it would be interesting to study this relation. In modern physics, HDE attracted many researchers in recent past due to its success in solving numerous cosmological issues (Susskind 1995; Huang and Li 2004; Elizalde et al. 2005; Zhang and Wu 2005, 2007; t’Hooft 2006).

In literature, two modifications of this entropy relation are proposed, namely power law entropy correction (Das et al. 2008a, 2008b, 2010; Radicella and Pavon 2010) and logarithmic correction (Banerjee and Majhi 2008a, 2008b; Modak 2009; Jamil and Farooq 2010). The appearance of these corrections in the entropy of black hole emerges from the subject of loop quantum gravity and some well-proposed fluctuations like thermal, quantum, mass or charge etc. The logarithmic correction stems from the field fluctuations in loop quantum gravity while the power law correction arises when the wave function of the field is taken as a superposition of ground and exciting states. In the later case, only the excited state results this correction term as it does not agree with Hawking area law (Das et al. 2008a, 2008b). Since the Bekenstein entropy bound (\(S=\frac{A}{4G}\) with horizon area A∼L 2) is regarded as a basic ingredient in the derivation of HDE model, therefore two modified versions of HDE based on these entropy corrections have been presented in the literature (Wei 2009; Sheykhi and Jamil 2011; Karami et al. 2011; Khodam-Mohammadi and Malekjani 2012; Pasqua and Khomenko 2013). These HDE candidates with correction terms are proved to be much promising in the early inflationary cosmic stage (that corresponds to small length scale L) however, for late cosmic stages, these corrections disappear and simple HDE model can be recovered in both cases.

There is a large body of literature available on cosmic expansion using different HDE models along with different choices of IR cut-off. The future event, particle and apparent horizons are three most frequently used choices for IR cut-offs (Cataldo et al. 2001; Guberina et al. 2005; Pavon and Zimdahl 2005). Another useful IR cut-off, based on the Hubble parameter and its first-order derivative, is suggested by Granda and Oliveros (2008). The BD theory fascinated many researchers due to its natural appearance in string theory and extra dimensional theories in low energy limit. In BD field scenario, a lot of work has been done by considering HDE models with different IR cut-offs (Setare 2007; Banerjee and Pavon 2007; Sheykhi 2010; Setare and Jamil 2010). Jamil et al. (2012) argued that the combination of HDE model with Granda-Oliveros cut-off and BD field leads to phantom crossing easily. Sheykhi et al. (2012a, 2012b) studied the entropy corrections (logarithmic and power-law) using future event horizon as IR cut-off and concluded that these models yield phantom crossing more easily as compared to Einstein’s gravity.

The concept of chameleon field is motivated from the quintessence model of DE as it determines a non-minimal interaction of matter with gravity and is in agreement with equivalence principle (Khoury and Weltman 2004). Chameleon mechanism is described by a potential V(ϕ) and an arbitrary function f(ϕ) providing the interaction of matter and scalar fields. It is found that chameleon fields do not violate the existing planetary orbit constraints like lunar laser ranging constraints etc. (Waterhouse 2006). Farajollahi and Salehi (2010a) examined FRW chameleon cosmological model that can give rise to cosmic acceleration by using some cosmological tests. Cannata and Kamenshchik (2011) investigated the exact solutions in FRW cosmology using scalar field ansatz that determine the crossing of phantom divide line. Farajollahi and Salehi (2012a, 2012b) explored the recent universe expansion of FRW chameleonic cosmology with low mass chameleon field using both exponential and power law potentials. Bisabr (2012) investigated the role of power law potential function on cosmic acceleration in BD chameleonic cosmology. Stability of FRW universe in BD chameleonic cosmology has been discussed through both perturbation and phase space analysis (Farajollahi and Salehi 2010b; Saaidi 2012).

Farajollahi et al. (2012) found the bouncing solutions and phantom crossing for FRW universe in generalized BD chameleonic cosmology and also checked their viability by cosmological red shift drift and distance modulus tests of cosmology. Some other interesting studies in this regard include the papers (Banerjee and Das 2008; Jamil et al. 2011; Sharif and Waheed 2012a, 2012b, 2012c). The combination of BD gravity and chameleon mechanism yields chameleonic BD gravity that resolves some cosmological issues effectively. In this scenario, Sheykhi and Jamil (2011) discussed interacting HDE as well as NADE and concluded that phase transition from quintessence to phantom regions can be described by choosing suitable values of parameters.

In a recent paper (Sharif and Waheed 2012a), we have considered BD gravity with perfect fluid matter contents and power law ansatze. It is seen that the accelerated expansion of the universe model cannot be achieved in this gravity with such matter contents. However, this approach works and the cosmic expansion could be easily discussed by including the anisotropic effects. In another study (Sharif and Waheed 2012c), we have imposed the condition of cosmic acceleration (i.e., the deceleration parameter is taken to be negative) and checked the nature of unknowns in this framework. Thus it would be interesting to include some other type of models within this gravitational framework for the possible explanation of cosmic expansion.

The motivation for studying HDE models in the BD theory (BD gravity yields a dynamical gravitational framework) comes from the fact that both holographic and agegraphic DE models belong to a dynamical cosmological constant. Thus we need a dynamical frame to accommodate them instead of Einstein gravity. It is due to this fact that we can combine both these mechanisms together and study the behavior of different cosmological parameters within such framework. The purpose of choosing the chameleon field in BD gravity is just to study HDE in a dynamical gravitational framework involving non-minimal coupling of a massive scalar field with matter contents. Although the introduction of such a coupling leads to a quite complicated model but the resulting model includes more free variables as compared to simple BD gravity. An appropriate choice of these variables yields more accurate and consistent results.

We generalize the work (Radicella and Pavon 2010; Jamil et al. 2012; Sheykhi et al. 2012a, 2012b) to the case when interaction between matter and scalar fields is taken into account in BD formalism and discuss the effect of chameleon mechanism on phase transition. The present study investigates some distinct features of different HDE models in BD chameleon gravity. In the next section, the basic formulation of BD chameleonic gravity is given. In Sect. 3, we discuss the interacting as well as non-interacting cases of HDE model with Granda-Oliveros cut-off. Section 4 provides the discussion of HDE models with entropy corrections (power law and logarithmic) in interacting case only where Hubble scale is taken as IR cut-off. We also discuss the graphical behavior of some cosmological parameters for these models. In the last section, we present an outlook of the obtained results.

2 Chameleonic Brans-Dicke formulation

The chameleonic BD theory is specified by the action, \(S=\int\sqrt{-g}\textit{L}d^{4}x\), where L is the Lagrangian density defined by (Sen and Sen 2001; Nojiri and Odintsov 2004; Setare and Jamil 2010; Farajollahi et al. 2012)

$$\begin{aligned} &\textit{L}=\phi R-\frac{\omega}{\phi}\phi^{,\mu} \phi_{,\mu}-V(\phi)+f(\phi)L_{m}, \\ &\quad \mu=0,1,2,3, \end{aligned}$$
(1)

where ω, ϕ, f, R, V and L m are the constant BD parameter, the scalar field, interaction function, Ricci scalar, potential and matter part of Lagrangian, respectively. The variation of action S with respect to the metric tensor results

$$\begin{aligned} G_{\mu\nu}=\frac{1}{\phi}\bigl(f(\phi)T^{m}_{\mu\nu}+T^{BD}_{\mu\nu} \bigr). \end{aligned}$$
(2)

Here \(T^{BD}_{\mu\nu}\) denotes the energy-momentum tensor with BD scalar field as a source defined by

$$\begin{aligned} T^{BD}_{\mu\nu}&=\frac{\omega}{\phi}\biggl[ \phi_{,\mu}\phi_{,\nu} -\frac{1}{2}g_{\mu\nu} \phi_{,\mu}\phi^{,\mu}\biggr]+\phi_{,\mu;\nu} \\ &\quad {}-g_{\mu\nu} \Box\phi-\frac{V(\phi)}{2}g_{\mu\nu}, \end{aligned}$$
(3)

where □ represents the d’Alembertian operator and the term \(T^{m}_{\mu\nu}\) represents the energy-momentum tensor describing the matter part of the Lagrangian density and is given by

$$ T^m_{\mu\nu}=(\rho+p)u_\mu u_\nu-pg_{\mu\nu}. $$
(4)

Further, we assume that it is a combination of dust matter and DE (which we describe by taking different HDE models), i.e., ρ=ρ m +ρ D , p=p D (dust case). Also, the variation of action with respect to ϕ provides the dynamical scalar wave equation

$$ \Box\phi=\frac{T}{3+2\omega}\biggl(f-\frac{1}{2}\phi f_{,\phi}\biggr)-\frac{1}{3+2\omega}\biggl[2V(\phi)-\phi\frac{dV(\phi)}{d\phi} \biggr]. $$
(5)

The non-flat, homogeneous and isotropic FRW universe model is specified by

$$ ds^{2}=dt^{2}-a^{2}(t) \biggl( \frac{dr^{2}}{1-kr^2}+r^2d\varOmega^2\biggr), $$
(6)

where a is the scale factor and k represents the curvature index. The corresponding field equations are

$$\begin{aligned} &H^2+\frac{k}{a^2}+H\frac{\dot{\phi}}{\phi}- \frac{\omega}{6}\biggl(\frac{\dot{\phi}}{\phi}\biggr)^2 = \frac{f(\phi)}{3\phi}(\rho_{m}+\rho_{D})+\frac{V(\phi)}{6\phi}, \end{aligned}$$
(7)
$$\begin{aligned} &2\frac{\ddot{a}}{a}+H^2+\frac{k}{a^2}+ \frac{\omega}{2}\biggl(\frac{\dot{\phi}}{\phi}\biggr)^2+ \frac{2H\dot{\phi}}{\phi} +\frac{\ddot{\phi}}{\phi} \\ &\quad =-\frac{p_{D}f(\phi)}{\phi}+\frac{V(\phi)}{2\phi}, \end{aligned}$$
(8)
$$\begin{aligned} &\ddot{\phi}+3H\dot{\phi} \\ &\quad =\frac{\rho-3P}{2\omega+3}\biggl(f- \frac{\phi f_{,\phi}}{2}\biggr)-\frac{2}{2\omega+3}\biggl(V-\frac{\phi V_{,\phi}}{2}\biggr). \end{aligned}$$
(9)

Equations (7)–(9) yields the following relationship

$$\begin{aligned} \dot{\rho}+3H(\rho+p)=-\frac{3}{4}(\rho+p)\frac{\dot{f}}{f} \end{aligned}$$

which can be further split into

$$\begin{aligned} &\dot{\rho}_{D}+3H\rho_{D}(1+ \omega_{D})=-\frac{3}{4}(1+\omega_{D}) \rho_{D}\frac{\dot{f}}{f}, \end{aligned}$$
(10)
$$\begin{aligned} &\dot{\rho}_{m}+3H\rho_{m}=-\frac{3}{4} \rho_{m}\frac{\dot{f}}{f}. \end{aligned}$$
(11)

The non-homogeneous part occurs due to the presence of interaction between matter and scalar fields.

For chameleon mechanism, there are various forms of interaction function and potential available in literature. A particular and viable choice for these functions is (Khoury and Weltman 2004)

$$ V(\phi)=\frac{M^{4+n}}{\phi^n}, \qquad f(\phi)=f_{0}e^{b_{0}\phi}, $$
(12)

where f 0, b 0, n and M are finite constants. Here the mass of the field depends on the local mass density and also, the potential has runaway form. We shall make use of this special choice for integration purposes in onward discussion. Since the modification of gravity from general relativity leads to some constraints coming from the local gravity tests like solar system tests, violation of equivalence principle etc. In this context, the Jordan frame can be transformed to Einstein frame by using a conformal mapping \(g_{\mu\nu}=e^{2Q\phi}\tilde{g}_{\mu\nu}\) with \(\frac{1}{2Q^{2}}=3+2\omega\). When such coupling is of order unity, the local gravity tests become inconsistent due to the propagation of fifth force between the field and the non-relativistic matter. The constraint for coupling factor for a massless scalar field is given by |Q|<2.5×10−3 that emerges from the experimental bound on BD parameter ω BD >40,000 while this coupling factor should be large, i.e., |Q|∼1, for massive scalar field.

In this case, chameleon mechanism provides a way to be consistent with the local gravity tests. Using thin shell parameter analysis for the potential V(ϕ)=M 4+n/ϕ n, the respective constraint on M has been given (Tsujikawa 2008; Tsujikawa et al. 2009; Mota and Winther 2011), i.e., M<10−(15n+130)/(n+4) M pl ; n>0. For n=2, this condition leads to M<10−4, i.e., the mass scale allowed by the fifth force constraint should be small. This is consistent with the recent DE scale and also it is compatible with the fifth force search as well as local tests of general relativity, e.g., equivalence principle violation (Khoury and Weltman 2004).

3 New HDE with Granda-Oliveros cut-off

We discuss here new HDE model with Granda-Oliveros cut-off with and without interaction between DE components. Since HDE includes the IR cut-off denoted by L which can be selected as the size of the current universe, e.g., the Hubble scale. Basically, the Hubble scale, the particle and event horizons are well-known choices for L. Using the Hubble scale, Hsu (2004) proved that this choice is incompatible with the accelerated expansion as the evolution of the dark energy is same with that of dark matter. If one chooses the particle horizon of the universe as the IR length scale, the resulting EoS parameter lies within range ω>−1/3 which is also incompatible with the recent observations. The selection of event horizon as the IR cut-off yields the recent observations of cosmic expansion, however, an obvious ambiguity related to the causality appears in this proposal. In fact, the existence of event horizon depends upon the final evolution of the universe and consequently valid only for a universe with forever accelerated expansion.

This motivated Granda and Oliveros (2008) to construct a new length scale L for the HDE. They proposed a new cut-off for HDE which is proportional to both the square of Hubble parameter and its time derivative (a model of DE that is proportional to the Ricci scalar) and is defined on purely dimensional grounds. Their model depends on local quantities and avoids the issue of causality. This new HDE model is given by

$$ \rho_{D}=3\phi\bigl(\gamma_{1}H^2+ \gamma_{2}\dot{H}\bigr) $$
(13)

with γ’s as constants. We consider a power law relationship for the scalar field in terms of scale factor, i.e., ϕ=ϕ 0 a m with m>0. Consequently, we have

$$ \dot{\phi}=mH\phi, \qquad \ddot{\phi}=m\phi\bigl(mH^2+ \dot{H}\bigr); \qquad \phi_{0}=1. $$
(14)

The rate of change of Hubble parameter, in terms of e-folding parameter x=lna, is given by

$$ \dot{H}=\frac{1}{2}\frac{dH^2}{dx}. $$
(15)

Moreover, the derivatives with respect to time and e-folding parameter are related by the following expression

$$ \frac{d}{dx}=\frac{1}{H}\frac{d}{dt}; \qquad \frac{dx}{dt}=H. $$

Integration of Eq. (11) implies ρ 0,m a −3 f −3/4, where ρ 0,m is the constant of integration. Equations (7), (12) and (13) with power-law ansatz for scalar field and \(\dot{H}\) given by (15) yield

$$\begin{aligned} &\frac{dH^2}{dx}-2\biggl[\frac{(m-\frac{\omega}{6}m^2+1)}{\gamma_{2} f_{0}}\bigl(1-b_{0}e^{mx} \bigr)-\frac{\gamma_{1}}{\gamma_{2}}\biggr]H^2 \\ &\quad =\frac{2k}{\gamma_{2} f_{0}}e^{-2x} \bigl[1-b_{0}\phi_{0}e^{mx}\bigr] \\ &\qquad {}-\frac{M^{4+n}}{3\gamma_{2} f_{0}}e^{-(n+1)mx}\bigl[1-b_{0} \phi_{0}e^{mx}\bigr] \\ &\qquad {}-\frac{2\rho_{0,m}f_{0}^{-3/4}}{3\gamma_{2}}\biggl(e^{-(m+3)x} -\frac{3}{4}b_{0}\phi_{0}e^{-3x}\biggr), \end{aligned}$$

where |b 0 f 0 e mx|≤1. The solution of this equation is

$$\begin{aligned} H^2 =&e^{A_{1}x} \bigl(1+A_{2}e^{mx} \bigr)^{-1}\biggl[\frac{2k}{\gamma_{2} f_{0}}\biggl(-\frac{e^{-(A_{1}+2)x}}{A_{1}+2} \\ &{}+ \frac{(A_{2}-b_{0})e^{(m-A_{1}-2)x}}{m-A_{1}-2} -\frac{A_{2}b_{0}e^{(2m-A_{1}-2)x}}{2m-2-A_{1}}\biggr) \\ &{}-\frac{M^{4+n}}{3f_{0}\gamma_{2}} \biggl(-\frac{e^{-(n+1)mx -A_{1}x}}{(n+1)m+A_{1}} \\ &{}-(A_{2}-b_{0}) \frac{e^{-(nm+A_{1})x}}{nm+A_{1}} -b_{0}A_{2}\frac{e^{(\alpha-A-n\alpha)x}}{\alpha-A-n\alpha} \biggr) \\ &{}- \frac{2\rho_{0,m}f_{0}^{-3/4}}{3\gamma_{2}} \biggl(-\frac{e^{-(A_{1}+3+m)x}}{A_{1}+3+m} \\ &{}- \biggl(A_{2}- \frac{3}{4}b_{0}f_{0} \biggr) \frac{e^{-(A_{1}+3)x}}{A_{1}+3} \\ &{} -\frac{3}{4}b_{0} \phi_{0}A_{2}\frac{e^{(m-3-A_{1})x}}{m-3-A_{1}} \biggr) \biggr] +c_{1}e^{A_{1}x} \bigl(1+A_{2}e^{mx} \bigr)^{-1}. \\ \end{aligned}$$
(16)

Here c 1 is the integration constant and constant parameters A 1 and A 2 are defined by

$$ \begin{aligned} A_{1}&=\frac{2}{\gamma_{2} f_{0}}\biggl(m-\frac{\omega}{6}m^2+1 \biggr)-\frac{2\gamma_{1}}{\gamma_{2}}, \\ A_{2}&=\frac{2b_{0}}{\gamma_{2}f_{0}m}\biggl(m- \frac{\omega}{6}m^2+1\biggr). \end{aligned} $$
(17)

The EoS parameter is determined by Eq. (10) as follows

$$\begin{aligned} \omega_{D}=-1-\biggl(\frac{m}{3}+\frac{2\gamma_{1}\dot{H}H+\gamma_{2}\ddot{H}}{3H(\gamma_{1}H^2+\gamma_{2}\dot{H})}\biggr) \biggl(1+\frac{b_{0}m\phi}{4}\biggr)^{-1}. \end{aligned}$$

By introducing the e-folding parameter x and Θ(x)=H 2 in the above equation, we obtain

$$\begin{aligned} \omega_{D} =&-1-\biggl(\frac{m}{3}+\frac{2\gamma_{1}\frac{d\varTheta}{dx} +\gamma_{2}\frac{d^2\varTheta}{dx^2}}{3(2\gamma_{1}\varTheta+\gamma_{2}\frac{d\varTheta}{dx})} \biggr) \\ &{}\times \biggl(1+\frac{b_{0}m\phi_{0}e^{mx}}{4}\biggr)^{-1}. \end{aligned}$$
(18)

Substituting Θ(x) and its derivatives, the EoS parameter turns out to be

$$\begin{aligned} \omega_{D} =&-1-\biggl[\frac{m}{3}+ \biggl[- \frac{2k}{\gamma_{2}f_{0}} \biggl(\frac{4(\gamma_{2}-\gamma_{1})e^{-2x}}{A_{1}+2} \\ &{}- \biggl(\frac{A_{2}}{A_{1}+2}+ \frac{A_{2}-b_{0}}{m-A_{1}-2} \biggr) (m-2) \\ &{}\times \bigl(2\gamma_{1}+(m-2)\gamma_{2} \bigr)e^{(m-2)x}+ \biggl(\frac{A_{2}(A_{2}-b_{0})}{m-A_{1}-2} \biggr) \\ &{}\times (2m-2) \bigl(2 \gamma_{1}+\gamma_{2}(2m-2) \bigr) \\ &{}\times e^{(2m-2)x}+\frac{A_{2}^2b_{0}}{2m-2-A_{1}}e^{(3m-2)x}(3m-2) \\ &{}\times \bigl(2\gamma_{1}+\gamma_{2}(3m-2) \bigr) \biggr) +\frac{M^{4+n}}{3\phi_{0}^{n+1}f_{0}\gamma_{2}} \\ &{}\times \biggl(\frac{e^{-(n+1)mx}}{(n+1)m+A_{1}} (n+1)m \bigl((n+1)m \gamma_{2}-2 \gamma_{1} \bigr) \\ &{}+ \biggl(\frac{A_{2}-b_{0}}{nm+A_{1}} -\frac{A_{2}}{(n+1)m+A_{1}} \biggr)nm(nm \gamma_{2}-2 \gamma_{1}) \\ &{}+ \biggl(\frac{b_{0}\phi_{0}A_{2}}{m-A_{1} -nm}-\frac{A_{2}(A_{2}-b_{0})}{nm+A_{1}} \biggr) \\ &{}\times e^{(1-n)mx}(1-n)m \bigl((1-n)m\gamma_{2}+2 \gamma_{1} \bigr) \\ &{}-\frac{A_{2}^2b_{0}}{m-nm-A_{1}}e^{(2-n)mx}(2-n) \\ &{}\times m \bigl(2\gamma_{1}+m(2-n)\gamma_{2} \bigr) \biggr)+\frac{\rho_{0,m}f_{0}^{-3/4}}{ \phi_{0}\gamma_{2}} \\ &{}\times \biggl(\frac{2e^{-(m+3)x}}{3(A_{1}+3+m)}(m+3) \bigl((m+3) \gamma_{2} -2\gamma_{1} \bigr) \\ &{}+ \biggl(\frac{A_{2}-3/4b_{0}f_{0}}{A_{1}+3}- \frac{A_{2}}{A_{1}+3+m} \biggr) \\ &{}\times 2e^{-3x}(3\gamma_{2}-2 \gamma_{1}) +(m-3) \bigl(2\gamma_{1} +\gamma_{2}(m-3) \bigr) \\ &{}\times \biggl(\frac{b_{0}A_{2}}{2(m-3-A_{1})} - \frac{2A_{2}(A_{2}-3/4b_{0}f_{0})}{3(A_{1}+3)} \biggr) \\ &{}\times e^{(m-3)x}-(2m-3) \bigl(2\gamma_{1}+ \gamma_{2}(2m-3) \bigr) \\ &{}\times \frac{b_{0}A_{2}^2}{2(m-3-A_{1})}e^{(2m-3)x} \biggr) \\ &{}+c_{1}e^{A_{1}x} \bigl(2A_{1}\gamma_{1}+\gamma_{2}A_{1}^2 \bigr) \\ &{}-c_{1}A_{2}e^{(A_{1}+m)x}(A_{1}+m) \bigl(2\gamma_{1}+\gamma_{2}(A_{1}+m) \bigr) \biggr] \\ &{}\times \biggl[-\frac{4k}{\gamma_{2}f_{0}} \biggl(\frac{(\gamma_{2}-\gamma_{1})e^{-2x}}{A_{1}+2} - \biggl(\frac{A_{2}}{A_{1}+2} \\ &{}+\frac{A_{2}-b_{0}}{m-A_{1}-2} \biggr) \bigl(2 \gamma_{1} +(m-2)\gamma_{2} \bigr)e^{(m-2)x} \\ &{}+ \biggl( \frac{A_{2}(A_{2}-b_{0})}{m-A_{1}-2} \biggr) \bigl(2\gamma_{1} \gamma_{2}(2m-2) \bigr)e^{(2m-2)x} \\ &{}+ \frac{A_{2}^2b_{0}}{2m-2-A_{1}}e^{(3m-2)x} \bigl(2\gamma_{1}+ \gamma_{2}(3m-2) \bigr) \biggr) \\ &{}+\frac{M^{4+n}}{3\phi_{0}^{n+1}f_{0}\gamma_{2}} \biggl(\frac{e^{-(n+1)mx}}{(n+1)m+A_{1}} \bigl((n+1)m \gamma_{2}-2\gamma_{1} \bigr) \\ &{}+ \biggl(\frac{A_{2}-b_{0}}{nm+A_{1}} -\frac{A_{2}}{(n+1)m+A_{1}} \biggr) (-nm\gamma_{2}+2 \gamma_{1}) \\ &{}+ \biggl(\frac{b_{0}\phi_{0}A_{2}}{m-A_{1}-nm} -\frac{A_{2}(A_{2}-b_{0})}{nm+A_{1}} \biggr) \\ &{}\times e^{(1-n)mx} \bigl((1-n)m\gamma_{2}+2 \gamma_{1} \bigr) \\ &{}-\frac{A_{2}^2b_{0}}{m-nm-A_{1}}e^{(2-n)mx} \bigl(2 \gamma_{1}+m(2-n)\gamma_{2} \bigr) \biggr) \\ &{}+\frac{\rho_{0,m}f_{0}^{-3/4}}{\phi_{0}\gamma_{2}} \biggl(\frac{2e^{-(m+3)x}}{3(A_{1}+3+m)} \bigl((m+3) \gamma_{2} -2\gamma_{1} \bigr) \\ &{}+ \biggl(\frac{A_{2}-3/4b_{0}f_{0}}{A_{1}+3} -\frac{A_{2}}{A_{1}+3+m} \biggr) \\ &{}\times e^{-3x}(-3\gamma_{2}+2 \gamma_{1}) + \bigl(2\gamma_{1}+\gamma_{2}(m-3) \bigr) \\ &{}\times \biggl(\frac{b_{0}A_{2}}{2(m-3-A_{1})} -\frac{2A_{2}(A_{2}-3/4b_{0}f_{0})}{3(A_{1}+3)} \biggr) \\ &{}\times e^{(m-3)x} -(2m-3) \bigl(2 \gamma_{1} +\gamma_{2}(2m-3) \bigr) \\ &{}\times \frac{b_{0}A_{2}^2}{2(m-3-A_{1})}e^{(2m-3)x} \biggr)+c_{1}e^{A_{1}x}(2 \gamma_{1} +\gamma_{2}A_{1}) \\ &{}-c_{1}A_{2}e^{(A_{1}+m)x} \bigl(2\gamma_{1}+\gamma_{2}(A_{1}+m) \bigr)\biggr]^{-1}\biggr] \\ &{}\times \biggl(1+\frac{b_{0}m\phi_{0}e^{mx}}{4} \biggr)^{-1}. \end{aligned}$$
(19)

Since the chameleon mechanism is described by potential V and interaction functions f, therefore the simple BD theory can be obtained by substituting V(ϕ)=0 and f(ϕ)=1 which further implies f 0=1, b 0=0 and M=0. In this limit, the above EoS parameter reduces to that for the simple BD theory (Jamil et al. 2012). Moreover, in the limit m=0 (constant scalar field), ρ 0,m →0 (absence of matter) and f(ϕ)=1 with V(ϕ)=0, this leads to EoS parameter in Einstein’s gravity (Karami and Fehri 2010) given by

$$\begin{aligned} \omega_{D} =&-\frac{1}{3} \\ &{}\times\biggl[\mbox{\small$\displaystyle\frac{k(\gamma_{1}-\gamma_{2})e^{-2x}-c_{1}(3 +\frac{2\delta}{\gamma_{2}})(\gamma_{2}+\delta)(\gamma_{1}+\delta)e^{\frac{2x\delta}{\gamma_{2}}}}{ k(\gamma_{1}-\gamma_{2})e^{-2x}-c_{1}(\gamma_{1}+\delta) (\gamma_{2}+\delta)e^{\frac{2x\delta}{\gamma_{2}}}}$} \biggr]. \end{aligned}$$
(20)

The comparison of both these expressions of EoS parameter (Eqs. (19) and (20)) indicates that in the presence of chameleon mechanism, phase transition from quintessence (ω D >−1) to phantom phase (ω D <−1) can be described more substantially as compared to Einstein’s gravity. It is due to the fact that chameleon mechanism induces more free parameters and the suitable choice of these parameters yields more significant results. The graphical behavior of EoS parameter for DE (19) is given in Figs. 1 and 2(a). Here the choice of constants is arbitrary but γ 1=0.93 and γ 2=0.5 is a suitable choice given in Granda and Oliveros (2008) which we shall use here. Both the parameters ϕ 0 and m are related with the power law ansatz for scalar field ϕ=ϕ 0 a m. For the sake of simplicity, we have taken ϕ 0=1 while some small values satisfying m>0 have been taken into account which correspond to the expanding positive scalar field with the passage of time (with the increase of scale factor). Here ω is the BD parameter and we have used a suitable range of this parameter that is compatible with the cosmic expansion given by −2<ω<−3/2 (Setare and Jamil 2010).

Fig. 1
figure 1

Plots (a) and (b) show EoS parameter for DE versus x for flat and open universe models, respectively. Here we have taken f 0=1, b 0=0.9, ρ 0,m =1 and c 1=1. Red, green and blue colors indicate the plots for (m,n,M,ω)=(3,2,0.0006,−1.7),(4,3,0.0007,−1.8) and (5,4,0.0008,−1.9), respectively

Full size image
Fig. 2
figure 2

Plots (a) shows EoS parameter for DE versus x for closed universe and (b) indicates the deceleration parameter versus time x. In plot (b), we have taken γ 1=0.93, γ 2=0.5, m=0.4, f 0=0.2, b 0=0.2, ρ 0,m =1, ω=−1.8, M=0.0009, c 1=1 and n=2. Here red, green and blue lines show flat, open and closed universes, respectively

Full size image

The parameters f 0 and b 0 are related with the coupling function of matter and scalar field given by f(ϕ)=f 0exp(b 0 ϕ) (Farajollahi and Salehi 2010a, 2012a). Here we take positive coupling of matter and scalar fields that is ensured by the condition f 0>0. The parameter b 0 plays an important role in determining the strength of this coupling, e.g., b 0<0 indicates small coupling (if b 0→−∞, we get f(ϕ)=0) while b 0>0 yields the increasing large coupling and b 0=0 implies no interaction. In this paper, we have taken small positive values of b 0. Further, ρ 0,m and k are the integration constant and the curvature index, respectively. We have normalized this constant, i.e., ρ 0,m =1 while k=−1,0 and 1 in separate cases. Figures 1(a), 1(b) and 2(a) show the graphical behavior of ω D for flat, open and closed universes, respectively. It is obvious that by increasing the values of parameters m and n and a suitable choice of other parameters, phantom crossing (ω D ≤−1) can be achieved easily in each case. For the future evolution of the universe, i.e., x→∞, the EoS parameter (19) approaches to −∞ which shows that dark energy era is dominating and the universe will be in phantom region for its future evolution.

The deceleration parameter, q, and the statefinder parameters, r and s are of much importance in describing the expansion history of the universe and characterizing different DE models. The parameters (s,r) were firstly proposed by Sahni et al. (2003) and then widely used in literature (Yi and Zhang 2007; Setare et al. 2007; Zhang et al. 2008). The deceleration and statefinder parameters are described by the relations

$$ q=-\frac{\ddot{a}}{aH^2}, \qquad r=\frac{\dddot{a}a^2}{\dot{a}^3}, \qquad s=\frac{r-1}{3(q-1/2)}. $$

In terms of deceleration parameter and total density parameter Ω, the statefinder parameters become

$$ r=2q^2+q-\frac{\dot{q}}{H}, \qquad s=\frac{r-\varOmega}{3(q-\varOmega/2)}. $$

Equation (8) (division by H 2) leads to

$$\begin{aligned} q =&-\frac{\ddot{a}}{aH^2} \\ =&\frac{1}{\varTheta(x)e^{2x}(2+3\omega_{D}f_{0} \gamma_{2}(1+b_{0}e^{mx})+m)} \\ &{}\times \biggl[\varTheta(x)e^{2x} \biggl((1+m)^2 +\frac{\omega}{2}m^2-m\biggr)+k \\ &{}+3(\gamma_{1}- \gamma_{2}) \omega_{D}f_{0}\bigl(1+b_{0}e^{mx} \bigr)e^{2x}\varTheta(x) \\ &{}-\frac{M^{4+n}}{2}e^{(2-(n+1)m)x}\biggr]. \end{aligned}$$
(21)

The red shift parameter is related with the scale factor a by the relation \(z=\frac{1}{a}-1\) which can be written in terms of e-folding parameter as e x=(1+z)−1. In this way, the above expression can be rewritten in terms of red-shift parameter z. Figure 2(b) indicates the deceleration parameter versus x. This shows that the combination of chameleonic BD field and HDE model yields a signature flip from positive to negative and for later times, it turns out to be negative in each case (flat, open and closed). The dimensionless DE parameter evolution is determined by the relation

$$\begin{aligned} \varOmega'_{D} =&\frac{\dot{\varOmega}}{H} \\ =& \frac{-3(1+\omega_{D})}{H^2} -\frac{m}{H^2}\bigl(\gamma_{1}H^2+ \gamma_{2}\dot{H}\bigr) \\ &{}-2\gamma_{1}\frac{\dot{H}}{H^2} -2 \gamma_{2}\biggl(\frac{\dot{H}}{H^2}\biggr)^2, \end{aligned}$$
(22)

where \(\frac{\dot{H}}{H^{2}}\) can be calculated from Eq. (16) and prime indicates the derivative with respect to e-folding parameter.

In our case, statefinder parameters take the following form

$$\begin{aligned} r =&2\biggl(\frac{1}{\varTheta(x)e^{2x}(2+3\omega_{D}f_{0}\gamma_{2}(1+b_{0}e^{mx})+m)} \\ &{}\times \biggl[\varTheta(x)e^{2x} \biggl((1+m)^2+\frac{\omega}{2}m^2 -m\biggr)+k \\ &{}+3(\gamma_{1}-\gamma_{2}) \omega_{D}f_{0} \bigl(1+b_{0}e^{mx} \bigr)e^{2x}\varTheta(x) \\ &{}-\frac{M^{4+n}}{2}\times e^{(2-(n+1)m)x} \biggr]\biggr)^2 \\ &{}+\biggl(\frac{1}{\varTheta(x)e^{2x}(2+3\omega_{D}f_{0}\gamma_{2} (1+b_{0}e^{mx})+m)} \\ &{}\times \biggl[\varTheta(x)e^{2x} \biggl((1+m)^2+\frac{\omega}{2}m^2-m\biggr) \\ &{}+k+3(\gamma_{1}-\gamma_{2}) \omega_{D}f_{0}\bigl(1+b_{0}e^{mx} \bigr)e^{2x}\varTheta(x) \\ &{}-\frac{M^{4+n}}{2}e^{(2-(n+1)m)x}\biggr] \biggr) -\biggl[\bigl(2 +3\omega_{D}f_{0}\gamma_{2} \\ &{}\times \bigl(1+b_{0}e^{mx}\bigr)+m\bigr) \biggl(\frac{-2ke^{-2x}}{\theta(x)} - \frac{k\theta(x)_{,x}e^{-2x}}{\theta(x)^2} \\ &{}+3f_{0}b_{0}(\gamma_{1}- \gamma_{2})\omega_{D} me^{mx}+3f_{0}(\gamma_{1}- \gamma_{2}) \\ &{}\times \bigl(1+b_{0}e^{mx}\bigr) \omega_{D,x}\biggr) -\biggl((1+m)^2+\frac{ke^{-2x}}{\theta(x)}+ \frac{\omega}{2}m^2 \\ &{}-m+2f_{0}\omega_{D}(\gamma_{1}- \gamma_{2}) \bigl(1+b_{0}e^{mx}\bigr) \\ &{}- \frac{M^{4+n}e^{-(n+1)mx}}{2\theta(x)}\biggr) \bigl(3f_{0}\gamma_{2}\bigl(1 +b_{0}e^{mx}\bigr)\omega_{D,x} \\ &{}+3f_{0} \omega_{D}\gamma_{2}me^{mx}\bigr)\biggr] \bigl[2+3 \omega_{D}f_{0}\gamma_{2} \\ &{}\times \bigl(1+b_{0}e^{mx} \bigr)+m\bigr]^{-2}, \end{aligned}$$
(23)
$$\begin{aligned} s =&[r-1] \biggl[\frac{3}{\varTheta(x)e^{2x}(2+3\omega_{D}f_{0}\beta(1+b_{0}e^{\alpha x})+\alpha)} \\ &{}\times \biggl[\varTheta(x)e^{2x} \biggl((1+\alpha)^2 +\frac{\omega}{2}\alpha^2-\alpha\biggr)+k \\ &{}+3( \gamma_{1}-\gamma_{2})\omega_{D}f_{0} \bigl(1+b_{0}e^{\alpha x}\bigr)e^{2x}\varTheta(x)- \frac{M^{4+n}}{2} \\ &{}\times e^{(2-(n+1)\alpha)x}\biggr]-1.5 \biggr]^{-1}. \end{aligned}$$
(24)

Clearly, these parameters depend upon the EoS parameter and the parameters induced by chameleon mechanism. The graphical behavior of these parameters for flat and closed universe models is given in Figs. 3(a) and 3(b), respectively. We know that (s,r)=(0,1) corresponds to standard ΛCDM model of DE. Both graphs show that parameter r increases as s increases. This is obvious from the s–r trajectories that the model does not follow ΛCDM model. Figure 3(a) indicates that by decreasing the BD parameter and the mass scale M, the trajectories of s–r plane passes through the point (s,r)=(0,1) showing the correspondence with ΛCDM model. For the closed universe model, the distance from the fixed point (s,r)=(0,1) decreases as we increase the value of parameter M and decrease the value of parameter n (these parameters come from chameleon potential and interaction function) as shown in Fig. 3(b).

Fig. 3
figure 3

Plots (a) and (b) show s–r plane trajectory versus x for flat and closed universe models. In plot (a), we have taken γ 1=0.93, γ 2=0.5, m=1, k=0, f 0=2, b 0=2, ρ 0,m =1, n=2, c 1=1 and red, green and blue lines indicate (ω,M)=(−1.7,0.0006),(−1.8,0.0004) and (−1.9,0.0002), respectively. For (b), we have considered m=2, f 0=2, b 0=2, ρ 0,m =1, ω=−1.8 and c 1=1. The red line indicates the curve for M=0.0004 and n=2, green corresponds to M=0.0006, n=1.5 and blue represents M=0.0008, n=1

Full size image

Now we consider the more general case in which the interaction between the HDE and DM is taken. The energy conservation equations with interaction term are

$$\begin{aligned} &{\dot{\rho_{D}}+3H\rho_{D}(1+ \omega_{D})=-\frac{3\dot{f}}{4f}(1+\omega_{D}) \rho_{D}-Q,} \end{aligned}$$
(25)
$$\begin{aligned} &{\dot{\rho_{DM}}+3H\rho_{DM}=-\frac{3\dot{f}}{4f} \rho_{DM}+Q,} \end{aligned}$$
(26)
$$\begin{aligned} &{\dot{\rho_{BM}}+3H\rho_{BM}=-\frac{3\dot{f}}{4f} \rho_{BM}.} \end{aligned}$$
(27)

The interaction is expected between the dominant quantities therefore, it is interesting to consider the interaction between dark matter and dark energy. As both these are dominant quantities with mysterious nature therefore, the interaction term would be constructed hypothetically. One has to assume a specific interaction term from the outset or formulate it from the phenomenological requirements (Das et al. 2006; Amendola et al. 2006). The positivity of the interaction term, i.e., Q>0 yields the transfer of energy from dark energy to dark matter and affirms the validity of second law of thermodynamics (Pavon and Wang 2009). It has been argued that the interaction function should depend on the energy densities multiplied by a quantity with units of inverse of time. Three different forms of interaction term have been proposed in literature. These are Q∝Hρ D (Pavon and Zimdahl 2005), Q∝Hρ m (Amendola et al. 2007) and Q∝H(ρ D +ρ m ) (Wang et al. 2005a, 2005b). Here the Hubble parameter is multiplied for dimensional consistency (Jamil and Farooq 2010). Since the scale factor a and function f have the same dimensions and consequently, same does the functions H and \(\frac{\dot{f}}{f}\), therefore we propose that in chameleonic BD theory, the interaction term Q takes the form

$$ Q=\frac{3\dot{f}}{4f}b^2(\rho_{D}+ \rho_{DM}), $$
(28)

where b 2 is an arbitrary coupling constant.

With the help of critical energy density ρ cr =3ϕH 2 and curvature energy density ρ k =3kϕ/a 2, the dimensionless energy density parameters are defined as follows

$$\begin{aligned} &{\varOmega_{DM}=\frac{\rho_{DM}}{\rho_{cr}}=\frac{\rho_{DM}}{3\phi H^2}, \qquad \varOmega_{D}=\frac{\rho_{D}}{\rho_{cr}}=\frac{\rho_{D}}{3\phi H^2},} \end{aligned}$$
(29)
$$\begin{aligned} &{\varOmega_{BM}=\frac{\rho_{BM}}{\rho_{cr}}=\frac{\rho_{BM}}{3\phi H^2}, \qquad \varOmega_{k}=\frac{\rho_{k}}{\rho_{cr}}=\frac{k}{a^2H^2},} \end{aligned}$$
(30)
$$\begin{aligned} &{\varOmega_{\phi}=\frac{\rho_{\phi}}{\rho_{cr}}, \qquad \varOmega_{V}=\frac{\rho_{V}}{\rho_{cr}},} \end{aligned}$$
(31)

where \(\rho_{\phi}=mH^{2}\phi(\frac{m\omega}{2}-3)\) and \(\rho_{V}=\frac{M^{4+n}}{\phi^{n}}\). Introducing these parameters in Friedmann equation (7), it follows that

$$ f(\phi) (\varOmega_{D}+\varOmega_{BM}+ \varOmega_{DM})+\varOmega_{\phi}=1+\varOmega_{k}- \frac{\varOmega_{V}}{2}. $$
(32)

In terms of ratio of DM energy density to DE density, i.e., \(u=\frac{\rho_{DM}}{\rho_{D}}\), the interaction term becomes

$$\begin{aligned} Q=\frac{3b^2\dot{f}\rho_{D}}{4f^2\varOmega_{D}}\biggl[1+\varOmega_{k}- \frac{\varOmega_{V}}{2}-\varOmega_{\phi}-f(\phi)\varOmega_{BM} \biggr]. \end{aligned}$$
(33)

From Eqs. (25) and (33), the EoS parameter turns out to be

$$\begin{aligned} \omega_{D} =&-1-\biggl(\frac{m}{3}+ \frac{2\gamma_{1}\dot{H}H +\gamma_{2}\ddot{H}}{3H(\gamma_{1}H^2+\gamma_{2}\dot{H})} \\ &{}+ \frac{b^2[1+\varOmega_{k}-\frac{\varOmega_{V}}{2}-\varOmega_{\phi}-f(\phi)\varOmega_{BM}]}{ 4(b_{0}m\phi)^{-1}f(\phi)\varOmega_{D}}\biggr) \\ &{}\times \biggl(1+\frac{1}{4}b_{0}m \phi_{0}e^{mx}\biggr)^{-1} \end{aligned}$$
(34)

or equivalently, we can write

$$\begin{aligned} \omega_{D} =&\omega_{D} \text{(without interaction)}-\frac{b^2b_{0}m\phi}{4f(\phi)\varOmega_{D}} \\ &{}\times \biggl[1+\varOmega_{k} - \frac{\varOmega_{V}}{2}-\varOmega_{\phi}-f(\phi)\varOmega_{BM} \biggr] \\ &{}\times \biggl(1+\frac{1}{4}b_{0}m \phi_{0}e^{mx}\biggr)^{-1}. \end{aligned}$$

Clearly, due to the presence of interaction term, the EoS parameter gets modified. Since, in the above expression, the interaction term appears with negative sign, therefore it yields more rapid phantom crossing as compared to the expression found in the absence of interaction term. The deceleration parameter turns out to be

$$\begin{aligned} q =&\frac{1}{\varTheta(x)e^{2x}(2+3\omega_{D}f_{0} \gamma_{2}(1+b_{0}e^{mx})+m)} \\ &{}\times \biggl[\varTheta(x)e^{2x} \biggl((1+m)^2+\frac{\omega}{2}m^2-m\biggr) \\ &{}+k-\frac{M^{4+n}}{2}e^{(2-(n+1)m)x}\biggr] \\ &{}- \frac{3(\gamma_{1}-\gamma_{2})f_{0}(1+b_{0}e^{mx})e^{2x}\varTheta(x)}{\varTheta(x)e^{2x}(2+3\omega_{D}f_{0} \gamma_{2}(1+b_{0}e^{mx})+m)} \\ &{}\times \biggl\{-1 -\biggl(\frac{m}{3}+\frac{2\gamma_{1}\dot{H}H +\gamma_{2}\ddot{H}}{3H(\gamma_{1}H^2+\gamma_{2}\dot{H})} \\ &{}+ \frac{b^2[1+\varOmega_{k}-\frac{\varOmega_{V}}{2}-\varOmega_{\phi}-f(\phi)\varOmega_{BM}]}{ 4(b_{0}m\phi)^{-1}f(\phi)\varOmega_{D}} \biggr) \\ &{}\times \biggl(1+\frac{1}{4}b_{0}m \phi_{0}e^{mx}\biggr)^{-1}\biggr\}. \end{aligned}$$
(35)

It is clear that the deceleration parameter contains negative contribution of the interaction term. Thus it leads to a more clear picture of accelerated expansion of the universe model as compared to non-interacting case. Likewise, the evolution of DE density parameter can be calculated in the presence of interaction term by using Eq. (34).

4 Entropy-corrected HDE models

In this section, we discuss the power-law entropy corrected (PLEC) and logarithmic entropy corrected (LEC) HDE models (Wei 2009; Radicella and Pavon 2010; Sheykhi et al. 2012a, 2012b) in BD chameleon cosmology with IR cut-off as Hubble radius. Since an interacting HDE model, being a more general case, yields more interesting results, so we discuss here the interacting case only. The PLECHDE in BD cosmology takes the form

$$\begin{aligned} \rho_{D}=\frac{3c^2\phi}{L^2}-\frac{\beta\phi}{L^{\alpha}}, \quad \alpha\neq2, \end{aligned}$$

while the logarithmic corrected HDE model is given by

$$\begin{aligned} \rho_{D}=\frac{3c^2\phi}{L^2}-\frac{\alpha}{L^{4}}\ln\bigl(\phi L^2\bigr)+\frac{\beta}{L^4}. \end{aligned}$$

Here L denotes the IR cut-off, α and β are any dimensionless constants. In case of PLECHDE, the value of parameter α should satisfy 2<α<4 for being consistent with thermodynamics laws. We discuss some cosmological parameters when IR cut-off is taken as Hubble scale, i.e., L=H −1. Thus the above expressions for HDE turn out to be

$$\begin{aligned} \rho_{D} =&3c^2\phi H^2-\beta\phi H^{\alpha}, \end{aligned}$$
(36)
$$\begin{aligned} \rho_{D} =&3c^2\phi H^2+\alpha H^{4}\ln\biggl(\frac{\phi}{H^2}\biggr)+\beta H^4. \end{aligned}$$
(37)

The time derivatives of PLECHDE and LECHDE densities are

$$\begin{aligned} \dot{\rho}_{D} =&3c^2m\phi H^3-\beta m\phi H^{\alpha+1} \\ &{}+\bigl(6c^2\phi H-\alpha\beta\phi H^{\alpha-1} \bigr)\dot{H}, \end{aligned}$$
(38)
$$\begin{aligned} \dot{\rho}_{D} =&3c^2\phi mH^3+m \alpha H^{5}+\biggl(6c^2\phi H+4\alpha H^3\ln \biggl(\frac{\phi}{H^2}\biggr) \\ &{}-2\alpha H^3+4\beta H^3 \biggr)\dot{H}, \end{aligned}$$
(39)

where \(\dot{H}\) is determined by solving the field equations (7) and (8) given by

$$\begin{aligned} \dot{H} =&\frac{1}{(m+2)}\biggl[H^2\biggl(-3- \frac{\omega}{2}m^2-2m-m^2 \\ &{}-\frac{3\omega_{D}}{1+u}\biggl(1 - \frac{\omega}{6}m^2+m\biggr)\biggr) \\ &{}-\frac{k}{a^2}\biggl(1+\frac{3\omega_{D}}{1+u}\biggr)+ \frac{M^{4+n}}{2\phi^{n+1}}\biggl(1+\frac{\omega_{D}}{1+u}\biggr)\biggr], \end{aligned}$$
(40)

where ω D is the EoS parameter for DE given in Eq. (34). Moreover, \(u=\frac{\rho_{DM}}{\rho_{D}}\) denotes the ratio of dark matter to HDE. Substituting the respective values in Eq. (34), the EoS parameter expression for PLECHDE becomes

$$\begin{aligned} \omega_{D} =&(1+u)\biggl[-(1+b_{0}m\phi)- \frac{b^2b_{0}m\phi}{f_{0}\varOmega_{D} (1+b_{0}\phi)} \\ &{}\times \biggl(1+\varOmega_{k}+m\biggl(1-\frac{\omega}{6}m \biggr) \\ &{}-f_{0}\varOmega_{BM}(1+b_{0}\phi)- \frac{\varOmega_{V}}{2}\biggr)-\frac{3mc^2H-\beta mH^\alpha}{3(3c^2H^2-\beta H^\alpha)} \\ &{}-\frac{H^2(6c^2-\alpha\beta H^{\alpha-2})}{3(m+2)(3c^2H^2-\beta H^\alpha)} \\ &{}\times \biggl(-3-\frac{\omega}{2}m^2-2m-m^2- \varOmega_{k} +\frac{M^{4+n}}{2H^2\phi^{n+1}}\biggr)\biggr] \\ &{}\times \biggl[(1+u) (1+b_{0}m\phi) +\frac{(6c^2H^2-\alpha\beta H^{\alpha})}{3(m+2)(3c^2H^2-\beta H^\alpha)} \\ &{}\times \biggl(-3-3m+\frac{\omega}{2}m^2 \biggr)+\frac{6c^2-\alpha\beta H^{\alpha-2}}{(m+2)(3c^2H^2-\beta H^\alpha)} \\ &{}\times \biggl(\frac{M^{4+n}}{2\phi^{n+1}}-\frac{3k}{a^2}\biggr) \biggr]^{-1}. \end{aligned}$$
(41)

Likewise, the EoS parameter using logarithmic corrected HDE model is

$$\begin{aligned} \omega_{D} =&(1+u)\biggl[-(1+b_{0}m\phi)- \frac{b^2b_{0}m\phi}{f_{0}\varOmega_{D} (1+b_{0}\phi)} \\ &{}\times \biggl(1+\varOmega_{k}+m\biggl(1-\frac{\omega}{6}m \biggr) -f_{0}\varOmega_{BM}(1+b_{0}\phi) \\ &{}- \frac{M^{4+n}}{6H^2\phi^{n+1}}\biggr)-\frac{3mc^2H^2\phi+m\alpha H^4}{3(3c^2H^2\phi+\beta H^4+\alpha H^4\ln(\frac{\phi}{H^2}))} \\ &{}-\biggl(H^2\biggl(-3-\frac{\omega}{2}m^2-2m-m^2 \biggr)-H^2\varOmega_{k} \\ &{}+\frac{M^{4 +n}}{2\phi^{n+1}}\biggr) \biggl( \biggl(6c^2\phi+4\alpha H^2\ln\biggl(\frac{\phi}{H^2} \biggr)\biggr) \biggl(3(m+2) \\ &{}\times \biggl(3c^2\phi H^2+\beta H^4+\alpha H^4\ln\biggl(\frac{\phi}{H^2}\biggr)\biggr) \biggr)^{-1} \\ &{}+\bigl(-2\alpha H^2+4\beta H^2\bigr) \biggl(3(m+2) \\ &{}\times \biggl(3c^2\phi H^2+\beta H^4+\alpha H^4\ln\biggl(\frac{\phi}{H^2}\biggr)\biggr) \biggr)^{-1}\biggr)\biggr] \\ &{}\times \biggl[\biggl(H^2\biggl(-3-3m+ \frac{\omega}{2}m^2\biggr) \\ &{}+\frac{M^{4+n}}{2\phi^{n+1}}-\frac{3k}{a^2}\biggr) \biggl((1+u) (1+b_{0}m\phi) \\ &{}+\frac{(6c^2\phi+4\alpha H^2\ln(\frac{\phi}{H^2}))(m+2)^{-1}}{3(3c^2H^2\phi+\beta H^4+\alpha H^4\ln(\frac{\phi}{H^2}))} \\ &{}-\frac{(2\alpha H^2-4\beta H^2)(m+2)^{-1}}{3(3c^2H^2\phi+\beta H^4+\alpha H^4\ln(\frac{\phi}{H^2}))}\biggr)\biggr]^{-1}. \end{aligned}$$
(42)

The deceleration parameter can be obtained from Eq. (8) as

$$\begin{aligned} q =&\frac{1}{(m+2)}\biggl[\varOmega_{k}+(1+m)^2+m \biggl(\frac{\omega}{2}m-1\biggr) \\ &{}+3\varOmega_{D} \omega_{D}f_{0}(1+b_{0}\phi)-\frac{M^{4+n}}{2H^2\phi^{n+1}} \biggr]. \end{aligned}$$
(43)

In deceleration parameter, we use the EoS parameter defined by Eq. (41) for PLECHDE while Eq. (42) will be used to discuss the LECHDE model. Also, we use the present value of Hubble parameter that is constant and is estimated to lie within the range 73.60±3.18 or 74.2±3.6 km s−1 Mpc−1 in the literature (Freedman and Madore 2010; Chimento and Richarte 2011). From these ranges, we have used H 0=70.5 km s−1 Mpc−1 for graphical illustration of these parameters. Figure 4 indicates the EoS and deceleration parameters for PLECHDE. In these figures, red, blue and green lines correspond to flat, open and closed universe models, respectively. It is seen from Fig. 4(a) that for the coupling parameter 0<b 2≤1, the universe will be in phantom region. Thus phantom crossing can be achieved with the suitable choice of parameters that appear due to power-law entropy corrections and chameleon mechanism. Figure 4(b) indicate that the deceleration parameter remains negative for 0<b 2≤1 yielding accelerated phases of the universe. Figure 5(a) indicates that for LECHDE model, the universe lies in phantom region for small values of coupling parameter b 2. However, the deceleration parameter exhibits a similar behavior for logarithmic corrected HDE model as shown in Fig. 5(b).

Fig. 4
figure 4

Plot (a) shows the EoS parameter for DE versus b 2 with H 0=70.5,α=3, f 0=2, b 0=2, n=4, c=1,ϕ 0=1,β=2. Here red line represents flat universe with ω=−1.7, M=0.0009, green line indicates closed universe with ω=−1.8, M=0.0007 and blue corresponds to ω=−1.9, M=0.0005 while (b) shows deceleration parameter versus b 2 for PLECHDE using H 0=70.5, α=3, f 0=2, b 0=2, M=0.0006, n=2, c=1, ϕ 0=1, β=0.2, and m=1. Here red, green and blue lines indicate flat universe with ω=−1.7, closed universe with ω=−1.8 and open universe with ω=−1.9, respectively

Full size image
Fig. 5
figure 5

Plot (a) shows the EoS parameter versus b 2 with H 0=70.5, α=1, f 0=0.5, b 0=0.5, M=0.0006, n=1, c=1, ϕ 0=1, β=1, m=10 and (b) shows deceleration parameter versus b 2 for LECHDE with H 0=70.5, α=3, f 0=2, b 0=2, M=0.0009, n=4, c=1, ϕ 0=1, β=2 and m=1. In both plots, red, green and blue lines indicate the flat universe with ω=−1.7, closed universe with ω=−1.8 and open universe with ω=−1.9, respectively

Full size image

The evolution of energy density parameter for PLECHDE and LECHDE respectively, are given by

$$\begin{aligned} \varOmega'_{D} =&-\frac{(1+\omega_{D})}{a^mH^2}-\biggl( \frac{m}{3H^2}+\frac{2\dot{H}}{3a^mH^4}\biggr) \\ &{}\times\bigl(3c^2\phi H^2-\beta\phi H^{\alpha}\bigr), \end{aligned}$$
(44)
$$\begin{aligned} \varOmega'_{D} =&-\frac{(1+\omega_{D})}{a^mH^2}-\biggl( \frac{m}{3H^2}+\frac{2\dot{H}}{3a^mH^4}\biggr) \\ &{}\times\biggl(3c^2\phi H^2+\alpha H^{4}\ln\biggl(\frac{\phi}{H^2}\biggr) \beta H^4\biggr). \end{aligned}$$
(45)

The statefinder parameters for entropy modifications (logarithmic and power-law) generally turn out to be

$$\begin{aligned} r =&2\biggl(\frac{1}{(m+2)}\biggl[\varOmega_{k}+(1+m)^2+m \biggl(\frac{\omega}{2}m-1\biggr) \\ &{}+3\varOmega_{D} \omega_{D}f_{0}(1+b_{0}\phi) -\frac{M^{4+n}}{2H^2\phi^{n+1}}\biggr]\biggr)^2 \\ &{}+\frac{1}{(m+2)} \biggl[\varOmega_{k}+(1+m)^2+m\biggl(\frac{\omega}{2}m-1 \biggr) \\ &{}+3\varOmega_{D}\omega_{D}f_{0}(1 +b_{0}\phi)-\frac{M^{4+n}}{2H^2\phi^{n+1}}\biggr] \\ &{}-\frac{1}{m+2} \biggl[-2\varOmega_{k} -2\varOmega_{k}\frac{\dot{H}}{H^2}+3 \omega_{D}f_{0}\bigl(1+ba^m\bigr) \frac{\dot{\varOmega_{D}}}{H} \\ &{}+\frac{3\varOmega_{D}f_{0}(1+b_{0}a^m)}{H}+3\varOmega_{D}\omega_{D}f_{0}b_{0}ma^m \\ &{}+\frac{M^{4+n}\dot{H}}{a^{(n+1)m}H^4}+\frac{m(n+1)M^{4+n}}{2H^2} a^{-(n+1)m}\biggr], \end{aligned}$$
(46)
$$\begin{aligned} s =&(r-1)\biggl[3\biggl(\frac{1}{(m+2)}\biggl[\varOmega_{k}+(1+m)^2+m \biggl(\frac{\omega}{2}m-1\biggr) \\ &{}+3\varOmega_{D}\omega_{D}f_{0}(1 +b_{0}\phi) -\frac{M^{4+n}}{2H^2\phi^{n+1}}\biggr]-0.5\biggr)\biggr]^{-1}. \\ \end{aligned}$$
(47)

In case of PLECHDE, EoS parameter ω D is given by Eq. (41) while for LECHDE, it is given by Eq. (42) and \(\dot{H}\) is given by Eq. (40) in both cases. The graphical behavior of these parameters for flat universe is shown in Fig. 6. It is seen from Fig. 6(a) that by decreasing BD parameter and mass scale M, the s–r trajectories may pass through the point (s,r)=(0,1) yielding the ΛCDM model of the universe. Likewise, Fig. 6(b) represents s–r trajectories of LECHDE model which may represent ΛCDM model for increasing mass scale M and decreasing ω. From both these figures, it is obvious that the parameter r increases with the increase of the parameter s.

Fig. 6
figure 6

Plots (a) shows s–r plane trajectories versus coupling parameter 0<b 2<0.8 using PLECHDE model for flat universe model. Here we have taken H 0=70.5, α=2.5, f 0=2, ρ 0,m =1, b 0=1, n=1, a=1, c=1, ϕ 0=1, β=0.02 and m=2. Moreover, red, green and blue lines indicate (ω,M)=(−1.6,0.0001), (−1.65,0.00005) and (−1.7,0.00001), respectively. Plot (b) shows s–r plane trajectories versus b 2 using LECHDE model for flat universe model. Here H 0=70.5, α=2, f 0=0.05, ρ 0,m =1, b 0=0.05, n=0.05, a=1, c=1, ϕ 0=1, β=2 and m=0.5. The red, green and blue lines represent (ω,M)=(−1.9,0.0001),(−1.8,0.00005) and (−1.7,0.00001), respectively

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5 Outlook

Chameleon mechanism offers a non-minimal interaction between scalar and matter fields and plays a significant role in cosmic evolution. In this paper, we have discoursed some HDE models by introducing chameleonic mechanism within BD formulation. For this purpose, we have taken FRW universe model with Hubble scale as IR cut-off and its natural modification known as Granda-Oliveros cut-off scale. We have described the dynamics of FRW universe by examining the cosmological parameters like deceleration parameter, energy density parameter, statefinders and EoS parameter. This is done by setting the correspondence between HDE models and BD chameleon field.

Firstly, we have discussed the HDE model with Granda-Oliveros cut-off scale as IR cut-off length. We have considered both the interacting and non-interacting cases and a power law relationship for the scalar field. We have used a special form of interaction function and chameleon potential already available in literature. In non-interacting case, it is seen that both the crossing of phantom divide and the cosmic expansion at an accelerating rate are easily accommodated within this formalism. The graphs of these parameters are shown by using some specified values of free parameters to clarify their behavior. In the interacting case, it is concluded that such behavior of the cosmological parameters can be discussed more easily due to negative contribution of the interaction term. It is interesting to mention here that in the absence of chameleon mechanism, our results reduce to the simple BD theory and consequently to GR. We have also determined the statefinder parameters for this HDE model and demonstrated through graphs. It is shown that the model corresponds to ΛCDM model for decreasing values of the BD parameter as well as mass scale M that appear due to special example of chameleon potential and interaction function.

Secondly, we have studied the entropy-corrected HDE models by assuming power law form for scalar field and interaction between DE components. These modifications arise due to quantum effects of loop quantum gravity. We have considered both the power-law entropy and logarithmic corrections to HDE models in BD chameleon framework. The graphical behavior of EoS parameters for both these corrected HDE models indicate that the phase transition from quintessence to phantom era can be obtained more easily by a suitable selection of free parameters with a particular range of coupling parameter b 2. Likewise, the deceleration parameters having negative sign can lead to accelerated expanding universe in both cases. We have also evaluated statefinders for both these models. It is interesting to mention here that although these models have features distinct from ΛCDM model, however, a correspondence can be obtained by decreasing the values of BD parameter as well as the mass scale for PLEHDE while for LECHDE, this can be achieved by increasing mass scale and decreasing BD parameter.

In literature (Hsu 2004), it is argued that the choice of IR cut-off as Hubble scale implies EoS, i.e., ω D =0 which cannot yield an accelerated expanding universe. However, in the present study, we have discussed some HDE models with Hubble length as IR cut-off. In each case, it is seen that both phantom crossing as well as cosmic acceleration are achieved very easily due to presence of many free parameters (due to chameleon mechanism). It would be interesting to extend this study with these HDE models in chameleonic BD theory of gravity using future event horizon as IR cut-off.