𝑓(𝑅) in Holographic and Agegraphic Dark Energy Models and the Generalized Uncertainty Principle

Abstract

We studied a unified approach with the holographic, new agegraphic, and 𝑓(𝑅) dark energy model to construct the form of 𝑓(𝑅) which in general is responsible for the curvature driven explanation of the very early inflation along with presently observed late time acceleration. We considered the generalized uncertainty principle in our approach which incorporated the corrections in the entropy-area relation and thereby modified the energy densities for the cosmological dark energy models considered. We found that holographic and new agegraphic 𝑓(𝑅) gravity models can behave like phantom or quintessence models in the spatially flat FRW universe. We also found a distinct term in the form of 𝑓(𝑅) which goes as 𝑅^3/2 due to the consideration of the GUP modified energy densities. Although the presence of this term in the action can be important in explaining the early inflationary scenario, Capozziello et al. recently showed that 𝑓(𝑅)∼𝑅^3/2 leads to an accelerated expansion, that is, a negative value for the deceleration parameter 𝑞 which fits well with SNeIa and WMAP data.

1. Introduction

Observations of type IA supernovae confirm that our present universe is expanding at an accelerating rate [1]. Present observational cosmology has provided enough evidence in favour of the accelerated expansion of the universe [2–4]. Theoretical aid came in the form of exotic dark energy (DE) which can generate sufficient negative pressure and is believed to account nearly 70% of present energy of the universe. Researchers in theoretical physics have proposed many DE models but they face problems while incorporating the history of the universe. The models generally have many free parameters and face serious constraints from observational data. Recent reviews [5–9] are useful for a brief knowledge of DE models.

The holographic DE is one of the promising DE models and the model is based on the holographic principle [10–14]. Bekenstein's entropy bound suggests that quantum field theory breaks down at large volumes. This can be reconciled by using a relation between UV and IR cut-offs such that 𝐿³Λ⁴≤𝐿𝑚²_𝑝, where 𝑚_𝑝 is the reduced Planck Mass (𝑚_𝑝⁻²=8𝜋𝐺). In this situation an effective local quantum field theory will give a good approximate description [15]. The holographic DE was first proposed in [16] following the line of [15] where the infrared cut-off is taken to be the size of event horizon for DE. The problem of cosmic coincidence can be resolved by the inflationary paradigm with minimal e-foldings in this model. Later this holographic DE was studied in detail by many authors [17–26]. Clearly it can be mentioned that black hole entropy bound played an important role in the interpretation of holographic dark energy model. Various theories of quantum gravity (e.g., [27–33]) have predicted the following form for the entropy of a black hole: 𝐴𝑆=4𝑙²_𝑝+𝑐₀𝐴ln4𝑙²_𝑝+const. (1)

𝑐₀ is a model dependent parameter and 𝑙_𝑝 is the Planck length. Many researchers have expressed a vested interest in fixing 𝑐₀ (the coefficient of the subleading logarithmic term) [27]. Recent rigorous calculations of loop quantum gravity predict the value of 𝑐₀ to be −1/2 [33]. A entropy corrected holographic DE model (ECHDE) was proposed recently in [34] where the inflation was driven by ECHDE. The curvature perturbation may be generated through the curvaton and the only requirement remain as 𝐻≃const [35, 36].

Another promising DE candidate is the agegraphic DE and was proposed in [37]. Considering the quantum fluctuations of spacetime Károlyházy and his collaborators [38–40] argued that in Minkowski spacetime any distance 𝑡 cannot be known to a better accuracy than 𝛿𝑡∼𝑡_𝑝^2/3𝑡^1/3, where 𝑡_𝑝 is the reduced Planck time. Based on the arguments of Károlyházy it can be shown that for Minkowski spacetime the energy density of metric fluctuations is given by 𝜌_Λ∼𝑚²_𝑝/𝑡²[41, 42]. The agegraphic DE model considers spacetime and matter field fluctuations responsible for DE. If conformal time is considered in place of the age of the universe the model can describe the matter dominated epoch [43] with a natural solution to the coincidence problem [44] and is known as the new agegraphic DE model. The conformal time 𝑒𝑡𝑎 is defined by 𝑑𝑡=𝑎𝑑𝜂, where 𝑡 is the cosmic time and 𝑎 the scale factor. Many authors did some detailed study of this new agegraphic DE model [45–48].

Also we have other possible explanations for the cosmic acceleration, the different being the approach with 𝑓(𝑅) gravity, where 𝑅 is the scalar curvature. Other forms of 𝑅 along with 𝑅 in the Lagrangian can explain the observed acceleration without considering other additional components (the review [49] is useful). Among other existing theories 𝑓(𝑅) gravity models can be shown to be compatible with a matter dominated epoch transiting into an accelerating phase [50]. Also the forms of 𝑓(𝑅) with positive powers of curvature support the inflationary epoch and forms with negative powers of curvature serving as the effective DE responsible for cosmic acceleration and compatible with solar system experiments [51]. Also it is worth mentioning that these models face some challenges in the line of argument discussed in [52–56].

The idea that the uncertainty principle could be affected by gravity was given by Mead [57]. In the regime when the gravity is strong enough, conventional Heisenberg uncertainty relation is no longer satisfactory (though approximately but perfectly valid in low gravity regimes). Later modified commutation relations between position and momenta, commonly known as generalized uncertainty principle (GUP), were given by candidate theories of quantum gravity (String Theory, doubly special relativity theory and Black Hole Physics) with the prediction of a minimum measurable length [58–67]. Similar kind of commutation relation can also be found in the context of Polymer Quantization in terms of polymer mass scale [68]. Importance of the GUP can also be realized on the basis of simple gedanken experiments without any reference to a particular fundamental theory [65, 66]. So we can think of the GUP as a model-independent concept, ideally perfect for the study of black hole entropy. The authors in [69] proposed a GUP which is consistent with DSR theory, string theory, and black hole physics. This is approximately covariant under DSR transformations but not Lorentz covariant [67]. With the GUP as proposed by the authors in [69] we can arrive at the corrected entropy-area relation for a black hole which can be written in the following expansive form [70, 71]: 𝐴𝑆≃4𝑙²_𝑝+𝑎𝐴4𝑙²_𝑝𝐴+𝑏ln4𝑙²_𝑝+^∞_{𝑚=1/2,3/2,…}𝑑_𝑚𝐴4𝑙²_𝑝^−𝑚+^∞_𝑛=1,2,…𝑐_𝑛𝐴4𝑙²_𝑝^−𝑛+const. (2)

In this paper we will try to predict the form of 𝑓(𝑅) in holographic and new agegraphic DE models in the light of the generalized uncertainty principle. In [72] it has been argued that the holographic theory does not retain its good features by considering minimal length in quantum gravity. But here we will try to avoid the issue and hope to present the discussion in some future work. We will use (2) to calculate the energy density for the models. Later we will construct the form of 𝑓(𝑅) and the equation of state parameter 𝜔 for each of these DE models. Although an earlier attempt is present in the literature for the reconstruction of 𝑓(𝑅) [73] but we will later conclude with a brief comparison of the results.

2. 𝑓(𝑅) from Holographic DE Model with GUP

In 𝑓(𝑅) gravity the action is written as [74–76] 𝑆=√−𝑔𝑑⁴𝑥𝑅+𝑓(𝑅)16𝜋𝐺+𝐿_matter. (3)

The considerations lie in the fact that the higher order modifications of the Ricci curvature 𝑅 in the form of 𝑅² or 𝑅_𝜇𝜈𝑅^𝜇𝜈 could give rise to inflation at the very early universe. The term 𝑅_𝜇𝜈𝑅^𝜇𝜈 does not lead to any new kind of inflation different from that produced by the 𝑅² term since the combination 𝑅_𝜇𝜈𝑅^𝜇𝜈−(1/3)𝑅² does not contribute to the de Sitter solution at all. So this leads to a notion that whether inverse powers in 𝑅 dominant in late time universe can give an explanation to the recent predicted acceleration of the universe. But this type of models faces problems of stability [77]. The variation of the action with respect to the metric gives the field equations as 𝑅_𝜇𝜈−12𝑅𝑔_𝜇𝜈=8𝜋𝐺𝑇^(𝑅)_𝜇𝜈+𝑇^(𝑚)_𝜇𝜈, (4) where 8𝜋𝐺𝑇^(𝑅)_𝜇𝜈=12𝑔_𝜇𝜈𝑓(𝑅)−𝑅_𝜇𝜈𝑓^′(𝑅)+∇_𝜇∇_𝜈−𝑔_𝜇𝜈□𝑓^′(𝑅). (5)

Here 𝑓(𝑅)=𝜕𝑓(𝑅)/𝜕𝑅, 𝑅_𝜇𝜈 is the Ricci tensor, 𝑇^(𝑚)_𝜇𝜈 is the energy momentum tensor of matter, and 𝑅 denotes the curvature contribution.

For a spatially flat FRW universe the modified Friedmann equation can be written as 𝐻²=8𝜋𝐺3𝜌_𝑚+𝜌_𝑅,2𝐻̇+3𝐻²=−8𝜋𝐺𝑝_𝑚+𝑝_𝑅, (6) where 𝜌_𝑅=1−18𝜋𝐺2𝑓(𝑅)+3𝐻̇+𝐻²𝑓^′(𝑅)−184𝐻²𝐻̇+𝐻𝐻̈𝑓^′′(𝑅),𝑝_𝑅=118𝜋𝐺2𝑓(𝑅)−𝐻̇+3𝐻²𝑓^′(𝑅)+68𝐻²𝐻̇+6𝐻𝐻̈+4𝐻̇²+𝐻⃛𝑓^′′(𝑅)+36𝐻̈+4𝐻𝐻̇²𝑓^′′′(𝑅),𝑅=6𝐻̇+2𝐻². (7)(8)(9)

Here the Hubble parameter is 𝐻=𝑎/̇𝑎 and the overdot denotes derivative with respect to cosmic time 𝑡. We can show that the curvature contribution will have its own equation of state and it can be written as [78] 𝜔_𝑅=𝑝_𝑅𝜌_𝑅=1−4𝐻̇𝑓^′(𝑅)+33𝐻𝐻̈−4𝐻²𝐻̇+4𝐻̇²+𝐻⃛×𝑓^′′(𝑅)+18(𝐻̈+4𝐻𝐻̇)²𝑓^′′′(𝑅×𝑓(𝑅)̇)−6𝐻+𝐻²𝑓^′(𝑅)+364𝐻²𝐻̇+𝐻𝐻̈𝑓^′′(𝑅)⁻¹. (10)

In 𝑓(𝑅) gravity theories we usually encounter three types of scale factors for accelerating and inflationary cosmological solutions. We will follow the details of [73, 79]. Here we will study phantom, quintessence, and de Sitter scale factors which are given by 𝑎𝑎=⁰𝑡_𝑠−𝑡^−ℎ,𝑡≤𝑡_𝑠,ℎ>0phantom,𝑎₀𝑡^ℎ,ℎ>0quintessence,𝑎₀𝑒^𝐻𝑡,𝐻=constant(deSitter). (11)

With the phantom scale factor and (9) we get ℎ𝐻=6()𝑅2ℎ+1^1/2 (12) and also 𝐻̇=𝐻²2. (13)

Recent observations constrain the value of ℎ for the phantom scale factor to be −∞>ℎ≥7.81 [80]. Similarly with the quintessence scale factor and (9) we get ℎ𝐻=𝑅6(2ℎ−1)^1/2 (14) and also 𝐻̇=−𝐻²2. (15)

For this quintessence scale factor the value of ℎ is very close to unity [80]. For de Sitter solution we have 𝐻=constant. This scale factor is used to describe the early inflationary scenario. For this case we get 𝑅𝐻=12^1/2. (16)

Now we will try to evaluate the form of 𝑓(𝑅) for each of the scale factor mentioned above in the light of the generalized uncertainty principle (GUP). For our purpose we need to solve (7) and we borrow the energy density from the holographic and agegraphic dark energy models, respectively.

Considering the leading order terms of (2) and following the arguments of [34, 81] we can write the GUP motivated energy density for the holographic DE model as 𝜌_Λ=3𝑛²𝑚²_𝑝𝐿²+𝑎𝑚_𝑝𝐿³+𝑏𝐿⁴ln𝐿²𝑚²_𝑝+𝑐𝐿⁴. (17)

Here 𝑛,𝑎,𝑏, and 𝑐 are constants and 𝐿 is the future event horizon. If 𝑎=𝑏=𝑐=0 we get the usual holographic DE model. Though 𝑛 is a constant, its value can be constrained from the latest observational data [82]. The future event horizon is defined as 𝐿=𝑎^∞_𝑡𝑑𝑡𝑎. (18)

For the phantom scale factor the future event horizon is 𝐿=𝑎^𝑡𝑠_𝑡𝑑𝑡𝑎=1ℎ+16ℎ(2ℎ+1)𝑅. (19)

Putting the value of 𝐿 in (17) we get the form of energy density as 𝜌_Λ=3𝑛²𝑚²_𝑝(ℎ+1)²6ℎ(2ℎ+1)𝑅+𝑎𝑚_𝑝(ℎ+1)³(6ℎ)^3/2(2ℎ+1)^3/2𝑅^3/2+(ℎ+1)⁴𝑅²(6ℎ)²(2ℎ+1)²()𝑚𝑏ln6ℎ2ℎ+1²_𝑝(ℎ+1)²𝑅+𝑐. (20)

Now (7) can be written in terms of 𝑅 as 𝑅²𝑓^′′(𝑅)−(ℎ+1)2𝑅𝑓^′(𝑅)+(2ℎ+1)2𝑓(𝑅)𝑛=−²(ℎ+1)²𝑎2ℎ𝑅−𝑚_𝑝(ℎ+1)³(6ℎ)^3/2(2ℎ+1)^1/2𝑅^3/2+𝑏𝑚²_𝑝(ℎ+1)⁴(6ℎ)²()𝑅2ℎ+1²[𝑅]−𝑏ln𝑚²_𝑝(ℎ+1)⁴(6ℎ)²()𝑚2ℎ+1×ln^2𝑝6ℎ(2ℎ+1)(ℎ+1)²𝑅²+𝑐𝑚²_𝑝(ℎ+1)⁴(6ℎ)²()𝑅2ℎ+1², (21) where 𝑚²_𝑝=1/8𝜋𝐺. This equation is a nonhomogeneous Euler differential equation and the solution can be written as 𝑓(𝑅)=𝐶₁𝑅^𝑞+𝐶₂𝑅^𝑟+𝛿𝑅+𝛼𝑅^3/2+𝛽𝑅²+𝛾𝑅²ln[𝑅], (22) where 1𝑞=43+ℎ+ℎ²,1−10ℎ+1𝑟=43+ℎ−ℎ²,𝑛−10ℎ+1𝛿=−²(ℎ+1)²ℎ²,1𝛼=−𝑚_𝑝4𝑎(ℎ+1)³(ℎ+2)(6ℎ)^3/2(2ℎ+1)^1/2,1𝛽=𝑚²_𝑝−𝑏10+3ℎ−ℎ²(ℎ+1)⁴162ℎ²()−𝑏()ℎ+2)(2ℎ+1ℎ+1⁴54ℎ²()𝑚2ℎ+1×ln^2𝑝6ℎ(2ℎ+1)(ℎ+1)²+𝑐(ℎ+1)⁴54ℎ²(),𝑏2ℎ+1𝛾=𝑚²_𝑝(ℎ+1)⁴54ℎ²(2ℎ+1) (23) with 𝐶_1,2 as integration constants whose value can be predicted by the boundary conditions. The boundary conditions are 𝑓(𝑅)∣_𝑅=𝑅0=−2𝑅₀,𝑓^′(𝑅)∣_𝑅=𝑅0∼0, (24) where 𝑅=𝑅₀ is the present value of 𝑅 which is a small constant. The value of 𝑅₀ is of the order of (10⁻³³eV)². If we apply the boundary conditions we get the values of 𝐶_1,2 as 𝐶₁=𝑅₀^1−𝑞22()𝑞−𝑟2𝑟−𝛿+𝑟𝛿−𝛼𝑅₀(3−2𝑟)−2𝛽𝑅₀(2−𝑟)−2𝛾𝑅₀×1+2ln𝑅₀−𝑟ln𝑅₀,𝐶₂𝑅=−^01−𝑟22𝑞−𝑟2𝑞−𝛿+𝑞𝛿−𝛼𝑅₀3−2𝑞−2𝛽𝑅₀2−𝑞−2𝛾𝑅₀×1+2ln𝑅₀−𝑞ln𝑅₀. (25)

In general the equation of state of (10) will be a function of 𝐻 and hence time in this case and so it can explain the transition from quintessence (𝜔_𝑅>−1) to phantom dominated regime (𝜔_𝑅<−1) as predicted by recent observations [83–85]. If we see (22) we can infer that there is a contribution from 𝑅^3/2. This is interesting from the fact that we can have contributions from fractional powers of 𝑅. We will discuss this in a later part of this study.

For the quintessence scale factor the future event horizon is at 𝐿=𝑎^∞_𝑡𝑑𝑡𝑎=1ℎ−16ℎ(2ℎ−1)𝑅 (26) with the condition ℎ>1. Putting the value of 𝐿 from (26) in (17) we get the form of energy density as 𝜌_Λ=𝑛²𝑚²_𝑝(ℎ−1)²2ℎ(2ℎ−1)𝑅+𝑎𝑚_𝑝(ℎ−1)³(6ℎ)^3/2(2ℎ−1)^3/2𝑅^3/2+(ℎ−1)⁴𝑅²(6ℎ)²(2ℎ−1)²()𝑚𝑏ln6ℎ2ℎ−1²_𝑝(ℎ−1)²𝑅+𝑐. (27)

So for the quintessence scale factor equations (14) and (15) we can rewrite (7) with (27) as 𝑅²𝑓^′′(𝑅)+(ℎ−1)2𝑅𝑓^′(𝑅)−(2ℎ−1)2𝑓(𝑅)=𝑛²(ℎ−1)²𝑎2ℎ𝑅+𝑚_𝑝(ℎ−1)³(6ℎ)^3/2(2ℎ−1)^1/2𝑅^3/2−𝑏𝑚²_𝑝(ℎ−1)⁴(6ℎ)²()𝑅2ℎ−1²[𝑅]+𝑏ln𝑚²_𝑝(ℎ−1)⁴(6ℎ)²()𝑚2ℎ−1×ln^2𝑝6ℎ(2ℎ−1)(ℎ−1)²𝑅²+𝑐𝑚²_𝑝(ℎ−1)⁴(6ℎ)²()𝑅2ℎ−1², (28) where 𝑚²_𝑝=1/8𝜋𝐺. Similarly like the phantom case the solution can be written as 𝑓(𝑅)=𝐶₁𝑅^𝑞+𝐶₂𝑅^𝑟+𝛿𝑅+𝛼𝑅^3/2+𝛽𝑅²+𝛾𝑅²ln[𝑅], (29) where 1𝑞=43−ℎ+ℎ²,1+10ℎ+1𝑟=43−ℎ−ℎ²,𝑛+10ℎ+1𝛿=−²(ℎ−1)²ℎ²,𝛼=2𝑎3𝑚_𝑝(ℎ−1)³ℎ(2−ℎ)(2ℎ−1)^1/2,1𝛽=𝑚²_𝑝𝑏10−3ℎ−ℎ²(ℎ−1)⁴162ℎ²()+𝑏()2−ℎ)(2ℎ−1ℎ−1⁴54ℎ²()𝑚2ℎ−1×ln^2𝑝6ℎ(2ℎ−1)(ℎ−1)²+𝑐(ℎ−1)⁴54ℎ²(),𝑏2ℎ−1𝛾=−𝑚²_𝑝(ℎ−1)⁴54ℎ²().2ℎ−1 (30)

The boundary conditions will give 𝐶₁=𝑅₀^1−𝑞22()𝑞−𝑟2𝑟−𝛿+𝑟𝛿−𝛼𝑅₀×(3−2𝑟)−2𝛽𝑅₀(2−𝑟)−2𝛾𝑅₀1+2ln𝑅₀−𝑟ln𝑅₀,𝐶₂𝑅=−^01−𝑟22𝑞−𝑟2𝑞−𝛿+𝑞𝛿−𝛼𝑅₀3−2𝑞−2𝛽𝑅₀2−𝑞−2𝛾𝑅₀×1+2ln𝑅₀−𝑞ln𝑅₀. (31)

For the scale factor in de Sitter space 𝐻 is constant. The future event horizon is located at 𝐿=𝑎𝐿=𝑎^∞_𝑡𝑑𝑡𝑎=1𝐻=12𝑅, (32) where 𝐻 is given by (16). So we can write the GUP motivated energy density from (17) as 𝜌_Λ=𝑛²𝑚²_𝑝4𝑅+𝑎𝑚_𝑝12^3/2𝑅^3/2+𝑏144𝑚²_𝑝𝑅²×ln12𝑚²_𝑝𝑅+𝑐144𝑚²_𝑝𝑅². (33)

So (7) takes the form 𝑅𝑓^′(𝑅)−2𝑓(𝑅)=4𝜌_Λ𝑚²_𝑝. (34)

The solution of this equation can be written in the form 𝑓(𝑅)=−𝑛²𝑅+𝐶₁𝑅²−𝑎3√3𝑚_𝑝𝑅^3/2−𝑏72𝑚²_𝑝𝑅²ln12𝑚²_𝑝𝑅²+𝑐36𝑚²_𝑝𝑅²ln(𝑅), (35) where 𝐶₁ is the arbitrary integration constant to be fixed by boundary conditions. The GUP motivated terms in 𝑓(𝑅) are important for inflationary scenario. We have instances for the 𝑅² term in the literature to explain early time inflation [74] as a curvature driven phenomenon. Here we have a new term 𝑅^3/2 in 𝑓(𝑅) which can be important for curvature driven inflation. We will discuss more about this term later in the Discussion section.

3. 𝑓(𝑅) from New Agegraphic DE Model with GUP

With the corrections due to the generalized uncertainty principle to the entropy area relation we can frame the energy density of the new agegraphic DE model [34, 81] as 𝜌_Λ=3𝑛²𝑚²_𝑝𝜂²+𝑎𝑚_𝑝𝜂³+𝑏𝜂⁴ln𝜂²𝑚²_𝑝+𝑐𝜂⁴, (36) where 𝜂 is the conformal time. 𝑎=𝑏=𝑐=0 will give back the usual new agegraphic DE model. The parameter 𝑛 is constrained by present observations and its best fit value is around 2.716^+0.111_−0.109 with 1𝜎 uncertainty [44]. The numerical factor 3𝑛² was introduced for a parameterization of some uncertainties such as the effect of curved spacetime, (as the Károlyházy relation considered only the metric quantum fluctuations of Minkowski spacetime) and the species of quantum fields in the universe.

For the phantom scale factor the conformal time can be evaluated as 𝜂=^𝑡𝑠_𝑡𝑑𝑡𝑎=1𝑎₀()()ℎ+16ℎ2ℎ+1𝑅^(ℎ+1)/2,ℎ>0. (37)

Substituting this in (36) we get 𝜌_Λ=3𝑛²𝑚²_𝑝𝑎²₀(ℎ+1)²(6ℎ)^ℎ+1(2ℎ+1)^ℎ+1𝑅^ℎ+1+𝑎𝑚_𝑝𝑎³₀(ℎ+1)³(6ℎ)^(3ℎ+3)/2(2ℎ+1)^(3ℎ+3)/2𝑅^(3ℎ+3)/2+𝑏𝑎⁴₀(ℎ+1)⁴(6ℎ)^2ℎ+2(2ℎ+1)^2ℎ+2()×ln6ℎ^ℎ+1(2ℎ+1)^ℎ+1𝑚²_𝑝𝑎²₀(ℎ+1)²+𝑐𝑎⁴₀(ℎ+1)⁴(6ℎ)^2ℎ+2(2ℎ+1)^2ℎ+2𝑅^2ℎ+2−𝑏𝑎⁴₀(ℎ+1)⁴(6ℎ)^2ℎ+2(2ℎ+1)^2ℎ+2𝑅^2ℎ+2ln𝑅^ℎ+1. (38)

Solving the inhomogeneous Euler differential equation (7) with (38) we get the form of 𝑓(𝑅) as 𝑓(𝑅)=𝐶₁𝑅^𝑞+𝐶₂𝑅^𝑟+𝛿𝑅^ℎ+1+𝛼𝑅^(3/2)(ℎ+1)+𝛽𝑅^2ℎ+2+𝛾𝑅^2ℎ+2ln𝑅^ℎ+1, (39) where 1𝑞=43+ℎ+ℎ²,1−10ℎ+1𝑟=43+ℎ−ℎ²−10ℎ+1,𝛿=−3𝑛²𝑎²₀(ℎ+1)²ℎ(6ℎ)^ℎ+1(2ℎ+1)^ℎ×288+3360ℎ+14816ℎ²+31360ℎ³+33408ℎ⁴+17280ℎ⁵+3456ℎ⁶×288+3504ℎ+16496ℎ²+38768ℎ³+49088ℎ⁴+33984ℎ⁵+12096ℎ⁶+1728ℎ⁷⁻¹,𝛼=−𝑎𝑎³₀(ℎ+1)³𝑚_𝑝(6ℎ)^(3ℎ+3)/2(2ℎ+1)^(3ℎ+1)/2×576+4128ℎ+10624ℎ²+12032ℎ³+6144ℎ⁴+1152ℎ⁵×288+3504ℎ+16496ℎ²+38768ℎ³+49088ℎ⁴+33984ℎ⁵+12096ℎ⁶+1728ℎ⁷⁻¹,𝛽=𝑏𝑎⁴₀(ℎ+1)⁴𝑚²_𝑝(6ℎ)^2ℎ+2(2ℎ+1)^2ℎ+1()×ln6ℎ^ℎ+1(2ℎ+1)^ℎ+1𝑚²_𝑝𝑎²₀(ℎ+1)²+𝑐𝑎⁴₀(ℎ+1)⁴𝑚²_𝑝(6ℎ)^2ℎ+2(2ℎ+1)^2ℎ+1×320+2528ℎ+6432ℎ²+7168ℎ³+3616ℎ⁴+672ℎ⁵×288+3504ℎ+16496ℎ²+38768ℎ³+49088ℎ⁴+33984ℎ⁵+12096ℎ⁶+1728ℎ⁷⁻¹+𝑏𝑎⁴₀(ℎ+1)⁴𝑚²_𝑝(6ℎ)^2ℎ+2(2ℎ+1)^2ℎ+1×192+1696ℎ+4960ℎ²+5920ℎ³+3072ℎ⁴+576ℎ⁵×288+3504ℎ+16496ℎ²+38768ℎ³+49088ℎ⁴+33984ℎ⁵+12096ℎ⁶+1728ℎ⁷⁻¹,𝛾=𝑏𝑎⁴₀(ℎ+1)⁴𝑚²_𝑝(6ℎ)^2ℎ+2(2ℎ+1)^2ℎ+1×192+1696ℎ+4960ℎ²+5920ℎ³+3072ℎ⁴+576ℎ⁵×288+3504ℎ+16496ℎ²+38768ℎ³+49088ℎ⁴+33984ℎ⁵+12096ℎ⁶+1728ℎ⁷⁻¹. (40)

The boundary conditions will give 𝐶₁=𝑅₀^1−𝑞2()𝑅𝑞−𝑟4𝑟−𝛼3+3ℎ−2𝑟₀^(3ℎ+1)/2−22𝛽+2ℎ𝛽−𝑟𝛽+𝛾+ℎ𝛾×𝑅₀^2ℎ+1−2𝛿(1+ℎ−2𝑟)𝑅^ℎ₀−2𝛾(2+2ℎ−𝑟)𝑅₀^2ℎ+1ln𝑅₀^ℎ+1,𝐶₂=𝑅₀^1−𝑟2𝑅𝑟−𝑞4𝑞−𝛼3+3ℎ−2𝑞₀^(3ℎ+1)/2−22𝛽+2ℎ𝛽−𝑞𝛽+𝛾+ℎ𝛾×𝑅₀^2ℎ+1−2𝛿1+ℎ−2𝑞𝑅^ℎ₀−2𝛾2+2ℎ−𝑞𝑅₀^2ℎ+1ln𝑅₀^ℎ+1. (41)

In general the equation of state of (10) will be a function of 𝐻 and hence time in this case and so it can explain the transition from quintessence (𝜔_𝑅>−1) to phantom dominated regime (𝜔_𝑅<−1) as predicted by recent observations [83–85]. For the quintessence scale factor (11) the conformal time can be evaluated with (14) and (15) as 𝜂=^𝑡₀𝑑𝑡𝑎=1𝑎₀()()1−ℎ6ℎ2ℎ−1𝑅^(1−ℎ)/2,12<ℎ<1. (42)

For a real finite conformal time it is necessary to have 1/2<ℎ<1. Substituting this in (36) we get 𝜌_Λ𝑚²_𝑝=3𝑛²𝑎²₀(1−ℎ)²(6ℎ)^1−ℎ(2ℎ−1)^1−ℎ𝑅^1−ℎ+𝑎𝑎³₀(1−ℎ)³𝑚_𝑝(6ℎ)^(3−3ℎ)/2(2ℎ−1)^(3−3ℎ)/2𝑅^(3−3ℎ)/2+𝑎⁴₀(1−ℎ)⁴𝑚²_𝑝(6ℎ)^2−2ℎ(2ℎ−1)^2−2ℎ×()𝑏ln6ℎ^1−ℎ(2ℎ−1)^1−ℎ𝑚²_𝑝𝑎²₀(1−ℎ)²+𝑐×𝑅^2−2ℎ−𝑏𝑎⁴₀(1−ℎ)⁴𝑚²_𝑝(6ℎ)^2−2ℎ(2ℎ−1)^2−2ℎ×𝑅^2−2ℎln𝑅^1−ℎ. (43)

Solving the inhomogeneous Euler differential equation (7) with (43), (14), and (15) we get the form of 𝑓(𝑅) as 𝑓(𝑅)=𝐶₁𝑅^𝑞+𝐶₂𝑅^𝑟+𝛿𝑅^1−ℎ+𝛼𝑅^{(3/2)(1−ℎ)}+𝛽𝑅^2−2ℎ+𝛾𝑅^2−2ℎln𝑅^1−ℎ, (44) where 1𝑞=43−ℎ+ℎ²,1+10ℎ+1𝑟=43−ℎ−ℎ²+10ℎ+1,𝛿=3𝑛²𝑎²₀(1−ℎ)²ℎ(6ℎ)^1−ℎ(2ℎ−1)^−ℎ×288−3360ℎ+14816ℎ²−31360ℎ³+33408ℎ⁴−17280ℎ⁵+3456ℎ⁶×−288+3504ℎ−16496ℎ²+38768ℎ³−49088ℎ⁴+33984ℎ⁵−12096ℎ⁶+1728ℎ⁷⁻¹,𝛼=𝑎𝑎³₀(1−ℎ)³𝑚_𝑝(6ℎ)^(3−3ℎ)/2(2ℎ−1)^(1−3ℎ)/2×−576+4128ℎ−10624ℎ²+12032ℎ³−6144ℎ⁴+1152ℎ⁵×−288+3504ℎ−16496ℎ²+38768ℎ³−49088ℎ⁴+33984ℎ⁵−12096ℎ⁶+1728ℎ⁷⁻¹,𝑎𝛽=⁴⁰(1−ℎ)⁴𝑚²_𝑝(6ℎ)^2−2ℎ(2ℎ−1)^1−2ℎ×()𝑏ln6ℎ^1−ℎ(2ℎ−1)^1−ℎ𝑚²_𝑝𝑎²₀(1−ℎ)²×+𝑐−192+1696ℎ−4960ℎ²+5920ℎ³−3072ℎ⁴+576ℎ⁵×−288+3504ℎ−16496ℎ²+38768ℎ³−49088ℎ⁴+33984ℎ⁵−12096ℎ⁶+1728ℎ⁷⁻¹−𝑏𝑎⁴₀(1−ℎ)⁴𝑚²_𝑝(6ℎ)^2−2ℎ(2ℎ−1)^1−2ℎ×320−2528ℎ+6432ℎ²−7168ℎ³+3616ℎ⁴−672ℎ⁵×−288+3504ℎ−16496ℎ²+38768ℎ³−49088ℎ⁴+33984ℎ⁵−12096ℎ⁶+1728ℎ⁷⁻¹,𝛾=−𝑏𝑎⁴₀(1−ℎ)⁴𝑚²_𝑝(6ℎ)^2−2ℎ(2ℎ−1)^1−2ℎ×−192+1696ℎ−4960ℎ²+5920ℎ³−3072ℎ⁴+576ℎ⁵×−288+3504ℎ−16496ℎ²+38768ℎ³−49088ℎ⁴+33984ℎ⁵−12096ℎ⁶+1728ℎ⁷⁻¹. (45)

The boundary conditions 𝑓(𝑅)∣_𝑅=𝑅0=−2𝑅₀ and 𝑓^′(𝑅)∣_𝑅=𝑅0∼0 will give 𝐶₁=𝑅₀^{1−2ℎ−𝑞}2𝑞−𝑟4𝑟𝑅₀^2ℎ−𝛼(3−3ℎ−2𝑟)𝑅₀^(ℎ+1)/2−22𝛽−2ℎ𝛽−𝑟𝛽+𝛾−ℎ𝛾×𝑅₀−2𝛿(1−ℎ−𝑟)𝑅^ℎ₀−2𝛾(2−2ℎ−𝑟)𝑅₀ln𝑅₀^1−ℎ,𝐶₂=𝑅₀^{1−2ℎ−𝑟}2𝑟−𝑞4𝑞𝑅₀^2ℎ−𝛼3−3ℎ−2𝑞𝑅₀^(ℎ+1)/2−22𝛽−2ℎ𝛽−𝑞𝛽+𝛾−ℎ𝛾×𝑅₀−2𝛿1−ℎ−𝑞𝑅^ℎ₀−2𝛾2−2ℎ−𝑞𝑅₀ln𝑅₀^1−ℎ. (46)

For the scale factor 𝑎(𝑡)=𝑎₀𝑒^𝐻𝑡 with 𝐻=constant (de Sitter) we write the conformal time as 𝜂=^∞₀𝑑𝑡𝑎=1𝑎₀𝐻=12𝑎²₀𝑅. (47)

Here we have set the upper limit of the integration to 𝑡→∞ to express 𝜂 in terms of 𝑅. The relevant modification to the energy density (36) will be 𝜌_Λ=3𝑛²𝑚²_𝑝𝜂²+𝑎𝑚_𝑝𝜂³+𝑏𝜂⁴ln𝜂²𝑚²_𝑝+𝑐𝜂⁴. (48)

The solution of (7) with (47) and (48) yields the form of 𝑓(𝑅) as 𝑓(𝑅)=−𝑛𝑎²₀𝑅+𝐶₁𝑅²−𝑎𝑎³₀3√3𝑚_𝑝𝑅^3/2−𝑏𝑎⁴₀72𝑚²_𝑝𝑅²×ln12𝑚²_𝑝𝑎₀𝑅²+𝑐𝑎⁴₀36𝑚²_𝑝𝑅²ln[𝑅], (49) where 𝐶₁ is the integration constant to be fixed with boundary conditions. Here also like the holographic DE model we have a new term 𝑅^3/2 in 𝑓(𝑅) which can be important for curvature driven inflation.

4. Discussion

In this study we considered the generalized uncertainty principle motivated forms of the holographic and the new agegraphic DE models to reconstruct the form of 𝑓(𝑅) suitable to explain the unification of early time inflation and late time acceleration. The idea that the Heisenberg uncertainty principle could be affected by gravity was given by Mead [57]. In the regime when the gravity is strong enough, conventional Heisenberg uncertainty relation is no longer satisfactory (though approximately but perfectly valid in low gravity regimes). Modified commutation relations between position and momenta, commonly known as the generalized uncertainty principle (or GUP), were given by candidate theories of quantum gravity like string theory, doubly special relativity, and black hole physics with the prediction of a minimum measurable length. Importance of the GUP can also be realized on the basis of simple gedanken experiments without any reference of a particular fundamental theory [65, 66]. So we can think of the GUP as a model-independent concept suitable for the study of black hole entropy at least phenomenologically.

According to the holographic principle the number of degrees of freedom of a bounded system should be finite and is related to the area of its boundary. As an application of the principle the upper bound of the entropy of the universe can be obtained. The total energy of a system of size 𝐿 should not exceed the mass of a black hole of the same size, otherwise it would decay into a black hole. The saturation of the inequality means 𝜌_Λ=(3𝑛²𝑚²_𝑝)/𝐿² where 𝑚_𝑝 is the reduced Planck Mass (𝑚_𝑝⁻²=8𝜋𝐺). The UV cut-off is related to the vacuum energy and the IR cut-off is related to the large scale of the universe. The holographic dark energy scenario is viable if we set the IR cut-off by the future event horizon and also make a concrete prediction about the equation of state of the DE [16]. On the other hand the new agegraphic DE model is based on the Károlyházy relation which considers energy density of quantum fluctuations of the metric and matter in the universe. The energy density of the new agegraphic DE model has the same form as the holographic dark energy but the conformal time takes care of the IR cut-off instead of considering the future event horizon of the universe. The model not only accounts the observed value of DE in the universe but also predicts an accelerated expansion. Among various theoretical approaches to explain the present cosmic accelerated expansion of the universe only the holographic and the new agegraphic DE model is somehow based on the entropy-area relation. The entropy-area relation on the other hand can have quantum corrections through various approaches of quantum gravity.

As no single theoretical proposal for DE enjoys a pronounced supremacy over the others in terms of having a strong field theoretic support as well as being able to explain all the present observational data. This state of art explores another possibility of whether geometry in its own right could explain the presently observed accelerated expansion. The idea stems from the fact that higher order modifications of the Ricci curvature 𝑅 along with 𝑅 in the Einstein-Hilbert action could generate inflation in the very early universe. As the curvature is expected to fall off with the cosmic evolution it is then obvious whether inverse powers of 𝑅 in the action dominant during the later stages could drive a late time acceleration. In general this alternative theory is coined as 𝑓(𝑅) gravity.

In this paper we studied a unified approach with the holographic, new agegraphic, and 𝑓(𝑅) DE model to construct the form of 𝑓(𝑅) which in general is responsible for the curvature driven explanation of the very early inflation along with presently observed late time acceleration. We considered the generalized uncertainty principle in our approach which incorporated the corrections in the entropy area relation which thereby modified the energy densities for the cosmological DE models considered here. In the context of modified theories of gravity we should be cautious with the Wald entropy [86, 87] and not the Bekenstein-Hawking entropy. The Wald entropy is defined in terms of quantities on the Killing horizon and it depends on the variation of the Lagrangian density of the modified gravity theory with respect to the Riemann tensor. The Wald entropy is a local quantity and in 𝑓(𝑅) gravity it is given by 𝑆_𝑊=𝐴𝑓^′(𝑅)/4𝐺 [88]. However, here we have just reconstructed 𝑓(𝑅) from the energy densities of other DE models. We found that the GUP motivated holographic and new agegraphic 𝑓(𝑅) gravity models can behave like phantom or quintessence models in the spatially flat FRW universe. A similar study was also carried out by authors in [73]. We reproduced all the result and conclusion of [73] but in addition we also found a distinct term in the form of 𝑓(𝑅) which goes as 𝑅^3/2 due to the consideration of the GUP modified energy densities. This is really very interesting if we consider the phenomenological consequence of our study. Although the presence of this term in the action can have its importance for inflation, Capozziello et al. [89, 90] introduced an action with 𝑓(𝑅)∼𝑅^𝑚 and showed that it leads to an accelerated expansion, that is, a negative value for the deceleration parameter 𝑞 for 𝑚≈3/2 which fits well with SNeIa and WMAP data. Apart from the 𝑅^3/2 term we also found the other possible contributions of 𝑅 like 𝑅² and 𝑅²ln[𝑅] which also have importance in the inflationary scenario. We should also mention here that in the latter case one needs not only quasi-exponential expansion but a metastable (i.e. slowly rolling) one. In 𝑓(𝑅) gravity this may occur only if 𝑓(𝑅) is close to 𝑅² over some range of 𝑅 [91].

Acknowledgments

The author was partly supported by the Excellence-in-Research Fellowship of IIT Gandhinagar. The author would like to thank an anonymous referee for enlightening comments which immensely helped in improving the paper.