Fractional Action Cosmology: Some Dark Energy Models in Emergent, Logamediate and Intermediate Scenarios of the Universe
aa r X i v : . [ phy s i c s . g e n - ph ] S e p Fractional Action Cosmology: Some Dark Energy Models in Emergent,Logamediate and Intermediate Scenarios of the Universe
Ujjal Debnath, ∗ Surajit Chattopadhyay, † and Mubasher Jamil ‡ Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah-711 103, India. Department of Computer Application (Mathematics Section),Pailan College of Management and Technology, Bengal Pailan Park, Kolkata-700 104, India. Center for Advanced Mathematics and Physics (CAMP),National University of Sciences and Technology (NUST), H-12, Islamabad, Pakistan.
Abstract
In the framework of Fractional Action Cosmology, we have reconstructed the scalar po-tentials and scalar fields, namely, quintessence, phantom, tachyon, k-essence, DBI-essence,Hessence, dilaton field and Yang-Mills field. To get more physical picture of the variationof the scalar field and potential with time, we express scale factor in emergent, logamediateand intermediate scenarios, under which the Universe expands differently. ∗ Electronic address: [email protected] , [email protected] † Electronic address: surajit˙[email protected], [email protected] ‡ Electronic address: [email protected] , [email protected]
I. INTRODUCTION
Fractional action cosmology (FAC) is based on the principles and formalism of the fractional calculus appliedto cosmology. The fractional derivative and fractional integrals are the main tools in fractional calculus, wherethe order of differentiation or integration is not an integer. The fractional calculus is immensely useful in variousbranches of mathematics, physics and engineering [1]. In doing FAC, one can proceed in two different ways[2]: the first one is quite easy as one has to replace the partial derivatives in the Einstein field equations withthe corresponding fractional derivatives; the second technique involves deriving the field equations and geodesicequations from a more fundamental way, namely starting with the principle of least action and replacing the usualintegral with a fractional integral. This later technique is more useful in giving extra features of the FAC [3]:Rami introduced the FAC by introducing the fractional time integral, S = − m ξ ) Z ˙ x µ ˙ x ν g µν ( x )( t − τ ) ξ − dτ. (Ia)Here Γ( ξ ) = R ∞ t ξ − e − t dt is the Gamma function, 0 < ξ ≤
1, 0 < τ < t , m = constant and ˙ x µ = dx µ dτ . Thevariation yields an extra term in the field equations which he termed as ‘variable gravitational constant G ’.Moreover, when the weight function in the fractional time integral is replaced with a sinusoidal function, then thesolution of the corresponding field equations yield a variable cosmological constant and an oscillatory scale factor[4] S = m Z τ ˙ x µ ˙ x ν g µν ( x ) e − χ sin( βt ) dt, (Ib)where χ = 0 reduces to the standard action. In [5], the authors extended the previous study by working out witha general weight function: S = m τ Z g µν ( x ) ˙ x µ ˙ x ν µ ( χ, t ) dt, (Ic)Several examples were studied and cosmological parameters were calculated in there. An interesting feature ofFAC is that it yields an expanding Universe whose scale factor goes like power law form or exponential formdepending on the choice of the weight function. Hence cosmic acceleration can be modeled in FAC.Reconstruction of potentials has been done by several authors in various cases. Capozziello et al [8] consideredscalar-tensor theories and reconstruct their potential and coupling by demanding a background ΛCDM cosmol-ogy. In the framework of phantom quintessence cosmology, [9] used the Noether Symmetry Approach to obtaingeneral exact solutions for the cosmological equations.In this paper, we are going to reconstruct the potentialsand scalar fields, namely, quintessence, phantom, tachyonic, k-essence, DBI-essence, Hessence, dilaton field andYang-Mills field. Such reconstructions have been studied previously in other gravitational setups [6]. To get morephysical insight into the model, we express scale factor in three useful forms [7] namely emergent, logamediateand intermediate scenarios, under which the Universe expands differently. Such expansion scenarios are consistentwith the observations with some restrictions on their parameters [7]. II. FRACTIONAL ACTION COSMOLOGICAL MODEL
For a FRW spacetime, the line element is ds = − dt + a ( t ) (cid:20) dr − kr + r ( dθ + sin θdφ ) (cid:21) , (1)where a ( t ) is the scale factor and k (= 0 , ±
1) is the curvature scalar. We consider the Universe contains normalmatter and dark energy. From Eq. (Ia), the Einstein equations for the space-time given by equation (1) are [3] H + 2( ξ − T H + ka = 8 πG ρ, (2)˙ H − ( ξ − T H − ka = − πG ( ρ + p ) , (3)where T = t − τ , ρ = ( ρ m + ρ φ ) and p = ( p m + p φ ). Here ρ m and p m are the energy density and pressure of thenormal matter connected by the equation of state p m = w m ρ m , − ≤ w m ≤ ρ φ and p φ are the energy density and pressure due to the dark energy.Now consider there is an interaction between normal matter and dark energy. Dark energy interacting withdark matter is a promising model to alleviate the cosmic coincidence problem. In Ref. [10], the authors studied thesignature of such interaction on large scale cosmic microwave background (CMB) temperature anisotropies. Basedon the detail analysis in perturbation equations of dark energy and dark matter when they are in interaction,they found that the large scale CMB, especially the late Integrated Sachs Wolfe effect, is a useful tool to measurethe coupling between dark sectors. It was deduced that in the 1 σ range, the constrained coupling between darksectors can solve the coincidence problem. In Ref. [11], a general formalism to study the growth of dark matterperturbations when dark energy perturbations and interactions between dark sectors were presented. They showedthat the dynamical stability on the growth of structure depends on the form of coupling between dark sectors.Moreover due to the influence of the interaction, the growth index can differ from the value without interactionby an amount up to the observational sensibility, which provides an opportunity to probe the interaction betweendark sectors through future observations on the growth of structure.Due to this interaction, the normal matter and dark energy are not separately conserved. The energy conser-vation equations for normal matter and dark energy are˙ ρ m + 3 H ( p m + ρ m ) = − δHρ m , (5)and ˙ ρ φ + 3 H ( p φ + ρ φ ) = 3 δHρ m , (6)where H = ˙ a/a is the Hubble parameter.From equation (5) we have the expression for energy density of matter as ρ m = ρ a − w m + δ ) , (7)where ρ is the integration constant. III. EMERGENT, LOGAMEDIATE AND INTERMEDIATE SCENARIOS • Emergent Scenario : For emergent Universe, the scale factor can be chosen as [26] a ( T ) = a (cid:0) λ + e µT (cid:1) n (8)where a , µ, λ and n are positive constants. (1) a > a to be positive; (2) λ >
0, to avoidany singularity at finite time (big-rip); (3) a > n > a < n < t = −∞ .So the Hubble parameter and its derivatives are given by H = nµe µT ( λ + e µT ) , ˙ H = nλµ e µT ( λ + e µT ) , ¨ H = nλµ e µT ( λ − e µT )( λ + e µT ) (9)Here H and ˙ H are both positive, but ¨ H changes sign at T = µ log λ . Thus H, ˙ H and ¨ H all tend to zero as t → −∞ . On the other hand as t → ∞ the solution gives asymptotically a de Sitter Universe. • Logamediate Scenario : Consider a particular form of Logamediate Scenario, where the form of the scalefactor a ( t ) is defined as [7] a ( T ) = e A (ln T ) α , (10)where Aα > α >
1. When α = 1, this model reduces to power-law form. The logamediate form is motivatedby considering a class of possible cosmological solutions with indefinite expansion which result from imposing weakgeneral conditions on the cosmological model. Barrow has found in their model, the observational ranges of theparameters are as follows: 1 . × − ≤ A ≤ . × − and 2 ≤ α ≤
50. The Hubble parameter H = ˙ aa and itsderivative become, H = AαT (ln T ) α − , ˙ H = AαT (ln T ) α − ( α − − ln T ) (11) • Intermediate Scenario : Consider a particular form of Intermediate Scenario, where the scale factor a ( t ) ofthe Friedmann universe is described as [7], a ( t ) = e BT β , (12)where Bβ > B > < β <
1. Here the expansion of Universe is faster than Power-Law form, where thescale factor is given as, a ( T ) = T n , where n > β = 1. The Hubble parameter H = ˙ aa and its derivative become, H = BβT β − , ˙ H = Bβ ( β − T β − (13) IV. VARIOUS CANDIDATES OF DARK ENERGY MODELSA. Quintessence or Phantom field
Quintessence is described by an ordinary time dependent and homogeneous scalar field φ which is minimallycoupled to gravity, but with a particular potential V ( φ ) that leads to the accelerating Universe. The action forquintessence is given by [27] S = Z d x √− g (cid:20) − g ij ∂ i φ∂ j φ − V ( φ ) (cid:21) . The energy momentum tensor of the field is: T ij = − √− g δSδg ij , which gives T ij = ∂ i φ∂ j φ − g ij (cid:20) g kl ∂ k φ∂ l φ + V ( φ ) (cid:21) . The energy density and pressure of the quintessence scalar field φ are as follows ρ φ = − T = 12 ˙ φ + V ( φ ) ,p φ = T ii = 12 ˙ φ − V ( φ ) . The EoS parameter for the quintessence scalar field is given by ω φ = p φ ρ φ = ˙ φ − V ( φ )˙ φ + 2 V ( φ ) . For ω φ < − /
3, we find that the Universe accelerates when ˙ φ < V ( φ ) . The energy density and the pressure of the quintessence (phantom field) can be represented by the minimallycoupled spatially homogeneous and time dependent scalar field φ having positive (negative) kinetic energy termgiven by ρ φ = ǫ φ + V ( φ ) (14)and p φ = ǫ φ − V ( φ ) (15)where V ( φ ) is the relevant potential for the scalar field φ , ǫ = +1 represents quintessence while ǫ = − ρ ( t ) ofthe phantom field increases with the expansion of the Universe; it can be used as a source to form and stabilizetraversable wormholes [14–17]; the phantom energy can disrupt all gravitationally bound structures i.e fromgalaxies to black holes [18–23]; it can produce infinite expansion of the Universe in a finite time thus causing the‘big rip’ [24].From above equations, we get˙ φ = − (1 + w m ) ǫ ρ m + 14 πǫG (cid:20) − ˙ H + ( ξ − T H + ka (cid:21) , (16)and V = ( w m − ρ m + 18 πG (cid:20) ˙ H + 3 H + 5( ξ − T H + 2 ka . (cid:21) (17) • For emergent scenario, we get the expressions for φ and V as φ = Z vuut − (1 + w m ) ρ a − w m + δ )0 ǫ ( λ + e µT ) n (1+ w m + δ ) + 14 πǫG ( − nλµ e µT ( λ + e µT ) + ( ξ − nµe µT T ( λ + e µT ) + k a − ( λ + e µT ) n ) dT , (18)and V = ( w m − ρ a − w m + δ )0 λ + e µT ) n (1+ w m + δ ) + 18 πG ( nµ e µT ( λ + 3 ne µT )( λ + e µT ) + 5( ξ − nµe µT T ( λ + e µT ) + 2 k a − ( λ + e µT ) n ) . (19) • For logamediate scenario, we get the expressions for φ and V as φ = Z s − (1 + w m ) ρ ǫ e − A (1+ w m + δ )(ln T ) α + 14 πǫG (cid:26) AαT (ln T ) α − (1 − α + ξ ln T ) + k e − A (ln T ) α (cid:27) dT (20) Φ V Φ V Fig.1 Fig.2 Φ V Fig.3Figs.1-3 show the variations of V against quintessence or phantom field φ in the emergent, logamediate and intermediatescenarios respectively. Solid, dash and dotted lines represent k = − , +1 , ǫ = +1) and phantom field ( ǫ = −
1) respectively. and V = ( w m − ρ e − A (1+ w m + δ )(ln T ) α + 18 πG (cid:20) AαT (ln T ) α − { α − ξ −
6) ln T + 3 Aα (ln T ) α } + 2 k e − A (ln T ) α (cid:21) . (21) • For intermediate scenario, we get the expressions for φ and V as φ = Z r − (1 + w m ) ρ ǫ e − B (1+ w m + δ ) T β + 14 πǫG n Bβ ( ξ − β ) T β − + k e − BT β o dT , (22)and V = ( w m − ρ e − B (1+ w m + δ ) T β + 18 πG h BβT β − (5 ξ + β + 3 BβT β ) + 2 k e − BT β i . (23)In figures 1, 2 and 3, we have plotted the potentials against the scalar fields for the quintessence and phantomfields in emergent, logamediate and intermediate scenarios of the universe respectively in fractional action cosmol-ogy. It has been observed in figure 1 that after gradual decay, the potential starts increasing with scalar field forquintessence as well as phantom field models of dark energy in the emergent scenario of the universe irrespective ofits type of curvature. On the contrary, when logamediate scenario is considered, the figure 2 exhibits a continuousdecay in the potential V with increase in the scalar field φ . A different behavior is observed in figure 3 that depictsthe behavior of the potential V against scalar field φ in the case of intermediate scenario of the universe. Theblue lines in this figure show a continuous decay in V with increase in φ for quintessence model. However, the redlines exhibit an increasing pattern of V with scalar field φ . B. Tachyonic field
A rolling tachyon has an interesting equation of state whose state parameter smoothly interpolates between − V ( φ ) from some dark energy models. An action for tachyon scalar φ is given byBorn-Infeld like action S = − Z d x √− gV ( φ ) q − g ij ∂ i φ∂ j φ (24)where V ( φ ) is the tachyon potential. Energy-momentum tensor components for tachyon scalar φ are obtained as T ij = V ( φ ) " ∂ i φ∂ j φ p − g ij ∂ i φ∂ j φ + g ij p − g kl ∂ k φ∂ l φ (25)The energy density ρ φ pressure p φ due to the tachyonic field φ have the expressions ρ φ = V ( φ ) q − ǫ ˙ φ , (26) p φ = − V ( φ ) q − ǫ ˙ φ , (27)where V ( φ ) is the relevant potential for the tachyonic field φ . It is to be seen that p φ ρ φ = − ǫ ˙ φ > − < − ǫ = +1) or phantom tachyon ( ǫ = − φ = (cid:20) − (1 + w m ) ǫ ρ m + 14 πǫG (cid:26) − ˙ H + ( ξ − T H + ka (cid:27)(cid:21) × (cid:20) − ρ m + 38 πG (cid:26) H + 2( ξ − T H + ka (cid:27)(cid:21) − (28)and V = (cid:20) w m ρ m + 18 πG (cid:26) H + 3 H + 4( ξ − T H + ka (cid:27)(cid:21) × (cid:20) − ρ m + 38 πG (cid:26) H + 2( ξ − T H + ka (cid:27)(cid:21) (29) • For emergent scenario, we get the expressions for φ and V as φ = Z " − (1 + w m ) ρ a − w m + δ )0 ǫ ( λ + e µT ) n (1+ w m + δ ) + 14 πǫG ( − nλµ e µT ( λ + e µT ) + ( ξ − nµe µT T ( λ + e µT ) + k a − ( λ + e µT ) n ) × " − ρ a − w m + δ )0 ( λ + e µT ) n (1+ w m + δ ) + 38 πG ( n µ e µT ( λ + e µT ) + 2( ξ − nµe µt T ( λ + e µT ) + k a − ( λ + e µT ) n ) − dT (30)and V = " w m ρ a − w m + δ )0 ( λ + e µT ) n (1+ w m + δ ) + 18 πG ( nµ e µT (2 λ + 3 ne µT )( λ + e µT ) + 4( ξ − nµe µT T ( λ + e µT ) + k a − ( λ + e µT ) n ) × " − ρ a − w m + δ )0 ( λ + e µT ) n (1+ w m + δ ) + 38 πG ( n µ e µT ( λ + e µT ) + 2( ξ − nµe µT T ( λ + e µT ) + k a − ( λ + e µT ) n ) . (31) • For logamediate scenario, we get the expressions for φ and V as φ = Z (cid:20) − (1 + w m ) ρ ǫ e − A (1+ w m + δ )(ln T ) α + 14 πǫG (cid:26) AαT (ln T ) α − (1 − α + ξ ln T ) + k e − A (ln T ) α (cid:27)(cid:21) × (cid:20) − ρ e − A (1+ w m + δ )(ln T ) α + 38 πG (cid:26) AαT (ln T ) α − { Aα (ln T ) α − + 2( ξ − } + k e − A (ln T ) α (cid:27)(cid:21) − dT (32)and V = (cid:20) − ρ e − A (1+ w m + δ )(ln T ) α + 38 πG (cid:26) AαT (ln T ) α − { Aα (ln T ) α − + 2( ξ − } + k e − A (ln T ) α (cid:27)(cid:21) × (cid:20) w m ρ e − A (1+ w m + δ )(ln T ) α + 18 πG (cid:26) Aαt (ln T ) α − { α −
1) + 2( ξ −
3) ln t + 3 Aα (ln T ) α } + k e − A (ln T ) α (cid:27)(cid:21) (33) • For intermediate scenario, we get the expressions for φ and V as φ = Z (cid:20) − (1 + w m ) ρ ǫ e − B (1+ w m + δ ) T β + 14 πǫG n Bβ ( ξ − β ) T β − + k e − BT β o(cid:21) Φ V Φ V Fig.4 Fig.5 Φ V Fig.6Figs.4-6 show the variations of V against tachyonic field φ in the emergent, logamediate and intermediate scenariosrespectively. Solid, dash and dotted lines represent k = − , +1 , ǫ = +1) and phantom tachyonic field ( ǫ = −
1) respectively. × (cid:20) − ρ e − B (1+ w m + δ ) T β + 38 πG n BβT β − (2( ξ −
1) +
BβT β ) + k e − BT β o(cid:21) − dT (34)and V = (cid:20) − ρ e − B (1+ w m + δ ) T β + 38 πG n BβT β − (2( ξ −
1) +
BβT β ) + k e − BT β o(cid:21) × (cid:20) w m ρ e − B (1+ w m + δ ) T β + 18 πG n BβT β − (2(2 ξ + β −
3) + 3
BβT β ) + k e − BT β o(cid:21) . (35)In figure 4, the V - φ plot for normal tachyon and phantom tachyon models of dark energy is presented foremergent scenario of the universe. Potential of normal tachyon exhibits decaying pattern. However, it showsincreasing pattern for phantom tachyonic field φ . It happens irrespective of the curvature of the universe. In thelogamediate scenario (figure 5) the potentials for normal tachyon and phantom tachyon exhibit increasing anddecreasing behavior respectively with increase in the scalar field φ . From figure 6 we see a continuous decay inthe potential for normal tachyonic field in the intermediate scenario. However, in this scenario, the behavior ofthe potential varies with the curvature of the universe characterized by interacting phantom tachyonic field. For k = − ,
1, the potential increases with phantom tachyonic field and for k = 0, it decays after increasing initially. C. k-essence
In the kinetically driven scalar field theory, we have non-canonical kinetic energy term with no potential.Scalars, modelling this theory, are popularly known as k-essence . Motivated by Born-Infeld action of StringTheory, it was used as a source to explain the mechanism for producing the late time acceleration of the universe.This model is given by the action [31] S = Z d x √− g ˜ L ( ˜ φ, ˜ X ) , (36)with ˜ L ( ˜ φ, ˜ X ) = K ( ˜ φ ) ˜ X + L ( ˜ φ ) ˜ X , (37)ignoring higher order terms of ˜ X = 12 g ij ∂ i ˜ φ∂ j ˜ φ. (38)Using the following transformations, φ = R d ˜ φ q | L ( ˜ φ ) | /K ( ˜ φ ) , X = | L | K ˜ X and V ( φ ) = K / | L | , the action can berewritten as S = Z d x √− gV ( φ ) L ( X ) , (39)with L ( X ) = X − X . (40)From the action, the energy-momentum tensor components can be written as T ij = V ( φ ) h d L dX ∂ i φ∂ j φ − g ij L i . (41)2The energy density and pressure of k-essence scalar field φ are given by ρ k = V ( φ )( − X + 3 X ) , (42)and p k = V ( φ )( − X + X ) , (43)where φ is the scalar field having kinetic energy X = ˙ φ and V ( φ ) is the k-essence potential.From above, we get ˙ φ = (cid:20) w m − ρ m + 12 πG (cid:26) ˙ H + 3 H + 5( ξ − T H + 2 ka (cid:27)(cid:21) × (cid:20) (3 w m − ρ m + 34 πG (cid:26) ˙ H + 2 H + 3( ξ − T H + ka (cid:27)(cid:21) − , (44)and V = (cid:20) (3 w m − ρ m + 34 πG (cid:26) ˙ H + 2 H + 3( ξ − T H + ka (cid:27)(cid:21) × (cid:20) w m − ρ m + 12 πG (cid:26) ˙ H + 3 H + 5( ξ − T H + 2 ka (cid:27)(cid:21) − . (45) • For emergent scenario, we have φ = Z " w m − ρ a − w m + δ )0 ( λ + e µT ) n (1+ w m + δ ) + 12 πG ( nµ e µT ( λ + 3 ne µT )( λ + e µT ) + 5( ξ − nµe µT T ( λ + e µT ) + 2 k a − ( λ + e µT ) n ) × " (3 w m − ρ a − w m + δ )0 ( λ + e µT ) n (1+ w m + δ ) + 34 πG ( nµ e µT ( λ + 2 ne µT )( λ + e µT ) + 3( ξ − nµe µT T ( λ + e µT ) + k a − ( λ + e µT ) n ) − dt. (46)and V = " (3 w m − ρ a − w m + δ )0 ( λ + e µT ) n (1+ w m + δ ) + 34 πG ( nµ e µT ( λ + 2 ne µT )( λ + e µT ) + 3( ξ − nµe µT T ( λ + e µT ) + k a − ( λ + e µT ) n ) × " w m − ρ a − w m + δ )0 ( λ + e µT ) n (1+ w m + δ ) + 12 πG ( nµ e µT ( λ + 3 ne µT )( λ + e µT ) + 5( ξ − nµe µT T ( λ + e µT ) + 2 k a − ( λ + e µT ) n ) − . (47)3 • For logamediate scenario, we get the expressions for φ and V as φ = Z (cid:20) w m − ρ e − A (1+ w m + δ )(ln T ) α + 12 πG (cid:26) AαT (ln T ) α − ( α − ξ −
6) ln T + 3 Aα (ln T ) α ) + 2 k e − A (ln T ) α (cid:27)(cid:21) × (cid:20) (3 w m − ρ e − A (1+ w m + δ )(ln T ) α + 34 πG (cid:26) AαT (ln T ) α − ( α − ξ −
4) ln T + 2 Aα (ln T ) α ) + k e − A (ln T ) α (cid:27)(cid:21) − dT (48)and V = (cid:20) (3 w m − ρ e − A (1+ w m + δ )(ln T ) α + 34 πG (cid:26) AαT (ln T ) α − ( α − ξ −
4) ln T + 2 Aα (ln T ) α ) + k e − A (ln T ) α (cid:27)(cid:21) × (cid:20) w m − ρ e − A (1+ w m + δ )(ln T ) α + 12 πG (cid:26) AαT (ln T ) α − ( α − ξ −
6) ln T + 3 Aα (ln T ) α ) + 2 k e − A (ln T ) α (cid:27)(cid:21) − (49) • For intermediate scenario, we get the expressions for φ and V as φ = Z (cid:20) w m − ρ e − B (1+ w m + δ ) T β + 12 πG n Bβ (5 ξ + β − BβT β ) T β − + 2 k e − BT β o(cid:21) × (cid:20) (3 w m − ρ e − B (1+ w m + δ ) T β + 34 πG n Bβ (3 ξ + β − BβT β ) T β − + k e − BT β o(cid:21) − dT , (50)and V = (cid:20) (3 w m − ρ e − B (1+ w m + δ ) T β + 34 πG n Bβ (3 ξ + β − BβT β ) T β − + k e − BT β o(cid:21) × (cid:20) w m − ρ e − B (1+ w m + δ ) T β + 12 πG n Bβ (5 ξ + β − BβT β ) T β − + 2 k e − BT β o(cid:21) − . (51)From figures 7, 8 and 9 we see that for interacting k-essence the potential V always decreases with increase inthe scalar field φ in all of the three scenarios and it happens for open, closed and flat universes. D. DBI-essence
Consider that the dark energy scalar field is a Dirac-Born-Infeld (DBI) scalar field. In this case, the action ofthe field be written as [33] S D = − Z d xa ( t ) T ( φ ) s − ˙ φ T ( φ ) + V ( φ ) − T ( φ ) , (52)4 Φ V Φ V Fig.7 Fig.8 Φ V Fig.9Figs.7-9 show the variations of V against k-essence field φ in the emergent, logamediate and intermediate scenariosrespectively. Red, green and blue lines represent k = − , +1 , where T ( φ ) is the warped brane tension and V ( φ ) is the DBI potential. The energy density and pressure of theDBI-essence scalar field are respectively given by ρ D = ( γ − T ( φ ) + V ( φ ) , (53)5and p D = γ − γ T ( φ ) − V ( φ ) , (54)where γ is given by γ = 1 q − ˙ φ T ( φ ) . (55)Now we consider here particular case γ = constant. In this case, for simplicity, we assume T ( φ ) = T ˙ φ ( T > γ = q T T − . In this case the expressions for φ , T ( φ ) and V ( φ ) are given by˙ φ = r T − T (cid:20) − (1 + w m ) ρ m + 14 πG (cid:18) − ˙ H + ξ − T H + ka (cid:19)(cid:21) . (56) T = p T ( T − (cid:20) − (1 + w m ) ρ m + 14 πG (cid:18) − ˙ H + ξ − t H + ka (cid:19)(cid:21) . (57)and V = h(cid:16) T − p T ( T − (cid:17) (1 + w m ) − w m i ρ m − πG h(cid:16) − T + p T ( T − (cid:17) ˙ H + 3 H +2 (cid:16) T − p T ( T −
1) + 2 (cid:17) ξ − T H + (cid:16) T − p T ( T −
1) + 1 (cid:17) ka (cid:21) . (58) • For emergent scenario, we get the expressions for φ , T and V as φ = (cid:18) T − T (cid:19) Z " − (1 + w m ) ρ a − w m + δ )0 ( λ + e µT ) n (1+ w m + δ ) + 14 πG ( − nλµ e µT ( λ + e µT ) + ( ξ − nµe µT T ( λ + e µT ) + k a − ( λ + e µT ) n ) dT (59) T = p T ( T − " − (1 + w m ) ρ a − w m + δ )0 ( λ + e µT ) n (1+ w m + δ ) + 14 πG ( − nλµ e µT ( λ + e µT ) + ( ξ − nµe µt T ( λ + e µT ) + k a − ( λ + e µT ) n ) (60)and V = h(cid:16) T − p T ( T − (cid:17) (1 + w m ) − w m i ρ a − w m + δ )0 ( λ + e µT ) n (1+ w m + δ ) − πG "(cid:16) − T + p T ( T − (cid:17) nλµ e µT ( λ + e µT ) + 3 n µ e µT ( λ + e µT ) + 2 (cid:16) T − p T ( T −
1) + 2 (cid:17) ( ξ − T nµe µT ( λ + e µT ) + (cid:16) T − p T ( T −
1) + 1 (cid:17) k a − ( λ + e µT ) n . (61)6 • For logamediate scenario, we get the expressions for φ , T and V as φ = (cid:18) T − T (cid:19) Z (cid:20) − (1 + w m ) ρ e − A (1+ w m + δ )(ln T ) α + 14 πG (cid:26) AαT (ln T ) α − (1 − α + ξ ln T ) + k e − A (ln T ) α (cid:27)(cid:21) dT (62) T = p T ( T − (cid:20) − (1 + w m ) ρ e − A (1+ w m + δ )(ln T ) α + 14 πG (cid:26) AαT (ln T ) α − (1 − α + ξ ln T ) + k e − A (ln T ) α (cid:27)(cid:21) , (63)and V = h(cid:16) T − p T ( T − (cid:17) (1 + w m ) − w m i ρ e − A (1+ w m + δ )(ln T ) α − πG (cid:20) (cid:16) T − p T ( T −
1) + 2 (cid:17) ( ξ − AαT (ln T ) α − + 3 A α T (ln T ) α − + (cid:16) − T + p T ( T − (cid:17) AαT (ln T ) α − ( α − − ln T ) + (cid:16) T − p T ( T −
1) + 1 (cid:17) k e − A (ln T ) α (cid:21) . (64) • For intermediate scenario, we get the expressions for φ , T and V as φ = (cid:18) T − T (cid:19) Z (cid:20) − (1 + w m ) ρ e − B (1+ w m + δ ) T β + 14 πG n Bβ ( ξ − β ) T β − + k e − BT β o(cid:21) dT . (65) T = p T ( T − (cid:20) − (1 + w m ) ρ e − B (1+ w m + δ ) T β + 14 πG n Bβ ( ξ − β ) T β − + k e − BT β o(cid:21) (66)and V = h(cid:16) T − p T ( T − (cid:17) (1 + w m ) − w m i ρ e − B (1+ w m + δ ) T β − πG h(cid:16) − T + p T ( T − (cid:17) Bβ ( β − T β − +3 B β T β − + 2 (cid:16) T − p T ( T −
1) + 2 (cid:17) ( ξ − T BβT β − + (cid:16) T − p T ( T −
1) + 1 (cid:17) k e − BT β (cid:21) (67)When we consider an interacting DBI-essence dark energy, we get decaying pattern in the V - φ plot for emergentand intermediate scenarios in the figures 10 and 12. However, from figure 11 we see an increasing plot of V - φ forfor interacting DBI-essence in the logamediate scenario. E. Hessence
Wei et al [32] proposed a novel non-canonical complex scalar field named “hessence” which plays the role ofquintom. In the hessence model the so-called internal motion ˙ θ where θ is the internal degree of freedom of7 Φ V Φ V Fig.10 Fig.11 Φ V Fig.12Figs.10-12 show the variations of V against DBI field φ in the emergent, logamediate and intermediate scenariosrespectively. Solid, dash and dotted lines represent k = − , +1 , hessence plays a phantom like role and the phantom divide transitions is also possible. The Lagrangian densityof the hessence is given by L h = 12 [( ∂ µ φ ) − φ ( ∂ µ θ ) ] − V ( φ ) . (68)The pressure and energy density for the hessence model are given by p h = 12 ( ˙ φ − φ ˙ θ ) − V ( φ ) , (69)8and ρ h = 12 ( ˙ φ − φ ˙ θ ) + V ( φ ) , (70)with Q = a φ ˙ θ = constant, (71)where Q is the total conserved charge, φ is the hessence scalar field and V is the corresponding potential.From above we get, ˙ φ − Q a φ = − (1 + w m ) ρ m + 14 πG (cid:18) − ˙ H + ξ − T H + ka (cid:19) , (72)and V = 12 ( w m − ρ m + 18 πG (cid:18) ˙ H + 3 H + 5( ξ − T H + 2 ka (cid:19) . (73) • For emergent scenario, we get the expressions for φ and V as˙ φ − Q a ( λ + e µT ) n φ = − (1 + w m ) ρ a − w m + δ )0 ( λ + e µT ) n (1+ w m + δ ) + 14 πG ( − nλµ e µT ( λ + e µT ) + ( ξ − nµe µt T ( λ + e µT ) + k a − ( λ + e µT ) n ) , (74)and V = ( w m − ρ a − w m + δ )0 λ + e µT ) n (1+ w m + δ ) + 18 πG ( nµ e µT ( λ + 3 ne µT )( λ + e µT ) + 5( ξ − nµe µT T ( λ + e µT ) + 2 k a − ( λ + e µT ) n ) . (75) • For logamediate scenario, we get the expressions for φ and V as˙ φ − Q e − A (ln T ) α φ = − (1 + w m ) ρ e − A (1+ w m + δ )(ln T ) α + 14 πG (cid:26) AαT (ln T ) α − (1 − α + ξ ln T ) + k e − A (ln T ) α (cid:27) (76)and V = ( w m − ρ e − A (1+ w m + δ )(ln T ) α + 18 πG (cid:20) AαT (ln T ) α − { α − ξ −
6) ln T + 3 Aα (ln T ) α } + 2 k e − A (ln T ) α (cid:21) . (77) • For intermediate scenario, we get the expressions for φ and V as˙ φ − Q e − BT β φ = − (1 + w m ) ρ e − B (1+ w m + δ ) T β + 14 πG n Bβ ( ξ − β ) T β − + k e − BT β o , (78)and V = ( w m − ρ e − B (1+ w m + δ ) T β + 18 πG h BβT β − (5 ξ + β + 3 BβT β ) + 2 k e − BT β i . (79)9 Φ V Φ V Fig.13 Fig.14 Φ V Fig.15Figs.13-15 show the variations of V against hessence field φ in the emergent, logamediate and intermediate scenariosrespectively. Red, green and blue lines represent k = − , +1 , For interacting hessence dark energy, figure 13 shows increase in the potential with scalar field and figures 14and 15 show decay in the potential with scalar field. This means the potential for interacting hessence increasesin the emergent universe and decays in logamediate and intermediate scenarios.
F. Dilaton Field
The energy density and pressure of the dilaton dark energy model are given by [27] ρ d = − X + 3 Ce λφ X , (80)and p d = − X + Ce λφ X , (81)0where φ is the dilaton scalar field having kinetic energy X = ˙ φ , λ is the characteristic length which governs allnon-gravitational interactions of the dilaton and C is a positive constant.We get, φ = Z (cid:20)
12 (3 w m − ρ m + 38 πG (cid:18) ˙ H + 2 H + 3( ξ − T H + ka (cid:19)(cid:21) dT . (82) • For emergent scenario, we have φ = Z " (3 w m − ρ a − w m + δ )0 λ + e µT ) n (1+ w m + δ ) + 38 πG ( nµ e µT ( λ + 2 ne µT )( λ + e µT ) + 3( ξ − nµe µT T ( λ + e µT ) + k a − ( λ + e µT ) n ) dT . (83) • For logamediate scenario, we get φ = Z (cid:20) πG (cid:26) AαT (ln T ) α − ( α − ξ −
4) ln T + 2 Aα (ln T ) α ) + k e − A (ln T ) α (cid:27) + 12 (3 w m − ρ e − A (1+ w m + δ )(ln T ) α (cid:21) dT (84) • For intermediate scenario, we get φ = Z (cid:20)
12 (3 w m − ρ e − B (1+ w m + δ ) T β + 38 πG n Bβ (3 ξ + β − BβT β ) T β − + k e − BT β o(cid:21) dT . (85)For interacting dilaton field, the scalar field φ always increases with cosmic time T irrespective of the scenarioof the universe we consider. This is displayed in figures 16, 17 and 18 for emergent, logamediate and intermediatescenarios respectively. G. Yangs-Mills Dark Energy
Recent studies suggest that Yang-Mills field can be considered as a useful candidate to describe the dark energyas in the normal scalar models the connection of field to particle physics models has not been clear so far andthe weak energy condition cannot be violated by the field. In the effective Yang Mills Condensate (YMC) darkenergy model, the effective Yang-Mills field Lagrangian is given by [34], L Y MC = 12 bF (ln (cid:12)(cid:12)(cid:12)(cid:12) FK (cid:12)(cid:12)(cid:12)(cid:12) − , (86)1 Φ Φ Fig.16 Fig.17 Φ Fig.18Figs.16-18 show the variations of dilaton field φ against time T in the emergent, logamediate and intermediate scenariosrespectively. Red, green and blue lines represent k = − , +1 , where K is the re-normalization scale of dimension of squared mass, F plays the role of the order parameter ofthe YMC where F is given by, F = − F aµν F aµν = E − B . The pure electric case we have, B = 0 i.e.F = E .From the above Lagrangian we can derive the energy density and the pressure of the YMC in the flat FRWspacetime as ρ y = 12 ( y + 1) bE , (87)and p y = 16 ( y − bE , (88)where y is defined as, y = ln (cid:12)(cid:12)(cid:12)(cid:12) E K (cid:12)(cid:12)(cid:12)(cid:12) . (89)2We get, E = (cid:20) b (3 w m − ρ m + 38 πGb (cid:18) ˙ H + 2 H + 3( ξ − T H + ka (cid:19)(cid:21) . (90) • For emergent scenario, we have E = " (3 w m − ρ a − w m + δ )0 b ( λ + e µT ) n (1+ w m + δ ) + 38 πbG ( nµ e µT ( λ + 2 ne µT )( λ + e µT ) + 3( ξ − nµe µT T ( λ + e µT ) + k a − ( λ + e µT ) n ) . (91) • For logamediate scenario, we get E = (cid:20) πbG (cid:26) AαT (ln T ) α − ( α − ξ −
4) ln T + 2 Aα (ln T ) α ) + k e − A (ln T ) α (cid:27) + 12 b (3 w m − ρ e − A (1+ w m + δ )(ln T ) α (cid:21) . (92) • For intermediate scenario, we get E = (cid:20) b (3 w m − ρ e − B (1+ w m + δ ) T β + 38 πbG n Bβ (3 ξ + β − BβT β ) T β − + k e − BT β o(cid:21) . (93)When we consider Yang-Mills dark energy, we find that E is always increasing with cosmic time T . This isdisplayed in figures 19, 20 and 21 for emergent, logamediate and intermediate scenarios respectively. V. CONCLUSION
This paper is dedicated to the study of reconstruction of scalar fields and their potentials in a newly developedmodel of Fractional Action Cosmology by Rami [3]. The fields that we used are quintessence, phantom, tachyonic,k-essence, DBI-essence, Hessence, dilaton field and Yang-Mills field. We assumed that these fields interact with thematter. These fields are various options to model dark energy which is varying in density and pressure, so calledvariable dark energy. Different field models possess various advantages and disadvantages. The reconstructionof the field potential involves solving the Friedmann equations in the FAC model with the standard energydensities and pressures of the fields, thereby solving for the field and the potential. For simplicity, we expressedthese complicated expressions explicitly in time dependent form. We plotted these expressions in various figuresthroughout the paper.In plotting the figures for various scenarios, we choose the following values: Emergent scenario: ξ = . n = 4, λ = 8, µ = . a = . G = 1 (all DE models); Logamediate: ξ = . α = 3, A = 5, G = 1 (all DE models);3 E E Fig.19 Fig.20 E Fig.21Figs.19-21 show the variations of E against time T in the emergent, logamediate and intermediate scenarios respectively.Red, green and blue lines represent k = − , +1 , Intermediate: ξ = . β = . B = 2, G = 1 (all DE models). Moreover in all cases δ = . w m = .
01. Infigures 1 to 3, we show the variations of V against φ in the emergent, logamediate and intermediate scenariosrespectively for phantom and quintessence field. In the first two cases, the potential function is a decreasingfunction of the field. For the quintessence field, the potential is almost constant while for the phantom field, thepotential increases for different field values. Figures (4-6) show the variations of V against φ in the emergent,logamediate and intermediate scenarios respectively for the tachyonic field. In figure 4, the V - φ plot for normaltachyon and phantom tachyon models of dark energy is presented for emergent scenario of the universe. Potentialof normal tachyon exhibits decaying pattern. However, it shows increasing pattern for phantom tachyonic field φ . It happens irrespective of the curvature of the universe. In the logamediate scenario (figure 5) the potentialsfor normal tachyon and phantom tachyon exhibit increasing and decreasing behavior respectively with increasein the scalar field φ . From figure 6 we see a continuous decay in the potential for normal tachyonic field inthe intermediate scenario. However, in this scenario, the behavior of the potential varies with the curvature of4the universe characterized by interacting phantom tachyonic field. For k = − ,
1, the potential increases withphantom tachyonic field and for k = 0, it decays after increasing initially.Similarly figures (7-9) show the reconstructed potentials for the k-essence field. We have seen that for interactingk-essence the potential V always decreases with increase in the scalar field φ in all of the three scenarios and ithappens for open, closed and flat universes. When we consider an interacting DBI-essence dark energy, we getdecaying pattern in the V - φ plot for emergent and intermediate scenarios in the figures 10 and 12. However,from figure 11 we see an increasing plot of V - φ for for interacting DBI-essence in the logamediate scenario. Forinteracting hessence dark energy, figures 13 shows increase in the potential with scalar field and figures 14 and 15show decay in the potential with scalar field. This means the potential for interacting hessence increases in theemergent universe and decays in logamediate and intermediate scenarios. Figures (16-18) discuss the dilaton fieldwhile figures (19-21) show the behavior of the Yang-Mills field in the FAC. For interacting dilaton field, the scalarfield φ always increases with cosmic time T irrespective of the scenario of the universe and when we considerYang-Mills dark energy, we find that E in always increasing with cosmic time T . [1] I. Podlubny, An Introduction to Fractional Derivatives, Fractional Differential Equations, to methods of their solutionand some of their Applications, (Academic Press, New York, 1999);R. Hilfer, Editor, Applications of Fractional Calculus in Physics, (World Scientific Publishing, Singapore, 2000)[2] M. Robert, arXiv:0909.1171 [gr-qc];V. K. Shchigolev, arXiv:1011.3304v1 [gr-qc].[3] R.A. EL.Nabulsi, Romm. Rep. Phys. 59 (2007) 763;R.A. EL.Nabulsi, Fizika B 19 (2010) 103.[4] R.A. EL.Nabulsi, Commun. Theor. Phys 54 (2010) 16.[5] M. Jamil, D. Momeni, M.A. Rashid, arXiv:1106.2974[6] M. U. Farooq, M. Jamil, U Debnath, arXiv:1104.3983;U. Debnath, M. Jamil, arXiv:1102.1632;K. Karami, M. S. Khaledian, M. Jamil, Phys. Scr. 83 (2011) 025901;M. R. Setare, M. Jamil, Europhys. Lett.92 (2010) 49003;A. Sheykhi, M. Jamil, Phys. Lett. B 694 (2011) 284;M. Jamil, K. Karami, A. Sheykhi, arXiv:1005.0123;M. U. Farooq, M.A. Rashid, M. Jamil, Int. J. Theor. Phys. 49 (2010) 2278;M. Jamil, A. Sheykhi, M. U. Farooq, Int. J. Mod. Phys. D 19 (2010) 1831;M. R. Setare, E. N. Saridakis, Phys. Lett. B 670 (2008) 1.[7] J. D. Barrow, N. J. Nunes, Phys. Rev. D 76 (2007) 043501;C. Campuzano et al, Phys. Rev. D 80 (2009) 123531; B. C. Paul, P. Thakur, S. Ghose, arXiv:1004.4256;G. F.R. Ellis, J. Murugan, C. G. Tsagas, Class. Quant. Grav. 21 (2004) 233;R. B. Laughlin, Int. J. Mod. Phys. A 18 (2003) 831;P. B. Khatua, U. Debnath, Int. J. Theor. Phys. 50 (2011) 799.[8] S. Capozziello, S. Nesseris, L. Perivolaropoulos, JCAP 0712 (2007) 009.[9] S. Capozziello, E. Piedipalumbo, C. Rubano, P. Scudellaro, Phys. Rev. D 80 (2009) 104030.[10] J-H. He, B. Wang, P. Zhang, Phys. Rev. D 80 (2009) 063530.[11] J-H. He, B. Wang, Y.P. Jing, JCAP 0907 (2009) 030.[12] L. Amendola, Phys. Rev. Lett. 93 (2004) 181102.[13] K.A. Bronikov, Acta Phys. Polon. B 4 (1973) 251.[14] H.G. Ellis, J. Math. Phys. 14 (1973) 104.[15] C.A. Pic´on, Phys. Rev. D 65 (2002) 104010.[16] F. Rahaman et al, Phys. Scr. 76 (2007) 56.[17] P.K. Kuhfittig, Class. Quantum Grav. 23 (2006) 5853.[18] E. Babichev et al, Phys. Rev. Lett. 93 (2004) 021102.[19] S. Nesseris and L. Perivolaropoulos, Phys. Rev. D 70 (2004) 123529.[20] D.F. Mota and C. van de Bruck, Astron. Astrophs. 421 (2004) 71.[21] D.F. Mota and J.D. Barrow, Mon. Not. Roy. Astron. Soc. 358 (2005) 601.[22] E. Babichev, S. Chernov, V. Dokuchaev and Y. Eroshenko, arXiv:0806.0916[gr-qc].[23] E. Babichev, S. Chernov, V. Dokuchaev and Y. Eroshenko, Phys. Rev. D 78 (2008) 104027.[24] R. R. Caldwell et al, Phys. Rev. Lett. 91 (2003) 071301.[25] L. Xu, JCAP 09 (2009) 016.[26] S. Chattopadhyay, U. Debnath, arXiv:1102.0707v1 [physics.gen-ph];S. Mukherjee, B. C. Paul, N. K. Dadhich, S. D. Maharaj and A. Beesham (2006) Class. Quantum Grav. 23 6927.[27] E.J. Copeland, M. Sami, S. Tsujikawa, Int. J. Mod. Phys. D , 1753 (2006).[28] G. W. Gibbons, Phys. Lett. B 537 (2002) 1.[29] A. Mazumdar, S. Panda and A. Perez-Lorenzana, Nucl. Phys. B 614, 101 (2001);A. Feinstein, Phys. Rev. D 66, 063511 (2002);Y. S. Piao, R. G. Cai, X. M. Zhang and Y. Z. Zhang, Phys. Rev. D 66, 121301 (2002).[30] T. Padmanabhan, Phys. Rev. D 66, 021301 (2002);J.S. Bagla, H.K.Jassal, T. Padmanabhan, Phys. Rev. D 67 (2003) 063504;Z. K. Guo and Y. Z. Zhang, JCAP 0408, 010 (2004);E. J. Copeland, M. R. Garousi, M. Sami and S. Tsujikawa, Phys. Rev. D 71, 043003 (2005).[31] Armendariz-Picon C., Damour T., Mukhanov V.F., Phys. Lett. B 458, 209 (1999);Armendariz-Picon C., Mukhanov V.F., Steinhardt P.J., Phys. Rev. D 63, 103510 (2001);Armendariz-Picon C., Mukhanov V.F., Steinhardt P.J., Phys. Rev. Lett. 85, 4438 (2000);Chiba T., Okabe T., Yamaguchi M., Phys. Rev. D 62, 023511 (2000);6