Tachyonic matter cosmology with exponential and hyperbolic potentials
aa r X i v : . [ h e p - t h ] S e p Tachyonic matter cosmology with exponentialand hyperbolic potentials
B. Pourhassan a, ∗ , and J. Naji b, † a, School of Physics, Damghan University, Damghan, Iran b, Physics Department, Ilam University, Ilam, IranP.O. Box 69315-516, Ilam, Iran
September 9, 2018
Abstract
In this paper we consider tachyonic matter in spatially flat FRW universe, and obtainbehavior of some important cosmological parameters for two special cases of poten-tials. First we assume the exponential potential and then consider hyperbolic cosinetype potential. In both cases we obtain behavior of the Hubble, deceleration and EoSparameters. Comparison with observational data suggest the model with hyperboliccosine type scalar-field potentials has good model to describe universe.
Keywords:
Dark Energy, Cosmology.
PACs:
Recent cosmological observations [1-3] confirmed that our universe expanded while accel-erated. Dark energy models come to explain nature of this accelerating expansion. Thesimplest model to describe the dark energy is the cosmological constant which has two fa-mous problems as the fine-tuning and cosmic coincidence [4]. Hence, another models of thedark energy proposed such as Chaplygin gas model and its extensions [5-12].There are also other interesting models such as quintessence [13-15], phantom [16, 17] andquintom [18-20]. These are based on the scalar fields which plays an important role in cos-mology. One of the first major mechanisms where scalar fields are thought to be responsibleis the inflationary scenario [21, 22]. Single scalar field is the underlying dynamics in many ∗ Email: [email protected] † Email: [email protected]
Tachyon field described by the following energy density and pressure respectively [28, 29], ρ T = V ( T ) p − ˙ T . (1)and, p T = − V ( T ) p − ˙ T , (2)Therefore, the equation of state (EoS) of tachyon field obtained as follow, ω T = p T ρ T = ˙ T − . (3)Also, tachyon potential is given by, V ( T ) = √− p T ρ T . (4)The effective action of the tachyon field which minimally coupled to the gravitational fieldin the Born-Infeld form is given by, S = Z d x √− g (cid:20) R V ( T ) p g µν ∂ µ T ∂ ν T (cid:21) , (5)2here R is the curvature scalar, T is the tachyon field, V ( T ) is the self-interaction potential,and we used 8 πG = k = c = 1 units.The tachyonic matter in this model described by a fluid with EoS parameter − ≤ ω T ≤ The spatially flat Friedmann-Robertson-Walker (FRW) Universe is described by the followingmetric, ds = dt − a ( t ) ( dr + r d Ω ) , (6)where d Ω = dθ + sin θdφ . Also, a ( t ) represents time-dependent scale factor. Therefore,field equations obtained as follows, 3 H = ρ T , (7)and 2 ˙ H + 3 H = − p T , (8)where H = ˙ a/a is Hubble expansion parameter, also ρ T and p T are given by the equations(1) and (2) respectively. Then, the evolution equation for the scalar field is given by [30],¨ T − ˙ T + 3 H ˙ T + V ′ ( T ) V ( T ) = 0 , (9)where the over dot denotes the derivative with respect to the time-coordinate t , while theprime denotes the derivative with respect to the scalar field T , respectively.Combining the equations (1), (2), (7) and (8) we can obtain the following equation describingevolution of Hubble expansion parameter,˙ H = V ( T )6 H − H . (10)Also, combining the equations (1), (7) and (9) we can obtain the following equation describingevolution of the scalar field,¨ T − ˙ T + s V ( T ) p − ˙ T ˙ T + dV ( T ) dT V ( T ) − = 0 . (11)The deceleration parameter q is an important observational quantity which is given by, q = − ˙ HH − T − . (12)In the case of constant T we yield q = − The exponential potential scalar field
The number of known exact solutions for cosmological models based on scalar fields is ratherlimited. One of such models is the flat Friedmann universe filled with a minimally coupledscalar field with exponential potential. This solution describes a power-law expansion of theuniverse. The tachyon potential is given by [31], V = V e αT , (13)where V and α (tachyon mass) are arbitrary constants. It order to obey V → T → ∞ we need to choose negative α . In that case we have a condition as, V ′ V = const. (14)These types of exponential potential are important in four-dimensional effective Kaluza-Kleintype theories from compactification of the higher-dimensional supergravity or superstringtheories [32] and may arise due to non-perturbative effects such as gaugino condensation[33]. In that case, the equation (11) rewritten as the following form,¨ T − ˙ T + s V e αT p − ˙ T ˙ T + α = 0 . (15)In the Ref. [30], above equation considered and behavior of T in terms of ˙ T obtained to seeattractor behavior which is important in inflationary scenario. Here we would like to obtainevolution of tachyon field and try to obtain time dependent T . Numerically, we find timeevolution of the tachyon field in the Fig. 1. We can see that it is increasing function of time.Also, it is find that increasing | α | increases value of the tachyon field.Figure 1: Tachyon field in terms of t for V = 1. α = − . α = − α = − α = − T (0) = ˙ T (0) = 0.4t is general behavior of the tachyon field. In order to obtain other important cosmologicalparameter we need an analytical expression of the tachyon field.We suggest the following form of time dependent tachyon field, T = At + B + Ce − βt , (16)where A , B , C and β are arbitrary constant. We can fix these constant to obtain behaviorcorresponding to Fig. 1. For example choosing B = − . C = 0 . β = 2 and A = 1 or A = 0 . H = √ s V e α ( At + B + Ce − βt ) p − ( A − Cβe − βt ) . (17)In the Fig. 2 we can see behavior of the Hubble expansion parameter in terms of cosmictime. We can see that | α | > α .Figure 2: Hubble expansion parameter in terms of t for V = 1, A = 1, B = − . C = 0 . β = 2. α = − . α = − α = − α = − q = − (cid:0) A − Cβe − βt (cid:1) . (18)It is easy to find that (selecting constant like C = 0 . β = 2 and A = 0 .
4) the decelerationparameter is decreasing function of time which yields to -1 at the late time. Correspondingto the value of the constant C ( | C | > ω = − (cid:0) A − Cβe − βt (cid:1) , (19)which guarantee that ω ≥ − | α | the tachyon potential behaves as a constant. Generally we have a maximumfor the potential and it yields to zero at the late time.Figure 3: Tachyon potential in terms of t for V = 1, A = 1, B = − . C = 0 . β = 2. α = − . α = − α = − α = − It is also possible to consider the hyperbolic cosine type potential given by, V = V cosh ν [ γ ( T − T )] , (20)where V , ν , γ and T are arbitrary constants. T is initial value of tachyon field which maybe considered zero as before, however we assume small initial value to avoid divergency ofsolution. In that case the condition (14) is no longer valid. Therefore, the equation (11)rewritten as the following form,¨ T − ˙ T + s V cosh ν [ γ ( T − T )] p − ˙ T ˙ T + dV ( T ) dT V ( T ) − = 0 . (21)General solution of the equation (21) presented in Fig. 4 numerically. Similar to the previouscase we can see that the scalar field is increasing function of time for negative ν . It isillustrated that the scalar field yields to a constant at the late time for ν = 0.6igure 4: Tachyon field in terms of t for T = 0 . V = 1, and γ = 1. ν = 1 (dotted green), ν = 0 (dashed orange), ν = − . ν = − φ ≪ γ ≪
1, then the explicit form of the scalar field will be available, T = C e − √ (3 √ V − √ V − V γ ) t + C e − √ (3 √ V + √ V − V γ ) t , (22)where C and C are integration constants. Using this approximation we can obtain othercosmological parameters such as the deceleration parameter. It is easy to find that the de-celeration parameter is decreasing function of time yields to -1 at the late time similar tothe previous model. It is also possible to see accelerating to decelerating phase transition.Also we can obtain equation of state parameter and find that ω ≥ −
1. It is clear that theEoS parameter yields to -1 at the late time.We can perform numerical analysis on the Hubble expansion parameter. In the Fig. 5 wedraw Hubble expansion parameter versus cosmic time. We can see that it is decreasingfunction of time which yields to a constant value at the late time which is expected. It isillustrated that the value of the Hubble parameter decreased by γ .Finally, we can see behavior of the scalar potential in the Fig. 6. For the very smallvalues of parameter γ the scalar potential behaves as a constant. Generally, the potential isdecreasing function of time similar the previous model.We found reasonable behavior of our models, however we have to add ordinary matter tothe models and compare our results with appropriate observational data such as H ( z ) data[34]. In order to have comparison with observational data we should consider all matter contri-bution in our model. In that case we have the following conservation equation,˙ ρ + 3 H ( p + ρ ) = 0 , (23)7igure 5: Hubble expansion parameter in terms of t for V = 1, C = 5, C = 5, γ = 0 . γ = 0 . γ = 0 .
01 (solid line).Figure 6: Potential in terms of t for V = 1, C = 5, C = 5, γ = 0 . γ = 0 . γ = 0 .
075 (solid line).where ρ = ρ φ + ρ m and p = p φ + p m , with matter density ρ m and pressure p m . Now, theequations (7) and (8) extended to the following relations,3 H = ρ, (24)and, 2 ˙ H = − p − ρ. (25)We assume non-interacting case, therefore conservation equation (23) separates as follow,˙ ρ T + 3 H ( p T + ρ T ) = 0 , (26)and ˙ ρ m + 3 H ( p m + ρ m ) = 0 , (27)with ω m = p m /ρ m as EoS of matter. An important parameter to compare with observationaldata is H ( z ). We will calculate H ( z ) for two different cases of exponential and Hyperbolicscalar potential. 8 .1 Exponential potential In this case we use relation (13) and investigate behavior of Hubble expansion parameterversus redshift. We can see good agreement with observational data for 0 . ≤ | α | ≤ . . < | α | < . z for V = 1, C = 5, B = 5, α = − . α = − . α = − .
01 (solid line). Big dots denote observationaldata.
In this case we use relation (20) and investigate behavior of Hubble expansion parameterversus redshift. We can see good agreement with observational data for 0 . ≤ γ ≤ . ν = − In this work, we considered tachyon scalar field in flat FRW universe and proposed two differ-ent models based on tachyon field potential. In the first model, we assumed the exponentialpotential scalar field and solved the equation describing the evolution of the tachyon fieldnumerically. We found that the tachyon field in this model has totally positive value andincreased by time. It means that at the late time the scalar field takes infinite value. In orderto obtain other cosmological parameters we fit tachyon field and found explicit expressionfor that, then deceleration and EoS parameters obtained. Also, we discussed numericallyabout tachyon potential and Hubble expansion parameter. We found that the decelerationparameter as well as EoS parameter is decreasing function of time and yields to -1 at thelate time. We have shown that the Hubble expansion parameter is decreasing function of9igure 8: Hubble expansion parameter in terms of z for V = 1, C = 5, C = 5, γ = 0 . γ = 0 . γ = 0 .