Warm Modified Chaplygin Gas Shaft Inflation
aa r X i v : . [ g r- q c ] F e b Warm Modified Chaplygin Gas ShaftInflation
Abdul Jawad ∗ , Amara Ilyas † and Shamaila Rani ‡ Department of Mathematics, COMSATS Institute ofInformation Technology, Lahore-54000, Pakistan.
Abstract
In this paper, we examine the possible realization of a new familyof inflation called “shaft inflation” by assuming the modified Chaply-gin gas model and tachyon scalar field. We also consider the specialform of dissipative coefficient as Γ = a T φ and calculate the variousinflationary parameters in the scenario of strong and weak dissipativeregimes. In order to examine the behavior of inflationary parameters,the planes of n s − φ, n s − r and n s − α s (where n s , α s , r and φ represent spectral index, its running, tensor to scalar ratio and scalarfield respectively) are being developed which lead to the constraints: r < . n s = 0 . ± .
025 and α s = − . ± . Keywords:
Inflationary Cosmology; Tachyon field model; Modified Chap-lygin Gas; Shaft potential; Inflationary Parameters.
PACS:
Inflation is the most acceptable paradigm that describes the physics of thevery early universe. Besides of solving most of the shortcomings of the hot ∗ [email protected]; [email protected] † amara [email protected] ‡ [email protected], [email protected] R ≡ Γ / H . The weak dissipative regime for warm inflation is for R ≪ R ≫
1, it is the strong dissipative regime. Following Refs.[25, 26],a general parametrization of the dissipative coefficient depending on boththe temperature of the thermal bath T and the inflaton scalar field φ can bewritten as Γ( T, φ ) = C φ T m φ m − , (1)where the parameter C φ is related with the dissipative microscopic dynamics,the exponent m is an integer whose value is depends on the specifics of themodel construction for warm inflation and on the temperature regime of thethermal bath. Typically, it is found that m = 3 (low temperature), m = 12high temperature) or m = 0 (constant dissipation).Later on, Linde [27] introduced the chaotic inflation by realizing that theinitial conditions for scaler field driving inflation which may help in solvingthe persisting inflationary problems. A plethora of works in the subject ofwarm inflation along with chaotic potential have been done. For instance,Herrera [28] investigated the warm inflation in the presence of chaotic poten-tial in loop quantum cosmology and found consistencies of results with obser-vational data. Del campo and Herrera [29] discussed the warm inflationarymodel in the presence of standard scalar field, dissipation coefficient of theform Γ ∝ φ n and generalized Chaplygin gas (GCG) and extract various infla-tionary parameters. Setare and Kamali investigated warm tachyon inflationby assuming intermediate [30] and logamediate scenarios [31]. Bastero-Gill etal. [32] obtained the expressions for the dissipation coefficient in supersym-metric (SUSY) models and their result provides possibilities for realizationof warm inflation in SUSY field theories. Bastero-Gill et al. [33] have alsoexplored inflation by assuming the quartic potential. Herrera et al. [34] stud-ied intermediate inflation in the context of GCG using standard and tachyonscalar field.Panotopoulos and Vidaela [35] discussed the warm inflation by assumingquartic potential and decay rate proportional to temperature and found thatthe results of inflationary parameters are compatible with the latest Planckdata. Moreover, many authors have investigated the warm inflation in variousalternative as well as modified theories of gravity [36, 37]. Recently, a newfamily of inflation models is being developed named as shaft inflation [38].The idea of this inflation was that the inflationary flatness is effected byshaft i.e; when the scalar field found itself nearest to one of them, it slow-rolls inside the shaft, until inflation ends and gives way to the hot big bangcosmology. The generalized form of shaft potential is V ( φ ) = M p φ n − ( φ n + m n ) − n , (2)where M p , m, n are massless constants.In the present, we discuss the warm inflation by assuming shaft potential,modified Chaplygin gas model and tachyon scalar field. We will extract theinflationary parameters. The formate of the paper is as follows: In thenext section, we will discuss the detailed inflationary scenario with tachyonfield and generalized dissipative coefficient. Section and contains the3nformation about disordered parameters for shaft potential in strong andweak dissipative regimes, respectively. In section , the results are given insummarized form. The universe undergoes an accelerated expansion of the universe The respon-sible for this acceleration of the late expansion is an exotic component havinga negative pressure, usually known as dark energy (DE). Several models havebeen already proposed to be DE candidates, such as cosmological constant[39], quintessence [40]-[42], k-essence [43]-[45], tachyon [46]-[48], phantom[49]-[51], Chaplygin gas [52], holographic dark energy [53], among others inorder to modify the matter sector of the gravitational action. Despite theplenty of models, the nature of the dark sector of the universe, i.e. DE anddark matter, is still unknown. There exists another way of understandingthe observed universe in which dark matter and DE are described by a singleunified component. Particularly, the Chaplygin gas [52] achieves the unifi-cation of DE and dark matter. In this sense, the Chaplygin gas behaves asa pressureless matter at the early times and like a cosmological constant atlate times. The original Chaplygin gas is characterized by an exotic equationof state with negative pressure p = − βρ , (3)whit β being a constant parameter. The original Chaplygin gas has beenextended to the so-called generalized Chaplygin gas (GCG) with the followingequation of state [54] p gcg = − βρ σ , (4)with 0 ≤ σ ≤
1. For the particular case λ = 1, the original Chaplygin gas isrecovered. The main motivation for studying this kind of model comes fromstring theory. The Chaplygin gas emerges as an effective fluid associatedwith D-branes which may be obtained from the Born-Infeld action [55]. Atbackground level, the GCG is able to describe the cosmological dynamics [56],however the model presents serious issues at perturbative level [57]. Thus, amodification to the GCG, results in the modified Chaplygin gas (MCG) with4n equation of state given by [58] p = ωρ − βρ σ , (5)where ω is a constant parameters with 0 ≤ σ ≤
1, is suitable to describe theevolution of the universe [59, 60] which is also consistent with perturbativestudy [61].The energy conservation equation for MCG model turns out to be ρ mcg = (cid:18) β ω + υa σ )(1+ ω ) (cid:19) σ , (6)where υ is constant of integration. In spatially flat FRW model, Friedmannequation described as H = 13 M p ( ρ m + ρ γ ) , (7)where ρ m is the energy density of matter field and ρ γ is the energy densityof radiation field. The warm MCG model modifies first Friedmann equationwhich has been used in Eq.(7) reduces to H = 13 M p (cid:20)(cid:18) β ω + υρ (1+ σ )(1+ ω ) φ (cid:19) σ + ρ γ (cid:21) , (8)which is named as Chaplygin gas inspired inflation. The energy density andpressure of tachyon scalar field are defined as follows [62], ρ φ = V ( φ ) q − ˙ φ , p φ = − V ( φ ) q − ˙ φ . (9)The inflaton and imperfect fluid energy densities according to the Eq. (9)are conserved as ˙ ρ φ + 3 H ( ρ φ + p φ ) = − Γ ˙ φ , (10)˙ ρ γ + 4 Hρ γ = Γ ˙ φ , (11)where Γ is the dissipation factor that evaluates the rate of decay of ρ φ into ρ γ . It is also important to note that this decay rate can be used as a function5f the temperature of the thermal bath and the scalar field, i.e., Γ( T, φ ) ora function of only temperature of thermal bath Γ( T ), or a function of scalarfield only Γ( φ ), or simply a constant.The second law of thermodynamics indicates that Γ must be positive, sothe inflaton energy density decompose into radiation density. The secondconservation equation is given by¨ φ − ˙ φ + 3 H ˙ φ + V ′ ( φ ) V ( φ ) = − Γ ˙ φV q − ˙ φ , ⇒ ¨ φ + (cid:18) H + Γ V (cid:19) ˙ φ = − V ′ ( φ ) V ( φ ) , ˙ φ ≪ ⇒ H (1 + R ) ˙ φ = − V ′ ( φ ) V ( φ ) , where ¨ φ ≪ (3 H + Γ V ) ˙ φ , (12)where R = Γ3 HV . In weak dissipative epoch, R ≪ ≪ H while R ≫ ρ φ ≈ V ( φ ), slow-roll limit, V ( φ ) ≫ ˙ φ , (3 H + Γ) ˙ φ ≫ ¨ φ , quasi-stable decay of ρ φ into ρ γ , 4 Hρ γ ≫ ˙ ρ γ andΓ ˙ φ ≫ ˙ ρ γ . As we know that the energy density of scalar field is much greaterthan the energy density of radiations but also at the same time, the energycan be larger than the expansion rate with ρ γ > H . This is approximatelyequal to T > H by considering thermalization, which is true condition totake place in warm inflation. With the help of all these limits Eqs. (7), (11)and (12) become H = 13 M p (cid:18) β ω + υρ (1+ σ )(1+ ω ) φ (cid:19) σ , (13)4 Hρ γ = Γ ˙ φ , (14)3 H (1 + R ) ˙ φ = − V ′ ( φ ) V ( φ ) , (15)where prime represents the derivative with respect to φ .The energy density of radiation can be used as C γ T when we have takenthe thermalization. Here C γ = π g ∗ /
30, where g ∗ shows the degree of free-dom. This expression gives the value as C γ ≃
70 with g ∗ = 228 .
75. Thetemperature of thermal bath can be obtained by merging the Eqs. (14) and615) T = (cid:18) Γ V ′ C γ H V (1 + R ) (cid:19) , (16)where Γ = a T q φ q − , which is the general form of dissipative coefficient, while a and q are constant parameters associated with dissipative microscopicdynamics. The consequences of radiation are studied during inflation throughthis kind of dynamic which is suggested first time in warm inflation withtheoretical basis of supersymmetry (SUSY) [63, 64].A set of dimensionless slow-roll parameters must be satisfied for the oc-currence of warm inflation which are defined in the form of Hubble parameteras [65] ǫ = − ˙ HH , η = − ¨ HH ˙ H .
The slow-roll parameters can also be deduced in the form of scalar field andthermalization according to the tachyon field along with modified Chaplygingas, which are defined as ǫ = υ (1 + ω ) M p V (1+ σ )(1+ ω ) − V ′ R ) (cid:0) β ω + υV (1+ σ )(1+ ω ) (cid:1) σ ,η = M p (1 + R ) (cid:0) β ω + υV (1+ σ )(1+ ω ) (cid:1) σ (cid:20) ((1 + σ )(1 + ω ) − V ′ V + 2 V ′′ V (cid:21) − σ ) ǫ . We can describe number of e-folds in terms of Hubble parameter as well asinflaton N ( φ ) = Z t f t i Hdt = Z φ f φ i H ˙ φ dφ , (17)where φ i and φ f can be calculated with the help of first and second slow-rollparametric conditions, i.e., ǫ = 1 + R and | η | = 1 + R .Next, we will calculate some inflationary parameters such as tensor andscalar power spectra ( P R , P g ), tensor and scalar spectral indices ( n R , n s ). Theform of scalar power spectrum can be estimated as P R ( k ) ≡ δ H ( k ), wheredensity disorders δ H ( k ) ≡ k F ( T R )2 π and k F = √ Γ H . However, the amplitudeof the tensor and scalar power spectrum of the curvature perturbation are7iven by P R ≃ (cid:18) H π (cid:19) (cid:18) H V ′ (cid:19) (cid:18) TH (cid:19) (1 + R ) , P g ≃ κ (cid:18) H π (cid:19) . (18)The tensor-to-scalar ratio can be computed by using the relation r = P R P g .However, the spectral index and running of spectral index are defined as [66] n s = 1 + d ln P R d ln k , α s = dn s d ln k . (19)Here, the interval in wave number k is referred to the number of e-folds N ,through the expression as d ln k = − dN . (20)In the following, we will evaluate the inflationary parameters for weak andstrong dissipative regimes. The special case of shaft potential where n = 2 is considered for which Eq.(2)takes the form as V ( φ ) = M p φ ( φ + m ) . The temperature of the radiation forpresent model with the help of shaft potential, Eq. (16) takes the followingform T = m M p s (cid:18) β ω + υ (cid:16) M p φ m + φ (cid:17) (1+ ω )(1+ σ ) (cid:19) − σ a ( m + φ ) (cid:16) M p − M p φ m + φ (cid:17) C γ . (21)The number of e-folds can be calculated by Eq. (17) with ˙ φ = − V ′ HV R forstrong regime as N = a / m M p Z φ f φ i vuut β ω + υ (cid:18) M p φ m + φ (cid:19) (1+ ω )(1+ σ ) ! σ (cid:18) ( m + φ ) C / γ φ (cid:19) dφ . (22)For the strong epoch, φ i and φ f can be described by considering ǫ = R and | η | = R respectively. The power spectrum attains the value from Eq.(18) asfollows P R = a / (cid:16) m M p m + φ (cid:17) / / π C / γ (cid:18) β ω + υ (cid:16) M p φ g + φ (cid:17) (1+ ω )(1+ σ ) (cid:19) σ ) M p / × (cid:16) m M p φ ( m + φ ) (cid:17) / m / (cid:16) M p φ m + φ (cid:17) / . The scalar power spectrum is given by P g = 29 πM p (cid:18) β ω + υV (1+ σ )(1+ ω ) (cid:19) σ . (23)The tensor-to-scalar ratio can be found by using expression (23) which yields r = 32 m M p υφ (cid:16) m M p φ ( m + φ ) (cid:17) / (cid:16) M p φ m + φ (cid:17) + ω + σ + ωσ / (1 + ω ) a C / γ (cid:16) a m M p m + φ (cid:17) / (cid:18) υ (cid:16) M p φ m + φ (cid:17) (1+ σ )(1+ ω ) + β ω (cid:19) σ ) × (cid:18) υ (cid:16) M p φ m + φ (cid:17) (1+ σ )(1+ ω ) + β ω (cid:19) σ M p / . Figure shows the plot of tensor-to-scalar ratio versus spectral index withinstrong regime. This ratio is being plotted for three different values of m withthe condition m < φ . Red line has been plotted for m = 0 .
2, green dashedline for m = 0 . m = 0 .
9. According to the plot,ratio is not satisfied with spectral index when m = 0 . .3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0000.0010.0020.0030.0040.0050.006 n s r Figure 1: Plot of tensor-to-scalar ratio verses spectral index in strong epochwith a = 2 × .However, the spectral index and it’s running attained the values by usingEqs. (19) and (20) as n s = 1 − / φ C / γ M p (cid:16) m a M p m + φ (cid:17) / m + φ ) a (cid:16) m φM p ( m + φ ) (cid:17) / (cid:18) β ω + υ (cid:16) φ M p m + φ (cid:17) (1+ σ )(1+ ω ) (cid:19) × (cid:18) β ( m +23 φ ) ω − υ ( − φ + m ( −
39 + 10 ω )) (cid:16) φ M p m + φ (cid:17) (1+ σ )(1+ ω ) (cid:19) φ M p (cid:18) β ω + υ (cid:18) φ M pm φ (cid:19) (1+ σ )(1+ ω ) (cid:19) σ m + φ / . We plot spectral index n s versus scalar field φ in Figure and notice thatred line which represents the behavior of spectral index with respect to φ for m = 0 . φ to reach in the range of spectralindex. The other two different values i.e., m = 0 . m = 0 . φ ∈ [1 , r ) remains less than 0 . . < n s < .
97 in the strong dissipativeepoch. 10
10 20 30 40 500.40.50.60.70.80.91.0
Figure 2: Plot of spectral index number w.r.t inflaton in strong epoch with a = 2 × .The running of spectral index becomes α s = − (cid:20) / φC / γ (cid:18) m φM p ( m + φ ) (cid:19) / (cid:18) m a M p m + φ (cid:19) / (cid:18) (cid:18) β ω (cid:19) (cid:0) m φ + 23 φ (cid:1) + 2 υβ ω (cid:0) φ + m φ (214 − ω ) + 7 m (1 + ω )(12 + 5 ω + 5 σ × (1 + ω )) (cid:1) (cid:18) φ M p m + φ (cid:19) (1+ σ )(1+ ω ) + υ (cid:0) φ + m φ (197 − ω ) − m × (1 + ω )( −
39 + 10 ω ) (cid:1) (cid:18) φ M p m + φ (cid:19) σ )(1+ ω ) (cid:19)(cid:21) (cid:0) m a M p (cid:1) − (cid:18) β ω + υ (cid:18) φ M p m + φ (cid:19) (1+ σ )(1+ ω ) (cid:19) − φ M p (cid:18) β ω + υ (cid:16) φ M p m + φ (cid:17) (1+ σ )(1+ ω ) (cid:19) σ m + φ − / . The plot of running of spectral index with respect to scalar field is shown inFigure . The suggested values for running of spectral index by WMAP7 [67,68] and WMAP9 [69] are approximately equal to − . ± .
019 and − . ± . m = 0 . m = 0 .
9. However for m = 0 .
2, the plotof running of spectral index is not compatible with the required range ofspectral index. 11igure 3: Plot for running of spectral index versus spectral index in strongepoch with a = 2 × . Here we study the tachyon model in weak epoch ( R ≪ T = a m M p (cid:18) β ω + υ (cid:16) φ M p m + φ (cid:17) (1+ σ )(1+ ω ) (cid:19) − σ ) √ φ ( m + φ ) C γ . (24)The number of e-folds can be calculated by Eq. (17) with ˙ φ = − V ′ HV as N = 12 m M p Z φ f φ i β ω + υ (cid:18) φ M p m + φ (cid:19) (1+ σ )(1+ ω ) ! σ (cid:0) m + φ (cid:1) φdφ , where φ i and φ f can be found by taking ǫ = 1 and | η | = 1 respectively.The power spectrum attains the value from Eq.(18) as P r = ( m + φ ) a m π φ C γ M p (cid:20) M p − φ M p m + φ (cid:21) " β ω + υ (cid:18) φ M p m + φ (cid:19) (1+ σ )(1+ ω ) σ . The scalar power spectrum remains same as for the strong regime. Thetensor-to-scalar ratio is obtained by using expressions of power spectrum12 .90 0.92 0.94 0.96 0.98 1.0002. 10
4. 10
6. 10
8. 10
1. 10 n s r Figure 4: Plot of tensor-to-scalar ratio verses spectral index in weak epochwith a = 2 × .and scalar spectrum, which is given by r = 8 M p (1 + ω ) υ (cid:16) M p φ m + φ (cid:17) − (1+ ω )(1+ σ ) (cid:18) M p φm + φ − M p φ ( m + φ ) (cid:19) × β ω + υ (cid:18) M p φ m + φ (cid:19) (1+ ω )(1+ σ ) ! − − σ . Figure shows the plot of tensor-to-scalar ratio versus spectral index withinweak regime. Tensor-to-scalar ratio is being plotted for three different valuesof m with the condition m < φ . Red line has been plotted for m = 0 .
2, greendashed line for m = 0 . m = 0 .
9. According to theplot, there is no change in the behavior of tensor-to-scalar ratio for spectralindex while tensor -to-scalar ratio is compatible with the spectral index forall values of m .The value of spectral index is found with the help of above motionedpower spectrum along with first part of Eq.(19) and Eq.(20). It is given asfollows n s = 1 + β ω + υ (cid:18) M p φ m + φ (cid:19) (1+ ω )(1+ σ ) ! − σ (cid:20) m M p (1 + ω ) υφ ( m + φ ) × (cid:18) M p φ m + φ (cid:19) (1+ ω )(1+ σ ) β ω + υ (cid:18) M p φ m + φ (cid:19) (1+ ω )(1+ σ ) ! − Figure 5: Plot of spectral index number w.r.t inflaton in weak epoch with a = 2 × . − M p ( m + φ ) (cid:0) m (cid:0) M p (cid:1) + 2 m φ + φ (cid:1) (cid:21) . (25)Figure represents the spectral index versus scalar field for m = 0 . , m = 0 . m = 0 .
9. According to WMAP7 [67, 68], WMAP9 [69] and Planck 2015[70], the value of spectral index lies in the ranges 0 . ± . , . ± . . ± . α s = 32 m M p φ ( m + φ ) β ω + υ (cid:18) M p φ m + φ (cid:19) (1+ σ )(1+ ω ) ! − σ )1+ σ (cid:20) (cid:18) β ω (cid:19) × φ (cid:0) m + φ (cid:1) (cid:0) m (cid:0) M p (cid:1) + 2 m φ + φ (cid:1) + (cid:18) M p φ m + φ (cid:19) − σ + ω + σω × (cid:0) φ + m (cid:0) M p (cid:1) (1 + ω ) + m φ (9 + ω ) + 4 m φ (cid:0) M p + ω (cid:1) + m φ (cid:0) φ + M p (1 + ω )(3 + 2 ω + 2 σ (1 + ω )) + 4 φ (cid:0) ω + M p (4+ ω ) (cid:1)(cid:1) + m φ (cid:0) − M p (1 + ω ) + 2 φ (cid:0) ω + M p (13 + ω ) (cid:1)(cid:1) (cid:1) υβM p φ ω + M υ φ (cid:18) M p φ m + φ (cid:19) σ + ω + σω ) (cid:2) φ + m (cid:0) M p (cid:1) (1 + ω ) + m φ × (5 + ω ) + 2 m φ (cid:0) M p + 2 ω (cid:1) + m φ (cid:0) − M p (1 + ω ) + 2 φ (cid:0)
5+ 3 ω + M p (7 + ω ) (cid:1)(cid:1) + m φ (cid:2) − M p (1 + ω )(1 + 2 ω ) + φ (cid:0) ω + 2 M p × (5 + 2 ω ) (cid:1)(cid:3)(cid:3)(cid:21) . .70 0.75 0.80 0.85 0.90 0.95 1.0005. 10
1. 10
2. 10
3. 10 n s Α s Figure 6: Plot for running of spectral index versus spectral index in weakepoch with a = 2 × .The plot of running of spectral index with respect to scalar field is shownin Figure . It can be observed that the running of spectral index is com-patible with observational data for m = 0 . , m = 0 . m = 0 . The warm MCG inflationary scenario is being investigated with shaft poten-tial for tachyon scalar field. We have discussed this inflationary scenario forboth (weak and strong) dissipative regimes in flat FRW universe. We havealso examined the results for some of necessary inflationary parameters suchas the slow-roll parameters, number of e-folds, scalar-tensor power spectra,spectral indices, tensor-to-scalar ratio and running of scalar spectral index.We have analyzed these parameters for strong epoch as well as weak regimeby using the special case of shaft potential. We have restricted constantparameters of the models according to WMAP7 results for examining thephysical behavior of n s − φ, n s − R and n s − α s trajectories in both cases.We have analyzed the behavior of inflationary parameters according totwo dimensionless parameters ( a , m ) where the value of a = 2 × remainssame for all necessary parameter. All the trajectories are plotted for threedifferent values i.e., m = 0 . , m = 0 . m = 0 .
9. The case for m = 0 . r < . , . , .
11, the spectral index n s = 0 . ± . , . ± . , . ± . and ). Also, Figure and clearly showed the compatibility of spectralindex for it’s running with observational data since observational values ofrunning of spectral index are α s = − . ± . , − . ± . , − . ± .
025 according to Planck 2015 [70], WMAP7 [67, 68] and WMAP9[69], respectively.