Chaplygin gas inspired non-canonical scalar field warm inflation

Astroparticle Physics 117 (2020) 102402

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Chaplygin gas inspired non-canonical scalar field warm inflation Abdul Jawad∗, Azmat Rustam Department of Mathematics, COMSATS University, Islamabad, Lahore-Campus, Lahore 54000, Pakistan

a r t i c l e

i n f o

Article history: Received 4 July 2019 Revised 21 September 2019 Accepted 30 October 2019 Available online 11 November 2019 Keywords: Non-canonical scalar field model Warm inflation Inflationary parameters Chaplygin gas models

a b s t r a c t We consider a warm inflation universe model in the presence of generalized Chaplygin gas and modified Chaplygin gas by taking Dirac-Born-Infeld non-canonical scalar field. For developing various inflationary parameters, we assume quadratic potential and quartic potential and Anti-de-Sitter warp factor for the Dirac-Born-Infeld inflation. We develop the number of e-folds, scalar as well as tensor power spectrum, scalar spectral index and tensor-scalar ratio in the terms of φ under slow-roll approximation by assuming the constant sound speed. We establish the graphical behavior of r − N, ns − N and α − N for both .02 .001 .0 0 04 Chaplygin gas models. The results of r − N, ns − N and α − N are 0.972+0 , −0.0 0 05+0 , 0.0 0 04+0 −0.02 −0.001 −0.0 0 04 .05 +0.0055 +0.0045 (for generalized Chaplygin gas with quadratic potential) and 0.97+0 , 0 . 0 045 , 0 . 0 045 (gener−0.05 −0.0055 −0.0045 alized Chaplygin gas with quartic potential), respectively. Also, the results of inflationary parameters are +0.02 +0.0025 +0.125 .05 0.97−0.02 , 0.0015−0.0025 , 0.125−0.125 (for modified Chaplygin gas with quadratic potential) and 0.94+0 , −0.05 .015 +0.0185 0.01+0 , 0 . 0185 (modified Chaplygin gas with quartic potential), respectively. It is found that these −0.015 −0.0185 results are well matched with 68% and 95% CL constraints of the Planck 2018 TT,TE,EE+lowP+lensing data. © 2019 Published by Elsevier B.V.

1. Introduction It is well know that warm inflation gives the important characteristics, as conflicted to the conventional cool inflation that it passes through the reheating period [1,2]. During the inflationary era, in these models dissipative effects are important, therefore radiation production eventuate coincidentally together with cosmic expansion. In inflationary scenario if radiation field is in a highly excited state and in the inflaton dynamics it has strong damping effect, then result is in the form of strong regimen of the warm inflation. Moreover, effect of dissipation appear from a friction phrase due to the processes of scalar field dissipate into a thermal bath through its communication with the other fields. In these type of models density fluctuations occur from a thermal comparatively than the quantum fluctuations [3–7]. The components of universe matter occur from decay of the inflation field or dominant radiation field [8–14]. The standard cosmology describes observation of CMB radiations in an effective way but initial era of universe is facing some problems like horizon issues, flatness, numerical density of the monopoles and origin of fluctuations [15,16]. The models of inflation provide a good description of early universe that provide the most of these problems. Inflation can create a better mechanism, which will explain the actual ∗

Corresponding author. E-mail addresses: [email protected], [email protected] (A. Jawad), [email protected] (A. Rustam). https://doi.org/10.1016/j.astropartphys.2019.102402 0927-6505/© 2019 Published by Elsevier B.V.

interpretation of the observed anisotropy of CMB and formatting structure. The scalar field model consists of the kinetic and potential terms that combine the gravity and create dynamic frameworks and behave as a source for the inflation. These models are capable of interpreting Large-Scale-Structure(LSS) distribution and CMB radiations of observed anisotropy exhaustively in the period of inflation [17]. The universe expands rapidly. This long-term expansion acceleration is due to an exotic component, which has negative pressure and generally called DE. Many models have already introduced to become a candidate of DE and they are cosmological constant [18] quintessence [19], chaplygin gas (CG) [20] and holographic DE [21]. Instead of many models, the nature of the dark sector is still unknown, i.e., DE and DM. For explanation of cosmic acceleration DE with negative pressure should be a core candidate. DE can be explained with the help of CG due to its negative pressure. In cosmology, it was first introduce by Kamenshchik et al. [20], nevertheless, in its original form, this fluid presents inconsistency with some observational data such that a generalization is necessary to improve consistency with data. A first attempt was made considering the generalized Chaplygin gas (GCG). Further details can bee seen in [20]. The models of cosmos of the CG has three important characteristics: they explain smooth transition from a suddenly change in the universe, they try to express an collaborative macroscopic phosphology which is known DE and DM and finally, they represent the easiest deformation of the traditional CDM models. The CG solution is related to the fact that the

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A. Jawad and A. Rustam / Astroparticle Physics 117 (2020) 102402

corresponding Euler equation is a huge group of symmetry, which means their integrability. Recently the corresponding symmetry group has been explained in modern terms. Different models of CG have been proposed [22] till now because it is very effective to describe the expansion of the universe. Its expansion form is referred to as generalized CG (GCG). Pure CG and GCG are good fluid which initially stays in pressure without fluid-like pressure and later a cosmological constant. The interesting feature of CG is related to the string theory. String theory is related to high dimensional universe. Monerat et al. [23] discussed the initially universe cosmology and early condition in a model with the radiation for inflation and CG. Antonella et al. [24] studied the warm inflationary scenario on brane. Del campo and Herrera [25] discribe the model of warm inflation with GCG and then expand the model with quadratic potential. Setare and Kamali examined warm tachyon inflation considering intermediate [26] and logamediate framework [27]. Bastero-Gill et al. determined the expressions for the dissipation coefficient in supersymmetric (SUSY) models in [28]. This result gave us opportunity for realization of warm inflation in SUSY field theories. Benaoum has been introduced an exotic fluid, titled Modified CG (MCG). MCG as a cosmic model has the necessary features. The CG models are generally applied to described the late-time acceleration of the universe. The main inspiration for investigation of this kind of the model related to the string theory. The original and GCG models are usually applied to explain the late-time acceleration of our universe as a possible candidate of dark energy. On the other hand, the MCG is also a model that mimics the behavior of matter at early times and that of a cosmological constant at late times. We conclude that warm intermediate inflation inspired by the Chaplygin gas models is compatible with current data. The main goal of the present work is to investigate the dynamics of warm inflation driven by a standard scalar field in the CG scenario. It was Einstein, who first gained the gravity basic law. The branch of the cosmic physics is very dependent on the maximum scale of the entire universe where the effect of gravity power is important. Inflation is the defensible model that express early physics of the universe. Our universe has recently started a high-speed expansion phase according to observational data. Apart from solving many defects of hot big bang scenes inflation give us accurate solution to explanation of universal LSS. The inflationary scenario is also responsible for creating the root of all the LSS in universe. This phase of the inflation demand a condition of the negative pressure that may be easily attained by the scalar field. Including remarkable progress in the model independent analysis from productive field theories [29–31], inflation is still generally a model dependent phenomenon. In inflationary cosmology the list of models is thus practically incredible [32]. In the warm inflationary framework, which has the necessary characteristics that a reheating juncture overcome at the end of cosmic expansion due to decomposition of the inflaton into radiation and particles throughout the slow-roll phase [33,34]. Such ideas can help in the period of non-canonical inflation that is easily commonly connected to slow- roll regime. A type of inflation which is non-canonical version of ultra-slow roll inflation, for which many methods are not frozen on horizontal crossing, but it is rapidly developed on super horizon scales, we pay attention on the simplest quadratic inflation scenario in the non-canonical generalization [35]. This setup suggests that the inflation might be the successful moving into a wrapped area (throat) developed by a D3-brane radial coordinate in compact space. Brane works as vision and according to its movement in the wrapping area, in which there are two versions of brane inflation Infrared (IR) Model [36,37] and Ultraviolet (UV) Model [38,39]. In UV model, inflation runs towards the IR side from the UV side in wrapped space, while in IR model, it runs in opposite direction. The DBI inflation may be

considered in category of the kinflation models so that it is run by a non-canonical scalar field [40–46]. Moreover, many authors have investigated the warm inflation in various alternative as well as modified theories of gravity [47,48]. In this paper, we extend the work of Rasouli et al. [49] by considering CG models with quartic and quadratic potentials [50]. This paper is categorize as follows: firstly, we present the slow-roll parameters, e-folds number, scalar and tensor power spectrum, scalar spectral index and tensor-to-scalar ratio. In Section 3, we consider the quadratic potential and quartic potential in the context of DBI inflation using different CG models and determined our results which are comparable with planck data 2018 [51]. In the last chapter, we finalize the results and discussion. 2. Background of warm inflationary scenario In warm inflationary scenario, in the appearance of standard scalar field for the flat FRW universe first Friedmann equation determine as

1 ( ρφ + ρR ) . 3M2p

H2 =

(1)

where ρ R and ρ φ are the radiation and energy density of inflaton. The inflaton scalar field connect with the other fields in the warm inflation and dissipation plays a dynamic behavior during inflation. Due to dissipative results, the vacuum energy converted in the form of radiation energy. In the presence of energy density and radiation, we have the following conservation equations

ρ˙φ + 3H (ρφ + pφ ) = −ϒ (ρφ + pφ ), ρ˙R + 3H (ρR + pR ) = ϒ (ρφ + pφ ).

(2)

Here ϒ is the dissipation parameter which depends upon inflaton field as well as temperature and it can be resolute in quantum field theory [52]. It is noticed that radiation energy satisfy the black body equation ρR = α T 4 in warm inflation. Here, α = π30g∗ , present the Stefan–Boltzmann constant while g∗ = 228.75 is relativistic degree of the freedom [53] also T is the temperature of thermalize bath. Therefore, energy density as well as pressure regarding DBI scalar field are constitute as 2

ρφ ≡ 2X L,X − L =

γ −1 + V ( φ ), f (φ )

pφ ≡ L =

γ −1 − V ( φ ), γ f (φ ) (3)

where V(φ ) and f(φ ) represents the potential energy and the warp factor of inflaton, respectively. Also, L,X ≡ ∂∂ XL and γ called the Lorentz factor defined as

γ ≡ 

1 1 − f (φ )φ˙ 2

.

(4)

The slow-roll parameters take the following forms [54–56]

≡−

H˙ , H2

η≡

˙ , H

κ≡

c˙s . Hcs

(5)

Moreover, the speed of sound in the scenario of DBI scalar field can be given as

cs2 ≡

∂ pφ /∂ X 1 = = 1 − f (φ )φ˙ 2 . ∂ ρφ /∂ X γ 2

(6)

It illustrated that speed of propagation of density perturbations should be real and subluminal, 0 < cs2  1. In the onward work, we consider the squared speed to be constant. In the context of squared speed of sound, Eq. (3) can be re-written as

ρφ =

φ˙ 2

cs ( 1 + cs )

+ V ( φ ),

pφ =

φ˙ 2 1 + cs

− V ( φ ).

(7)

A. Jawad and A. Rustam / Astroparticle Physics 117 (2020) 102402

By substituting the derivative of Eq. (2) in Eq. (7), one can get

2φ¨ + (3H + ϒ )φ˙ + csV (´φ ) = 0. 1 + cs

(8)

Also, throughout the warm inflationary scenario the quasi-stable production of photons is, i.e. ρ˙ R  4H ρR . Moreover, we realized that the energy density of inflation control the density of radiation energy at the initial stage of inflation, i.e. ρ φ  ρ R . In the perspective of these ideas, one can obtain the following expressions from Eqs. (2) and (8) as

(3H + ϒ )φ˙ + csV  (φ ) ≈ 0,

ρR ≈

3Q φ˙ 2 . 4 cs

(9) (10)

It is easy to implement the Hamilton–Jacobi formalism [57–59], where the Hubble parameter H is consider the function of inflation field φ . This form was firstly developed [38] in the setting of DBI inflation and it was investigated in [60]. According to this formalism the slow roll parameters reduce to the following form [54–56]

˜ ≡ η˜ ≡ κ˜ ≡

2M2p  H  2

γ

H

H

γ

γ H γ

(11)

(12)

.

ϒ  H σ˜ ≡ 2M2p cs . ϒ H

(13)

e

a

,

(15)

dN = −Hdt d ln κ ≈ Hdt = −dN.

(16)

The scalar spectral index is define to measure the scaledependence of the scalar power spectrum as

d ln Ps . d ln κ

(17)

In DBI model, the tensor power spectrum is written as [62,63]

Pt =

2H 2 . π 2 M2p

(18)

In the tensor-to-scalar ratio, an important inflationary observable is defined as

r≡

Pt . Ps

(20)

Also, the tensor perturbation are disconnect from the thermal perturbations. Moreover, in cold and warm inflation the tensor power spectra is same. Therefore, tensor-to-scalar ratio r = PPts becomes smaller in the warm inflation than in cold and Q is dissipation ratio given as

Q≡

ϒ

where ϒ = 3HQ.

3H

(21)

Also, running of scalar spectral index can be defined as follows

α=

d dns =− ( ns ). dlnκ dN

f (φ ) =

(22)

f0

φ4

,

(23)

where f0 represents a positive dimensionless constant. Due to above relation Eq. (23), Eq. (6) leads to

φ˙ = −

1 − cs2 2 φ . f0

(24)



Q =

(14)

where ae at the end of inflationary era known as the scale factor of universe. It is sure that in the observable universe, the largest scale at N∗ ≈ 50 − 60 e-folds left the horizon from the end of inflation. We use the following expression to obtain the number of e-folding

ns − 1 ≡

3 HT 5 (1 + Q ) 2 . 3 16π 2 M2p cs ˜

Putting this into Eqs. (9) and (10), respectively, we obtain

During the inflation, universe size is usually represented by number of e-fold

a 

√

Ps =



Note that the stability of the above analysis rely on the behavior of a new slow-roll parameter [61]

N ≡ ln

powerful criterion to discriminate between the usable inflationary models. In the DBI warm inflation, beyond the sound horizon scales, the curvature perturbations can be calculated and the power spectrum of the curvature perturbations can be gained after complement to the perturbations eliminate at the smaller thermal noise scales,

We assume the following AdS warp factor in our work [38,39]

,

2M2p H  , γ H 2M2p

3

(19)

This amount has a special importance in distinguish around nonidentical inflationary models. In fact, in the light of observational data, combination of observational bounds of r and ns provide a

T =

−csV (φ ) − 1, 3H φ˙

(25)

 3Q ( 1 − c 2 )φ 4  1 s

4 cs α f 0

4

.

(26)

If we solve the differential Eq. (28), inflationary field evolution can be determine as



φ=

f0 1 , 1 − cs2 t

(27)

where we take the integration constant to be zero. 3. Chaplygin gas models with quadratic potential We consider the CG models using the quadratic potential V (φ ) = (m2φ ) , here φ is the scalar field and m denotes mass of the scalar field [50]. It is a simplest model of a scalar field. Quadratic inflationary scenario resolved the many problems of old and new inflation. Due to this scenario, in the early universe inflationary era may begin even if there was no thermal equilibrium occur. Various generalization of CG has been suggested as it is very effective for exploring the cosmic expansion. In our work we use two models of CG which are • GCG • MCG We discuss a recently suggested class of the models of cosmos which is based on the use of static perfect fluid [64]. Commonly, we work on the simple model of the universe filled with the CG, which is a good fluid given by the equation of state (EoS) as following 2

P=−

A

ρ

,

(28)

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A. Jawad and A. Rustam / Astroparticle Physics 117 (2020) 102402

A is the positive constant. The interesting feature of the CG is related to string theory. String theory is related to high dimensional universe. Monerat et al. [65] discussed the initially universe cosmology and the early condition for inflation and CG. It is noticed that CG is an alternate description of accelerating expansion. In (d + 1, 1 ) space time CG appears as a powerful fluid of generalized D-brane and the action can be given in the form of generalized born-infield [66]. Kamenshchik et al. [67] noticed that FRW universe consisted of the CG which express that the universe is in good agreement according to recent observation of cosmic acceleration.

 × 12

6 − 6cs2 3 M p φ ( 4A + m4 φ 4 ) 4 f0

(35)



ns = 1 − 3(−1 +

cs2



)M p φ −62Acs m M p −3cs m M p φ + 2

 × 30Aφ (4A + m4 φ 4 ) + 1 4





× ( 4A + m

The expanded term of CG known as GCG and its equation of state (EoS) is given as

Pgcg = − λ , ρ

.

We can determine the scalar spectrum index by using Eqs. (16),(17) and (34) as

3.1. Generalized chaplygin gas

A

−1

4

φ ) 4

5 4

cs

6

4

6 − 6cs2 f0

6 − 6cs2 4 5 1 m φ ( 4A + m4 φ 4 ) 4 f0



6 − 6cs2 1 f 0 m2 M p − 3φ ( 4A + m4 φ 4 ) 4 f0



+

(29)

3cs2

φ ( 4A + m φ ) 4

4

1 4

−1

.

(36)

gcg

where Pgcg and ρ gcg represent the pressure and energy density and A is the positive constant and 0 < λ ≤ 1 respectively. GCG’s energy density can be derive using continuity equation which is given as

ρgcg = (A +

B

a

) 3(1+λ )

1 1+λ

,

(30)

By using the Eq. (22) we can get the value of running of scalar spectral index as



α = 18(−1 + )M φ



−2Am



1 1 (A + ρφ1+λ ) 1+λ + ρR . 3M2p

(31)

ρgcg = (A + ρm1+λ )

→ (A + ρφ1+λ )

1 1+λ

,

2

H =

3M2p

1  A + ρφ ∼ A + V2 3M2p

+ 8A

− 45φ

⎛

×(4A + m φ ) ⎝φ 3 4



4

9 4

3(−1 + cs2 )φ cs f 0

× 482 4 3 4 cs m8 M5p φ 2 φ 6 3

1

3



−1

 +

6−6cs2 m2 M p f0

( 4A + m4 φ 4 ) 4

1

.

⎞

1 4

× (α ) ⎠ −1

(34)

Similarly we can drive the expression for number of e-fold in term of φ by using the Eqs. (28) and (42) as



N = − −12(4A + m

4

φ

4

√ m4 φ ) + 2m4 φ 4 4 + A

 × Hyper geometr ic2F 1

m φ 3 3 7 , , ,− 4 4 4 4A 4

4

 34 4



−30cs

2



4A + m4 φ 4 )

6 − 6cs2 1 f 0 m2 M p φ ( 4A + m4 φ 4 ) 4 f0

4A + m4 φ 4 + cs2 (31 f0 m4 M2







4A +

m4

φ )

× f 0 ( 4A + m

4

 4

φ ) 4

5 4

6 − 6cs2 f0

cs

× f0 m2 M p − 3φ (4A + m4 φ 4 ) + 3cs2 φ (4A + m4 φ 4 ) 1 4

1 4

2

−1

.

(37) By inserting the Eqs. (18) and (34) into Eq. (19) expression for tensor-to-scalar ratio turns out. In this way, we can get the following form

f0



6 − 6cs2 1 f 0 m2 M p φ ( 4A + m4 φ 4 ) 4 f0

4A + m4 φ 4





6 − 6cs2 1 f0 m10 M p φ 9 (4A + m4 φ 4 ) 4 f0



Now, we construct the inflationary parameters for the case of GCG. Considering Eqs. (11), (29), (30), (42) and quadratic potential in terms of φ and inserting the results in Eq. (20) we can calculate the power spectrum of curvature perturbation as



−1 ⎞ 52 2 1 − cs 1 ⎠ Ps = γ ⎝(cs m2 M p ) × φ ( 4A + m4 φ 4 ) 4

2

+ 45φ 2

(33)

⎛

−99cs



cs





(32)

2

φ

4

2

+ 2cs2 (47 f0 m4 M2 − 75φ 2

where ρ m represent the matter energy density. In the duration of inflationary period, energy density of scalar field dominated by the energy density of the radiation field as we mention above, i.e., ρ φ  ρ R and ρ φ ∼ V. Here, λ = 1 so that, the Friedmann equation becomes

 1

4

+150φ 2

Due to an extrapolation of Eq. (39) this modification is feasible so that 1 1+λ

2



where a represents the scale factor and B consider the positive integration constant. Due to GCG, Friedmann equation written as

H2 =



cs2

r=



16 × 2 4 3 4 cs m8 M p φ 6 3

1



×



γ ( cs m2 M p ) ×

× ( 4A + m4 φ 4 ) 4

7

 +



1 − cs2 1 φ ( 4A + m4 φ 4 ) 4 f0



−1

5 2

 3(−1 + cs2 )φ φ3 cs f 0

6−6cs2 m2 M p f0

( 4A + m4 φ 4 ) 4

1

⎞ ⎞

(α )−1 ⎠ 4 ⎠−1 . 1

(38)

Fig. 1 comprises the graph of ns − N in warm DBI inflationary scenario in the presence of quadratic potential. The interval of red trajectory at cs = 0.3 for ns lies in the interval [0.9525, 0.98] for the desired e-folds number. At cs = 0.4, the trajectory of ns lies in

A. Jawad and A. Rustam / Astroparticle Physics 117 (2020) 102402

5

in the set up of inflationary model. The intervals [0, 0.00525], [0, 0.0 0675], [0, 0.0 085] for different choices of cs represent the behavior of r − N plane which is comparable with the constraints of Planck TT,TE,EE+lowP+lensing data 2018 [51]. 3.2. Modified Chaplygin gas model For MCG EoS is given as [65]

Pmcg = μρmcg −

Fig. 1. Plot of ns versus N for the different values of cs . The values of other parameters are M p = 1, λ = 1, A = 4, f 0 = 12095.1, α = 0.5 and m = 2. We can find the best range of ns where 60 < N using quadratic potential model for GCG.

ν , λ ρmcg

(39)

where Pmcg and ρ mcg shows the pressure and energy density and 0 ≤ λ ≤ 1, μ, ν considered positive constants respectively. We determine the energy conservation equation also, shows the MCG density, as following

 ρmcg = A +



c a3(1+λ)(1+μ)

1+λ

,

(40)

where c denote the constant of integration and A = 1+νμ . Firstly, we discuss the modified Friedmann equation which is given as follows

H2 =



1 (A + ρφ(1+λ)(1+μ) )1+λ + ρR 3M2p



(41)

the above relation is occurred because of an extrapolation of Eq. (40) so that

    ρmcg = A + ρm(1+λ)(1+μ) 1+λ → A + ρφ(1+λ)(1+μ) 1+λ

(42)

where ρ m represent the matter energy density, so that Friedmann equation adopt the following expression

Fig. 2. Plot of α versus N for the different values of cs . The values of other parameters are M p = 1, λ = 1, A = 4, f 0 = 3095.1, α = 0.5 and m = 2. We can find the best range of α where 60 < N using quadratic potential model for GCG.

H2 =



1 (1+λ )(1+μ ) A + ρφ 3M2p



1+λ

∼



1 A + V (1+λ)(1+μ) 3M2p



− 7 +2(λ+μ+λμ ) 2

 −2(1+λ)(1+φ ) γ m2 φ m2 φ 2

× (2−(1+λ)(1+μ) (m2 φ 2 )(1+λ)(1+μ) + A )2

⎛



×⎝cs M p m2

1 − cs2 f0

⎛

⎛

⎜ ⎝

⎜ ⎝

×⎜(1 − cs2 )φ 4 × ⎜ 

−1

× ( cs f 0 α )

−1 ⎞ 1 2 (1+λ )(1+μ ) 1+λ

(A + 2−(1+λ)(1+μ) (m2 φ )

3−3cs2 f0

3

⎠2

)

c s M p m2



1

(A + 2−(1+λ)(1+μ) (m2 φ 2 )(1+λ)(1+μ) ) 1+λ 



−1



1 4

×

3

1 − cs2 4 3 M p π 2 ( 1 + μ )2 f0

−1

(44)

.

The number of e-folds is given as



(A + 2−(1+λ)(1+μ) (m2 φ 2 )(1+λ)(1+μ) ) 1+λ

 2(1+1 λ) 2−(λ)(1+μ) (m2 φ 2 )(1+λ)(1+μ) 1

N= interval [0.955, 0.98125]. Similarly, at cs = 0.5, the trajectory of ns lies in the interval [0.9575, 0.9825] for the desired e-folds number. For 60 < N, we can see the spectral index is compatible with constraints Planck TT,TE,EE+lowP+lensing data 2018 range which is 0.9649 ± 0.0042 [51]. Fig. 2 shows the behavior of α versus N for quadratic potential. At cs = 0.3, α lies in the interval of [−0.0 015, 0 0 0 05625]. At cs = 0.4, α lies in the interval [−0.001375, 0, 0005]. At cs = 0.5 for α lies in the interval [−0.00125, 0, 0004375]. These interval are well matched with constraints Planck TT,TE,EE+lowP+lensing data 2018 [51]. Fig. 3 indicates the r − N plane for quadratic potential

(43)

Now, we can construct the inflationary parameters for the case of MCG. Considering Eqs. (11), (29), (30), (43) and quadratic potential in terms of φ and inserting the results in Eq. (20) we get the power spectrum of curvature perturbation as Ps = 2

Fig. 3. Plot of r versus N for the different values of cs . The values of other parameters are M p = 1, λ = 1, A = 4, f 0 = 3095.1, α = 0.5 and m = 2. We can find the best range of r using quadratic potential for GCG model where 60 < N.

1+λ

× 1+

A

+1



×Hyper geometr ic2F 1 × − ×1 −

 ×

1 2(1 + λ )(1 + μ ) 3 − 3cs2 × M pφ f0

,−

−1 .

1 1 ,− , 2(1 + λ )(1 + μ ) 2 (1 + λ )

2−(λ)(1+μ) (m2 φ 2 )(1+λ)(1+μ) A



(45)

6

A. Jawad and A. Rustam / Astroparticle Physics 117 (2020) 102402

Fig. 4. Plot of ns versus N for different values of cs . We have considered cs = 0.3, cs = 0.4, cs = 0.5, M p = 1, λ = 1, A = 4, f 0 = 28095.1, α = 0.5, μ = 0.7 and m = 2. We can find the best range of ns using quadratic potential for MCG model where 60 < N.

Fig. 5. Plot of α versus N for different values of cs . Using quadratic potential for MCG model we can find the best range of α where N > 60.

We can find the value of scalar spectrum index by using Eqs. (16), (17) and (44) as





ns = 1 − 3 ( 1 −

cs2

2 (1+λ )(1+μ )

)M p φ ((m φ ) 2

2

3 − 3cs2 ( 2 + 3μ )φ f0

×(A + 2−(1+λ)(1+μ) (m2 φ 2 )(1+λ)(1+μ) ) 1+λ − cs m2 M p (6 + 7μ ) 1





×

(A + 

×A 2

2−(1+λ)(1+μ)

(

m2

φ )

2 (1+λ )(1+μ )

+ 2(1+λ)(1+μ)

1 1+λ

)

3 − 3cs2 (7 + 8λ + 8(1 + λ )μ )φ × (A + 2−(1+λ)(1+μ) f0

×(m2 φ 2 )(1+λ)(1+μ) ) 1+λ 1

Fig. 6. Plot of r versus N for different values of cs . using quadratic potential for MCG model we can find the best range of r where 60 < N.



(A + 2−(1+λ)(1+μ) (m2 φ 2 )(1+λ)(1+μ) )  × cs m2 M p (15 + 16λ + 16(1 + λ )μ )  × 4(2(1+λ)(1+μ) A(m2 × φ 2 )(1+λ)(1+μ) )  1  × (A + 2−(1+λ)(1+μ) (m2 φ 2 )(1+λ)(1+μ) ) 1+λ 3(1 − cs2 )  1 1+λ

−

3 − 3cs2 f0

×φ (A + 2−(1+λ)(1+μ) (m2 φ 2 )(1+λ)(1+μ) ) 1+λ + cs 1

 (A +

×

2−(1+λ)(1+μ)

(

m2

φ )

2 (1+λ )(1+μ )

)

1 1+λ



2

× f0 m M p

.

(46)

Moreover, using the Eq. (22) the value of the running of the scalar spectral index obtain as we calculate in above section. The value of the tensor-to-scalar ratio obtain by plugging the Eqs. (18) and (44) into Eq. (19). In this way, we can obtain the following expression r = 2−

9 2

−2(λ+μ+λμ )

(m2 φ 2 )−2(1+λ)(1+φ )

×



√ 1 − cs2 (1 + μ )2 m2 πγ φ cs M p m2 × f0





1 − cs2 f0

   (A + 2−(1+λ)(1+μ) (m2 φ 2 )(1+λ)(1+μ) ) 1+λ 1

×

−1

3 2

−1



(1 − cs2 )φ 4

⎛ × ⎝−1 + 

⎞ 3−3cs2 f0



V (φ ) = λφ4 which is known as simple Higgs potential [68], where λ is consider a dimensionless parameter. A simple quadratic inflation can be determine by this potential [50] and for the theoretical point of view it has a significant importance, due to its appearance in all renormalizable gauge field theories [68]. Quartic potential have many stunning reheating properties [69,70]. 4

1



4. Models of Chaplygin gas with quartic potential In particle physics theories, we examine a quartic potential

×(2−(1+λ)(1+μ) (m2 φ 2 )(1+λ)(1+μ) + A )−2+ 1+λ



0.99], [0.94875, 0.99125], [0.9525, 0.9925] for cs = 0.3, cs = 0.4, cs = 0.5 respectively, indicate the behavior of our graphs which is comparable with constraints of Planck TT,TE,EE+lowP+lensing data 2018 [51]. Fig. 5 describe the graph of α versus N for quadratic potential. The intervals [−0.0 0 075, 0.0 04125], [−0.0 0 0875, 0.0 0375], [−0.0 01, 0.0 035] for different values of cs show the behavior of α which is compatible with constraints of Planck TT,TE,EE+lowP+lensing data 2018 [51]. Fig. 6 indicates the r − N plane for quadratic potential in inflationary model. In r − N plane, the intervals [0, 0.25], [0, 0.2625], [0, 0.2625] for different values of cs show the values of r − N which is well matched with constraints of Planck TT,TE,EE+lowP+lensing data 2018 [51].

cs M p m2

⎞ ⎞

⎠(cs f0 α ) ⎠ 14 ⎠−1 . −1

(A + 2−(1+λ)(1+μ) (m2 φ 2 )(1+λ)(1+μ) ) 1+λ 1

4.1. Generalized Chaplygin gas

(47) Fig. 4 show the graph of ns − N in warm DBI inflationary scenario in the presence of quadratic potential. The intervals [0.945,

With the help of Eqs. (11), (29), (30), (42) and quartic potential and plugging these results in Eq. (20) power spectrum can be obtained as

A. Jawad and A. Rustam / Astroparticle Physics 117 (2020) 102402

Ps = 8γ

A+

λφ 2

⎛



8

⎛

×⎝

9 4

16

⎞ cs M p λφ



⎛

×⎝(1 − cs2 )φ 4 ⎝−1 +

(16A + λ2 φ 8 ) 4 ⎠ 2 1

1−cs2 f0

⎞

2cs M p λφ



3−3cs2 f0

5

(16A + λ2 φ 8 ) 4

1

⎞

  ⎠ × (cs f0 α )−1⎠ 14 3cs M5p π 32 λ4 φ 14 −1 .

(48)

The e-fold number in term of φ are calculated by using the Eqs. (28) and (42) as



N = − −28(16A + λ

2

 × 56

φ ) + +λ φ 8

2

+λ2 φ 8 16 + A

8

3 − 3cs2 3 M p φ (16A + λ2 φ 8 ) 4 f0

 34



+λ2 φ 8 7 3 15 Hyper geometr ic2F 1 , , ,− 8 4 8 16A

−1

.

(49)



ns = 1 − 3 ( 1 −

)

cs2 2 M p

2





+ λ2 φ 8 −9cs M p λφ + 5

⎛ ×





φ (16A + λ φ ) 16A 41cs M p λφ − 21 5

8

3 − 3cs2 1 (16A + λ2 φ 8 ) 4 f0

⎝1024cs (−1 + cs2 ) f0 α A + λ φ 2

8

 52

16



⎛⎛





Eqs. (16),(17) and (48) gives the expression for scalar spectral index as follows 3 4

7



3 − 3cs2 1 (16A + λ2 φ 8 ) 4 f0

⎛⎛

⎞

⎝⎝−1 + 

⎞

2cs M p λφ 3−3cs2 f0

⎞ ⎞

(16A + λ φ ) 2

8

1 4



⎞

⎠(1 − cs2 )φ 4 (cs f0 α )−1 ⎠ 14

cs 2 3−3 M p λφ f0 3(−1 + cs2 ) ⎠φ 4 (α )−1 ⎠ 34 ⎠−1 . × ⎝⎝ + 1 2 8 cs f 0 (16A + λ φ ) 4 2

(50)

In addition, the value of running of the scalar spectral index turns out to be





cs2 2 M2p

α = 18(−1 + )

φ





− 32Aλ2 φ 8 458cs

× 256A

2

 82cs





3λ

2

4

φ

16



6 cs

  3 − 3cs2 1 f0 M p λφ (16A + λ2 φ 8 ) 4 − 5 16A + λ2 φ 8 + cs2 (−6 f0 M2p λ2 φ 2 + 5 16A + λ2 φ 8 ) f0

  3 − 3cs2 1 f0 M p λφ (16A + λ2 φ 8 ) 4 − 351 16A + λ2 φ 8 + 9cs2 (−50 f0 M2p λ2 φ 2 + 39 16A + λ2 φ 8 ) f0

  3 − 3cs2 1 f0 M p λφ (16A + λ2 φ 8 ) 4 − 63 16A + λ2 φ 8 + 9cs2 (−82 f0 M2p λ2 φ 2 + 39 16A + λ2 φ 8 ) f0





× f0 (16A + λ2 φ 8 ) 48A(−1 + cs2 ) + λφ 3(−1 + cs2 ) × λφ 7 + 2cs

3 − 3cs2 3 f0 M p (16A + λ2 φ 8 ) 4 f0











2

−1

.

(51)

By inserting the Eqs. (18) and (48) into Eq. (19) the expression for tensor-to-scalar ratio takes the following form

r=



cs M5p λ4 φ 14

⎛





⎛

√

64 + φ 8 × ⎝16γ

π A+

λ2 φ 8 16

⎛

× ⎝(1 − cs2 )φ 4 × ⎝−1 +

⎛

 9 4

⎞ 

2cs M p λφ 3−3cs2 f0

(16A + λ φ ) 2

8

1 4

×⎝

⎞

cs M p λφ



1−cs2 f0

5 (16A + λ2 φ 8 ) ⎠ 2 1 4

⎞ ⎞

⎠(cs f0 α )−1 ⎠ 14 ⎠−1 .

(52)

In Fig. 7 we have plotted the graph of ns − N in warm DBI inflationary scenario in the presence of GCG with quartic potential. It is observed that the trajectories of ns − N lie in the intervals [0.935, 1.0125], [0.9275, 1.0125], [0.9325, 1.0125] for cs = 0.3, cs = 0.4, cs = 0.5, respectively. Fig. 8 describes the range of α versus N for quartic potential in the warm DBI inflation which lies in the intervals [−0.00175, 0.010], [−0.0015, 0.00925], [−0.00125, 0.00825] cs = 0.3, cs = 0.4, cs = 0.5 respectively. Fig. 9 leads to r − N plane for quartic potential in the current set up of inflationary model. The trajectories of r − N remain in the intervals [0, 0.009], [0, 0.00725], [0, 0.00575] for cs = 0.3, cs = 0.4, cs = 0.5, respectively. It is found that the results of all inflationary parameters for GCG with quartic potential are well matched with observational TT,TE,EE+lowP+lensing data 2018 [51].

8

A. Jawad and A. Rustam / Astroparticle Physics 117 (2020) 102402

Fig. 7. Plot of ns versus N for different values of cs . We have assumed cs = 0.3, cs = 0.4, cs = 0.5, M p = 1, λ = 1, A = 4, f 0 = 30095.1, α = 0.5. We can find the best range of ns using quatric potential for GCG model where N > 60.

Fig. 8. Plot of α versus N for different values of cs . The constants parameters are cs = 0.3, cs = 0.4, cs = 0.5, M p = 1, λ = 1, A = 2, f 0 = 8095.1 and α = 0.5. Using quatric potential for GCG model we can find the best range of α where 60 < N.

Fig. 9. Plot of r versus N for different values of cs . The values of parameters are cs = 0.3, cs = 0.4, cs = 0.5, M p = 1, λ = 1, A = 2, f 0 = 3095.1, α = 0.5 and γ = find the best range of r where 60 < N using quatric potential for GCG model.

1 cs

. We can

4.2. Modified Chaplygin gas model We can construct the inflationary parameters for the case of MCG. Expression for Ps , ns , α and r are given below which is obtain as we calculated these expression in the above section.

Ps = 2− 2 +4(λ+μ+λμ) γ (λφ 4 )−3(λ+μ+λμ) (A + 4−(1+λ)(1+μ) (λφ 4 )(1+λ)(1+μ) )2 7





× cs M p λφ ×

⎛

⎛

×⎝φ 4 × ⎝−1 +

1 − cs2 f0



3−3cs2 f0





(A +

4−(1+λ)(1+μ)

(λφ )

4 (1+λ )(1+μ )

)

1 1+λ

−1

⎞ 

1 4

cs M p λφ

(A + 4−(1+λ)(1+μ) (λφ 4 )(1+λ)(1+μ) ) 1+λ 1

3 2

⎞



⎠(1 − cs2 )(cs f0 α )−1 ⎠ × 3 1 4

1 − cs2 4 3 M p π 2 ( 1 + μ )2 φ f0

−1

. (53)

A. Jawad and A. Rustam / Astroparticle Physics 117 (2020) 102402

 (A +

N=

2−(1+λ)(1+μ)



(λφ )

)

4 (1+λ )(1+μ )

2−(λ)(1+μ) (λφ 4 )(1+λ)(1+μ) 1+ +1 A

1 1+λ

−

1 Hyper geometr ic2F 1 2 (1 + λ )

1 1 2−(λ)(1+μ) (λφ 4 )(1+λ)(1+μ) × − ,− ,1 − ,− 4(1 + λ )(1 + μ ) 2 (1 + λ ) 4(1 + λ )(1 + μ ) A 1

ns = 1 −

3(−1 + cs2 )M p

 ×

3 − 3cs2 f0

(A + V (1+λ)(1+μ) )



1

(A + V (1+λ)(1+μ) ) 1+λ c

 32



3 − 3cs2 f0

sMp

λφ

(1+λ )(1+μ )

)

1 1+λ

(A + V (1+λ)(1+μ) )

3(−1 +

) + cs

A + 2−2(1+λ)(1+μ) (λφ 4 )(1+λ)(1+μ)





α = 3(−1 + ) cs2

+2



cs2





 1 1+λ

1 1+λ

3 − 3cs2 f0

−1



3 − 3cs2 × f0

 2(1+λ )(1+μ )

 

× 4 (A + V



φ (λφ 4 )(1+λ)(1+μ) −6

1 1+λ



× ( 1 + μ )φ +

×





+ 32

A 2

3 − 3cs2 f0

9





×

1

 (A + V (1+λ)(1+μ) ) 1+λ (5 + 16μ ) − 32cs M p λ2 1





(A + V (1+λ)(1+μ) )

1 1+λ

μ − 9cs M p φ − 32cs M p μφ



(A + V (1+λ)(1+μ) )

1 1+λ

λφ f0 M p 





(55)



2 1 3 − 3cs2 (A + V (1+λ)(1+μ) ) 1+λ (1 + 2μ ) × (−3 + 8μ ) + (A + V (1+λ)(1+μ) ) 1+λ f0



3

1



2

1 3 − 3cs2 (A + V (1+λ)(1+μ) ) 1+λ (1 + 2μ ) f0

2 1 3 − 3cs2 3 (A + V (1+λ)(1+μ) ) 1+λ (5 + 16λ + 16(1 + λ )μ ) − 6cs M p ((A + V (1+λ)(1+μ) ) 1+λ ) 2 λ(9 + 32λ + 32(1 + λ )μ ) f0

6cs3 M p

((A + V

(1+λ )(1+μ )

)

1 1+λ

) λ(9 + 32λ + 32(1 + λ )μ )φ + 3 2

cs2

1 3 − 3cs2 (A + V (1+λ)(1+μ) ) 1+λ f0

× −6(A + V (1+λ)(1+μ) ) 1+λ (5 + 16λ + 16(1 + λ )μ ) + f0 M2p λ2 (9 + 32λ + 32(1 + λ )μ )φ 2 1



×2

3

+ 24(1+λ)(1+μ) A2





(54)

(A + V (1+λ)(1+μ) ) 1+λ + 7cs M p λφ + 16μ



× −6(A + V (1+λ)(1+μ) ) 1+λ (−3 + 8μ ) + f0 M2p λ2 (−7 + 16μ )φ 2

×φ +

.



3 − 3cs2 f0

1

× 6

−1



×6cs M p λ(−6 + μ(3 + 32μ ))φ − 6cs3 M p ((A + V (1+λ)(1+μ) ) 1+λ ) 2 λ(−6 + μ(3 + 32μ ))φ − cs2





41+λ+μ+λμ A(λφ 4 )(1+λ)(1+μ) .

3 − 3cs2 2 2 M p φ (λφ 4 )2(1+λ)(1+μ) −6 f0



3 − 3cs2 M pφ f0

3+2(λ+μ+λμ )

4 (1+λ )(1+μ )

A(λφ )

−6



2 3 − 3cs2 (A + V (1+λ)(1+μ) ) 1+λ (20 + 32λ2 (1 + μ )2 + 8λ(1 + μ )(7 + 8μ ) + μ(41 + 32μ ))3cs M p f0

× ((A + V (1+λ)(1+μ) ) 1+λ ) 2 λ(81 + 163μ + 32(4μ2 + 4λ2 (1 + μ )2 + λ(1 + μ )(7 + 8μ )))φ + 3cs3 M p ((A + V (1+λ)(1+μ) ) 1+λ ) 2 λ 1

1

3



×(81 + 163μ + 32(4μ2 + 4λ2 (1 + μ )2 + λ(1 + μ )(7 + 8μ ))) × φ + cs2

3

1 3 − 3cs2 (A + V (1+λ)(1+μ) ) 1+λ f0

× (−6(A + V (1+λ)(1+μ) ) 1+λ × (20 + 32λ2 (1 + μ )2 + 8λ(1 + μ )(7 + 8μ ) + μ(41 + 32 )) + f0 M2p λ2 1



× (40 + 64λ2 (1 + μ )2 + 16λ(1 + μ )(7 + 8μ ) + μ(81 + 64μ ))φ 2 )

× 4 3(−1 +

cs2

)(A + V

(1+λ )(1+μ )



 )

1 1+λ

+

(A + V (1+λ)(1+μ) )

1 1+λ

λφ cs

3 − 3cs2 f0 M p f0

 

A + 2−2(1+λ)(1+μ) (λφ 4 )(1+λ)(1+μ)

×(41+λ+μ+λμ A(λφ 4 )(1+λ)(1+μ) ).

× cs M p λφ

3 − 3cs2 f0

⎛

 1

(A + V (1+λ)(1+μ) ) 1+λ

−1

√

π ((A + V (1+λ)(1+μ) ) 1+λ ) 2 γ ⎞ ⎞ cs M p λφ 5 ⎠(cs f0 α )−1 ⎠ 14 . 2 ⎝ (1 − c2 )φ 4 ⎝−1 +   s 1 3−3cs2 (A + V (1+λ)(1+μ) ) 1+λ f0 2

9



1 1+λ

−1

(56)

r = 2 2 −4(λ+μ+λμ) cs M p λ2 (1 + μ2 )2 φ 6 (λφ 4 )2(λ+μ+λμ) (A + 4(−1−λ)(1+μ) (λφ 4 )(1+λ)(1+μ) )−2+ 1+λ (





1

3

⎛

(57)

10

A. Jawad and A. Rustam / Astroparticle Physics 117 (2020) 102402 Table 1 Combinations of Planck power spectra, Planck lensing. Parameter

TT+lowE

TT,TE,EE+lowE

TT,TE,EE+lowE+lensing

ns r

0.9626 ± 0.0057 < 0.102 −0.004 ± 0.015

0.9649 ± 0.0044 < 0.107 −0.006 ± 0.013

0.9649 ± 0.0042 < 0.101 −0.005 ± 0.013

α

set up of inflationary model whose trajectoris lie in the intervals [0, 0.036875], [0, 0.036875], [0, 0.036875] different values of cs respectively. It can be seen that all the trajectories of inflationary parameters lie within suggested constraints of the Planck TT,TE,EE+lowP+lensing data 2018 [51]. Fig. 10. Plot of ns versus N for different values of cs . We have use the values of constant are cs = 0.3, cs = 0.4, cs = 0.5, M p = 1, λ = 1, A = 1.5, f 0 = 93095.1, α = 0.5 and μ = 0.7. We can find the best range of ns where 60 < N using quatric potential for MCG model.

5. Concluding remarks In this work, we have discussed GCG and MCG inflationary models in flat FRW geometry by considering non-canonical scalar field model in the presence of specific form of AdS warp factor and potentials (quadratic and quartic scalar field). We have modified the first Friedmann equation under slow-roll approximation to study the CG inflationary cosmology. We have studied the behavior of these inflationary parameters through planes such as ns − N, r − N and α − N. These are summarized as follows: .02 • It is found that ns , α and r lie in the range 0.972+0 , −0.02

Fig. 11. Plot of α versus N for different values of cs . The constants parameters are cs = 0.7, cs = 0.8, cs = 0.9, M p = 1, λ = 1, A = 4, f 0 = 3095.1, α = 0.5 and μ = 0.5. We can find the best range of α using quatric potential for MCG model where 60 < N.

Fig. 12. Plot of r versus N for different values of cs in strong epoch. The constants parameters are cs = 0.3, cs = 0.4, cs = 0.5, M p = 1, λ = 1, A = 4, f 0 = 3095.1, α = 0.5 and μ = 0.5. Using quatric potential for MCG model we can find the best range of r where 60 < N.

We have plotted the graph of ns − N which is shown in Fig. 10 for MCG with quartic potential. The trajectories of ns − N lie in the intervals [0.885, 0.995], [0.89, 0.995], [0.895, 0.995] for cs = 0.3, cs = 0.4, cs = 0.5, respectively. Fig. 11 indicates the behavior of α versus N for quartic potential in the scenario of warm DBI. The trajectories of α lie in the intervals [−0.004375, 0.02625], [−0.00375, 0.02375], [−0.00125, 0.01625] for different values of cs . The constants parameters have the same values as we use in Fig. 12 leads to r − N plane for quartic potential in the current

.001 .0 0 04 −0.0 0 05+0 and −0.0 0 04+0 for the desired e-folds number, −0.001 −0.0 0 04 respectively, for GCG Model with quadratic potential (Figs. 1–3). • The inflationary parameters for MCG with quadratic poten.02 .0025 .125 tial lead to 0.97+0 (ns ), 0.0015+0 (α ) and 0.125+0 (r) −0.02 −0.0025 −0.125 (Figs. 4–6). • For GCG with quartic potential, the inflationary parameters re.05 .0055 .0045 main in the intervals 0.97+0 , 0.0045+0 and 0.0045+0 , −0.05 −0.0055 −0.0045 respectively (Figs. 7–9). • For MCG with quartic potential, the trajectories of inflation.05 .015 ary parameters correspond to intervals 0.94+0 , 0.01+0 and −0.05 −0.015 .0185 0.0185+0 , respectively (Figs. 10–12). −0.0185 It is worth mentioning that above results of inflationary parameters show consistency with observational data as given in Table 1. Zhang and Zhu [71] introduced the non-canonical warm inflationary scenario and reviewed the dynamics in the new scenario. They take the DBI inflation as a concrete example to find how the sound speed and the thermal dissipation strength to decide the non-Gaussianity and to get a lower bound of the sound speed constrained by Planck. Cai et al. [72] proposed a warm inflationary model in the context of relativistic D-brane inflation in a warped throat, which has DBI kinetic term. Various authors have examined warm inflation by considering the Chaplygin gas, standard and tachyon scalar field models in general relativity as well as in brane-world scenario with different expressions for the dissipative coefficient [73–79]. They found the consistency of their results with observational data, i.e., BICEP2, WMAP (7+9) and Planck data. Moreover, many authors have investigated warm inflation in various alternative as well as modified theories of gravity [80]. Recently, Rasouli et al. [49] investigated the inflation with the DBI non-canonical scalar field in both the cold and warm scenarios by assuming ADS warp factor for the DBI inflation and check viability of the quartic potential in light of the Planck 2015 observational results. They found good results of r − ns for both cold and warm inflation in agreement with Planck data. We extended this work by assuming different Chaplygin gas models in the presence of quadratic and quartic potential. We develop the trajectories of r − N, ns − N and α − N and found that our results are in agreement with Planck 2018.

A. Jawad and A. Rustam / Astroparticle Physics 117 (2020) 102402

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