Reheating constraints on a two-field inflationary model

JID:NUPHB AID:114827 /FLA

[m1+; v1.304; Prn:30/10/2019; 8:54] P.1 (1-12)

Available online at www.sciencedirect.com

ScienceDirect 1

1 2

2

Nuclear Physics B ••• (••••) ••••••

3

www.elsevier.com/locate/nuclphysb

3

4

4

5

5

6

6 7

7 8

Reheating constraints on a two-field inflationary model

10

Kosar Asadi a , Kourosh Nozari a,b,∗

11 12 13

8 9

9

a Department of Physics, Faculty of Basic Sciences, University of Mazandaran, P. O. Box 47416-95447, Babolsar, Iran b Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), P. O. Box 55134-441, Maragha, Iran

14

Received 4 March 2019; received in revised form 25 October 2019; accepted 26 October 2019

15 16

Editor: Clay Córdova

10 11 12 13 14 15 16

17

17

18

18 19

19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

Abstract Once inflation comes to an end, the Universe enters a short stage named reheating. Making coherent oscillations about the minimum of its potential, the inflaton field transforms into the particles that occupy the Universe at later times. Yet, there is no direct cosmological observables corresponding to the reheating epoch. However, one can achieve some indirect limits, simply by taking into account cosmological evolution for observable CMB scales between the Hubble crossing time during inflation and the present time, for example. Moreover, since the reheating period takes place before the radiation dominated era, it can be modeled by an effective equation-of-state parameter, wreh . There is a limited range of reasonable values for wreh which can be regarded as the reheating period. In this paper, considering a two-field model, we investigate the reheating epoch so as to impose some constraints on the parameter space of the model. To do so, we are first required to obtain the reheating e-folds number as well as its temperature. These parameters can be written in terms of the scalar spectral index which let us perform numerical analysis in this model. Therefore, we can set some constraints on the reheating parameters which correspond to observationally viable values of the scalar spectral index. © 2019 Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .

20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

35

35

36

36

37

37

38

38

39

39

40

40 41

41 42 43 44 45 46 47

* Corresponding author at: Department of Physics, Faculty of Basic Sciences, University of Mazandaran, P. O. Box

42

47416-95447, Babolsar, Iran. E-mail addresses: [email protected] (K. Asadi), [email protected] (K. Nozari).

43

https://doi.org/10.1016/j.nuclphysb.2019.114827 0550-3213/© 2019 Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .

45

44

46 47

JID:NUPHB AID:114827 /FLA

2

1

[m1+; v1.304; Prn:30/10/2019; 8:54] P.2 (1-12)

K. Asadi, K. Nozari / Nuclear Physics B ••• (••••) ••••••

1. Introduction

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

1 2

2

Inflationary cosmology has become an extremely convincing paradigm to understand the early stage of the Universe evolution, since it addresses several open problems in cosmology, such as the question of the origin of cosmological structures. In its simplest realization, the expansion of the Universe is driven by a single, slowly-varying light scalar field named inflaton, whose potential energy dominates the expansion of the Universe. In this picture, the vacuum fluctuation of inflaton during inflation grows into super-Hubble density perturbations which are regarded to be the source of the structure formation [1–8]. Thus, the statistical properties of the Cosmic Microwave Background fluctuations as well as of the Large Scale Structures may contain some information about the physics of the early Universe. Moreover, inflation generates tensor perturbations, which result in a spectrum of primordial gravitational waves. Since these gravitational waves have effects on the CMB and other astronomical sources, they can reveal some information about inflation, too [9–13]. Despite all the advantages of a single field inflation model, it usually suffers from some fine tuning problems on the parameters of its potential, such as the mass and the coupling constant. However it has been shown that we can relax most of the limits on the single scalar field inflation by involving a number of fields [14]. None of these fields are able to terminate inflation separately though, they are able to work cooperatively to result an enough long inflationary stage [15–18]. Furthermore, there are some good reasons which can convince us that inflation might have been driven by more than one scalar field. Many theories beyond the standard model of particle physics (such as string theory, supergravity, and supersymmetry) include multiple scalar fields [19–27]. On the other hand, introducing more fields may offer more attractive features. For example, hybrid inflationary scenarios (which are driven by two separated scalar fields) can give sufficient inflationary expansion [28]. These models, having more natural values for their coupling constants, agree with the observed power spectrum of density perturbations very well. Moreover, in a single-field inflationary scenario, the evolving expectation value of the field specifies when the inflation phase ends and the standard model is resumed [29]. Taking two or more fields under consideration, however, evolution of each separated field can be influenced by the fluctuations in the other field(s). These are some reasons for the inflationary models containing two or more fields to become more important recently and thus many authors have studied such models in different literature [30–38]. As a recent work related to the multi-field (and more precisely two-field) inflationary model, we refer to [39] where universality of multi-field α-attractors has been studied in details. As another study in this regard, in Ref. [40] the authors constructed a model composed of two minimally coupled scalar fields where each of the fields have a potential of the α-attractor-type. However, we must emphasize that our study in this paper is not based on imposing a special kind of potential such as α-attractor, power-low, natural inflation and so on. In contrast, our focus is on obtaining the form of the potential corresponding to the model. In this regard, using the Hamilton-Jacobi formalism and deriving the potential, we attempt to study the reheating stage of the Universe evolution. Inflation is followed by a period that converts the stored energy density in the inflaton to the thermal bath (a plasma of relativistic particles) [41]. This transition from inflation to later stages of the Universe evolution, such as radiation and matter dominance, is regarded as reheating. During this period, the inflaton field loses its energy, which results in the production of ordinary matter. It is worth mentioning that, a variety of reheating models have been suggested so far. However, despite being very unconstrained and uncertain, reheating consists of a simple canonical scenario by which the inflaton performs coherent oscillations around the minimum of its

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:114827 /FLA

[m1+; v1.304; Prn:30/10/2019; 8:54] P.3 (1-12)

K. Asadi, K. Nozari / Nuclear Physics B ••• (••••) ••••••

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38

3

potential and decays into relativistic particles corresponding to the radiation dominated period [42–44]. Along with this simple canonical reheating model, there are some more intricate scenarios including the non-perturbative processes [45–47]. We note that various clocks for starting the normal radiation dominated period can be set according to different rates for different types of decays. There might be a short preheating stage, for example, which refers to the initial stage of the reheating [48,49]. Actually, when the decay occurs exponentially, it produces high occupation numbers in selected frequency bands. Therefore, right after the preheating stage, these frequency bands, on the contrary of the rest of the space, will have extremely high occupation numbers. So there will be a resonant production of particles that can boost up the decay rate of the inflaton field by scattering [47]. This scattering will lead to spreading out the distribution which results in a blackbody spectrum characterized by a final temperature, normally corresponding to the temperature at the beginning of the radiation-dominated era [50]. Generally, it is assumed that the temperature of the reheating, Treh , is above the electroweak transition (Treh > 100 GeV). Moreover, this temperature must also be above 10 MeV, which corresponds to the temperature of the big-bang nucleosynthesis (BBN). Therefore Treh can be considered as an important parameter characterizing the reheating epoch. Another parameter associated with this era is the number of e-folds, Nreh . So investigating these parameters in an inflationary model can help us to find some constraints on the model’s parameter space [50–57]. Along with the number of e-folds and temperature of the reheating era, studying the effective equation of state parameter during this stage, that is wreh , is an issue that can be taken into account [50,58]. The value of this parameter lies between −1 (potential energy domination) and +1 (kinetic energy domination). It is obvious that the effective equation of state parameter should have values larger than − 13 , which is needed to end inflation. This parameter is also assumed to be smaller than +1 so as not to violate causality. On the other hand, its value at the beginning of the radiation dominated era must be + 13 . Therefore it might be more rational to consider this parameter during the reheating phase of the Universe in the range − 13 < wreh < + 13 [53]. We should note that it is not easy to conceive wreh > 13 because it requires a potential which is dominated by higher dimensional operators near its minimum. This feature is somehow unnatural from a quantum field theoretical point of view. So far we learned that investigating the process by which the Universe reheats may provide additional constraints on an inflationary model. Therefore, in this paper, considering a two-field inflationary model, we study the reheating phase. Our strategy differs with the existing studies in the sense that instead of adopting a suitable potential, we construct the required potential. Then we calculate the number of e-folds during the reheating stage of the Universe as well as reheating temperature. We also analyze the effective equation of state parameter over this period and finally constraint the parameter space of the model by comparing the results with the observational data. To this end, we use the recently released Planck2018 data [59].

43 44

47

4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38

41

We consider an inflationary model driven by two ordinary scalar fields which are minimally coupled to gravity and described by the following action

45 46

3

40

2. Cosmological inflation

41 42

2

39

39 40

1

 S=

√

d x −g 4



2 Mpl



1 1 R − ∂μ φ∂ μ φ − ∂μ χ∂ μ χ − V (φ, χ) 2 2 2

42 43 44 45

,

(1)

46 47

JID:NUPHB AID:114827 /FLA

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

[m1+; v1.304; Prn:30/10/2019; 8:54] P.4 (1-12)

K. Asadi, K. Nozari / Nuclear Physics B ••• (••••) ••••••

4

where, R is the Ricci scalar and Mpl = (8πG)−1/2 is the reduced Planck mass. φ and χ are two ordinary scalar fields and V is the potential of the model which is a function of both the fields. We consider a spatially flat FRW spacetime ds 2 = −dt 2 + a 2 (t)δij dx i dx j ,

1 2 3 4

(2)

where a(t) is the scale factor whose evolution is given by the following Friedmann equations   1 1 1 H2 = (3) φ˙ 2 + χ˙ 2 + V (φ, χ) , 2 2 2 3Mpl  1  (4) −2H˙ = 2 φ˙ 2 + χ˙ 2 . Mpl a˙ a

Here, H = is the Hubble parameter with a dot denoting the derivative with respect to the cosmic time, t. The equations of motion for fields are given by

5 6 7 8 9 10 11 12 13 14 15

16

φ¨ + 3H φ˙ + V,φ = 0 ,

(5)

16

17

χ¨ + 3H χ˙ + V,χ = 0 ,

(6)

17

18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

18

where “ , ” represents the derivative with respect to the scalar field. Following the definition of ¨ ˙ the slow-roll parameters as  ≡ − HH2 and η ≡ − H1 H , we obtain these parameters in our setup as H˙ follows 3(φ˙ 2 + χ˙ 2 ) , (7) = 2 φ˙ + χ˙ 2 + 2V −2(φ˙ φ¨ + χ˙ χ¨ ) η= . (8) H (φ˙ 2 + χ˙ 2 ) To have inflationary phase, one can easily impose the slow-roll limit in which these parameters should satisfy the conditions   1 and η  1. However, it is worth mentioning that slow-roll approximation is not the only possibility for successfully performing models of inflation. In other words, solutions beyond the slow-roll approximation have been found in particular situations as well. Therefore, we have not applied this approximation in the calculation of the slow-roll parameters. Instead, our method is based on the first-order Hamilton-Jacobi formalism in which inflation is defined as a period of accelerated expansion, ( aa¨ > 0). During the slow-roll inflation the expansion of the universe is of the de Sitter form with a constant Hubble parameter, H , and exponentially increasing scale factor, a, (H  const, and a ∝ eH t ). However the Hubble parameter is not exactly a constant. More precisely, one can consider it evolving gently along the potential (just as the field(s) does). An appropriate approach to a more general case is to consider this parameter evolving directly as a function of the field(s) instead of time [62–69]. Therefore, following [61], we proceed our analysis with reformulating the main equations of the model as the first order Hamilton-Jacobi equations. Describing the dynamics of the inflation using the Hamilton-Jacobi formalism, we focus on solutions satisfying the sum separable Hubble parameter given by

45 46 47

20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43

43 44

19

H = H0 + H1 φ + H2 χ .

(9)

44

In the first order Hamilton-Jacobi system, the equations of motion for the fields can be written as

45

2 H,φ , φ˙ = −2Mpl

(10)

46 47

JID:NUPHB AID:114827 /FLA

[m1+; v1.304; Prn:30/10/2019; 8:54] P.5 (1-12)

K. Asadi, K. Nozari / Nuclear Physics B ••• (••••) ••••••

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

2 χ˙ = −2Mpl H,χ ,

5

(11)

where we have focused on a homogeneous universe. These first order equations of motion result in the following solutions 2 φ(t) = −2Mpl H1 t 2 χ(t) = −2Mpl H2 t

+ φ0 ,

(12)

+ χ0 .

(13)

Moreover, the scale factor of the model gets the following expression

2 2 2 2 a(t) = a0 exp (H0 + H1 φ0 + H2 χ0 )t − Mpl (H1 + H2 )t ,

6H1 H2 φχ + 6H0 H1 φ + 6H0 H2 χ

+ 3H12 φ 2

+ 3H22 χ 2

20 21

(14)

24 25 26

29 30 31 32 33 34 35 36

39

tend N=

(16)

H dt ,

φend 

N =− φ∗

3H 2 dφ − V,φ

χ end

13 14 15 16

19 20 21

23

χ∗

26 27

3H 2 dχ V,χ

28

(17)

29 30

where φ∗ and χ∗ refer to the value of the scalar fields when the universe scale crosses the Hubble horizon during inflationary era, whereas φend and χend indicate the value of both fields when the universe scale exits the inflationary phase. So, in order to calculate the perturbation parameters of the model, such as the scalar spectral index (ns ) and tensor-to-scalar ratio (r), we are required to use the power spectrum which is evaluated at the horizon crossing time, Cs k = aH (with k being the comoving wave number) and given by

31 32 33 34 35 36 37

H2 As = 2 8π Ws Cs3

(18) φ˙ 2 +χ˙ 2 2 H2 2Mpl

38 39 40

42

and Cs2 = 1. By using the given power spectrum, the scalar

spectral index can be obtained as follows

41 42 43

43

47

12

25

and takes the following form in our two-field setup

where by definition Ws ≡

46

10

24

t∗

41

45

7

22

40

44

6

17

.

37 38

5

18

27 28

4

11

After deriving the form of the potential in this two field scenario, we are in the position to explore the inflationary phase of the model more precisely. The number of e-folds during inflation is defined by

22 23

3

9

18 19

2

8

where the parameters φ0 and χ0 refer to the initial value of each ordinary fields (φ and χ respectively) at the time that observable scales exit the horizon. Besides, using the Friedmann equation (3) we obtain the corresponding potential of the scalar fields as follows  2 2 (15) V (φ, χ) = 3H02 − 2Mpl H12 − 2Mpl H22 +

17

1

1 d ns − 1 = −2 − ln  . (19) H dt Another important inflationary parameter is the tensor-to-scalar ratio, which in this model takes the following form

44 45 46 47

JID:NUPHB AID:114827 /FLA

1

r=

2 3 4 5 6 7 8

[m1+; v1.304; Prn:30/10/2019; 8:54] P.6 (1-12)

K. Asadi, K. Nozari / Nuclear Physics B ••• (••••) ••••••

6

AT = 16 . As

(20)

2

Up to this point we obtained the main equations of this inflationary setup. In what follows, to test this scenario more precisely, we study the reheating stage of the Universe evolution which takes place after the inflationary phase. To this end, we calculate the expressions of the number of e-folds and temperature during reheating. Performing numerical analysis on their behavior, we obtain some constraints on these parameters through confrontation with the observationally viable values of the scalar spectral index.

13 14 15 16 17 18 19 20 21 22 23

26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

As mentioned previously, the process of reheating happens in the early stages of the Universe evolution. This is considered to be necessary since as the universe expands, it cools down. Thus, right after inflation, there must be a period to make it prepare thermally for next steps. By taking this reheating period into account, we are able to find some constraints on the model’s parameter space. To this end, considering the Universe expansion between the time of the horizon crossing during inflation and the time of re-entering, can be regarded as a possibility for gaining information about the reheating period. In cosmology, perturbation modes can be observed on scales comparable to the size of horizon. The comoving Hubble scale can also be related to that of the present time as follows k ak aend areh aeq Heq Hk = , a0 H0 aend areh aeq a0 H0 Heq

6 7 8

12 13 14 15 16 17 18 19 20 21 22

(21)

23 24

where k can be chosen as the pivot scale for a specific experiment and parameters with this subscript are calculated at the time of horizon crossing. Other subscripts also show what epochs the regarded parameters are evaluated at. For example, subscripts “end”, “eq” and “reh” refer to the end of inflation, radiation-matter equality and reheating stage, respectively. Subscript “0” stands for the present time values. The constraint on the total amount of expansion can be obtained as

25 26 27 28 29

aeq Heq k Hk = −Nk − Nreh − Nrad + ln + ln , (22) a0 H0 a0 H0 Heq   where Nk = ln aaend indicates the e-folds number from the time that modes exit the horizon and k     aeq reh and Nrad = ln areh illustrate the number of e-folds the end of inflation, while Nreh = ln aaend

30

from the end of inflation to the end of reheating and from the end of reheating to the end of the radiation dominated period, respectively. The energy density over the reheating era is given by ρ ∼ a −3(1+wreh ) , with wreh being the effective equation of state parameter during this epoch. Therefore, one can write the following relation for the number of e-folds during reheating as a function of the energy density and effective equation of state parameter

 1 ρend Nreh = ln . (23) 3(1 + wreh ) ρreh

36

ln

ρreh =

π2 30

31 32 33 34 35

37 38 39 40 41 42 43 44

The final energy density specifies the temperature of reheating through the relation [50,56]

46 47

5

11

24 25

4

10

3. Reheating

11 12

3

9

9 10

1

45 46

4 greh Treh

(24)

47

JID:NUPHB AID:114827 /FLA

[m1+; v1.304; Prn:30/10/2019; 8:54] P.7 (1-12)

K. Asadi, K. Nozari / Nuclear Physics B ••• (••••) ••••••

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

where greh is the effective number of relativistic species upon thermalization. By assuming the conservation of entropy we have

  a0 3 7 3 3 gs,reh Treh = 2T03 + 6 × Tν0 , (25) areh 8

1

with T0 being the temperature of the CMB at the present time, that is, T0 = 2.725 K. Tν0 is the current neutrino temperature which gives

1 4 3 Tν0 = T0 . (26) 11

6

So, we can relate the reheating temperature to the present CMB temperature through the following relation

1 3 a Treh 43 0 aeq = . (27) T0 11gs,reh aeq areh Using equations (24) and (27) we find

18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

7

a0 = areh

11gs,reh 43

 1 3

π 2 greh 30ρreh

2 3 4 5

7 8 9 10 11 12 13 14 15 16 17



18

− 14

.

(28)

20

In what follows we are going to obtain the explicit expressions for the reheating number of e-folds and temperature. The energy density in this scenario during inflation is given by

−1 κ 2 ρ =V 1− , (29) 3

22 23 24 25

27

ρend = (1 + λ) Vend ,

(30) 3 −1 with λ being the ratio of kinetic energy to potential energy, which is defined as λ =  − 1 . From equations (23) and (30) we have

By using equations (28) and (32) we obtain

2 



a0 43 π greh 1 1 ln = − ln − ln − areh 3 11greh 4 30ρreh 3 1 ln T0 + ln [(1 + λ)Vend ] − (1 + wreh )Nreh . 4 4 Finally we obtain the following expression for the reheating number of e-folds

 4 khc Nreh = − Nk − ln − 1 − 3wreh a 0 T0

   1 40 11greh 1  2 1 3 A W C ln ln + ln 8π − s s s − 4 π 2 greh 3 43 2

1 ln [(1 + λ)Vend ] . 4

21

26

which can also be written as follows

ρreh = (1 + λ)Vend exp [−3Nreh (1 + wreh )] .

19

(31)

28 29 30 31 32 33 34 35 36 37

(32)

38 39 40 41 42 43 44 45 46

(33)

47

JID:NUPHB AID:114827 /FLA

[m1+; v1.304; Prn:30/10/2019; 8:54] P.8 (1-12)

K. Asadi, K. Nozari / Nuclear Physics B ••• (••••) ••••••

8

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14 15

15 16 17 18 19 20 21

Fig. 1. The number of e-folds during reheating, Nreh , and the temperature during reheating, Treh , versus the scalar spectral index ns , in a two-field model using the sum separable Hubble parameter are shown in the left and right panels, respectively. These figures are plotted for different values of the equation of state parameters, w. The pink region shows the values of the scalar spectral index released by Planck2018 TT+TE+EE+ LowP data whereas the horizontally highlighted regions demonstrate the temperatures below the electroweak scale, T < 100 GeV and below the big bang nucleosynthesis scale, T < 10 MeV, respectively. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)

The reheating temperature can also be obtained from equations (27), (28) and (30) as follows

24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

17 18 19 20 21 22

22 23

16

Treh =

30 π 2 greh

24

1

4

23

25

×

 1 3 [(1 + λ)Vend ] 4 exp − Nreh (1 + wreh ) . 4

26 27

(34)

Therefore, the number of e-folds as well as the temperature during the reheating epoch can be obtained in terms of the values of the scalar fields at the horizon crossing (φk and χk ) simply by substituting the form of the potential (15) in the equations (33) and (34). On the other hand, the values of the scalar fields at horizon crossing is related to the scalar spectral index, ns , through equation (19). So we can analyze the reheating parameters numerically. In our numerical calculations, we have set greh = gs,reh = 100 and k = 0.05 Mpc−1 . Also  = 1, since it is evaluated at the end of inflation. We should emphasize that the e-folds number during reheating era has only logarithmic dependence on , greh and gs,reh . So, it is not seriously influenced by the precise values of these parameters. However on the contrary, the expression depends linearly on scalar spectral index through Nk , and is sensitive to wreh . Furthermore, we have set H0 = 0.16, H1 = 0.006 and H2 = 0.0068, which are chosen so that for a defined number of efolds, the inflation phase can terminate. Note also that we considered the case where inflation is initially driven by not dominantly one of the scalar fields but both of them. The initial values for the two fields are assumed φ0 = χ0 = 0.7. After applying these initial values of the parameters and computing Nreh and Treh , we can study these two parameters related to the reheating era numerically. We proceed our numerical analysis by adopting some rational sample values for the equation of state parameter, − 13 < wreh < 1. The corresponding results are shown in Fig. 1. We note that the value of the scalar spectral index has been used as ns = 0.9649 ± 0.0044 from

28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:114827 /FLA

[m1+; v1.304; Prn:30/10/2019; 8:54] P.9 (1-12)

K. Asadi, K. Nozari / Nuclear Physics B ••• (••••) ••••••

1 2 3

Table 1 The ranges of the reheating number of e-folds and temperature for a two-field inflationary model which are consistent with observational data. Nreh

Treh

w = − 13

4.83 < Nreh < 33.87

w = − 16

6.18 < Nreh < 45.43

w=0

9.14 < Nreh < 67.47

w = 16

Nreh > 17.20

Treh < 9.3 2.98 < log10 GeV   Treh −1.96 < log10 GeV < 8.57   Treh < 7.26 log10 GeV   Treh log10 GeV < 3.42

w = 23 w=1

Not Consistent Not Consistent

Not Consistent Not Consistent

4 5 6 7 8 9 10 11 12

9





1 2 3 4 5 6 7 8 9 10 11 12

13

13

14

14

15

15

16

16

17

17

18

18

19

19

20

20

21

21

22

22

23

23

24

24

25

25

26

26

27

27

28

28 29

29 30 31 32

Fig. 2. The ranges of the parameters Nreh and w which lead to the observationally viable values of the scalar spectral index corresponding to Planck2018 TT+TE+EE+ LowP data, for a two-field inflationary model beyond the slow-roll approximation.

35 36 37 38 39 40 41 42 43 44 45 46 47

31 32 33

33 34

30

Planck2018 TT+TE+EE+ LowP data [59]. It is worth mentioning that this recently released value of ns is in good agreement with the former Planck2015 TT, TE, EE+lowP %68 CL result which is ns = 0.9645 ± 0.0049 [59,70]. The left panel of Fig. 1 is plotted for various values of the equation of state parameter. This helps us to find the viable ranges of the number of e-folds in each case that is consistent with the observational data. For example, our results for wreh = − 16 indicate that the number of e-folds should satisfy the condition 6.18 < Nreh < 45.43 to be consistent with observational data. However, taking wreh = 23 , the corresponding two-field inflationary model is not consistent with observation under any circumstances. Similarly, the right panel of this figure shows the numerical analysis of temperature in reheating epoch for different values of wreh . Again for wreh = 23 and wreh = 1 this model is not consistent with observation. But for other values of the equation of state parameter there is a range in which the model is consistent with observational data. The results are summarized in Table 1. It can be seen form both panels of Fig. 1 that there is a point that all curves, which refer to different values of wreh , meet each other. It is obvious that this point corresponds to Nreh = 0. Moreover, Fig. 2 shows the e-folds

34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:114827 /FLA

1 2

[m1+; v1.304; Prn:30/10/2019; 8:54] P.10 (1-12)

K. Asadi, K. Nozari / Nuclear Physics B ••• (••••) ••••••

10

number versus the equation of state parameter for observationally viable values of the scalar spectral index. 4. Summary

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

In this paper we have considered an inflationary model driven by two ordinary scalar fields which are minimally coupled to the gravity. Our main focus was on studying the reheating period which happens right after the end of inflation. So, by relating the end of the inflationary epoch to the beginning of the radiation-dominated era, one can make quantitative inferences about the physics of reheating. Moreover, investigating the corresponding parameters during reheating may well give some constraints on the model’s parameter space. The e-folds number, Nreh , as well as the reheating temperature, Treh , are two parameters that can be studied for a given model during reheating era. In order to study the reheating phase of a two-field model, we firstly derived the basic equations of the model. Then we have tested our model with recent observational data. We emphasize that we proceeded our analysis beyond the slow-roll approximation. In other words, since the slow-roll condition can be violated during the inflationary era, it is possible to go beyond this approximation. So, instead of adopting this approximation, sum separable Hubble parameter has been assumed; which means our analysis have been done based on the first order Hamilton-Jacobi formalism. This formalism allows us to express inflationary parameters of the model at hand without imposing the slow-roll limit. After obtaining the main equations, we expressed Nreh and Treh as a function of the model’s parameters. Then, regarding the relation of the scalar fields at horizon crossing time with the scalar spectral index, it would be possible to study the number of e-folds and temperature at the reheating versus the scalar spectral index. Performing numerical analysis and considering the values of the scalar spectral index released by Planck2018 TT, TE, EE+lowE data, we have plotted the regions of Nreh for various values of wreh to see whether they are observationally viable or not. Then we have found the ranges in which the model is consistent. Studying the temperature during reheating era also resulted in some more constraints on the parameter’s space of the scenario. All these constraints, which are based on the observationally viable values of ns , are presented in Table 1. As one can understand from the results shown by Figs. 1 and 2, the model is in well agreement with recent observational data for − 13 < wreh < 13 . Finally, we note that the analysis presented here is two-field inflation which has isocurvature perturbations and can be essentially non-Gaussian. The more rigorous approach would be to extend the simulation analysis presented in [71] for the two-field model. This issue is the subject of our forthcoming study. Declaration of competing interest There is no conflict of interest in this work. Acknowledgement

45 46 47

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

38

40

42 43

43 44

8

41

41 42

7

39

39 40

6

37

37 38

4 5

5 6

2 3

3 4

1

We would like to appreciate two anonymous referees for very insightful comments that have boosted the quality of the paper considerably. The work of K. Nozari has been supported financially by Research Institute for Astronomy and Astrophysics of Maragha (RIAAM) under research project number 1/6025-51.

44 45 46 47

JID:NUPHB AID:114827 /FLA

[m1+; v1.304; Prn:30/10/2019; 8:54] P.11 (1-12)

K. Asadi, K. Nozari / Nuclear Physics B ••• (••••) ••••••

1

11

Uncited references

3

[60] References

8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41]

42 43 44 45 46 47

5 6

6 7

3 4

4 5

1 2

2

[42] [43] [44] [45] [46] [47]

A. Guth, Phys. Rev. D 23 (1981) 347. A.D. Linde, Phys. Lett. B 108 (1982) 389. A. Albrecht, P. Steinhard, Phys. Rev. D 48 (1982) 1220. A.D. Linde, Harwood Academic Publishers, Chur, Switzerland, arXiv:hep-th/0503203, 1990. A. Liddle, D. Lyth, Cambridge University Press, 2000. J.E. Lidsey, et al., Rev. Mod. Phys. 69 (1997) 373. A. Riotto, arXiv:hep-ph/0210162. D.H. Lyth, A.R. Liddle, Cambridge University Press, Cambridge, England, 2009. T.J. Allen, B. Grinstein, M.B. Wise, Phys. Lett. B 197 (1987) 66. T. Falk, R. Rangarajan, M. Srednicki, Phys. Rev. D 46 (1992) 4232. A. Gangui, F. Lucchin, S. Matarrese, S. Mollerach, Astrophys. J. 430 (1994) 447. V. Acquaviva, N. Bartolo, S. Matarrese, A. Riotto, Nucl. Phys. B 667 (2003) 119–148, arXiv:astro-ph/0209156. J.M. Maldacena, J. High Energy Phys. 0305 (2003) 013. A.R. Liddle, A. Mazumdar, F.E. Schunck, Phys. Rev. D 58 (1998) 061301. K.A. Malik, D. Wands, Phys. Rev. D 59 (1999) 123501. P. Kanti, K.A. Olive, Phys. Rev. D 60 (1999) 043502. E.J. Copeland, A. Mazumdar, N.J. Nunes, Phys. Rev. D 60 (1999) 083506. A.M. Green, J.E. Lidsey, Phys. Rev. D 61 (2000) 067301. P. Nath, R. Arnowitt, Phys. Lett. B 56 (1975) 177. D.Z. Freedman, P. van Nieuwenhuizen, S. Ferrara, Phys. Rev. D 13 (1976) 3214. E. Cremmer, B. Julia, J. Scherk, Phys. Lett. B 76 (1978) 409. M.B. Green, J.H. Schwarz, Phys. Lett. B 149 (1984) 117. M. Duff, P. Howe, T. Inami, K. Stelle, Nucl. Phys. B 191 (1987) 70. E. Bergshoeff, E. Sezgin, P. Townsend, Phys. Lett. B 189 (1987) 75. P. Candelas, G. Horowitz, A. Strominger, E. Witten, Nucl. Phys. B 258 (1985) 46. M. Duff, Int. J. Mod. Phys. A 11 (1996) 5623. J. Maldacena, Adv. Theor. Math. Phys. 2 (1998) 231. A.D. Linde, Phys. Rev. D 49 (1994) 748. V.F. Mukhanov, P.J. Steinhardt, Phys. Lett. B 422 (1998) 52. A.D. Linde, Phys. Lett. B 38 (1991) 259. E.J. Copeland, A.R. Liddle, D.H. Lyth, E.D. Stewart, D. Wands, Phys. Rev. D 49 (1994) 6410. S. Tsujikawa, H. Yajima, Phys. Rev. D 62 (2000) 123512. P.R. Ashcroft, C. van de Bruck, A.-C. Davis, Phys. Rev. D 66 (2002) 121302. L.E. Allen, S. Gupta, D. Wands, J. Cosmol. Astropart. Phys. 0601 (2006) 006. D. Langlois, J. Phys. Conf. Ser. 140 (2008) 012004. J. Frazer, A.R. Liddle, arXiv:1111.6646 [astro-ph.CO]. A. Mazumdar, L. Wang, J. Cosmol. Astropart. Phys. 09 (2012) 005. J. White, M. Minamitsuji, M. Sasaki, J. Cosmol. Astropart. Phys. 09 (2013) 015. K. Maeda, S. Mizuno, R. Tozuka, arXiv:1810.06914 [hep-th]. A. Achúcarro, R. Kallosh, A. Linde, D.-G. Wang, Y. Welling, J. Cosmol. Astropart. Phys. 04 (2018) 028, arXiv: 1711.09478 [hep-th]. R. Allahverdi, R. Brandenberger, F.Y. Cyr-Racine, A. Mazumdar, Annu. Rev. Nucl. Part. Sci. 60 (2010) 27, arXiv: 1001.2600 [hep-th]. L.F. Abbott, E. Farhi, M.B. Wise, Phys. Lett. B 117 (1982) 29. A.D. Dolgov, A.D. Linde, Phys. Lett. B 116 (1982) 329. A.J. Albrecht, P.J. Steinhardt, M.S. Turner, F. Wilczek, Phys. Rev. Lett. 48 (1982) 1437. L. Kofman, A.D. Linde, A.A. Starobinsky, Phys. Rev. Lett. 73 (1994) 3195. J.H. Traschen, R.H. Brandenberger, Phys. Rev. D 42 (1990) 2491. L. Kofman, A.D. Linde, A.A. Starobinsky, Phys. Rev. D 56 (1997) 3258.

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:114827 /FLA

12

1 2 3 4

[48] [49] [50] [51] [52]

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

[53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71]

[m1+; v1.304; Prn:30/10/2019; 8:54] P.12 (1-12)

K. Asadi, K. Nozari / Nuclear Physics B ••• (••••) ••••••

M.A. Amin, M.P. Hertzberg, D.I. Kaiser, J. Karouby, arXiv:1410.3808 [hep-ph]. G.N. Felder, L. Kofman, A.D. Linde, Phys. Rev. D 59 (1999) 123523. J.L. Cook, E. Dimastrogiovanni, D. Easson, L.M. Krauss, J. Cosmol. Astropart. Phys. 04 (2015) 047. B.R. Greene, T. Prokopec, T.G. Roos, Phys. Rev. D 56 (1997) 6484, arXiv:hep-ph/9705357. R. Micha, I.I. Tkachev, Phys. Rev. Lett. 90 (2003) 121301; R. Micha, I.I. Tkachev, Phys. Rev. D 70 (2004) 043538. L. Dai, M. Kamionkowski, J. Wang, Phys. Rev. Lett. 113 (2014) 041302. J.B. Munoz, M. Kamionkowski, Phys. Rev. D 91 (2015) 043521. R.-G. Cai, Z.-K. Guo, S.-J. Wang, Phys. Rev. D 92 (2015) 063506. Y. Ueno, K. Yamamoto, Phys. Rev. D 93 (2016) 083524. K. Nozari, N. Rashidi, Phys. Rev. D 95 (2017) 123518. D.I. Podolsky, G.N. Felder, L. Kofman, M. Peloso, Phys. Rev. D 73 (2006) 023501, arXiv:hep-ph/0507096. Y. Akrami, et al., arXiv:1807.06211 [astro-ph.CO]. D.S. Salopek, J.R. Bond, Phys. Rev. D 42 (1990) 3936. K.A. Malik, D. Wands, Phys. Rev. D 59 (1999) 123501. W.H. Kinney, Phys. Rev. D 56 (1997) 2002. D.S. Salopek, J.M. Stewart, K.M. Croudace, Mon. Not. R. Astron. Soc. 271 (1994) 1005. D.S. Salopek, Class. Quantum Gravity 15 (1998) 1185. D.S. Salopek, Class. Quantum Gravity 16 (1999) 299. K. Skenderis, P.K. Townsend, Phys. Rev. D 74 (2006) 125008. K. Tzirakis, W.H. Kinney, arXiv:0810.0270 [astro-ph]. J. Emery, G. Tasinato, D. Wands, J. Cosmol. Astropart. Phys. 08 (2012) 005. C.T. Byrnes, G. Tasinato, J. Cosmol. Astropart. Phys. 0908 (2009) 016, arXiv:0906.0767. P.A.R. Ade, et al., arXiv:1502.02114 [astro-ph.CO]. K.D. Lozanov, M.A. Amin, Phys. Rev. Lett. 119 (2017) 061301.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

22

22

23

23

24

24

25

25

26

26

27

27

28

28

29

29

30

30

31

31

32

32

33

33

34

34

35

35

36

36

37

37

38

38

39

39

40

40

41

41

42

42

43

43

44

44

45

45

46

46

47

47