# Reconstructing warm inflation

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## Abstract

The reconstruction of a warm inflationary universe model from the scalar spectral index \(n_S(N)\) and the tensor to scalar ratio *r*(*N*) as a function of the number of e-folds *N* is studied. Under a general formalism we find the effective potential and the dissipative coefficient in terms of the cosmological parameters \(n_S\) and *r* considering the weak and strong dissipative stages under the slow roll approximation. As a specific example, we study the attractors for the index \(n_S\) given by \(n_{S}-1\propto N^{-1}\) and for the ratio \(r\propto N^{-2}\), in order to reconstruct the model of warm inflation. Here, expressions for the effective potential \(V(\phi )\) and the dissipation coefficient \(\Gamma (\phi )\) are obtained.

## 1 Introduction

It is well known that during the evolution of the early universe, it exhibited an accelerated expansion or an inflationary scenario commonly called the inflationary universe [1, 2]. A crucial characteristic of the inflationary universe is that this scenario explicates the Large-Scale Structure (LSS) of the universe, and also the source of the anisotropies observed in the Cosmic Microwave Background (CMB) radiation [3, 4, 5, 6]. Although, inflation originally was proposed to solve some problems of the standard hot bing-bang model such as; the flatness, horizon, among other [1, 2].

In the context of the different models that give account of the inflationary universe and its early evolution, we can distinguish the model of warm inflation. In the framework of warm inflation, the universe is described by a self-interacting radiation field and a field scalar or inflaton field. In contradiction to the standard cold inflation, the model of warm inflation has the attractive feature that it avoids the reheating period, because the radiation production takes place concurrently together with the inflationary expansion driven by the scalar field [7, 8, 9, 10, 11]. This is possible through a friction term enclosed on the dynamical equations and this term describes the processes of the scalar field dissipating into a thermal bath with other fields. In this sense, the scenario of warm inflation ends whenever the universe stops inflating and softly goes into the radiation epoch of the standard big-bang model.

Another difference of warm inflation in relation to the cold inflation are the initial fluctuations essential for the LSS formation. In fact, during the development of warm inflation the thermal fluctuations have a fundamental role in the LSS formation and the density fluctuations from the scalar field arise from thermal rather than quantum fluctuations [12, 13, 14, 15, 16]. Thus, from the background dynamics and initial fluctuations, the stage of warm inflation differs substantially from the cold inflation (or the standard inflation) [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27]. For a review of models of warm inflation, see e.g. Refs. [11, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36] and for a list of recent articles, see [37, 38, 39, 40, 41, 42, 43, 44, 45].

On the other hand, the reconstruction of the effective potential in the evolution of cold inflation from observational data such as the scalar spectrum, scalar spectral index \(n_S\) and the tensor to scalar ratio *r*, have been discussed by several authors [46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56]. Here, we mentioned that the reconstruction of inflationary potentials for the case of a single scalar field assuming the primordial scalar spectrum was first made in Ref. [46], in which the general slow-roll approximation and without considering any specific form of the scalar spectrum index were assumed. An interesting mechanism in order to construct the effective potential of inflation assuming the slow roll approximation, is through the parametrization of the cosmological parameters or attractors \(n_S(N)\) and *r*(*N*), where *N* corresponds to the number of e-folds. The observational tests from Planck data [57] are in good accord with the parametrization on the scalar spectral index given by \(n_S\sim 1-2/N\) and the tensor to scalar ratio \(r\propto N^{-2}\), assuming that the number \(N\simeq 50-70\) at the end of the inflationary epoch. For large *N* (\(N\gg 1\)) the attractor \(n_S(N)\sim 1-2/N\) together with different expressions for the tensor to scalar ratio *r*(*N*) can be deduced from different models in the case of cold inflation such as; the T-model [58], E-model [59], Staronbisky \(R^2\)-model [1], the chaotic model [60], the model of Higgs inflation with non minimal coupling [61, 62, 63] among other.

On the other hand, it is also possible to consider the slow-roll parameter \(\epsilon \) and its parametrization in terms of *N*, in order to obtain the effective potential, scalar spectral index and the tensor to scalar ratio in models of cold inflation [52, 64, 65, 66]. In particular in Ref. [52] was studied different types of slow-roll parameter \(\epsilon (N)\) and thereby reconstructing the effective potential. Also, from the two slow roll parameters \(\epsilon (N)\) and \(\eta (N)\) the effective potential was reconstructed in Ref. [67]. Analogously, in Refs. [68, 69] related results are obtained for the reconstruction.

The objective of this article is to reconstruct the model of the warm inflation, considering the parametrization of the cosmological parameters as the scalar spectral index and the tensor to scalar ratio in terms of the number of e-folds. In this context, we analyze how the background dynamics in which there is a self-interacting scalar field and radiation affects the reconstruction of the effective potential and the dissipative coefficient from the attractors. Under a general formalism, we will build the potential and dissipative coefficient during the scenario of the weak and strong dissipative regimes from the attractors \(n_S(N)\) and *r*(*N*). Also, for the reconstruction the model of warm inflation we will consider the weak and strong dissipative regimes assuming the slow roll approximation.

In order to reconstruct analytical quantities for the potential and dissipative coefficient, we will study a concrete example for the cosmological parameters \(n_S(N)\) and *r*(*N*). Here, we will consider the attractors \(n_s-1\propto 1/N\) and \(r\propto 1/N^2\), for two regimes during the stages of warm inflation. In both scenarios, we will find the potential and dissipative coefficient together with the constraints on the different parameters assuming the condition for the weak regime and the strong regime, respectively.

The outline of the article is a follows: the next section presents a short review of the basic equations during the stage of warm inflation. In the Sect. 3 we discuss the reconstruction in the framework of warm inflation. In Sects. 4 and 5 we obtain under a general formalism, explicit expressions for the effective potential and dissipative coefficient in terms of the number of e-folds *N* during the weak and strong dissipative regimens, respectively. In Sect. 6 we discuss a concrete example for our model, in which we consider the specific attractors for \(n_S(N)\) and *r*(*N*), in order to find the potential \(V(\phi )\) and the coefficient \(\Gamma (\phi )\). Here, we analyze the reconstruction in the weak and strong dissipative regimes, respectively. Finally, our conclusions are presented in Sect. 7. We chose units so that \(c=\hbar =8\pi G=1\).

## 2 Warm inflation: basic relations

*a*corresponds to the scale factor. During the scenario of warm inflation we assume a two-component system, a scalar field homogeneous \(\phi =\phi (t)\) with an energy density \(\rho _\phi \) and a radiation field of energy density \(\rho _\gamma \). Here, the total energy density \(\rho =\rho _\phi +\rho _\gamma \), where the energy density \(\rho _\phi \) in terms of the scalar field is defined by \(\rho _\phi =\dot{\phi }^2/2+V\), where

*V*denotes the effective potential. In the following, we will consider that the dots correspond to differentiation with respect to the time.

In this context, the quantity \(\Gamma \) refers to the dissipation coefficient and considering the second law of thermodynamics the coefficient \(\Gamma \) is defined as positive [7, 8, 12, 13, 14]. In this sense, from Eqs. (2) and (3) we interpret that the coefficient \(\Gamma \) gives origin to the decay of the scalar field into radiation during the inflationary epoch of the universe. The parameter \(\Gamma \) can be considered to be a constant or a function of the scalar field \(\phi \), or the temperature of the thermal bath *T*, or both i.e., \(\Gamma =\Gamma (\phi ,T)\) [7, 8].

In the following we will analyze the reconstruction of the model of warm inflation, assuming that the dissipation coefficient and the effective potential depend only of the scalar field, i.e., \(\Gamma =\Gamma (\phi )\) and \(V=V(\phi )\), respectively.

*R*corresponds to the ratio between the coefficient \(\Gamma \) and the Hubble parameter and is defined as

*N*between two different values of cosmological times

*t*and \(t_e\), where the time \(t_e\) corresponds to the end of inflation. Thus, the number of e-folds

*N*assuming the slow roll approximation can be written as

*r*in the scenario of warm inflation can be written as

*r*in the model of warm inflation, see Eq. (13), cannot be written only in terms of the slow-roll parameter \(\epsilon \) as it occurs in cold inflation, in which \(r=16\epsilon \). Also, we observe that Eq. (13) coincides with the ratio

*r*obtained in Ref. [74].

In the following we will study the reconstruction of the effective potential *V* and the dissipative coefficient \(\Gamma \) in the scenario of warm inflation, considering an attractor point from scalar spectral index \(n_S(N)\) and the tensor to scalar ratio *r*(*N*) in the \(r{-}n_S\) plane.

## 3 Reconstruction

In this section and following Ref. [53], we explicate the procedure to follow in the reconstruction of the effective potential and the dissipation coefficient as a function of the scalar field \(\phi \) in the framework of warm inflation, considering the scalar spectral index \(n_S(N)\) and the tensor to scalar ratio *r*(*N*) as attractors. Since we have two quantities \(V(\phi )\) and \(\Gamma (\phi )\) in the reconstruction, we need first of all to express the scalar spectral index and the tensor to scalar ratio in terms of the number of e-folds *N*. For this it is necessary to rewrite Eqs. (11) and (13) in terms of the potential and the dissipation coefficient as a function of the number of e-folds *N* and its derivatives. Thus, from these equations and giving \(n_S=n_S(N)\) and \(r=r(N)\) we should obtain the effective potential and the dissipation coefficient as a function of the number *N*. Posteriorly, from Eq. (9) we should find the e-folds *N* in terms of the scalar field in order to reconstruct the potential \(V(\phi )\) and the coefficient \(\Gamma (\phi )\), respectively.

*N*. In fact, the slow roll parameters can be rewritten in terms of the number of e-folds

*N*, considering that

*T*from Eq. (8) results

*N*and the scalar field can be written as

*N*, considering the Eqs. (11), (17) and (18) such that

*T*is given by Eq. (19). Here, we have considered Eqs. (13) and (17).

In the following, we will restrict ourselves to the weak and strong dissipation regimes in order to reconstruct under a general formalism the model of warm inflation from the cosmological parameters \(n_S(N)\) and *r*(*N*).

## 4 The weak dissipative regime

*N*and the scalar field, is given by

*r*during this scenario can be written as

*N*can be written as

Here, we mention that the Eqs. (23), (27) and (28) are the fundamental equations in order to reconstruct of the effective potential \(V(\phi )\) and of the dissipation coefficient \(\Gamma (\phi )\), during this regime from the attractors \(n_S(N)\) and *r*(*N*).

## 5 The strong dissipative regime

*N*and the scalar field during this regime, is given by

*r*results

*T*in the strong regime is given by \(T=(V_{,\,N}/4C_\gamma )^{1/4}\) and the constant \(\bar{C}_\gamma \) is defined as \(\bar{C}_\gamma =8^5\,3C_\gamma /2\).

*V*(

*N*) can be written as

*V*(

*N*) in the weak and strong dissipative regime have the same structure, i.e., \(V(N)\sim r \exp [\int (1-n_S)dN]\) , see Eqs. (27) and (34).

*N*can be expressed as

Again, we refer to that the Eqs. (29), (34) and (35) are the fundamental expressions in order to reconstruct of the effective potential \(V(\phi )\) and \(\Gamma (\phi )\) from the quantities \(n_S(N)\) and *r*(*N*), during the strong dissipative regime.

## 6 An example

*r*(

*N*), in order to reconstruct analytically the effective potential \(V(\phi )\) and dissipative coefficient \(\Gamma (\phi )\). Following, Refs. [1, 53, 58] we consider that the spectral index is given by

*N*before the end of inflationary epoch at the horizon exit corresponds to \(N \simeq 60\), then the scalar spectral index and the tensor to scalar ratio given by relations (36) and (37) are well corroborate by observational data if \(\xi >-1/72\) for \(r<0.1\) [57] and \(\xi >-4/315\) for the ratio \(r<0.07\) [75]. In particular from Ref. [76] in which the ratio \(r<0.04\) (at \(1-\sigma \) confidence level) we have \(\xi >-0.0096\). Recently, in Ref. [77] was obtained different constraints on the parameter \(\xi \) from observational data. In the following we will assume that the number of e-folds

*N*is large, for values of \(N\sim \,\,\mathcal {O} (10^2)\).

As we mentioned before, the attractors given by Eqs. (36) and (37) in the limit \(\xi N\gg 1\) (such that \(r\propto 1/N^2\)) in the framework of cold inflation can be obtained in the E-model [59] and also in the model of the Higgs inflation with the nonminimal coupling [61, 62], see also Ref. [63]. A generalization of the attractors *r*(*N*) and \(n_S(N)\) are given by \(r=12\sigma /N^2\) and \(n_S-1=-2/N\) or also called \(\sigma \) attractor (or usually called \(\alpha \) attractor) which was proposed in Ref. [78], see also Ref. [79].

### 6.1 The weak regime

*V*(

*N*) is only obtained from \(n_S(N)\). In this sense, from the potential (38) we identify that \(\alpha \,\tilde{C_\gamma }^{1/4}\) corresponds to \(\alpha \) and the parameter \(\xi \rightarrow \beta /\alpha \) from cold inflation [53].

*N*becomes

*R*as a function of the number of e-folds in this regime is given by

*R*(

*N*) does not depend of the constant \(\xi \), when it is expressed in terms of the number of e-folds

*N*. Also, in order to obtain a scenario of weak dissipation in which \(R\ll 1\), we find a lower bound for the parameter \(\alpha \), given by \(\alpha \gg \frac{\tilde{C}_\gamma ^{3/4}}{\sqrt{3}}\,N^2\). In particular for large

*N*in which \(N=60\), we obtain the lower limit \(\alpha \gg 8^3\times 41.338\sim \,\mathcal {O}(10^7).\) Here, we have used \(C_\gamma =70\) [7, 8, 11].

In this form, during the stage of warm inflation we obtain a lower bound for the integration constant \(\alpha \), considering the condition of the weak dissipative regime \(\Gamma \ll 3H\). Also, we mention that this lower bound for the integration constant \(\alpha \gg \frac{\tilde{C}_\gamma ^{3/4}}{\sqrt{3}}\,N^2\), can not be obtained in the case of the reconstruction of the standard cold inflation from the background level, and it is only possible to say that \(\alpha >0\) [53].

*N*and \(\phi \) is given by the integral

*r*. In this sense, we notice that utilizing the specific attractors for \(n_S\) and

*r*given by Eqs. (36) and (37), the reconstruction in the model of warm inflation during the stage of weak dissipative regime coincides with the stage of cold inflation and it is a mere coincidence. Besides, we mention that this potential can also adapted to provide the Starobinsky model and \(\alpha -\)attractor model, see Ref. [53].

*r*takes the upper bound from Planck \(r=0.1\) [57] and \(\xi =-4/315\) for \(r=0.07\) [75]. In this sense, assuming the case in which the constant \(\xi <0\) and considering that \(\mid \xi ^{-1}\mid >N\), then the integration given by Eq. (41) becomes

In Fig. 1 we show the evolution of the ratio \(R=\frac{\Gamma }{3H}\) versus the number of e-folds *N* (left panel) and the dependence of the dissipative coefficient \(\Gamma \) on the scalar field (right panel) during the weak dissipative regime \(R\ll 1\). The left panel shows the condition of the weak dissipative regime in which \(\Gamma \ll 3H\), for three different values of \(\alpha \). In order to write down the rate \(R=\Gamma /3H\) in terms of the e-folding *N* during this regime, we consider Eq. (40). The right panel shows the evolution of the dissipation coefficient \(\Gamma \) as a function of the scalar field. Also, in order to write down the coefficient \(\Gamma \) in terms of the scalar field, we consider Eqs. (44), (48) and (50) for three different values of \(\xi \gtreqqless 0\), in which we have fixed \(8^{-3}\alpha =413.380\,\sim \mathcal {O}(10^6)\). In both panels we have considered \(C_\gamma =70\). From the left panel, we observe that the condition for the weak dissipative regime (\(R\ll 1\)) is satisfied for the values of the integration constant \(8^{-3}\alpha \gg 41.338\sim \,\mathcal {O}(10^5)\). Also, we consider in this plot the limit case of the weak scenario in which the rate \(R=1\), corresponding to the values \(N=60\) and \(8^{-3}\alpha =41.338\), respectively (dotted line).

From right panel, we note that the behaviors of the different parameters \(\Gamma =\Gamma (\phi )\) for the values of \(\xi \gtreqqless 0\) are similar. Also, we mention that from the relation given by Eq. (37) and considering values \(\xi >0\) together with large *N* i.e., \(N\sim \,\,\mathcal {O} (10^2)\), the tensor to scalar ratio \(r\sim 0\).

### 6.2 The strong regime

*N*results

*R*during the strong dissipative regime is given by

Nevertheless, from these solutions we find a transcendental equation from Eq. (29) to express the number of e-folds in function of the scalar field. Hence, in order to obtain analytical expressions for \(V(\phi )\) and \(\Gamma (\phi )\) and therefore the reconstructions, we can study the potential and the dissipation coefficient in the limits \( N\gg 1/\xi \) and \( N\ll 1/\xi \).

*r*(

*N*) given by Eq. (37) in this limit is approximately

*T*-model when \(\xi \) takes the value \(\xi =1/12\) in cold inflation [58]. Also, we note that from Eqs. (36) and (37) we get \(2\xi =(1+n_S)[(1+n_S)/2r-1]\) and considering the limit \(\xi N\gg 1\), we have \((1-n_s)\gg 4r\) and then the ratio \(r\ll 0.008\) in this limit.

*V*(

*N*) given by Eq. (51) results

On the other hand, now we consider the limit in which \(\xi N\ll 1\) where \(r(N)\approx 1/N\). We note this the attractor for large *N* and in particular for \(N=60\) results \(r(N=60)\approx 1/60\simeq 0.02\), wherewith still this attractor is well supported by the Planck data.

*V*(

*N*) becomes

*N*(in particular \(N=60\)) given by \(V(N=60)\gg 10^{-7}\) (in units of \(m_p^4\), with \(m_p\) the Planck mass).

*V*(

*N*) and the dissipation coefficient \(\Gamma (N)\) do not depend on the parameter \(\xi \), since \(r(N)\approx 1/N\).

In this sense, we observed that considering the attractor \(r(N)\approx 1/N\) (together with \(n_S-1=-2/N\)), the effective potential and the dissipation coefficient present a power law behavior during the strong regime, and its dependencies with the scalar field (reconstruction) are given by \(V(\phi )\sim \phi ^3\) and \(\Gamma (\phi )\sim \phi ^{5/2}\), respectively.

## 7 Conclusions

In this paper we have studied the reconstruction from recent cosmological observations in the framework of the warm inflation. Under a general formalism of reconstruction, we have found expressions for the effective potential and dissipative coefficient in the context of the slow roll approximation, motivated by the cosmological observations of the scalar spectral index \(n_S\) and tensor to scalar ratio *r*. In this general analysis we have obtained from the cosmological quantities \(n_S(N)\) and *r*(*N*) (where *N* corresponds to the number of e-folds), integrable expressions for the effective potential and dissipative coefficient. For warm inflation and its reconstruction, we have considered two different regimes, called the weak and strong dissipative regimes.

As a concrete example and in order to obtain the reconstructions for the effective potential \(V(\phi )\) and dissipation coefficient \(\Gamma (\phi )\), we have considered the attractors \(n_S-1=-2/N\) and \(r=(N[1+\xi N])^{-1}\). Here, we have applied our general results considering the weak and strong dissipative regimes for these attractors.

For the weak regime in which \(\Gamma \ll 3H\) (or equivalently \(R \ll 1\)) and considering the example or the attractors given by Eqs. (36) and (37), we have obtained a lower bound for the integration constant \(\alpha \) given by \(\alpha \gg \frac{\tilde{C}_\gamma ^{3/4}}{\sqrt{3}}\,N^2\), from the condition of weak dissipative regime i.e., \(R(N)\ll 1\). In particular for the case in which \(N=60\) (large *N*), we have found that the lower bound for the integration constant \(\alpha \) given by \(\alpha \gg 2\times 10^{7}\sim \mathcal {O}(10^7)\). Also, we have obtained that during the weak regime the reconstruction on the effective potentials are given by Eqs. (43), (47) and (49), and it coincides with the obtained in the case of cold inflation [53]. Similarly, we have obtained that the construction of the dissipative coefficients \(\Gamma (\phi )\) depends on the sign of the parameter \(\xi \gtreqless 0\). In particular for the case \(\xi =0\) where the potential corresponds to the chaotic potential, we have found that the dissipative coefficient \(\Gamma (\phi )\propto \phi ^5\).

For the case of the strong dissipative regime (\(R\gg 1\)) we have obtained the potential and dissipative coefficient in terms of the number of e-folds. During this regime, we have found that the potential *V*(*N*) has the same structure that in the weak regime. However, we could not find analytical solutions in order to obtain the number of e-fold in terms of the scalar field in form to obtain the reconstruction of \(V(\phi )\) and \(\Gamma (\phi )\). In this sense, we have analyzed the potential and the dissipation coefficient in the limits \( N\gg 1/\xi \) and \( N\ll 1/\xi \), in order to obtain analytical solutions. In the case in which \(r\propto N^{-2}\) (limit \(N\gg 1/\xi \)), we have obtained that the potential \(V(N)=\) constant, and the reconstruction does not work. For the case in which \(r\propto N^{-1}\) (limit \(N\ll 1/\xi \)) we have obtained that the potential and the dissipative coefficient in terms of the scalar field are given by \(V(\phi )\propto \phi ^3\) and \(\Gamma (\phi )\propto \phi ^{5/2}\), respectively.

Finally in this paper, we have not addressed the reconstruction of warm inflation in which the effective potential and dissipative coefficient also depend of the temperature of the thermal bath *T*, i.e., \(V(\phi ,T)\) and \(\Gamma (\phi ,T)\) [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 37, 38, 39, 40, 41, 42, 43, 44, 45, 73]. We hope to return to this point in the near future.

## Notes

### Acknowledgements

The author thanks Prof. Øyvind Grøn by the comment on the tensor to scalar ratio. This work was supported by Proyecto VRIEA-PUCV N\(_{0}\) 123.748/2017.

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