# Warm DBI inflation with constant sound speed

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## Abstract

We study inflation with the Dirac–Born–Infeld (DBI) noncanonical scalar field in both the cold and warm scenarios. We consider the Anti-de Sitter warp factor \(f(\phi )=f_{0}/\phi ^{4}\) for the DBI inflation and check viability of the quartic potential \(V(\phi )=\lambda \phi ^{4}/4\) in light of the Planck 2015 observational results. In the cold DBI setting, we find that the prediction of this potential in the \(r-n_s\) plane is in conflict with Planck 2015 TT, TE, EE + low P data. This motivates us to focus on the warm DBI inflation with constant sound speed. We conclude that in contrary to the case of cold scenario, the \(r-n_s\) result of warm DBI model can be compatible with the 68% CL constraints of Planck 2015 TT, TE, EE + low P data in the intermediate and high dissipation regimes, whereas it fails to be observationally viable in the weak dissipation regime. Also, the prediction of this model for the running of the scalar spectral index \(dn_s/d\ln k\) is in good agreement with the constraint of Planck 2015 TT, TE, EE + low P data. Finally, we show that the warm DBI inflation can provide a reasonable solution to the swampland conjecture that challenges the de Sitter limit in the standard inflation.

## 1 Introduction

Consideration of a short period of rapid inflationary expansion before the radiation dominated era can provide reasonable explanation for the well-known puzzles of the Hot Big Bang cosmology [1, 2, 3, 4, 5, 6, 7]. Also, the quantum fluctuations during inflation can lead to the perturbations whose we can see the imprints on the large-scale structure (LSS) formation and the anisotropies of cosmic microwave background (CMB) radiation [8, 9, 10, 11]. Inflationary models generally predict an almost scale-invariant power spectrum for the primordial perturbations. This prediction has been confirmed by the recent observational data from the Planck satellite [12, 13]. Although, these observational data as well support the inflation paradigm, so far we cannot determine the dynamics of inflation exclusively.

In the standard inflationary scenario, a canonical scalar field minimally coupled to the Einstein gravity, is employed to explain the accelerated expansion of the universe during inflation. The scalar field responsible for inflation is called “inflaton”. Unfortunately, we don’t have a suitable candidate to play the role of inflaton in the standard canonical inflationary setting. After the discovery of the Higgs boson [14] at the Large Hadron Collider (LHC) at CERN [15, 16], it is conceivable to consider it as an inflaton candidate. The scalar potential of the Higgs boson in the standard model of particle physics behaves asymptotically like the self-interacting quartic potential \(V(\phi )=\lambda \phi ^{4}/4\) in renormalizable gauge field theories [17, 18]. But unfortunately, this potential suffers from some critical problems in the standard inflationary setting. First, it leads to large values for the tensor-to-scalar ratio which are in conflict with the current observational bounds imposed by the Planck 2015 data [12]. Second, the value of the dimensionless constant \(\lambda \) deduced from CMB normalization is of order \(\sim 10^{-13}\), which is anomalously far from the Higgs coupling \(\lambda \simeq 0.13\) coming from the experimental searches [15, 16]. Also, taking into account the experimental bound \(\lambda \simeq 0.13\), leads to rather large values for the second slow-roll parameter \(\eta \), which disrupts the slow-roll conditions in the standard inflationary scenario [19]. This is generally so-called the \(\eta \)-problem in the standard inflation [20]. However, so far some theoretical attempts have been done to resolve the problems of the Higgs boson to be regarded as the inflaton [19, 21, 22, 23, 24, 25].

In addition to the problems mentioned above, there is another subject proposed recently that challenges the consistency of the standard scenario of inflation. This problem is the swampland conjecture [26, 27] and it results from the string theory considerations in the early universe. The swampland conjecture implies that the validity of the de Sitter (dS) limit is in contrast with the slow-roll conditions in the standard inflation.

The celebrity of the string theory as a fundamental model for physical phenomena motivates us to search for an inflaton candidate in the prospect of this theory. An interesting inflaton candidate inspired from the string theory is suggested in the context of Dirac–Born–Infeld (DBI) inflation [28, 29]. This setup proposes that the role of inflaton can be accomplished by the radial coordinate of a D3-brane moving in a warped region (throat) of a compactification space. The brane behaves like a point-like object and according to the direction of its motion in the warped space, there are two versions of brane inflation, namely the “ultraviolet” (UV) model [28, 29], and the “infrared” (IR) model [30, 31]. In the UV model, the inflaton moves from the UV side of the warped space to the IR side, while in the IR model, it moves in the inverse direction. Additionally, there exist a speed limit on the inflaton motion in the warped space. The speed limit is affected by the brane speed and the warp factor of the throat. Because of the speed limit, a parameter \(\gamma \) is introduced and it is analogous to the Lorentz factor in special relativity. When the speed of brane approaches the speed limit, the parameter \(\gamma \) can increase to arbitrarily large values (\(\gamma \gg 1\)), that we call this case as the “relativistic” regime. On the other side, in the “non-relativistic” regime, the brane speed is much less than the speed limit, giving \(\gamma \rightarrow 1\).

The DBI inflation can be included in the class of *k*-inflation models in which inflation is driven by a noncanonical scalar field [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44]. An outstanding feature of *k*-inflation is that in this model, the sound speed \(c_s\) of the scalar perturbations can be less than the light speed. As a result, *k*-inflation models are capable to provide small tensor-to-scalar ratio favored by the latest observational data from the Planck 2015 collaboration [12]. In addition, this feature can provide large non-Gaussianities [29, 45, 46, 47, 48, 49] which discriminates between *k*-inflation and the standard canonical inflation that predicts undetectably small non-Gaussianities [50, 51, 52, 53]. In the DBI inflation, the sound speed is equal to inverse of the Lorentz factor, \(c_{s}=1/\gamma \). Thus, in the relativistic and non-relativistic regimes, we have \(c_{s}\ll 1\) and \(c_{s}\sim 1\), respectively.

In the standard inflationary scenario, the interaction of inflaton field with other fields is neglected during inflation. So, the universe remains cold during inflation, and an additional process called “reheating” should be introduced to the final stages of inflation to make possible for the universe, so that it can transit from the accelerating phase of the inflationary era to the decelerating phase of the radiation epoch [54]. The details of the reheating process are unknown to us so far.

Alternatively, in the warm inflation scenario, the interaction between the inflaton and other fields has a dynamical role during inflation [55, 56]. As a result, the energy density of the inflaton field can be transformed to the energy density of the radiation field, so that inflation can terminate without resorting to any additional reheating process [57]. Also, note that in the cold and warm inflations, respectively, the quantum fluctuations of the inflaton field [8, 9, 10, 11] and thermal fluctuations of the radiation field [58, 59] are responsible for the anisotropies of the CMB radiation as well as the LSS formation.

Dissipative effects of warm inflation modifies the scalar primordial power spectrum in various ways. The first modification appears in the equation of the inflaton fluctuations so that in the warm inflation scenario, it cast into a Langevin equation governing the dynamics of the perturbed inflaton field [55, 56, 58, 59, 60, 61, 62, 63]. The second modification stems from the fact that in enough high temperatures, the distribution of the inflaton particles may deviate from the vacuum phase space distribution and tends toward the excited Bose–Einstein distribution [64, 65]. Finally, the last one is related to the cases in which duo to the temperature dependence of the dissipation coefficient, the inflaton and radiation fluctuations are coupled to each other, and this modification specially becomes important in the high dissipative regime [66, 67].

The required condition for realization of the warm inflation scenario is that the temperature *T* of the universe should be larger than the Hubble expansion rate *H* (\(T>H\)). This is necessary in order for the dissipation to potentially affect both the inflaton background dynamics, and the primordial power spectrum of the field fluctuations [68]. A few years after the original proposal for warm inflation, it was realized that the condition \(T>H\) cannot be easily provided in conventional models [69, 70]. Indeed, the inflaton could not couple directly with light fields easily. Moreover, a direct coupling to light fields can give rise to significant thermal corrections to the inflaton mass, so that the slow-roll regime is disturbed if \(T>H\). However, it was shown in the next promotions that the indirect coupling of inflaton to light degree of freedoms can provide successful models of warm inflation. The required conditions for these class of warm inflation models can be realized in special scenarios such as the case of the brane models [71]. More recently, a new mechanism for warm inflation has been proposed in [66], where warm inflation can be driven by an inflaton field coupled directly to a few light fields. In this scenario, the role of inflaton is played by a “Little Higgs” boson which is a pseudo Nambu-Goldstone boson (PNGB) of a broken gauge symmetry [72, 73, 74, 75].

In this paper, within the framework of cold and warm DBI inflation we consider the Anti-de Sitter (AdS) warp factor \(f(\phi )=f_{0}/\phi ^{4}\), and check viability of the quartic potential \(V(\phi )=\lambda \phi ^{4}/4\) in light of the Planck 2015 data [12]. First, in Sects. 2 and 3 , we present the background equations valid in the slow-roll approximation in the framework of cold DBI scenario, and examine inflation with the quartic potential \(V(\phi )=\lambda \phi ^4/4\). We show that in the cold DBI inflation, like the standard scenario, the result of quartic potential cannot be compatible with the Planck 2015 constraints [12]. Then, in Sects. 4 and 5 , we investigate the warm DBI inflation and check the viability of the quartic potential to see whether this potential can be resurrected in light of the Planck 2015 results. Section 6 is devoted to some regards on the dissipation parameter in our model. In Sect. 7, we examine the possibility of resolving the swampland conjecture problem in our model. Finally, in Sect. 8 we summarize our concluding remarks.

## 2 Cold DBI inflation

*R*is the Ricci curvature scalar. Also, \({\mathcal {L}}(X,\phi )\) is the DBI Lagrangian which is a function of the inflaton scalar field \(\phi \) and the canonical kinetic term \(X=-\partial ^{\mu }\phi \partial _{\mu }\phi /2\). The DBI Lagrangian is in the form

*a*of the universe. Here, the dot denotes the derivative with respect to the cosmic time

*t*. In the cold inflation scenario, it is presumed that the inflaton field does not interact with other fields during inflationary period, and thus creation of particles is not possible during inflation. As a result, the energy conservation law implies

*H*is taken as a function of the inflaton field \(\phi \). The Hamilton–Jacobi formalism in the context of DBI inflation, at first was introduced in [28], and then it was discussed in some details in [82]. Applying this formalism, we can also define the Hamilton–Jacobi slow-roll parameters [76, 78] as

*e*-fold number

*e*-folds from the end of inflation [85, 86]. Using Eqs. (10) and (23) one can obtain the following relation for the number of

*e*-foldings

*k*denotes the comoving wavenumber. To quantify the scale-dependence of the scalar power spectrum, we define the scalar spectral index as

*H*and the sound speed \(c_s\) vary significantly slower than the scale factor

*a*[33], so the relation \(c_{s}k=aH\) yields

*r*and \(n_s\) provides a powerful criterion to distinguish between viable inflationary models in light of the observational data. Inserting Eqs. (25) and (29) into (32), we get

## 3 Cold DBI inflation with the quartic potential

*N*. Afterwards, we insert it in Eqs. (38) and (39) to evaluate \(n_s\) and

*r*at the epoch of horizon crossing with a given \(N_*\). Consequently, the \(r-n_s\) plot of this model is resulted in as shown by the dashed (\(N_*=50\)) and solid (\(N_*=60\)) green lines in Fig. 1. Note that in the standard inflation, the scalar spectral index and tensor-to-scalar of the quartic potential (35) are given by \(n_{s}=1-3/N\) and \(r=16/N\), respectively [89]. Using these equations, the \(r-n_s\) plot of this potential is shown by the blue line in Fig. 1, that lies quite outside of the favored region of the Planck 2015 observational results [12]. As it is apparent in Fig. 1, the result of the quartic potential (35) in the cold DBI inflation setting, like the standard canonical one, is completely inconsistent with the Planck 2015 observational results [12]. In Fig. 1, we have considered the parameter \({\tilde{f}}_{0}\ge 0\) as the varying parameter to plot the \(r-n_s\) diagram of the DBI model. As the parameter \({\tilde{f}}_{0}\) decreases to zero, the sound speed \(c_s\) approaches unity and consequently the \(r-n_s\) diagram of the DBI inflation goes toward the result of the quartic potential (35) in the standard framework.

## 4 Warm DBI inflation with constant sound speed

*T*is the temperature of the thermalized bath, and \(\alpha =\pi ^{2}g_{*}/30\) is the Stefan–Boltzmann constant in which \(g_*\) is the relativistic degrees of freedom, and in this paper we take it as \(g_{*}\simeq 228.75\) [64].

*Q*is the dissipation ratio defined as

^{1}[92]

At the level of nonlinear perturbations, as it has been discussed in [92], for sufficiently small values of \(c_s\), the contribution of the inflaton perturbations dominates over the thermal contribution in the cubic order Lagrangian, and hence the equilateral non-Gaussianity parameter is obtained as in the cold DBI inflation, which is given by Eq. (34). Therefore, again the Planck 2015 constraints on the primordial non-Gaussianities [13] lead to the bound \(c_{s}\ge 0.087\) on the sound speed of our model.

## 5 Warm DBI inflation with the quartic potential

As we saw in Sect. 3, the quartic potential (35) fails to be consistent with the Planck 2015 data in the framework of cold DBI inflation. Then, we are motivated to examine the consistency of this model in the warm DBI inflationary setting. In our investigation, we again adopt the AdS warp factor (36), but now we make the further assumption that the sound speed \(c_s\) to be constant during inflation. The idea of constant sound speed in study of the DBI inflation has been regarded in [84, 96, 97], and it has been shown that it leads to considerable simplification in calculations. We employ this assumption in our work, because the definite form of the dissipation parameter \(\Upsilon \) has not been determined so far, and to specify it, we require further studies for the inflaton interactions in the thermal bath.

*H*,

*Q*, and

*T*, in terms of time, and after inserting these quantities in Eq. (56), the scalar power spectrum is obtained as

*e*-fold number. We note that if we replace Eqs. (65) into (61), the Hubble parameter becomes

*e*-fold number in our model as

*e*-foldings as

*e*-fold number \(N_*\), we can readily determine the parameter \(\lambda \) as

*e*-fold number, we substitute Eqs. (71) into (67), and obtain

The ranges of parameter \({\tilde{f}}_{0}\) for which our warm DBI inflation model with several values of the constant sound speed \(c_s\) and with \(N_*=50,\,60\), verifies the 68% CL constraints of Planck 2015 TT, TE, EE + low P data [12] in the \(r-n_s\) plane

\(c_{s}\) | \(N_{*}\) | \({\tilde{f}}_{0}\) |
---|---|---|

0.1 | 50 | \({\tilde{f}}_{0}\gtrsim 1.95\times 10^{3}\) |

60 | \({\tilde{f}}_{0}\gtrsim 2.48\times 10^{3}\) | |

0.3 | 50 | \({\tilde{f}}_{0}\gtrsim 6.03\times 10^{3}\) |

60 | \({\tilde{f}}_{0}\gtrsim 7.75\times 10^{3}\) | |

0.7 | 50 | \({\tilde{f}}_{0}\gtrsim 1.46\times 10^{2}\) |

60 | \({\tilde{f}}_{0}\gtrsim 1.90\times 10^{2}\) | |

0.99 | 50 | \({\tilde{f}}_{0}\gtrsim 4.07\) |

60 | \({\tilde{f}}_{0}\gtrsim 5.30\) |

*e*-fold number \(N_*\), the solution of Eq. (81) for \(\lambda \) is

## 6 Some regards on the dissipation parameter

*MS*refers to Morikawa and Sasaki [99] who found for the first time a similar result for non-supersymmetric models, and their result was verified later by Berera and Ramos [90, 100]. For a special case, if we have \(m_{y}\gg \left| g\phi \right| \), then the second and third terms in the bracket in Eq. (95) can be ignored against the first term, and the dissipation parameter turns into \(\Upsilon \propto \phi \). This form for the dissipation parameter is also interesting at the phenomenological point of view. In [101], a general form \(\Upsilon =C_{\phi }\frac{T^{m}}{\phi ^{m-1}}\) was proposed phenomenologically for the dissipation parameter, where the constant \(C_\phi \) depends on the microphysics of the dissipation process, and the exponent

*m*is an integer. The special cases of this dissipation parameter can also be extracted from the interactions given by the superpotential

*W*. One special case is \(m=1\) which gives \(\Upsilon \propto T\) that corresponds to the high temperature supersymmetry model [98]. Also, other special cases are \(m=0\) (\(\Upsilon \propto \phi \)) and \(m=-1\) (\(\Upsilon \propto \phi ^2/T\)) which are related to an exponentially decaying propagator in the high temperature supersymmetric and non-supersymmetric models, respectively [70, 90]. Some further investigations on the dissipation parameter \(\Upsilon \propto \phi \) in the framework of warm inflation can be found in [60, 91, 101, 102, 103].

## 7 Examination of the swampland conjecture

*V*is the scalar field potential and \(c, \, c'>0\) are some universal constants of order 1. The left hand side of Eq. (97) is the minimum of the Hessian \(\nabla _{i}\nabla _{j}V\) in an orthonormal frame. In the standard supercold inflation, validity of the criteria (96) and (97) is in contrast with the slow-roll conditions, and therefore we encounter the so-called swampland problem. But, the authors of [104, 105] have shown that the validity of the swampland criteria does not lead to any contradiction in the setup of warm inflation. Here, we are also interested in examining the validity of the swampland criteria in our warm DBI inflation model. To this aim, we first introduce the potential slow-roll parameter that is conventionally defined as

*Q*is chosen large enough. In order to show this fact in our investigation more explicitly, we calculate the potential slow-roll parameter (98) by the use of Eqs. (35) and (72), and reach

## 8 Conclusions

The DBI inflation has well-based motivations from the sting theory, and in this model the role of inflaton is played by a radial coordinate of a D3-brane moving in a throat of a compactification space. Therefore, this scenario proposes a convincing candidate for the inflaton field. In addition, thus far some suggestions have been offered for the inflationary potential in the DBI inflation which can alleviate the eta problem theoretically in this setting relative to the conventional inflationary frameworks. In this paper, we study the DBI inflation in both the cold and warm scenarios. We first focused on the cold DBI inflation and reviewed shortly the basic equations of the background dynamics and the scalar and tensor primordial perturbations in this setting. At the level of nonlinear perturbations, we showed that the 95% CL observational constraint of Planck 2015 T + E data [13] on the primordial non-Gaussianity leads to the bound \(c_{s}\ge 0.087\) on the sound speed of the model. Then, we investigated the quartic potential \(V(\phi )=\lambda \phi ^{4}/4\) in the context of cold DBI inflation with the AdS warp factor \(f(\phi )=f_{0}/\phi ^{4}\), and checked its viability in light of the Planck 2015 results. Our examination implied that the result of the quartic potential in the cold DBI setting like the standard scenario is completely outside the allowed region of Planck 2015 TT, TE, EE + low P data [12] in \(r-n_s\) plane.

Subsequently, we turned to study the warm DBI inflation, and presented the basic equations of this model. In this framework, the nonlinear perturbations of the inflaton dominates over the thermal contribution, provided that the sound speed be small enough. Consequently, in this scenario we recover the same lower bound found in the cold DBI inflation for the sound speed from the 95% CL constraint of Planck 2015 T + E data on the primordial non-Gaussianity parameter.

In the warm DBI inflation, we examined the same quartic potential \(V(\phi )=\lambda \phi ^{4}/4\) with the AdS warp factor \(f(\phi )=f_{0}/\phi ^{4}\) and assumed the sound speed to be a constant quantity during inflation. In the next step, we displayed the result of our model in \(r-n_s\) plane in comparison with the observational results. We saw that our model with different values of \(c_s\) is consistent with the CMB data, and its prediction can enter within the 68% CL region of Planck 2015 TT, TE, EE + low P data. We also demonstrated that the \(dn_s/d\ln k-n_s\) diagram of our warm DBI inflationary model with \(c_s = 0.1\) can be located inside the 68% CL region favored by Planck 2015 TT, TE, EE + low P data.

We further specified the parameter space of \({\tilde{f}}_{0}-c_{s}\) (\({\tilde{f}}_{0}\equiv f_{0}\lambda \)) for which the \(r-n_s\) prediction of our model is consistent with the 68% CL constraints of Planck 2015 TT, TE, EE + low P data. This parameter space implies that our model is compatible with the observations provided that \({\tilde{f}}_0\gtrsim 10^2\). Using the parameter space of \({\tilde{f}}_{0}-c_{s}\), we also showed that the essential requirement of warm inflation, i.e. \(T>H\), is perfectly preserved in our model. From this parameter space, we also concluded that our model can be compatible with observation in the intermediate (\(0.1\lesssim Q_*\lesssim 10\)) and high (\(Q_*\gtrsim 10\)) dissipation regimes of warm inflation, and it fails to be consistent in the weak (\(Q_*\lesssim 0.1\)) dissipation regime. Nevertheless, if one consider the modifications duo to the high temperatures regime which alter the distribution of the inflaton particles from the vacuum phase space state to the excited Bose–Einstein state [64, 66], the results of this model may be compatible with the observational data even in the weak dissipation regime, and we leave this suggestion to the future investigations.

Furthermore, in order to estimate the value of \(\lambda \) following from fixing the scalar power spectrum at the horizon exit in our model, we presented the parameter space of \(\lambda -c_s\), and from it we concluded that our model is in consistency with the Planck 2015 results provided \(\lambda \lesssim 10^{-13}\). This is in direct contrast with the value \(\lambda \simeq 0.13\) measured by the LHC at CERN for the coupling constant of the Higgs boson of the standard model of particle physics. Nonetheless, it is not required that this bound is kept in our model, because here the role of inflaton is performed by the radial coordinate of a D3-brane [28, 29], not by the Higgs boson of the standard model of particle physics. Putting the constraint \(\lambda \lesssim 10^{-13}\) together with the bound \({\tilde{f}}_0\gtrsim 10^2\) inferred from the \({\tilde{f}}_0-c_s\) parameter space, we found the lower bound \(f_0\gtrsim 10^{15}\) in our warm DBI inflation model.

Moreover, we showed that the dissipation parameter in our warm DBI model behaves as \(\Upsilon \propto \phi \) which is supported by QFT as well as phenomenological implications. Finally, we examined the swampland criteria in our warm DBI model and concluded that in contrary to the standard inflation which suffers from the violation of the dS limit in the slow-roll regime, the swampland problem can be resolved in our model.

## Footnotes

## Notes

### Acknowledgements

The work of A. Abdolmaleki has been supported financially by Research Institute for Astronomy and Astrophysics of Maragha (RIAAM) under research project No. 1/5440-50.

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