Gauss-Bonnet inflation and swampland

Gauss-Bonnet inflation and swampland

Zhu Yi yizhu92@hust.edu.cn School of Physics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China    Yungui Gong yggong@hust.edu.cn School of Physics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China
Abstract

The two swampland criteria are generically in tension with the single field slow-roll inflation because the first swampland criterion requires small tensor to scalar ratio while the second swampland criterion requires large tensor to scalar ratio. The challenge to the single field slow-roll inflation imposed by the swampland criteria can be avoided by modifying the relationship between the tensor to scalar ratio and the slow-roll parameter. We show that the Gauss-Bonnet inflation with the coupling function inversely proportional to the potential overcomes the challenge by adding a constant factor in the relationship between the tensor to scalar ratio and the slow-roll parameter. For the Gauss-Bonnet inflation, while the swampland criteria are satisfied, the slow-roll conditions are also fulfilled, so the scalar spectral tilt and the tensor to scalar ratio are consistent with the observations. We use the potentials for chaotic inflation and the E-model as examples to show that the models pass all the constraints. The swampland criteria may imply Gauss-Bonnet coupling.

I introduction

Inflation solves the flatness and horizon problems in standard cosmology Guth (1981); Linde (1982); Albrecht and Steinhardt (1982); Starobinsky (1980); Sato (1981), and is usually modelled by a single slow-roll scalar field which is obtained from low-energy Effective Filed Theories. In order to embed such scalar fields in a quantum theory successfully, they have to satisfy two criteria Ooguri and Vafa (2007); Vafa (2005). These two Swampland criteria are

  • Swampland Criterion I (I) Ooguri and Vafa (2017): The scalar field excursion, normalized by the reduced planck mass, in field space is bounded from above

    (1)

    where the reduced planck mass , and the order 1 constant .

  • Swampland Criterion II (II) Obied et al. (2018): The gradient of the filed potential with should satisfy the lower bound

    (2)

    where and the order 1 constant .

Obviously, the second criterion (2) violates the slow-roll condition and poses a threat to the inflation model by requiring a large tensor to scalar ratio . Even if we chose Kehagias and Riotto (2018), it is still inconsistent with the observational constraint Akrami et al. (2018); Ade et al. (2018) because . As point out in Ref. Agrawal et al. (2018), a viable way to solve this problem is by using the models with the tensor to scalar ratio reduced by a factor while keeping the lower bound on the field excursion by required by the Lyth bound Lyth (1997), such as the warm inflation Berera (1995). See Ref. Brennan et al. (2017); Das (2018a, b); Motaharfar et al. (2018); Ashoorioon (2018); Lin et al. (2018); Lin (2018); Kinney et al. (2018); Achúcarro and Palma (2018); Andriot (2018); Park (2018); Garg and Krishnan (2018); Garg et al. (2018); Schimmrigk (2018); Dimopoulos (2018); Matsui and Takahashi (2018); Ben-Dayan (2018); Brahma and Wali Hossain (2018); Roupec and Wrase (2018); Blåbäck et al. (2018); Odintsov and Oikonomou (2018); Kawasaki and Takhistov (2018); Wang (2018) for the other papers about this issue.

In Ref. Yi et al. (2018), we find a powerful mechanism to reduce the tensor to scalar ratio . With the help of the Gauss-Bonnet term, for any potential, the tensor to scalar is reduced by a factor with the order 1 parameter , which may possibly solve the swampland problem. Generally, the predictions of the inflation, and , are calculated under the slow-roll condition, . If the second swampland criterion (2) is satisfied, then the slow-roll condition will be violated, and the predictions and may be unreliable. But in the case with Gauss-Bonnet coupling, the slow-roll condition is , so even the second swampland criterion (2) is satisfied, as long as , the model still satisfies the slow-roll condition, so the slow-roll results are applicable. In this paper, we show that with the help of the Gauss-Bonnet coupling, the inflation model satisfies not only the swampland criteria but also the observational constraints.

This paper is organized as follows. In Sec. II, we give a brief introduction to the Gauss-Bonnet inflation, and point out the reason why it is easy to satisfy the swampland criteria for the model. In Sec. III, we use the power-law potential and the E-model to show that all the constraints can be satisfied. We conclude the paper in Sec. IV.

Ii The Gauss-Bonnet inflation

The action for Gauss-Bonnet inflation is Rizos and Tamvakis (1994); Kanti et al. (1999); Nojiri et al. (2005)

(3)

where is the Gauss-Bonnet term which is a pure topological term in four dimensions, and is the Gauss-Bonnet coupling function. In this paper, we use van de Bruck et al. (2016); Yi et al. (2018)

(4)

where . The parameter added here is to avoid the reheating problem of Gauss-Bonnet inflation van de Bruck et al. (2016), and it can be ignored during inflation, so in this paper we neglect the effect of . In terms of the horizon flow slow-roll parameters Schwarz et al. (2001), the slow-roll conditions are

(5)

the scalar spectral tilt and the tensor to scalar ratio are Yi et al. (2018)

(6)
(7)

In terms of the potential, the slow-roll parameters are expressed as Guo and Schwarz (2010)

(8)

Due to the factor in Eqs. (8), even if the gradient of the potential is consistent with the second swampland criterion II, , the slow-roll conditions (5) can still be satisfied as long as is close to 1. So the slow-roll results (6), (7) and (8) are applicable to the case satisfying the second swampland criterion II. Substituting Eq. (8) into Eq. (7), we get

(9)

From Eq. (9), we see that while the second swampland criterion II is satisfied, the tensor to scalar ratio can still be very small, as long as is small enough.

Now we discuss the first swampland criterion I for the field excursion. The Lyth bound tells us that Lyth (1997); Yi et al. (2018)

(10)

Without the Gauss-Bonnet term, if II is satisfied, then it is impossible to satisfy I for single field slow-roll inflation with . With the help the Gauss-Bonnet term, it is very easy to satisfy both I and II conditions, as long as small enough, for example .

In summary, with the help of the Gauss-Bonnet term, as long as the order one parameter is close to 1, the two swampland criteria I and II can be easily satisfied, and the tensor to scalar ratio is also consistent with the observations Akrami et al. (2018). From Eq. (6), we see that the parameter have no effect on the scalar spectral tilt , so the constraint on can also be satisfied.

In the next section, we will use two inflationary models, the power-law potentials and the E-model, as examples to support the above discussion.

Iii The models

In the following we consider two inflation models, the chaotic inflation model and the E-model. We show that with the help of the Gauss-Bonnet term, the two swampland criteria (1) and (2) are satisfied for both models. Additionally, the models also satisfy the observational constraints Akrami et al. (2018),

(11)

iii.1 The power-law potential

For the chaotic inflation with the power-law potential Linde (1983)

(12)

the excursion of the inflaton is

(13)

where is the remaining number of -folds before the end of inflation, and

(14)

The scalar spectral tilt and the tensor to scalar ratio are

(15)
(16)

From Eq. (15), we see that is independent on . If we choose and , the scalar spectral tilt is , which is consistent with the observations (11). By varying the value of , the values of the gradient of the potential , inflaton excursion and tensor to scalar ratio are shown in Fig. 1.

Figure 1: The dependence on for the power-law potential with . The upper panel shows the tensor to scalar . The lower panel shows the gradient of the potential and the field excursion .

As the value of becomes smaller and smaller, and will become smaller and smaller too, but will become larger and larger. As long as is small enough, the two swampland criteria (1) and (2) as well as the observational constraints (11) are satisfied. For example, if we chose , we have

(17)
(18)

The predictions (17) are consistent with the observations (11), and both the field excursion and the gradient of the potential satisfy the two swampland criteria.

iii.2 The E-model

For the E-model Kallosh and Linde (2013); Carrasco et al. (2015)

(19)

the excursion of the inflaton for is

(20)

where

(21)

and , the function is the lower branch of the Lambert W function. The scalar spectral tilt and the tensor to scalar ratio are Yi and Gong (2016)

(22)
(23)

If we chose and , we get which is consistent with the observation. Varying , the values of , and are shown in Fig. 2.

Figure 2: The dependence on for the the E-model with . The upper panel shows the tensor to scalar . The lower panel shows the gradient of the potential and the field excursion .

Similar to the chaotic inflation, as the becomes smaller and smaller, and become smaller and smaller, and become larger and larger. As long as is small enough, the two swampland criteria (1) and (2) as well as the observational constraints (11) are satisfied. If we chose , we get

(24)
(25)

The predictions (24) are consistent with the observations (11), and both the field excursion and the gradient of the potential satisfy the two swampland criteria.

Iv conclusion

The two swampland criteria pose threat on the single filed slow-roll inflation. With the help of the Gauss-Bonnet coupling, the relationship between the and is described by Eq. (9), i.e., is reduced by a factor of compared with the result in standard single field slow-roll inflation. Because of the reduction in , the first swampland criterion is easily satisfied by requiring to be small. On the other hand, it is easy to satisfy the second swampland criterion by requiring to be small and keeping to be small.

For the chaotic inflation with , if we take , we get , , and . Therefore, for the chaotic inflation with and , the models satisfy not only the observational constraints, but also the two swampland criteria. For the E-model, If we chose , we get , , and . The model satisfies not only the observational constraints, but also the two swampland criteria. In conclusion, the Gauss-Bonnet inflation with the condition (4) satisfies not only the observational constraints, but also the swampland criteria. Therefore, the swampland criteria may imply the existence of Gauss-Bonnet coupling.

Acknowledgements.
This work was supported in part by the National Natural Science Foundation of China under Grant Nos. 11875136 and 11475065 and the Major Program of the National Natural Science Foundation of China under Grant No. 11690021.

References

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