Further refining Swampland Conjecture on inflation in general scalar-tensor theories of gravity

In the first case scenario, we consider the condition (10). If we take $V^{J}(\phi)=V_{0}\big{(}1-\Omega^{2}(\phi)\big{)}^{2}$, then we obtain the exponential form of the potential, named E-model, given in the Einstein frame of the form

The associated conventional slow-roll parameters are given by

where $U_{\psi}$ denotes derivative with respect to $\psi$, i.e., $U_{\psi}=dU/d\psi$ and $U_{\psi\psi}=d^{2}U/d\psi^{2}$. We obtain

Inflation ends when $\varepsilon=1$ and we find

The number of e-foldings during inflation is defined via

The above result can be combined with Eq.(17) allowing us to write:

To generate the proper amplitude of the density perturbations the potential must satisfy at WMAP the normalization condition Planck:2018jri :

corresponding to the initial value assumed by the inflaton. We therefore obtain

Ii is useful to write $\varepsilon$ and $\eta$ in terms of the number of e-foldings. Substituting Eq.(19) into Eq.(16), we have

The spectral index of curvature perturbation $n_{s}$ and the tensor-to-scalar ratio $r$ are given in terms of the e-foldings $N$:

Note that the above results still hold since $N\gg\alpha$. Setting $\alpha=1$, we obtain the same results to those of the models found in the existing references including Higgs inflation Bezrukov:2007ep , Starobinsky inflation Starobinsky:1980te , Higgs Starobinsky inflation Calmet:2016fsr ; Salvio:2015kka ; Salvio:2016vxi , and even composite models of inflation Channuie:2015ewa ; Channuie:2016iyy .

In the second model, under the condition (10), if we choose

then we get the hyperbolic tangent form of the potential, named it as T-model, in the Einstein frame

The associated slow-roll parameters are given by

where $U^{E}_{\psi}$ denotes derivative with respect to $\psi$, i.e., $U^{E}_{\psi}=dU^{E}/d\psi$ and $U^{E}_{\psi\psi}=d^{2}U^{E}/d\psi^{2}$. We obtain

Inflation ends when $\varepsilon=1$ and we find

The number of e-foldings during inflation is defined via

The above result can be combined with Eq.(29) allowing us to write:

To generate the proper amplitude of the density perturbations the potential must satisfy at WMAP the normalization condition Planck:2018jri :

corresponding to the initial value assumed by the inflaton. We therefore obtain

Ii is useful to write $\varepsilon$ and $\eta$ in terms of the number of e-foldings. Substituting Eq.(19) into Eq.(16), we have

The spectral index of curvature perturbation $n_{s}$ and the tensor-to-scalar ratio $r$ are given in terms of the e-foldings $N$:

With the results given above, there was a class of inflationary models so called cosmological $\alpha$-attractors has recently received considerable attention Kallosh:2013yoa ; Kallosh:2014rga ; Kallosh:2015lwa ; Roest:2015qya ; Linde:2016uec ; Terada:2016nqg ; Ueno:2016dim ; Odintsov:2016vzz . Note that the above results still hold since $N\gg\alpha$.

We start with a action of the scalar field non-minimally coupled to gravity with a general form of the effective potential $V(\phi)$. It takes the form

where the action $S_{J}$ stands for the gravitational action in the Jordan frame. While $M_{P}^{2}\equiv 1/8\pi G$ and $\xi$ are reduced Plank mass and the non-minimal coupling constant, respectively. It is more convenient to study the inflation dynamics of the non-minimal coupling in the Einstein frame, i.e., the gravitational sector of the action written in the Einstein-Hilbert form only. The Einstein frame can be achieved by using the conformal transformation via a re-defining metric tensor as,

Here all variables with tilde symbol represent the quantities in the Einstein frame and $f(\phi)=\phi^{2}$. Applying the conformal transformation to the action (37), the action in Einstein frame is given by,

We have used the re-definition of new scalar field, $\psi$ in the Einstein frame to obtain the canonical form of the kinetic term of the scalar field, $\psi$ as

where $\Omega_{\phi}\equiv d\Omega/d\phi$. Comparing Eq.(40) with Eq.(10), we find $\omega(\phi)=2$. The new effective potential in the Einstein frame, $U(\psi)$ is also given by,

In this section, we will consider the self-interacting potential with phenomenological quantum correction and this potential has been proposed by authors of Ref.Joergensen:2014rya in order to analyze the characters of the quantum correction in the self-interacting scalar field phenomenology. They wrote the potential in the Jordan frame given by

Here the quantum correction (real) parameter $\gamma$ is introduced and will be used to characterize the quantum behavior of the self-interacting potential. The $\Lambda$ parameter is the cut-off at a given energy scale. It was shown that the range of the $\gamma$ should be $\mathcal{O}(\gamma)\sim 0.1$ according to the constraint from observational data Joergensen:2014rya .

Before calculating the slow-roll parameters, we would like to express the form of the effective potential in the Einstein in Eq.(41) under the large field assumption during the inflation i.e., $\phi\gg{M_{P}/\sqrt{\xi}}$ . One finds,

Then the Einstein frame potential then takes the following form

The associated slow-roll parameters are given by

where $U^{E}_{\psi}$ denotes derivative with respect to $\psi$, i.e., $U^{E}_{\psi}=dU^{E}/d\psi$ and $U^{E}_{\psi\psi}=d^{2}U^{E}/d\psi^{2}$. We obtain in terms of the field $\phi$

Inflation ends when $\varepsilon=1$ and we find

The number of e-foldings during inflation is defined via

The above result can be combined with Eq.(49) allowing us to write:

It is evident that the $\gamma$-correction lifts up inflation to higher field values. Since we have assumed that inflation takes place in the large field regime, $\phi\gg{M_{p}/\sqrt{\xi}}$, it was found in Ref.Joergensen:2014rya that $\xi\sim{\cal O}(10^{4})$ is required to generate the proper amplitude of density perturbations featuring a generic behavior of non-minimally coupled theories of single-field inflation. We can further determine field values to obtain $\phi_{\rm ini}\sim 0.1\,M_{p}$ for $\gamma\sim 0.01,\,\xi\sim 10^{4}$ and $N=60$. Interestingly, its value is sub-Planckian during inflation. However, with the large field assumption, we encounter the large field deviation since $\Delta\psi>M_{p}$ given by Eq.(43). Similar to $R^{2}$ and eternal inflation scenarios, it is in general a sense of difficulty of constructing models of large field inflation in string theory where the inflaton undergoes a super-Planckian excursion in field space. Nevertheless, a quantitative impression of this tension was suggested in Refs.Palti:2019pca ; Matsui:2018bsy where their analysis is applicable to that of our model. Therefore, we do not intentionally repeat it here and recommend the readers to follow those works. To generate the proper amplitude of the density perturbations the potential must satisfy at WMAP the normalization condition Planck:2018jri :

corresponding to the initial value assumed by the inflaton:

We therefore obtain

Ii is useful to write $\varepsilon$ and $\eta$ in terms of the number of e-foldings. Substituting Eq.(52) into Eq.(45), we have

The spectral index of curvature perturbation $n_{s}$ and the tensor-to-scalar ratio $r$ are given in terms of the e-foldings $N$:

Here usual $\phi^{4}$ inflation refers to the results when setting $\gamma=0$, that is, non-minimally coupled $\phi^{4}$ inflation. Note that we have expanded results in $\gamma$ to clarify to what extend the results deviate from $\phi^{4}$ inflation. An expansion is, however, justified for tiny values of $\gamma$.

We can constrain values of $\alpha$ using a condition of $r<0.06$

to obtain

For $N=60$, we find $\alpha<13.5$. With the above result, we then choose $n_{s}=0.965$ and obtain from Eq.(23):

Using $N=60$ yields $\alpha=1.333$, and therefore this model predicts $n_{s}\simeq 0.965$ and $r\simeq 0.004$ which are consistent with the observed data Planck:2018jri . Inserting these values into Eq.(59) and Eq.(60), we obtain

Considering the refined swampland conjecture (4), we find

However, $c_{1}$ and $c_{2}$ are both not ${\cal O}(1)$, meaning that inflationary model with the E-form potential is in strong tension with the refined swampland conjecture. Let us examine whether the E-form model satisfies the refining swampland conjecture. Considering Eq.(5), we have

Substituting Eq.(69) into Eq.(80), we find

If we can find $a$ to satisfy the condition

then the further refining swampland conjecture can be satisfied. In this case, when $a=\nicefrac{{1}}{{1.01675}}$, we have $1-1.01675\,a=0$. Therefore, we can examine that when $a<\nicefrac{{1}}{{1.01675}}$, we can always find a $q$ whose value is larger than $2$. It is possible to give an example of values of the parameters $a,\,b,\,q$, which work for this model. From Eq.(72), we use $q=2.2$ which is satisfied by a condition $q>2$. We find for this particular case that $0.983296\leq a<1$ and choose $a=0.9834<\nicefrac{{1}}{{1.01675}}=0.983526$ and $1-a=1-0.9834=b=0.0166>0$.

We can constrain values of $\alpha$ using a condition of $r<0.06$

to obtain

For $N=60$, we find $\alpha<18$. With the above result, we then choose $n_{s}=0.965$ and obtain from Eq.(23):

Using $N=60$ yields $\alpha=4$, and therefore this model predicts $n_{s}\simeq 0.965$ and $r\simeq 0.0133$ which are consistent with the observed data Planck:2018jri . Inserting these values into Eq.(59) and Eq.(60), we obtain

Considering the refined swanpland conjecture (4), we find

However, neither $c_{1}$ nor $c_{2}$ is positive constant of the order of ${\cal O}(1)$, meaning that inflationary model with the E-form potential is in strong tension with the refined swampland conjecture. Let us examine whether the E-form model satisfies the refining swampland conjecture. Considering Eq.(5), we have

Substituting Eq.(78) into Eq.(79), we find

If we can find $a$ to satisfy the condition

then the further refining swampland conjecture can be satisfied. For the second model, when $a=\nicefrac{{1}}{{1.0150}}$, we have $1-1.0150\,a=0$. Hence, we can examine that when $a<\nicefrac{{1}}{{1.0150}}$, we can always find a $q$ whose value is larger than $2$. Similarly, we can give an example of values of the parameters $a,\,b,\,q$, which work for this model. From Eq.(81), we use $q=2.2$ which is satisfied by a condition $q>2$. We find for this particular case that $0.984356\leq a<1$ and choose $a=0.9844<\nicefrac{{1}}{{1.0150}}=0.985222$ and $1-a=1-0.9844=b=0.0156>0$.

We can constrain values of $\alpha$ using a condition of $r<0.06$

to obtain

For $N=60$, we find $\gamma<0.2125$. Using $N=60,\,\gamma=0.10$, this model predicts $n_{s}\simeq 0.965$ and $r\simeq 0.030$ which are consistent with the observed data Planck:2018jri . Inserting these values into Eq.(59) and Eq.(60), we obtain

Considering the refined swanpland conjecture (5), we find

Neither $c_{1}$ nor $c_{2}$ are of the order of ${\cal O}(1)$ which mean that inflationary model with the quantum corrected potential is in strong tension with the refined swampland conjecture. Let us examine if this model satisfies the refining swampland conjecture. Considering Eq.(5), we have

Substituting Eq.(86) into Eq.(87), we find

If we can find $a$ to satisfy the condition

then the further refining swampland conjecture can be satisfied. When $a=\nicefrac{{1}}{{1.011875}}$, we see that $1-1.011875\,a=0$. Hence, we can examine that when $a<\nicefrac{{1}}{{1.011875}}$, we can always find a $q$ whose value is larger than $2$. In the last model, we can also give an example of values of the parameters $a,\,b,\,q$, which work for our model. From Eq.(89), we use $q=2.2$ which is satisfied by a condition $q>2$. We find for this particular case that $0.986145\leq a<1$ and choose $a=0.9862<\nicefrac{{1}}{{1.011875}}=0.988264$ and $1-a=1-0.9862=b=0.0138>0$. Interestingly, for this model the scale $\Lambda$ can be constrained. For instance, substituting $N=60,\,\lambda=\nicefrac{{1}}{{4}},\,\gamma=0.01504$ and $\xi=10^{4}$, we find that

with $\lambda=\nicefrac{{1}}{{4}}$ being a standard value Joergensen:2014rya . This value is close to the typical grand unification scale $M_{\rm GUT}$ of $10^{16}$ GeV with the lower value obtained for the reduced Planck mass of $2.44\times 10^{18}$ GeV and the higher one for the standard one of $1.22\times 10^{19}$ GeV.

Jureeporn Yuennan: Writing – review & editing. Phongpichit Channuie: Formal analysis, Methodology, Writing – original draft, Writing – review & editing.

The authors declare no conflict of interest.