The upper bound on the tensor-to-scalar ratio consistent with quantum gravity
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Lina Wu1, Qing Gao2, Yungui Gong3, Yiding Jia,4,5,∗, Tianjun Li4,51School of Sciences, Xi'an Technological University, Xi'an 710021, China 2School of Physical Science and Technology, Southwest University, Chongqing 400715, China 3School of Physics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China 4CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China 5School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
First author contact:Author to whom any correspondence should be addressed. Received:2021-02-24Revised:2021-03-26Accepted:2021-04-15Online:2021-05-20
Abstract We consider the polynomial inflation with the tensor-to-scalar ratio as large as possible which can be consistent with the quantum gravity (QG) corrections and effective field theory (EFT). To get a minimal field excursion Δφ for enough e-folding number N, the inflaton field traverses an extremely flat part of the scalar potential, which results in the Lyth bound to be violated. We get a CMB signal consistent with Planck data by numerically computing the equation of motion for inflaton φ and using Mukhanov–Sasaki formalism for primordial spectrum. Inflation ends at Hubble slow-roll parameter ${\epsilon }_{1}^{H}=1$ or $\ddot{a}=0$. Interestingly, we find an excellent practical bound on the inflaton excursion in the format $a+b\sqrt{r}$, where a is a tiny real number and b is at the order 1. To be consistent with QG/EFT and suppress the high-dimensional operators, we show that the concrete condition on inflaton excursion is $\tfrac{{\rm{\Delta }}\phi }{{M}_{\mathrm{Pl}}}\lt 0.2\times \sqrt{10}\simeq 0.632$. For ns = 0.9649, Ne = 55, and $\tfrac{{\rm{\Delta }}\phi }{{M}_{\mathrm{Pl}}}\lt 0.632$, we predict that the tensor-to-scalar ratio is smaller than 0.0012 for such polynomial inflation to be consistent with QG/EFT. Keywords：polynomail inflation;Lyth bound;tensor-to-scalar ratio
PDF (506KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Lina Wu, Qing Gao, Yungui Gong, Yiding Jia, Tianjun Li. The upper bound on the tensor-to-scalar ratio consistent with quantum gravity. Communications in Theoretical Physics, 2021, 73(7): 075402- doi:10.1088/1572-9494/abf824
1. Introduction
Inflation provides a natural solution to the well-known flatness, horizon, and monopole problems, etc, in the standard big bang cosmology [1–4]. And the observed temperature fluctuations in the cosmic microwave background radiation (CMB) strongly indicates an accelerated expansion at a very early stage of our Universe evolution, i.e. inflation. In addition, the inflationary models predict the cosmological perturbations for the matter density and spatial curvature due to the vacuum fluctuations of the inflaton, and thus can explain the primordial power spectrum elegantly. Besides the scalar perturbation, the tensor perturbation is also generated. Especially, it has special features in the B-mode of the CMB polarization data as a signature of the primordial inflation.
The Planck satellite measured the CMB temperature anisotropy with an unprecedented accuracy. From the latest observational data [5, 6], the scalar spectral index ns, the running of the scalar spectral index ${\alpha }_{s}\equiv {{\rm{d}}{n}}_{s}/{\rm{d}}\mathrm{ln}k$, the tensor-to-scalar ratio r, and the scalar amplitude As for the power spectrum of the curvature perturbations are respectively constrained to be$\begin{eqnarray}\begin{array}{rcl}{n}_{s} & = & 0.9649\pm 0.0042(68 \% \,\mathrm{CL}),\\ {\alpha }_{s} & = & -0.0045\pm 0.0067(68 \% \,\mathrm{CL}),\\ {r}_{0.002} & \leqslant & 0.056(95 \% \mathrm{CL}),\\ & & \mathrm{ln}[{10}^{10}{A}_{s}]=3.044\pm 0.014.\end{array}\end{eqnarray}$There is no sign of primordial non-Gaussianity in the CMB fluctuations. On the other hand, from the analysis of BICEP2/Keck CMB polarization experiments [7, 8], the upper limits are set to r < 0.07 at 95% CL. The future QUBIC experiment [9, 10] targets to constrain the tensor-to-scalar ratio of 0.01 at 95% CL with two years of data. Therefore, the interesting question is how to construct the inflation models which can be consistent with the Planck results and have large tensor-to-scalar ratio. The energy scale of inflation is described by the ratio between the amplitude of the tensor mode and scalar mode of CMB [11]$\begin{eqnarray}{V}^{1/4}\simeq {\left(\displaystyle \frac{r}{0.01}\right)}^{1/4}\times {10}^{16}\,\mathrm{GeV}\,.\,\,\end{eqnarray}$
The inflationary models with r ∼ 0.01 are very interesting for two reasons: (1) from the above equation (2), the energy scale of inflation is close to the unification scale in the grand unified theory (GUT), which is around 2 × 1016 GeV; (2) they might be probed by the future experiments. On the other hand, the naive analysis of the Lyth bound gives ${\rm{\Delta }}\phi /{M}_{\mathrm{Pl}}\gt N\sqrt{r/8}$ [12, 13], where N is the number of e-folding, and MPl is the reduced Planck scale, i.e. ${M}_{\mathrm{Pl}}^{2}\equiv {\left(8\pi G\right)}^{-1}$. Thus, the inflaton excursion Δφ is larger than about 2MPl for r = 0.01 and N ∼ 55. And we will define the large tensor-to-scalar ratio as r > 0.01, and call it the large field inflation if Δφ > 2MPl. Therefore, if the Lyth bound is valid, the big challenge to the inflationary models with large r is the high-dimensional operators in the inflaton potential from the effective field theory (EFT) point of view, which may be generated by the quantum gravity (QG) effects, since such high-dimensional operators can not be suppressed by MPl and then are out of control. Three kinds of solutions to this the problem are: (1) The natural inflation since the QG corrections may be forbidden by the discrete symmetry [14, 15]. However, the QG does not preserve the global symmetry. (2) The phase inflation since the phase of a complex field may not play an important role in the non-renormalizable operators from QG effects [16–18]. (3) The polynomial inflation with the Lyth bound violation and small Δφ [19–32]. However, no one has constructed a concrete, solid, and interesting inflationary model with ${\rm{\Delta }}\phi \leqslant { \mathcal O }(0.1){M}_{\mathrm{Pl}}$ for the whole e-folding stretch. And also, the Gao–Gong–Li bound on inflaton excursion [23] is Δφ < 0.1 MPl with a modified tensor-to-scalar bound r < 0.02. It's too low and can not be saturated, and it is not clear whether there exists a new practical bound. Note that, the minimal field excursion Δφ plays an important role to determine the upper bound of r.
For Δφ < 0.1 MPl, we can expand the generic inflaton potential via the Taylor series. Thus, for simplicity, we shall study the polynomial inflation with the tensor-to-scalar ratio as large as possible which is consistent with the QG corrections and EFT. We do realize the small field inflation with a large r, and then the Lyth bound is violated obviously. Interestingly, we find an excellent practical bound on the inflaton excursion in the format $a+b\sqrt{r}$, where a is a small real number and b is at the order 1. To be consistent with QG/EFT and suppress the high-dimensional operators, we show that the concrete condition on inflaton excursion is $\tfrac{{\rm{\Delta }}\phi }{{M}_{\mathrm{Pl}}}\lt 0.2\times \sqrt{10}\simeq 0.632$. For ns = 0.9649, Ne = 55, and $\tfrac{{\rm{\Delta }}\phi }{{M}_{\mathrm{Pl}}}\lt 0.632$, we predict that the tensor-to-scalar ratio is smaller than 0.0012 for such polynomial inflation to be consistent with QG/EFT. The interesting question is whether our polynomial inflation can arise from a concise function or a more complete theory. From our study during the last seven years, we have not found it yet.
2. The polynomial inflation
We will consider the order 5 polynomial inflation for numerical study, i.e.$\begin{eqnarray}V(\phi )={V}_{0}\left[1+\sum _{m=1}{\lambda }_{m}{\left(\phi -{\phi }_{* }\right)}^{m}\right],\,m=1,\,2,\,\ldots ,\,5.\end{eqnarray}$The slow-roll parameters ε, η, and ξ2 are defined as$\begin{eqnarray}\begin{array}{rcl}\epsilon & = & {\epsilon }_{1}=\displaystyle \frac{{M}_{\mathrm{Pl}}^{2}{\left({V}^{{\prime} }\right)}^{2}}{2{V}^{2}},\\ \eta & = & {\epsilon }_{2}=\displaystyle \frac{{M}_{\mathrm{Pl}}^{2}{V}^{{\prime\prime} }}{V},\\ {\xi }^{2} & = & {\epsilon }_{3}=\displaystyle \frac{{M}_{\mathrm{Pl}}^{4}{V}^{{\prime} }{V}^{\prime\prime\prime }}{{V}^{2}},\end{array}\end{eqnarray}$where $X^{\prime} \equiv {\rm{d}}X(\phi )/{\rm{d}}\phi $. And the other two relevant slow-roll parameters [33] in terms of the order 5 polynomial inflaton potential are$\begin{eqnarray}\begin{array}{rcl}{\sigma }^{3} & = & {\epsilon }_{4}={M}_{\mathrm{Pl}}^{6}{\left({V}^{{\prime} }\right)}^{2}{V}^{{\prime\prime \prime\prime} }/{V}^{3},\\ {\delta }^{4}(\phi ) & = & {\epsilon }_{5}={M}_{\mathrm{Pl}}^{8}{\left({V}^{{\prime} }\right)}^{3}{V}^{{\prime\prime} \prime\prime\prime }/{V}^{4},\end{array}\end{eqnarray}$The number of e-folding before the end of inflation is$\begin{eqnarray}N(\phi )={\int }_{t}^{{t}_{e}}H{\rm{d}}t\approx \displaystyle \frac{1}{{M}_{\mathrm{Pl}}}{\int }_{{\phi }_{e}}^{\phi }\displaystyle \frac{{\rm{d}}\phi }{\sqrt{2\epsilon (\phi )}},\end{eqnarray}$where the inflaton value φe at the end of inflation is determined by $\ddot{a}=0$ or ε = 1, where a is the scale factor. If ε(φ) is a monotonic function during inflation, we have ε(φ) > ε(φ*) = ε = r/16, and then get the Lyth bound [12]$\begin{eqnarray}\displaystyle \frac{{\rm{\Delta }}\phi }{{M}_{\mathrm{Pl}}}\equiv | {\phi }_{* }-{\phi }_{e}| \gt \sqrt{2\epsilon }\,{N}_{e}=\sqrt{r/8}\,{N}_{e}\,,\end{eqnarray}$where the subscript ‘*' means the value at the horizon crossing, and ns, αs and r are evaluated at φ*. For example, the Lyth bound gives Δφ = 1.947MPl for r = 0.01 and Ne = 55. Thus, to realize the small field inflation with large r, the Lyth bound must be violated. In other words, ε(φ) should not be a monotonic function and has at least one minimum between φ* and φe [19].
2.1. Numerical results
To compute observable quantities for the CMB, we numerically evolve the scalar field according to the Friedman equation and equation of motion for φ:$\begin{eqnarray}\begin{array}{l}\ddot{\phi }+3H\dot{\phi }+V^{\prime} =0,\\ {H}^{2}\equiv {\left(\displaystyle \frac{\dot{a}}{a}\right)}^{2}=\displaystyle \frac{8\pi G}{3}\left(\displaystyle \frac{1}{2}\dot{\phi }+V\right).\end{array}\end{eqnarray}$For numerical purposes it is more convenient to rewrite the inflaton evolution as a function of conformal time τ rather than time t. Using ${\rm{d}}\tau =\tfrac{{\rm{d}}t}{a}$ the cosmological evolution equation becomes$\begin{eqnarray}\displaystyle \frac{{{\rm{d}}}^{2}\phi }{{\rm{d}}{\tau }^{2}}+2{aH}\displaystyle \frac{{\rm{d}}\phi }{{\rm{d}}\tau }+{a}^{2}V^{\prime} =0.\end{eqnarray}$For convenience, we use the slow-roll parameters defined via Hubble parameter:$\begin{eqnarray*}\begin{array}{rcl}{\epsilon }_{1}^{H} & = & -\displaystyle \frac{\dot{H}}{{H}^{2}}\approx \epsilon \,,\,{\epsilon }_{2}^{H}=\displaystyle \frac{{\dot{\epsilon }}_{1}^{H}}{H{\epsilon }_{1}^{H}}\approx 4\epsilon -2\eta ,\\ {\epsilon }_{i+1}^{H} & = & \displaystyle \frac{{\dot{\epsilon }}_{i}^{H}}{H{\epsilon }_{i}^{H}}\,(i=2,3,4),\end{array}\end{eqnarray*}$where dots denote derivatives respect to the cosmic time. The inflation ends at $\ddot{a}=0$ or ${\epsilon }_{1}^{H}=1$.
To get the minimal field excursion Δφ and enough e-folding number N for each r, the inflaton field must traverse an extremely flat part of the scalar potential. This is similar to ultra-slow-roll inflation (USR) [34–38]. There are 3 inflection points for the 5th degree polynomial. We find a set of parameters which make sure there is an inflection point φinfl during inflation and the derivative of V(φ) at the inflection point should be small, i.e.$\begin{eqnarray}\displaystyle \frac{{\rm{d}}V}{{\rm{d}}\phi }\simeq 0\,,\,\,\displaystyle \frac{{{\rm{d}}}^{2}V}{{\rm{d}}{\phi }^{2}}=0\,.\end{eqnarray}$In such situation, an exact scalar spectral index ns and tensor-to-scalar ratio r will numerically calculated by primordial spectrum using the Mukhanov–Sasaki formalism [39, 40]. The scalar mode ${u}_{k}=-z{ \mathcal R }$ and tensor mode vk of primordial perturbation are given by$\begin{eqnarray}\displaystyle \frac{{{\rm{d}}}^{2}{u}_{k}}{{\rm{d}}{\tau }^{2}}+\left({k}^{2}-\displaystyle \frac{1}{z}\displaystyle \frac{{{\rm{d}}}^{2}z}{{\rm{d}}{\tau }^{2}}\right){u}_{k}=0,\end{eqnarray}$$\begin{eqnarray}\displaystyle \frac{{{\rm{d}}}^{2}{v}_{k}}{{\rm{d}}{\tau }^{2}}+\left({k}^{2}-\displaystyle \frac{1}{a}\displaystyle \frac{{{\rm{d}}}^{2}a}{{\rm{d}}{\tau }^{2}}\right){v}_{k}=0,\end{eqnarray}$where ${ \mathcal R }$ is the comoving curvature perturbation, and $z=\tfrac{a}{{ \mathcal H }}\tfrac{{\rm{d}}\phi }{{\rm{d}}\tau }$. In the limit k → ∞ , the modes are in the Bunch–Davies vacuum ${u}_{k}\to {{\rm{e}}}^{-{\rm{i}}{k}\tau }/\sqrt{2k}$ and ${v}_{k}\to {{\rm{e}}}^{-{\rm{i}}{k}\tau }/\sqrt{2k}$. The scalar and tensor spectrum of primordial perturbations at CMB scales can be accurately expressed as$\begin{eqnarray}\begin{array}{rcl}{{ \mathcal P }}_{{ \mathcal R }}(k) & = & {A}_{s}{\left(\displaystyle \frac{k}{{k}_{* }}\right)}^{{n}_{s}-1+\tfrac{{\alpha }_{s}}{2}\mathrm{ln}\displaystyle \frac{k}{{k}_{* }}+\cdots },\\ {{ \mathcal P }}_{t}(k) & = & {A}_{t}{\left(\displaystyle \frac{k}{{k}_{* }}\right)}^{{n}_{t}+\cdots }.\end{array}\end{eqnarray}$Then the spectral index, running of the spectral index and tensor-to-scalar ratio at k* = aH = 0.05 MPc−1 can determined by$\begin{eqnarray}{n}_{s}-1=\displaystyle \frac{\mathrm{dln}{{ \mathcal P }}_{{ \mathcal R }}}{\mathrm{dln}k}\,,\,r=\displaystyle \frac{{{ \mathcal P }}_{{ \mathcal T }}}{{{ \mathcal P }}_{{ \mathcal R }}}.\end{eqnarray}$
To simplify the numerical study, we choose Ne = 55, as well as the best fit for ns and αs, i.e. ns = 0.9649, αs = − 0.0045 and As = 2.20 × 10−9 [7, 8]. Without loss of generality, we will take φ* = 0. r and Δφ for the 5th degree polynomial inflation are given in figure 1, where the red point-line is corresponding to the low bound on inflaton excursion. The inflation ends at ${\epsilon }_{1}^{H}=1$ or $\ddot{a}=0$.
Figure 1.
New window|Download| PPT slide Figure 1.r versus Δφ for the polynomial inflation. Here the e-folding number, the scalar spectral index and the relative running are fixed to be N = 55, ns = 0.9649, and αs = − 0.0045, respectively. The red line corresponds to the the low bound on the inflaton excursion. The three horizontal dashed lines correspond to Δφ = 0.632, 1.0, 2.0 MPl, respectively.
To be concrete, we present some examples. Taking several tensor-to-scalar ratios r = 0.01–0.056, we try our best to get the minimal inflation excursions numerically. Using the relation of the CMB observations to the potential parameters at the beginning of inflation, which will be shown in equation (18), we first obtain the values of parameters λ1,2,3 and V0. Then the last two parameters λ4 and λ5 can numerically acquired by solving the equation (10) and requiring that there be at least one inflection point during inflation. In this way, we get an USR inflationary model6(6This feature can ensure the formation of primordial black holes which deserves further study.). with a smaller field excursion since the inflaton slows down around the inflection point. The results are shown in table 1. To understand why our polynomial potential violates the Lyth bound but is consistent with the Planck results, we plot the hubble slow-roll parameters ${\epsilon }_{1}^{H}$ and ${\epsilon }_{2}^{H}$ in figure 2 for the model with r = 0.01. The potential V/V0 and its first derivative $V^{\prime} /{V}_{0}$ with the parameters λi = ( − 3.6188 × 10−2, − 10.0437 × 10−3, − 9.8543 × 10−3, 32.1351 × 10−2, − 35.5515 × 10−2) are shown in figure 3. There is an inflection point φinfl = 0.515 MPl, and $V^{\prime} /{V}_{0}({\phi }_{{\rm{infl}}})$ is close to zero. The evolution of inflaton is extremely slow around φinfl and the slow-roll parameters ${\epsilon }_{1}^{H}$ decreases several orders of magnitude at φinfl.
Figure 2.
New window|Download| PPT slide Figure 2.The evolution of Hubble flow slow-roll parameters ${\epsilon }_{1}^{H}$ (blue solid line) and ${\epsilon }_{2}^{H}$ (red dashed line).
Figure 3.
New window|Download| PPT slide Figure 3.The potential V/V0 and its first derivative $V^{\prime} /{V}_{0}$ with the parameters λi = ( − 3.6188 × 10−2, − 10.0437 × 10−3, − 9.8543 × 10−3, 32.1351 × 10−2, − 35.5515 × 10−2) . The vertical dashed lines correspond to the horizon crossing point φ* = 0, the inflection point φinfl = 0.516 MPl and the end of inflation φe = 1.313 MPl.
Table 1. Table 1.The low bound of inflaton excursion and the parameters for inflation potential. Here e-folding number, the scalar spectral index and the relative running are fixed to the central value, i.e. N = 55, ns = 0.9649, and αs = − 0.0045.
In the slow-roll inflation with inflaton potential V(φ), the observations are$\begin{eqnarray}\begin{array}{rcl}{n}_{s} & = & 1+2\eta -6\epsilon \,,\,r=16\epsilon ,\\ {\alpha }_{s} & = & 16\epsilon \eta -24{\epsilon }^{2}-2{\xi }^{2}\,,\\ {A}_{s} & = & \displaystyle \frac{V}{24{\pi }^{2}\epsilon },\end{array}\end{eqnarray}$Since the inflation potential has order 5 polynomial, we need take into account the higher order corrections for ns and r. The higher order corrections [33, 41–43] are given in$\begin{eqnarray}\begin{array}{rcl}{\rm{\Delta }}{n}_{s} & = & 2\left[\displaystyle \frac{1}{3}{\eta }^{2}-\left(\displaystyle \frac{5}{3}+12C\right){\epsilon }^{2}\right.\\ & & \left.+(8C-1)\epsilon \eta -(C-\displaystyle \frac{1}{3}){\xi }^{2}\right],\\ {\rm{\Delta }}r & = & \displaystyle \frac{32\epsilon }{3}(3C-1)(2\epsilon -\eta ),\end{array}\end{eqnarray}$where $C=-2+\mathrm{ln}2+\gamma \simeq -0.7296$ with γ the Euler–Mascheroni constant. The slow-roll parameters at the horizon crossing φ* = 0 are approximated as$\begin{eqnarray}\epsilon =\displaystyle \frac{{\lambda }_{1}^{2}}{2}\,,\,\,\eta =2{\lambda }_{2}\,,\,\,{\xi }^{2}=6{\lambda }_{1}{\lambda }_{3}.\end{eqnarray}$Then, the parameters V0 and λ1,2,3 can be determined by the observations in equation (15) as follows:$\begin{eqnarray}\begin{array}{rcl}{V}_{0} & = & \displaystyle \frac{3}{2}{\pi }^{2}{P}_{s}{r}^{(1)},{\lambda }_{1}=-\sqrt{{r}^{(1)}/8},\\ {\lambda }_{2} & = & \displaystyle \frac{{n}_{s}^{(1)}-1+3{\lambda }_{1}^{2}}{4},{\lambda }_{3}=-\displaystyle \frac{{\alpha }_{s}+6{\lambda }_{1}^{4}-16{\lambda }_{1}^{2}{\lambda }_{2}}{12{\lambda }_{1}}.\end{array}\end{eqnarray}$Therefore, we can get the higher order corrections of these observations$\begin{eqnarray}\begin{array}{rcl}{\rm{\Delta }}{n}_{s} & = & \displaystyle \frac{1}{384}\left[64\left({\alpha }_{s}(6C-2)+{\left({n}_{s}^{(1)}-1\right)}^{2}\right)\right.\\ & & \left.+88{r}^{(1)}({n}_{s}^{(1)}-1)+7{\left({r}^{(1)}\right)}^{2}\right],\\ {\rm{\Delta }}r & = & -\displaystyle \frac{{r}^{(1)}}{3}(3C-1)\left({n}_{s}^{(1)}-1+\displaystyle \frac{{r}^{(1)}}{8}\right).\end{array}\end{eqnarray}$Solving these equations with αs = − 0.0045, we find that the second-order corrections will give a tiny contribution Δr < − 1.5 × 10−3 for r < 0.056 and Δns ∼ 4.9 × 10−3 for ns = 0.9649. These corrections for CMB are still within 2 Σ range of Planck data.
2.3. The lower bound on inflaton excursion
For slow-roll inflation, we obtain ${\rm{\Delta }}\phi /{M}_{\mathrm{Pl}}=b\sqrt{r}$ if ε(φ) is a constant during inflation. However, in general, ε(φ) is a φ dependent varying function during inflation. Interestingly, for the slow-roll inflation with polynomial inflaton potential, we find that the generic lower bound on inflaton excursion is a linear function of $\sqrt{r}$, i.e. ${\rm{\Delta }}\phi /{M}_{\mathrm{Pl}}=a+b\sqrt{r}$ approximately. The results for a fixed αs = − 0.0042 and N = 55 are show in figure 4. The black line in figure 4(a) is the fitted linear equation $0.4569+8.2247\sqrt{r}$ for fixed ns = 0.9649. We also check the field excursion variation as ns increases. The different color points in figure 4 are corresponding to different central value of ns. For the variation of ns, the low bound of Δφ is pushed toward to a smaller value, which is consistent with previous results [28]. On the other hand, after fixing the tensor-to-scalar ratio r = 0.01, we compare the field excursion results from equation (8) and slow-roll approximation in equation (16) for ns = 0.9625, 0.9655, 0.9685. The results are concluded in table 2. We can find that the field excursion become smaller at order 10−3 as ns increases.
Figure 4.
New window|Download| PPT slide Figure 4.The low bounds on inflaton excursions. The black line corresponds to the fitted linear equation $a+b\sqrt{r}$. The different color points correspond to various central ns. The range of r is r < 0.056.
Table 2. Table 2.The low bound of inflaton excursion. Δφ is numerically calculated from equation (8), and δφ is calculated with high order corrections under the slow roll approximation. Here the tensor-to-scalar ratio is fixed to be r = 0.01.
To be consistent with QG/EFT and suppress the high-dimensional operators, we require that $\tfrac{\phi }{{M}_{\mathrm{Pl}}}$ should be at the order of 0.1, i.e. $\tfrac{0.1}{\sqrt{10}}\leqslant \tfrac{\phi }{{M}_{\mathrm{Pl}}}\leqslant 0.1\times {\sqrt{10}}^{\star }$7(7Here, we want the higher order coefficient to be one order of magnitude smaller than the lower order coefficient in the Taylor expansion.). To minimize the absolute value of $\tfrac{\phi }{{M}_{\mathrm{Pl}}}$, we can choose $-{\phi }_{i}={\phi }_{e}=\tfrac{{\rm{\Delta }}\phi }{2}$, and then $\tfrac{\phi }{{M}_{\mathrm{Pl}}}\leqslant \tfrac{{\rm{\Delta }}\phi }{2{M}_{\mathrm{Pl}}}$. Thus, to be consistent with QG and EFT, we require $\tfrac{{\rm{\Delta }}\phi }{{M}_{\mathrm{Pl}}}\lt 0.2\times \sqrt{10}\simeq 0.632$.
With QG/EFT effects, inflation excursions are smaller than Δφ < 0.632MPl. We predict that the upper bound on tensor-to-scalar ratio r for single field slow-roll inflation is r ≤ 0.0012. The corresponding parameters are also shown in table 1. Meanwhile, for the inflaton excursions Δφ ∼ 1MPl and Δφ ∼ 2MPl, we find that the maximal tensor-to-scalar ratios are r ∼ 0.0046 and r ∼ 0.0335, respectively for Ne = 55. While the Lyth bound gives r ∼ 0.0026 and r ∼ 0.0106, respectively. Thus, the Lyth bound is indeed violated in our polynomial inflation. Our results are consistent with the current cosmological observation data and might be probed by the future observed results of CMB polarization missions [44–49] and gravitational waves experiments [50], which will detect the tensor-to-scalar ratio at the order of 10−3.
3. Conclusions
We have considered the polynomial inflation with the tensor-to-scalar ratio as large as possible which is consistent with the QG corrections and EFT. We got the small field inflation with large r, and then the Lyth bound is violated obviously, since the evolution of the slow-roll parameters are very slow and the inflation enters USR around the inflection point. Interestingly, we found an excellent practical bound on the inflaton excursion in the format $a+b\sqrt{r}$, where a is a small real number and b is at the order 1. To be consistent with QG/EFT and suppress the high-dimensional operators, we show that the concrete condition on inflaton excursion is $\tfrac{{\rm{\Delta }}\phi }{{M}_{\mathrm{Pl}}}\lt 0.2\times \sqrt{10}\simeq 0.632$. For ns = 0.9649, Ne = 55, and $\tfrac{{\rm{\Delta }}\phi }{{M}_{\mathrm{Pl}}}\lt 0.632$, we predict that the tensor-to-scalar ratio is smaller than 0.0012 for such polynomial inflation to be consistent with QG/EFT.
Acknowledgments
This work was supported in part by the Projects 11875062, 11875136, and 11947302 supported by the National Natural Science Foundation of China, by the Major Program of the National Natural Science Foundation of China under Grant No. 11690021, by the Key Research Program of Frontier Science, CAS. This work was also supported in part by the Program 2020JQ-804 supported by Natural Science Basic Research Plan in Shanxi Province of China, and by the Program 20JK0685 funded by Shanxi Provincial Education Department.
,4,5,∗, Tianjun Li4,51School of Sciences, 
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