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 n_s, the running of the scalar spectral index ${\alpha }_s\equiv {\mathrm{d}n}_s/\mathrm{d}\mathrm{l}\mathrm{n}k$, the tensor-to-scalar ratio r, and the scalar amplitude A_s for the power spectrum of the curvature perturbations are respectively constrained to beThere 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]

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 $\mathrm{\Delta }\varphi /M_{\mathrm{P}\mathrm{l}}>N\sqrt{r/8}$ [12, 13], where N is the number of e-folding, and M_Pl is the reduced Planck scale, i.e. $M_{\mathrm{P}\mathrm{l}}^2\equiv {\left(8\pi G\right)}^{-1}$. Thus, the inflaton excursion Δφ is larger than about 2M_Pl 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 > 2M_Pl. 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 M_Pl 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 $\mathrm{\Delta }\varphi ⩽\mathcal{O}(0.1)M_{\mathrm{P}\mathrm{l}}$ for the whole e-folding stretch. And also, the Gao–Gong–Li bound on inflaton excursion [23] is Δφ < 0.1 M_Pl 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 M_Pl, 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 $\frac{\mathrm{\Delta }\varphi }{M_{\mathrm{P}\mathrm{l}}}<0.2\times \sqrt{10}\simeq 0.632$. For n_s = 0.9649, N_e = 55, and $\frac{\mathrm{\Delta }\varphi }{M_{\mathrm{P}\mathrm{l}}}<0.632$, 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.

We will consider the order 5 polynomial inflation for numerical study, i.e.The slow-roll parameters ε, η, and ξ² are defined aswhere $X^{\mathrm{\prime }}\equiv \mathrm{d}X(\varphi )/\mathrm{d}\varphi$. And the other two relevant slow-roll parameters [33] in terms of the order 5 polynomial inflaton potential areThe number of e-folding before the end of inflation iswhere the inflaton value φ_e 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]where the subscript ‘*' means the value at the horizon crossing, and n_s, α_s and r are evaluated at φ_*. For example, the Lyth bound gives Δφ = 1.947M_Pl for r = 0.01 and N_e = 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 φ_e [19].

To compute observable quantities for the CMB, we numerically evolve the scalar field according to the Friedman equation and equation of motion for φ:For numerical purposes it is more convenient to rewrite the inflaton evolution as a function of conformal time τ rather than time t. Using $\mathrm{d}\tau =\frac{\mathrm{d}t}{a}$ the cosmological evolution equation becomesFor convenience, we use the slow-roll parameters defined via Hubble parameter:where dots denote derivatives respect to the cosmic time. The inflation ends at $\ddot{a}=0$ or ${ϵ}_1^H=1$.

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 φ_infl during inflation and the derivative of V(φ) at the inflection point should be small, i.e.In such situation, an exact scalar spectral index n_s 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 v_k of primordial perturbation are given by where $\mathcal{R}$ is the comoving curvature perturbation, and $z=\frac{a}{\mathcal{H}}\frac{\mathrm{d}\varphi }{\mathrm{d}\tau }$. In the limit k → ∞ , the modes are in the Bunch–Davies vacuum $u_k\to {\mathrm{e}}^{-\mathrm{i}k\tau }/\sqrt{2k}$ and $v_k\to {\mathrm{e}}^{-\mathrm{i}k\tau }/\sqrt{2k}$. The scalar and tensor spectrum of primordial perturbations at CMB scales can be accurately expressed asThen the spectral index, running of the spectral index and tensor-to-scalar ratio at k_* = aH = 0.05 MPc⁻¹ can determined by

To simplify the numerical study, we choose N_e = 55, as well as the best fit for n_s and α_s, i.e. n_s = 0.9649, α_s = − 0.0045 and A_s = 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 ${ϵ}_1^H=1$ or $\ddot{a}=0$.

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 λ_1,2,3 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 ${ϵ}_1^H$ and ${ϵ}_2^H$ in figure 2 for the model with r = 0.01. The potential V/V₀ and its first derivative $V^{\mathrm{\prime }}/V_0$ with the parameters λ_i = ( − 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 φ_infl = 0.515 M_Pl, and $V^{\mathrm{\prime }}/V_0({\varphi }_{\mathrm{i}\mathrm{n}\mathrm{f}\mathrm{l}})$ is close to zero. The evolution of inflaton is extremely slow around φ_infl and the slow-roll parameters ${ϵ}_1^H$ decreases several orders of magnitude at φ_infl.

In the slow-roll inflation with inflaton potential V(φ), the observations areSince the inflation potential has order 5 polynomial, we need take into account the higher order corrections for n_s and r. The higher order corrections [33, 41–43] are given inwhere $C=-2+\mathrm{l}\mathrm{n}2+\gamma \simeq -0.7296$ with γ the Euler–Mascheroni constant. The slow-roll parameters at the horizon crossing φ_* = 0 are approximated asThen, the parameters V₀ and λ_1,2,3 can be determined by the observations in equation (15) as follows:Therefore, we can get the higher order corrections of these observationsSolving these equations with α_s = − 0.0045, we find that the second-order corrections will give a tiny contribution Δr < − 1.5 × 10⁻³ for r < 0.056 and Δn_s ∼ 4.9 × 10⁻³ for n_s = 0.9649. These corrections for CMB are still within 2 Σ range of Planck data.

For slow-roll inflation, we obtain $\mathrm{\Delta }\varphi /M_{\mathrm{P}\mathrm{l}}=b\sqrt{r}$ 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. $\mathrm{\Delta }\varphi /M_{\mathrm{P}\mathrm{l}}=a+b\sqrt{r}$ approximately. The results for a fixed α_s = − 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 n_s = 0.9649. We also check the field excursion variation as n_s increases. The different color points in figure 4 are corresponding to different central value of n_s. For the variation of n_s, 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 n_s = 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 n_s increases.

To be consistent with QG/EFT and suppress the high-dimensional operators, we require that $\frac{\varphi }{M_{\mathrm{P}\mathrm{l}}}$ should be at the order of 0.1, i.e. $\frac{0.1}{\sqrt{10}}⩽\frac{\varphi }{M_{\mathrm{P}\mathrm{l}}}⩽0.1\times {\sqrt{10}}^{\star }$ 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 $\frac{\varphi }{M_{\mathrm{P}\mathrm{l}}}$, we can choose $-{\varphi }_i={\varphi }_e=\frac{\mathrm{\Delta }\varphi }{2}$, and then $\frac{\varphi }{M_{\mathrm{P}\mathrm{l}}}⩽\frac{\mathrm{\Delta }\varphi }{2M_{\mathrm{P}\mathrm{l}}}$. Thus, to be consistent with QG and EFT, we require $\frac{\mathrm{\Delta }\varphi }{M_{\mathrm{P}\mathrm{l}}}<0.2\times \sqrt{10}\simeq 0.632$.

With QG/EFT effects, inflation excursions are smaller than Δφ < 0.632M_Pl. 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 ∼ 1M_Pl and Δφ ∼ 2M_Pl, we find that the maximal tensor-to-scalar ratios are r ∼ 0.0046 and r ∼ 0.0335, respectively for N_e = 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.

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 $\frac{\mathrm{\Delta }\varphi }{M_{\mathrm{P}\mathrm{l}}}<0.2\times \sqrt{10}\simeq 0.632$. For n_s = 0.9649, N_e = 55, and $\frac{\mathrm{\Delta }\varphi }{M_{\mathrm{P}\mathrm{l}}}<0.632$, we predict that the tensor-to-scalar ratio is smaller than 0.0012 for such polynomial inflation to be consistent with QG/EFT.