Small field models of inflation that predict a tensor-to-scalar ratio $r=0.03$

Future observations of the cosmic microwave background polarization are expected to set an improved upper bound on the tensor-to-scalar ratio of $r\lesssim 0.03$. Recently, we showed that small field models of inflation can produce a significant primordial gravitational wave signal. We constructed viable small field models that predict a value of $r$ as high as 0.01. Models that predict higher values of $r$ are more tightly constrained and lead to larger field excursions. This leads to an increase in tuning of the potential parameters and requires higher levels of error control in the numerical analysis. Here, we present viable small field models which predict $r=0.03$. We further find the most likely candidate among these models which fit the most recent Planck data while predicting $r=0.03$. We thus demonstrate that this class of small field models is an alternative to the class of large field models. The BICEP3 experiment and the Euclid and SPHEREx missions are expected to provide experimental evidence to support or refute our predictions.

Figure 1

Polynomial potentials. The blue (solid) line depicts a potential of a model that predicts $r\simeq 0.03$, while the red (dash-dotted) line depicts a potential that predicts $r\simeq 0.01$. The purple (dotted) line depicts a degree five polynomial potential of a model that predicts $r\simeq 0.001$. All models are variants of the hilltop model, with a flatter region in which most e-folds are generated.

Figure 2

Field excursions in (reduced) Planck units. Different predicted values of $r$ require different field excursions to generate the $\sim 8$ $e$-folds probed by the CMB. The model predicting $r\simeq 0.03$ (blue line) requires an excursion of ${(\mathrm{\Delta }\varphi )}_{\mathrm{CMB}}\simeq 0.28$ to generate the same amount of $e$-folds which the model predicting $r\simeq 0.01$ model (red dash-dotted) generates in ${(\mathrm{\Delta }\varphi )}_{\mathrm{CMB}}\simeq 0.2$. This means more tuning is required for models that predict $r\simeq 0.03$. The model predicting $r\simeq 0.001$ (purple dots) requires only ${(\mathrm{\Delta }\varphi )}_{\mathrm{CMB}}\simeq 0.1$ to generate the CMB window.

Figure 3

Covering the observable phase space with small field models that predict $r\simeq 0.03$. The roughly uniform cover of the 95% C.L. areas ensures an accurate likelihood transfer from MCMC results to models.

Figure 4

Likelihood analysis for model coefficients $a_2$, $a_3$, $a_4$, using Gaussian fits to recover mean values. The skewness of the results reflects properties of the model, not of the underlying MCMC analysis. The width of the Gaussian fits is given by (0.016, 0.08, 0.21) and is an indication of the tuning level required for these models.

Figure 5

Primordial power spectra of three models. A model with a degree six polynomial potential that predicts $r\simeq 0.03$ (diamonds and blue line). A model with a degree six polynomial potential that predicts $r\simeq 0.01$ (squares and red dash-dotted) and a model with a degree five polynomial potential that predicts $r=0.001$ (circles and purple dots). The pivot scale for all three is $k_0=0.05{\mathrm{Mpc}}^{-1}$ and the results are overlayed at that scale for ease of comparison.