Natural Inflation: Consistency with Cosmic Microwave Background Observations of Planck and BICEP2

TL;DR

This paper evaluates Natural Inflation (NI) models against Planck and BICEP2 data, testing cosine NI, low-scale NI, and axion-motivated variants. It finds that the original NI and related cosine forms remain consistent with observations when the width is trans-Planckian ($f \gtrsim m_{Pl}$) and the height satisfies $\Lambda \sim m_{GUT}$, while low-scale NI and axion monodromy with $V \propto φ^{2/3}$ or $V \propto φ$ are disfavored by the tensor-mode constraints from BICEP2. The analysis emphasizes that running of the scalar spectral index is negligible across models, predicting a small $|dn_s/d\ln k| \lesssim \mathcal{O}(10^{-3})$, and that BICEP2 significantly narrows viable single-field inflation scenarios. Overall, NI offers a theoretically well-motivated, testable description of early inflation compatible with current CMB observations, particularly in the large-field, PNGB framework with $f$ near or above the Planck scale. $r$ is tied to the energy scale of inflation, linking cosmology to high-energy physics at the GUT scale, and future measurements of tensor modes and running will further probe the NI parameter space.

Abstract

Natural inflation is a good fit to all cosmic microwave background (CMB) data and may be the correct description of an early inflationary expansion of the Universe. The large angular scale CMB polarization experiment BICEP2 has announced a major discovery, which can be explained as the gravitational wave signature of inflation, at a level that matches predictions by natural inflation models. The natural inflation (NI) potential is theoretically exceptionally well motivated in that it is naturally flat due to shift symmetries, and in the simplest version takes the form $V(φ) = Λ^4 [1 \pm \cos(Nφ/f)]$. A tensor-to-scalar ratio $r>0.1 $ as seen by BICEP2 requires the height of any inflationary potential to be comparable to the scale of grand unification and the width to be comparable to the Planck scale. The Cosine Natural Inflation model agrees with all cosmic microwave background measurements as long as $f \geq m_{\rm Pl}$ (where $m_{\rm Pl} = 1.22 \times 10^{19}\ {\rm GeV}$) and $Λ\sim m_{\rm GUT} \sim 10^{16}\ {\rm GeV}$. This paper also discusses other variants of the natural inflation paradigm: we show that axion monodromy with potential $V\propto φ^{2/3}$ is inconsistent with the BICEP2 limits at the 95\% confidence level, and low-scale inflation is strongly ruled out. Linear potentials $V \propto φ$ are inconsistent with the BICEP2 limit at the 95\% confidence level, but are marginally consistent with a joint Planck/BICEP2 limit at 95\%. We discuss the pseudo-Nambu Goldstone model proposed by Kinney and Mahanthappa as a concrete realization of low-scale inflation. While the low-scale limit of the model is inconsistent with the data, the large-field limit of the model is marginally consistent with BICEP2. All of the models considered predict negligible running of the scalar spectral index, and would be ruled out by a detection of running.

Figures (5)

• Figure 1: Original Natural Inflation (Cosine Potential): The band between the (solid/blue) lines running from approximately the lower left up to the middle of the plot are predictions for Natural Inflation for constant $N$ and varying $f$, where $N$ is the number of e-foldings prior to the end of inflation at which current modes of scale $k = 0.002$ Mpc$^{-1}$ were generated and $f$ is the width of the potential. The range of values of $N$ reflect uncertainties in reheating after inflation as described in the text. Filled red (nearly vertical) regions are the parameter spaces allowed by Planck plus other CMB data as indicated at 68% and 95% C.L.'s. Dotted regions are the parameter spaces allowed by Planck + WMAP Polarization + Lensing +BAO at 68% and 95% C.L.'s. Horizontal blue bands correspond to 1(2) $\sigma$ measurements of $r$ from BICEP2 for the case of no running. The predictions match the data for trans-Planckian $f$. Also shown are the axion monodromy potential $V \propto \phi^{2/3}$ and the linear potential $V \propto \phi$, which are inconsistent with the BICEP2 limit at $2\sigma$, and the power-law inflationary potential $V \propto \exp{\left(\phi/\mu\right)}$ .
• Figure 2: Low-scale models of natural inflation Low-scale models with $\mu < m_{\rm Pl}$ are shown relative to the constraints from Planck and BICEP2. These models are strongly ruled out by the BICEP2 detection of tensor modes. Labels same as in Figure 1 (roughly, solid lines are theoretical predictions; red is Planck data; blue is BICEP2 data).
• Figure 3: KM Model of natural inflation Labels same as in Figure 1 (roughly, solid lines are theoretical predictions; red is Planck data; blue is BICEP2 data).
• Figure 4: Higgs-like inflation. Labels same as in Figure 1 (roughly, solid lines are theoretical predictions; red is Planck data; blue is BICEP2 data).
• Figure 5: Cosine Natural Inflation: Joint likelihood for Planck + WMAP Polarization + Lensing + BICEP, including ACT/SPT and BAO, for comparison with Figure \ref{['fig:cosine']}. Inner contours are 68 % confidence limits, outer contours are 95 % confidence. Note that the linear potential $V \propto \phi$ is marginally consistent with the joint fit at 95% confidence, as is the power-law potential $V \propto e^{\phi/\mu}$.