Natural Inflation: Consistency with Cosmic Microwave Background
  Observations of Planck and BICEP2
  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 $V(\phi) = \Lambda^4 [1
\pm \cos(N\phi/f)]$. 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 $\Lambda \sim m_{\rm GUT} \sim 10^{16}\ {\rm GeV}$.
This paper also discusses other variants of the natural inflation paradigm: we
show that axion monodromy with potential $V\propto \phi^{2/3}$ is inconsistent
with the BICEP2 limits at the 95\% confidence level, and low-scale inflation is
strongly ruled out. Linear potentials $V \propto \phi$ 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.
