Inflation that runs naturally: Gravitational waves and suppression of
  power at large and small scales
  We point out three correlated predictions of the axion monodromy inflation
model: large amplitude of gravitational waves, suppression of power on horizon
scales and on scales relevant for the formation of dwarf galaxies. While these
predictions are likely generic to models with oscillations in the inflaton
potential, the axion monodromy model naturally accommodates the required
running spectral index through Planck-scale corrections to the inflaton
potential. Applying this model to a combined data set of Planck, ACT, SPT, and
WMAP low-$\ell$ polarization cosmic microwave background (CMB) data, we find a
best-fit tensor-to-scalar ratio $r_{0.05} = 0.07^{+0.05}_{-0.04}$ due to
gravitational waves, which may have been observed by the BICEP2 experiment.
Despite the contribution of gravitational waves, the total power on large
scales (CMB power spectrum at low multipoles) is lower than the standard
$\Lambda$CDM cosmology with a power-law spectrum of initial perturbations and
no gravitational waves, thus mitigating some of the tension on large scales.
There is also a reduction in the matter power spectrum of 20-30\% at scales
corresponding to $k = 10~{\rm Mpc}^{-1}$, which are relevant for dwarf galaxy
formation. This will alleviate some of the unsolved small-scale structure
problems in the standard $\Lambda$CDM cosmology. The inferred matter power
spectrum is also found to be consistent with recent Lyman-$\alpha$ forest data,
which is in tension with the Planck-favored $\Lambda$CDM model with power-law
primordial power spectrum.
