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.