Forecasts for CMB μ- and i-type spectral distortion constraints on the primordial power spectrum on scales 8 < k < 10⁴ Mpc^-1 with the future Pixie-like experiments

Silk damping at redshifts 1.5 x 10^4 < z < 2 x 10^6 erases CMB anisotropies on scales corresponding to the comoving wavenumbers 8 < k < 10^4 Mpc^-1 (10^5 < \ell < 10^8). This dissipated energy is gained by the CMB monopole, creating distortions from a blackbody in the CMB spectrum of the \mu-type and the i-type. We study, using Fisher matrices, the constraints we can get from measurements of these spectral distortions on the primordial power spectrum from future experiments such as Pixie, and how these constraints change as we change the frequency resolution and the sensitivity of the experiment. We show that the additional information in the shape of the $i$-type distortions, in combination with the \mu-type distortions, allows us to break the degeneracy between the amplitude and the spectral index of the power spectrum on these scales and leads to much tighter constraints. We quantify the information contained in both the \mu-type distortions and the i-type distortions taking into account the partial degeneracy with the y-type distortions and the temperature of the blackbody part of the CMB. We also calculate the constraints possible on the primordial power spectrum when the spectral distortion information is combined with the CMB anisotropies measured by the WMAP, SPT, ACT and Planck experiments.