Reconstructing the Primordial Spectrum from WMAP Data by the Cosmic Inversion Method

We reconstruct the primordial spectrum of the curvature perturbation, $P(k)$, from the observational data of the Wilkinson Microwave Anisotropy Probe (WMAP) by the cosmic inversion method developed recently. In contrast to conventional parameter-fitting methods, our method can potentially reproduce small features in $P(k)$ with good accuracy. As a result, we obtain a complicated oscillatory $P(k)$. We confirm that this reconstructed $P(k)$ recovers the WMAP angular power spectrum with resolution up to $\Delta \ell \simeq 5$. Similar oscillatory features are found, however, in simulations using artificial cosmic microwave background data generated from a scale-invariant $P(k)$ with random errors that mimic observation. In order to examine the statistical significance of the nontrivial features, including the oscillatory behaviors, therefore, we consider a method to quantify the deviation from scale-invariance and apply it to the $P(k)$ reconstructed from the WMAP data. We find that there are possible deviations from scale-invariance around $k \simeq 1.5\times10^{-2}$ and $2.6\times10^{-2}{\rm Mpc}^{-1}$.