Direct Wavelet Expansion of the Primordial Power Spectrum

In order to constrain and possibly detect unusual physics during inflation, we allow the power spectrum of primordial matter density fluctuations, P_in(k), to be an arbitrary function in the estimation of cosmological parameters from data. The multiresolution and good localization properties of orthogonal wavelets make them suitable for detecting features in P_in(k). We expand P_in(k) directly in wavelet basis functions. The likelihood of the data is thus a function of the wavelet coefficients of P_in(k), as well as the Hubble constant H₀, baryon density Ω_b h², cold dark matter density Ω_c h², and the reionization optical depth τ_ri in a flat ΛCDM cosmology. We derive constraints on these parameters from cosmic microwave background anisotropy data (WMAP, CBI, and ACBAR) and large-scale structure data (2dFGRS and PSCZ) using the Markov chain Monte Carlo (MCMC) technique. The direct wavelet expansion method is different from and complementary to the wavelet band power method of Mukherjee & Wang, and results from the two methods are consistent. In addition, as we demonstrate, the direct wavelet expansion method has the advantage that once the wavelet coefficients have been constrained, the reconstruction of P_in(k) can be effectively denoised, i.e., P_in(k) can be reconstructed using only the coefficients that, say, deviate from zero at greater than 1 σ. In doing so, we retain the essential properties of P_in(k). The reconstruction also suffers much less from the correlated errors of binning methods. The shape of the primordial power spectrum, as reconstructed in detail here, reveals an interesting new feature at 0.001 ≲ k/Mpc^-1 ≲ 0.005. It will be interesting to see whether this feature is confirmed by future data. The reconstructed and denoised P_in(k) is favored over the scale-invariant and power-law forms at ≳1 σ.