Category: CMB publications

Summary

The primordial power spectrum describes the initial perturbations in the Universe which eventually grew into the large-scale structure we observe today, and thereby provides an indirect probe of inflation or other structure-formation mechanisms. The primordial power spectrum is therefore linked to cosmological observables and in particular the Cosmic Microwave Background (CMB). However, due to degeneracies in the transfer function that links the primordial power spectrum to the measured CMB, recovering the primordial power spectrum constitutes a non-trivial ill-posed inverse problem. Furthermore, the multiplicative nature of the noise on the estimator of CMB power spectrum adds to the complexity of the problem. PRISM is a new sparsity-based, non-linear reconstruction technique which aims at reconstructing global and local features on the primordial power spectrum from CMB measurements. Features are detected and reconstructed based on their significance compared to noise. The PRISM software is part of the ISAP package and will be included in the next release of the package. The figure below shows reconstructions of a primordial power spectrum with a localized feature around 0.03 Mpc using PRISM.

The algorithm

The primordial power spectrum reconstruction problem is an ill-posed linear inverse problem which we address through the framework of sparse regularisation. This framework allows for accurate and robust recovery by leveraging a sparsity prior on the signal to recover (i.e. it is assumed that it can be represented using a small number of coefficients in an adapted dictionary). Formally, the observed pseudo-power spectrum $$\widetilde{C}_\ell$$ computed from masked CMB maps can be linked to the underlying primordial power spectrum $$P_k$$ through a linear relation of the form: \[\widetilde{C}_\ell = \left( \mathbf{M} P_k \right) Z_\ell \] where $$\mathbf{M}$$ is a linear operator that encodes the angular transfer function of CMB anisotropies and the effects of masks and beams, and $$\mathbf{Z_\ell}$$ is a multipicative noise term. The recovery of the primordial power spectrum is performed by solving an optimization problem of the form: \[\min\limits_{X} \quad \frac{1}{2} \underbrace{\parallel C_\ell - \mathbf{M} X \parallel^{2}_{2}}_{\mbox{data fidelity term}} \quad + \quad \lambda \underbrace{\parallel \mathbf{\Phi} X \parallel_{1}}_{\mbox{sparsity promoting term}} \] where $$\mathbf{\Phi}$$ is a wavelet transform. To estimate the unknown quantity $$C_\ell - \mathbf{M} X$$ from the noisy quantity $$\widetilde{C}_\ell - \mathbf{M} X$$, we use a variance stabilisation approach derived from the technique used in the TOUSI algorithm (Paykari et al. (2012)) and based on the Whaba variance stabilisation transform.

Results on simulations

To assess the performance of PRISM we produced a large number of Monte-Carlo CMB simulations for different test power spectra. We tested a near scale invariant primordial power spectra with ns=0.972, a power spectrum with a running of the spectral index with ns=0.972, $$\alpha_s$$= -0.017 (see figure below), and a power law spectrum with a localised feature around 0.3 Mpc (see above figure).

Summary

The LGMCA method has been used to reconstruct the Cosmic Microwave Background (CMB) image from WMAP 9 year and Planck-PR1 data. Based on the sparse modeling of signals - a framework recently developed in applied mathematics - the proposed component separation method is well-suited for the extraction of foreground emissions. A joint WMAP9 year and Planck PR1 CMB has been reconstructed for the first time and produce a very high quality CMB map, especially on the galactic center where it is the most difficult due to the strong foreground emissions of our Galaxy. We compare the LGMCA results with other CMB Planck PR1 maps:

Comparison on the Galactic centre centered at (l,b)=(37.7,0). Top, PR1 NILC and SEVEM CMB maps, and bottom, PR1 SMICA and WPR1 LGMCA CMB maps:

Quality maps:

It is usually hard to derive a discriminative criterion from real data. For that purpose, the quality map measures an excess of power in the estimated CMB map in the wavelet domain with the respect to the expected power of the CMB. The expected CMB power is computed from the Planck best-fit power spectrum assuming a fiducial cosmology.

The LGMCA CMB map does not contain noticeable tSZ residuals (coma cluster area):

The maps below shows the differences between HFI-217GHz and CMB maps, respectively PR1 NILC, SEVEM, SMICA and WPR1 LGMCA. As the SZ vanishes at 217GHz, the presence of residuals of tSZ in the difference reveals a tSZ contamination in the estimated CMB map.

Below we show the power spectra.  Left, estimated power spectrum of the WPR1 LGMCA map (red) and official PR1 power spectrum (green). The solid black line is the Planck-only best-fit C_l provided by the Planck consortium. Right, power spectrum in logarithmic scale. Error bars are set to 1 sigma.${}_{}$

And finally, the difference spectra. Left, difference between the power spectrum estimated from the WPR1 LGMCA map (red) (resp. official PR1 power spectrum (green)) and the Planck-only best-fit provided by the Planck consortium. Right, difference between the estimated and theoretical power spectra in logarithmic scale. Error bars are set to 1 sigma.