Estimation of Primordial Spectrum with post-WMAP 3 year data

TL;DR

This work addresses recovering the primordial power spectrum $P(k)$ directly from the CMB angular power spectrum $C_b{\ell}$ under a flat $\Lambda$CDM framework by employing an error-sensitive Richardson-Lucy deconvolution with a normalization scheme that suppresses tail artifacts. It further smooths the recovered spectra with a Discrete Wavelet Transform to maximize the full-sky likelihood, and it automates the recovery of an optimal $P(k)$ for several cosmological-parameter points. The results show that for many parameter choices, a free-form $P(k)$ yields substantially better fits (e.g., $\Delta\chi^2_{\rm eff}$ as large as about $-26$) than a simple power-law, with common horizon-scale cutoffs and post-horizon bumps emerging across models, including even standard CDM. The study demonstrates that CMB data retain strong discriminatory power in the cosmological-parameter space even when the primordial spectrum is unconstrained, and it lays groundwork for extended parameter estimation and joint analyses with polarization data (e.g., Planck) to further exploit this freedom.

Abstract

In this paper we implement an improved (error sensitive) Richardson-Lucy deconvolution algorithm on the measured angular power spectrum from the WMAP 3 year data to determine the primordial power spectrum assuming different points in the cosmological parameter space for a flat LCDM cosmological model. We also present the preliminary results of the cosmological parameter estimation by assuming a free form of the primordial spectrum, for a reasonably large volume of the parameter space. The recovered spectrum for a considerably large number of the points in the cosmological parameter space has a likelihood far better than a `best fit' power law spectrum up to Δχ^2_{eff} \approx -30. We use Discrete Wavelet Transform (DWT) for smoothing the raw recovered spectrum from the binned data. The results obtained here reconfirm and sharpen the conclusion drawn from our previous analysis of the WMAP 1st year data. A sharp cut off around the horizon scale and a bump after the horizon scale seem to be a common feature for all of these reconstructed primordial spectra. We have shown that although the WMAP 3 year data prefers a lower value of matter density for a power law form of the primordial spectrum, for a free form of the spectrum, we can get a very good likelihood to the data for higher values of matter density. We have also shown that even a flat CDM model, allowing a free form of the primordial spectrum, can give a very high likelihood fit to the data. Theoretical interpretation of the results is open to the cosmology community. However, this work provides strong evidence that the data retains discriminatory power in the cosmological parameter space even when there is full freedom in choosing the primordial spectrum.

Figures (4)

• Figure 1: Resultant $P(k)$ for a sample point in the cosmological parameter space is shown in the blue curve. The other curves show the P(k) recovered at different levels of DWT smoothing. The blue line which is the reconstructed result obtained by retaining all wavelet coefficients up to the 9th wavelet level has the best likelihood with $\Delta {\chi_{\rm eff}^2} = -18.76$ with respect to the best fit power-law primordial spectrum in the whole parameter space. We used $H_0=72, \Omega_{dm}=0.246, \Omega_b=0.05, \Omega_{\Lambda}=0.704, \tau=0.06$ as the cosmological parameters. The plot in the inset shows the resultant $\Delta {\chi_{\rm eff}^2}$ of the reconstructed results at different wavelet levels. The 'optimality' of the $n=9$ level DWT smoothing in this case is clearly demonstrated.
• Figure 2: Reconstructed primordial spectrum (top panel) and the resultant $C_{\ell}^{TT}$ (middle panel) and $C_{\ell}^{TE}$ (lower panel) angular power spectra are plotted for $6$ different points in the parameter space assuming a flat $\Lambda$CDM cosmological model. Cosmological parameters of Model A: $H_0=72, \Omega_{dm}=0.246, \Omega_b=0.05, \Omega_{\Lambda}=0.704, \tau=0.06$ and the recovered results for this model gives $\Delta \chi^2_{\rm eff}=-18.76$. Cosmological parameters of Model B: $H_0=63, \Omega_{dm}=0.251, \Omega_b=0.041, \Omega_{\Lambda}=0.708, \tau=0.06$ and the recovered results for this model gives $\Delta \chi^2_{\rm eff}=-4.38$. Cosmological parameters of Model C: $H_0=68, \Omega_{dm}=0.229, \Omega_b=0.052, \Omega_{\Lambda}=0.719, \tau=0.06$ and the recovered results for this model gives $\Delta \chi^2_{\rm eff}=-2.93$. Cosmological parameters of Model D: $H_0=72, \Omega_{dm}=0.229, \Omega_b=0.046, \Omega_{\Lambda}=0.725, \tau=0.06$ and the recovered results for this model gives $\Delta \chi^2_{\rm eff}=-14.52$. Cosmological parameters of Model E: $H_0=71, \Omega_{dm}=0.226, \Omega_b=0.044, \Omega_{\Lambda}=0.730, \tau=0.0$ and the recovered results for this model gives $\Delta \chi^2_{\rm eff}=-13.40$. Cosmological parameters of Model F: $H_0=50, \Omega_{dm}=0.904, \Omega_b=0.096, \Omega_{\Lambda}=0.0, \tau=0.06$ and the recovered results for this model gives $\Delta \chi^2_{\rm eff}=-26.70$. Model G is the reference model against which all calculated $\Delta \chi^2_{\rm eff}$s are with respect to this model. This represents the best fit power law primordial spectrum in the whole parameter space. The red error-bars in the middle and lower panels represents the binned angular power spectrum from WMAP 3 year data. The black error-bars at the middle panel at the high $\ell$, are from ACBAR experiment acbar. The excess of power and the bump in the recovered $P(k)$ at the high $k$ ($log k/k_h \approx$), seems to be related to the higher measurements of the angular power spectrum at high $\ell$'s in WMAP 3 year data in comparison with the other experiments such as ACBAR.
• Figure 3: A 1-D slice ($\Omega_m =constant$) through the cosmological parameter space demonstrates that the data retains strong discriminatory power in the cosmological parameter space even when there is full freedom in choosing the primordial power spectrum. Left panel: Plot of $\Delta \chi^2_{\rm eff}$ of the reconstructed results with respect to the reference likelihood of model G, by assuming free form of the primordial spectrum, for a flat $\Lambda$CDM model with $h_0 = 0.72$, $\tau = 0.06$ and $\Omega_{\Lambda} =0.70$ and $\Omega_m= \Omega_{b} + \Omega_{dm} = 0.30$ for different values of $\Omega_{b}$ (blue line). The red curve is for similar models except for $\Omega_m= \Omega_{b} + \Omega_{dm} = 0.27$. Clearly, 'optimizing' over the primordial power spectrum allows us to get significantly higher likelihood ($\Delta \chi^2=-19.65$) for $\Omega_m = 0.30$ compared to $\Omega_m = 0.27$ ($\Delta \chi^2=-11.55$). This demonstrates that even though we allow a free form of the primordial spectrum, the data does show very strong preference for particular values of cosmological parameters. Right panel: Reconstructed primordial spectrum, $P(k)$, for a flat $\Lambda$CDM model with $\Omega_{b}=0.050, \Omega_{dm}=0.25, h_0 = 0.72$, $\tau = 0.06$(blue line). For these parameters of $\Omega_{b}$ and $\Omega_{dm}$, we could get the best likelihood for $\Omega_m = 0.30$. The red line is the reconstructed $P(k)$ for a flat $\Lambda$CDM model with $\Omega_{b}=0.0460, \Omega_{dm}=0.224, h_0 = 0.72$, $\tau = 0.06$. For these parameters of $\Omega_{b}$ and $\Omega_{dm}$, we could get the best likelihood for the $\Omega_m = 0.27$. It is clear that the reconstructed $P(k)$ for these two points in the cosmological parameter space are very similar. However the resultant $\Delta \chi^2_{\rm eff}$ for these two points in the parameter space shows a big difference.
• Figure 4: A coarse resolution and limited volume exploration of the cosmological parameter space demonstrates that the data retains strong discriminatory power in the cosmological parameter space even when there is full freedom in choosing the primordial power spectrum. The resultant $-\Delta \chi^2_{\rm eff}$ is shown (in Z axis and also in color indicated by the tool bar in the upper panel) versus different values of Hubble parameter (X axis in both upper and lower panel) and $\Omega_b h^2$ (Y axis in both upper and lower panel). The lower panel shows the relative values of the $\Omega_{\Lambda}$ in our parameter space (indicated by color in the lower panel). We have assumed here $\tau=0.06$.