Estimation of primordial spectrum with post-WMAP 3-year data

Arman Shafieloo and Tarun Souradeep

Phys. Rev. D 78, 023511 – Published 9 July 2008

Abstract

In this paper we implement an improved (error-sensitive) Richardson-Lucy deconvolution algorithm on the measured angular power spectrum from the Wilkinson Microwave Anisotropy Probe (WMAP) 3 year data to determine the primordial power spectrum assuming different points in the cosmological parameter space for a flat $Λ CDM$ 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 $Δ χ_{eff}^{2} \approx - 30$ . 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 cold dark matter 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.

Received 27 September 2007

DOI:https://doi.org/10.1103/PhysRevD.78.023511

Authors & Affiliations

Arman Shafieloo and Tarun Souradeep

Inter-University Centre for Astronomy and Astrophysics (IUCAA), Ganeshkhind, Pune-411007, India

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Vol. 78, Iss. 2 — 15 July 2008

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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 $Δ χ_{eff}^{2} = - 18.76$ with respect to the best fit powerlaw primordial spectrum in the whole parameter space. We used $H_{0} = 72$ , $Ω_{dm} = 0.246$ , $Ω_{b} = 0.05$ , $Ω_{Λ} = 0.704$ , $τ = 0.06$ as the cosmological parameters. The plot in the inset shows the resultant $Δ χ_{eff}^{2}$ of the reconstructed results at different wavelet levels. The “optimality” of the $n = 9$ level DWT smoothing in this case is clearly demonstrated.Reuse & Permissions
Figure 2
Reconstructed primordial spectrum (top panel) and the resultant $C_{ℓ}^{TT}$ (middle panel) and $C_{ℓ}^{TE}$ (lower panel) angular power spectra are plotted for 6 different points in the parameter space assuming a flat $Λ CDM$ cosmological model. Cosmological parameters of Model A: $H_{0} = 72$ , $Ω_{dm} = 0.246$ , $Ω_{b} = 0.05$ , $Ω_{Λ} = 0.704$ , $τ = 0.06$ , and the recovered results for this model gives $Δ χ_{eff}^{2} = - 18.76$ . Cosmological parameters of Model B: $H_{0} = 63$ , $Ω_{dm} = 0.251$ , $Ω_{b} = 0.041$ , $Ω_{Λ} = 0.708$ , $τ = 0.06$ , and the recovered results for this model gives $Δ χ_{eff}^{2} = - 4.38$ . Cosmological parameters of Model C: $H_{0} = 68$ , $Ω_{dm} = 0.229$ , $Ω_{b} = 0.052$ , $Ω_{Λ} = 0.719$ , $τ = 0.06$ , and the recovered results for this model gives $Δ χ_{eff}^{2} = - 2.93$ . Cosmological parameters of Model D: $H_{0} = 72$ , $Ω_{dm} = 0.229$ , $Ω_{b} = 0.046$ , $Ω_{Λ} = 0.725$ , $τ = 0.06$ , and the recovered results for this model gives $Δ χ_{eff}^{2} = - 14.52$ . Cosmological parameters of Model E: $H_{0} = 71$ , $Ω_{dm} = 0.226$ , $Ω_{b} = 0.044$ , $Ω_{Λ} = 0.730$ , $τ = 0.0$ and the recovered results for this model gives $Δ χ_{eff}^{2} = - 13.40$ . Cosmological parameters of Model F: $H_{0} = 50$ , $Ω_{dm} = 0.904$ , $Ω_{b} = 0.096$ , $Ω_{Λ} = 0.0$ , $τ = 0.06$ , and the recovered results for this model gives $Δ χ_{eff}^{2} = - 26.70$ . Model G is the reference model against which all calculated $Δ χ_{eff}^{2}$ ’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 errorbars at the middle panel at the high $ℓ$ , are from ACBAR experiment 57. 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 $ℓ$ ’s in WMAP 3 year data in comparison with the other experiments such as ACBAR.Reuse & Permissions
Figure 3
A 1-D slice ( $Ω_{m} = constant$ ) 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 $Δ χ_{eff}^{2}$ of the reconstructed results with respect to the reference likelihood of model G, by assuming free form of the primordial spectrum, for a flat $Λ CDM$ model with $h_{0} = 0.72$ , $τ = 0.06$ , and $Ω_{Λ} = 0.70$ and $Ω_{m} = Ω_{b} + Ω_{dm} = 0.30$ for different values of $Ω_{b}$ (blue line). The red curve is for similar models except for $Ω_{m} = Ω_{b} + Ω_{dm} = 0.27$ . Clearly, “optimizing” over the primordial power spectrum allows us to get significantly higher likelihood ( $Δ χ^{2} = - 19.65$ ) for $Ω_{m} = 0.30$ compared to $Ω_{m} = 0.27$ ( $Δ χ^{2} = - 11.55$ ). 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 $Λ CDM$ model with $Ω_{b} = 0.050$ , $Ω_{dm} = 0.25$ , $h_{0} = 0.72$ , $τ = 0.06$ (blue line). For these parameters of $Ω_{b}$ and $Ω_{dm}$ , we could get the best likelihood for $Ω_{m} = 0.30$ . The red line is the reconstructed $P (k)$ for a flat $Λ CDM$ model with $Ω_{b} = 0.0460$ , $Ω_{dm} = 0.224$ , $h_{0} = 0.72$ , $τ = 0.06$ . For these parameters of $Ω_{b}$ and $Ω_{dm}$ , we could get the best likelihood for the $Ω_{m} = 0.27$ . It is clear that the reconstructed $P (k)$ for these two points in the cosmological parameter space are very similar. However the resultant $Δ χ_{eff}^{2}$ for these two points in the parameter space shows a big difference.Reuse & Permissions
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 $- Δ χ_{eff}^{2}$ 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 $Ω_{b} h^{2}$ (Y axis in both upper and lower panel). The lower panel shows the relative values of the $Ω_{Λ}$ in our parameter space (indicated by color in the lower panel). We have assumed here $τ = 0.06$ .Reuse & Permissions

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