Primordial power spectrum from WMAP

TL;DR

Problem: recover the primordial power spectrum from the CMB angular power spectrum without assuming a parametric form. Method: apply an error-sensitive Richardson–Lucy deconvolution to invert $C_l = \sum_i G(l,k_i) P(k_i) G(l,k_i) \Delta k_i/k_i$ under a concordance cosmology, followed by artifact removal and smoothing to maximize the WMAP likelihood. Key results: the recovered $P(k)$ shows a sharp infra-red cutoff near the horizon scale with a compensating excess below it, dramatically improving $\\ln\\mathcal{L}$ relative to scale-invariant or simple tilted spectra and robust to modest parameter variations; the shape is consistent with inflationary scenarios that produce localized features. Implications: the approach links CMB features to early-universe physics (e.g., Starobinsky kink, pre-inflationary epochs) and can be extended to polarization and large-scale structure probes, offering a direct window into primordial perturbations.

Abstract

The observed angular power spectrum of the cosmic microwave background temperature anisotropy, $C_l$, is a convolution of a cosmological radiative transport kernel with an assumed primordial power spectrum of inhomogeneities. Exquisite measurements of $C_l$ over a wide range of multipoles from the Wilkinson Microwave Anisotropy Probe (WMAP) has opened up the possibility to deconvolve the primordial power spectrum for a given set of cosmological parameters (base model). We implement an improved (error sensitive) Richardson-Lucy deconvolution algorithm on the measured angular power spectrum from WMAP assuming a concordance cosmological model. The most prominent feature of the recovered $P(k)$ is a sharp, infra-red cut off on the horizon scale. The resultant $C_l$ spectrum using the recovered spectrum has a likelihood far better than a scale invariant, or, `best fit' scale free spectra ($Δ\ln{\cal L}=25$ {\it w.r.t.} Harrison Zeldovich, and, $Δ\ln{\cal L}=11$ {\it w.r.t.} power law with $n_s=0.95$). The recovered $P(k)$ has a localized excess just below the cut-off which leads to great improvement of likelihood over the simple monotonic forms of model infra-red cut-off spectra considered in the post WMAP literature. The recovered $P(k)$, in particular, the form of infra-red cut-off is robust to small changes in the cosmological parameters. We show that remarkably similar form of infra-red cutoff is known to arise in very reasonable extensions and refinements of the predictions from simple inflationary scenarios. Our method can be extended to other cosmological observations such as the measured matter power spectrum and, in particular, the much awaited polarization spectrum from WMAP.

Figures (12)

• Figure 1: The curves are $\bar{G}(l,k)$ versus wavenumber $k$ used in our work. $\bar{G}(l,k)$ is averaged over $G(l,k)$ within multipole bins used by WMAP. The two vertical lines roughly enclose the region of $k$-space strongly probed by the kernel where the primordial spectrum can be expected to be recovered reliably. The $k$-space sampling used is indicated by the line of '$+$' symbols at the top of the plot.
• Figure 2: Each of the three rows in the panel of figures illustrates the recovery the primordial power for a test case using $C_l$ arising from a known non-scale invariant primordial spectrum. The first column compares the raw deconvolved spectrum with the input spectrum. Note the similar artifacts in the all the raw spectra at the low and high $k$ end discussed in the text. The feature is outside the range of $G(l,k)$ and is completely missed in the third case. The second, column is the differenced spectrum obtained by dividing out by a raw reference spectrum. The differenced spectrum resembles the input spectrum (top two cases) with small oscillations. The third column shows that the final recovered spectrum obtained by smoothing the differenced spectrum matched the input spectrum very well.
• Figure 3: The three stages leading to the final recovered spectrum for a base cosmological model ($\tau\,=\,0.0$, $h=0.71$, $\Omega_b\,h^2\,=0.0224$ and $\Omega_{\Lambda}\,=\,0.73$) is shown. The lower dashed line is the raw deconvolved power. The upper dashed line is the differenced spectrum obtained by dividing out by the reference spectrum shown as a dotted line. The solid line is the final result after smoothing that gives the best likelihood.
• Figure 4: The recovered $C_l$ corresponding to the raw $P(k)$ are shown in the upper row and that corresponding to the final smoothed $P(k)$ spectrum are shown in the lower row. The left panels show the full range of multipole, while the right hand panels zoom into the low multipoles. The $C_l$'s from the raw as well as final $P(k)$ fit the binned $C_l^D$ well ($\chi^2\sim 10$ and $\chi^2\sim 20$, respectively for $38$ points). However, the jagged form of the $C_l$ between the $l$ bins (apparent in the low multipoles on the right) leads to poor WMAP likelihood for the $C_l$ from the raw $P(k)$. The differencing and smoothing procedure irons out the jagged $C_l$ dramatically improving the WMAP likelihood for the final smoothed $P(k)$. The WMAP likelihood is more relevant since it incorporates the estimation of each $C_l$ and the full error covariance.
• Figure 5: The final recovered spectrum for the base cosmological model ($\tau\,=\,0.0$, $h=0.71$, $\Omega_b\,h^2\,=0.0224$ and $\Omega_{\Lambda}\,=\,0.73$) is compared with set of $P(k)$ with WMAP likelihood within $\sim 2~\sigma$. The thick line gives the best likelihood equal to $-478.2$ and the other lines gives the likelihood bigger than $-480$. We can see that the sharp infra-red cut off is common to all these recovered spectra. The infra-red cutoff is remarkably close to the horizon scale and appears to be a robust feature. Another significant and robust feature is the bump just above the cut-off (reminiscent of the oscillation from under-damped transient). The difference between these spectra are in the smoothing and removing the noises from the raw deconvolved spectrum.