Reconstructing the Primordial Spectrum from WMAP Data by the Cosmic Inversion Method

TL;DR

The paper develops and applies the cosmic inversion method to reconstruct the primordial curvature spectrum $P(k)$ from the WMAP angular power spectrum $C_ll$, achieving high $k$-space resolution and enabling the detection of oscillatory features that binning methods can miss. It derives a first-order differential equation for $P(k)$ using transfer functions $f(k)$ and $g(k)$ and solves it between singularities, calibrating with $b_\ell=C_\text{ex}/C_\text{app}$ and incorporating observational errors via Monte Carlo simulations. The reconstructed $P(k)$ shows $\sim$20–30% oscillations with a resolution of $\Delta kd \approx 5$ and suggests possible deviations from scale-invariance around $k \sim 1.5\times 10^{-2}$ to $2.6\times 10^{-2}\ \mathrm{Mpc}^{-1}$, though features near a singularity near $kd\sim 430$ are attributed to data noise. The study demonstrates a higher-resolution, model-independent approach to probing primordial fluctuations and outlines future improvements including CMB polarization to better constrain cosmological parameters.

Abstract

We reconstruct the primordial spectrum of the curvature perturbation, $P(k)$, from the observational data of the Wilkinson Microwave Anisotropy Probe (WMAP) by the cosmic inversion method developed recently. In contrast to conventional parameter-fitting methods, our method can potentially reproduce small features in $P(k)$ with good accuracy. As a result, we obtain a complicated oscillatory $P(k)$. We confirm that this reconstructed $P(k)$ recovers the WMAP angular power spectrum with resolution up to $Δ\ell \simeq 5$. Similar oscillatory features are found, however, in simulations using artificial cosmic microwave background data generated from a scale-invariant $P(k)$ with random errors that mimic observation. In order to examine the statistical significance of the nontrivial features, including the oscillatory behaviors, therefore, we consider a method to quantify the deviation from scale-invariance and apply it to the $P(k)$ reconstructed from the WMAP data. We find that there are possible deviations from scale-invariance around $k \simeq 1.5\times10^{-2}$ and $2.6\times10^{-2}{\rm Mpc}^{-1}$.

Figures (6)

• Figure 1: Primordial spectrum $P(k)$ reconstructed from the WMAP data for $h=0.72$, $\Omega_b=0.047$, $\Omega_\Lambda=0.71$, $\Omega_m=0.29$, and $\tau=0.17$. The solid curve and the dashed curves represent the mean and the $1 \sigma$ errors, respectively, of the reconstructed $P(k)$, while the dash-dotted curves represent the $1 \sigma$ from the scale-invariance. The horizontal axis $kd$ corresponds roughly to $\ell$. The singularities lie at $kd \simeq 70$ and $430$. Some prominent features are seen around $kd \simeq 200$ and $350$.
• Figure 2: Dependence on the cutoff scale $r_{{\rm cut}}$. Top: Reconstructed $P(k)$ spectra from the WMAP data for $r_{{\rm cut}} \simeq 0.5d$ and $0.3d$. We see that the resolution becomes better as the cutoff scale is made larger. Bottom: Recovered $C_\ell$ spectra from the obtained $P(k)$ spectra.
• Figure 3: Accuracy check of our reconstruction method. We show the cases of $r_{{\rm cut}} \simeq 0.5d$, which leads to $\Delta\ell \sim 6$ ( top), and $r_{{\rm cut}} \simeq 0.3d$, which leads to $\Delta\ell \sim 10$ ( bottom), from the relation $\Delta\ell \sim \Delta kd \simeq \pi d/r_{{\rm cut}}$. Left: Comparison of the binned WMAP data, $C^{{\rm bin}}_\ell$ ( plus signs with error bars), with the angular power spectrum, $C^{{\rm re}}_\ell$ ( solid curve), recovered from the reconstructed $P(k)$ shown in Fig. \ref{['CUT']}. Right: Relative errors, $(C^{{\rm bin}}_\ell-C^{{\rm re}}_\ell)/C^{{\rm re}}_\ell$. The relative errors are small for most of the bins except for those corresponding to the scales of the singularities.
• Figure 4: Reconstruction from the binned WMAP data. Top: Binned WMAP data whose bin sizes are $\Delta\ell=10, 20,$ and $50$; the solid curve represents the fiducial model for the scale-invariant spectrum. Bottom: The $P(k)$ spectra reconstructed from the binned WMAP data. As the bin size is made smaller, we can see more oscillations.
• Figure 5: Primordial spectra $P(k)$ reconstructed from artificial CMB data for four different realizations. The original spectrum is taken to be scale invariant, and each realization is generated by drawing a random number to each $C_\ell$ with the same error as the WMAP data, assuming that the cosmological parameters are known. The same oscillatory features as shown in Fig. \ref{['REWMAP']} are seen.