The Shape of the Primordial Power Spectrum: A Last Stand Before Planck

TL;DR

This paper assesses the shape of the primordial power spectrum by performing a minimally-parametric reconstruction of $P(k)$ using cross-validated smoothing splines, thereby avoiding strong theoretical priors and over-fitting. It integrates diverse data sets, including WMAP5, ground-based CMB, LRG clustering, and Ly$\alpha$ forest, within an MCMC framework to infer a smooth spectrum described by a spline with an optimally chosen complexity. The main finding is that there is no compelling evidence for significant departures from a power-law form, though exact scale invariance is disfavored; the strength of this conclusion is sensitive to data sets and priors. The work provides a robust, priors-light benchmark for Planck-era analyses and highlights the value of future data in decisively constraining the primordial spectrum’s shape.

Abstract

We present a minimally-parametric reconstruction of the primordial power spectrum using the most recent cosmic microwave background and large scale structure data sets. Our goal is to constrain the shape of the power spectrum while simultaneously avoiding strong theoretical priors and over-fitting of the data. We find no evidence for any departure from a power law spectral index. We also find that an exact scale-invariant power spectrum is disfavored by the data, but this conclusion is weaker than the corresponding result assuming a theoretically-motivated power law spectral index prior. The reconstruction shows that better data are crucial to justify the adoption of such a strong theoretical prior observationally. These results can be used to determine the robustness of our present knowledge when compared with forthcoming precision data from Planck.

Figures (4)

• Figure 1: (Left) WMAP 5 year data NoltaWMAP5 and (right) external CMB data from ACBAR and QUaD ACBAR09Quad09, showing knot placement (triangles, arbitrary normalization) and the cross-validation set-up. CV$_A$ is red and CV$_B$ is blue. We show only the temperature data here (in $\mu$K$^2$) as the constraints on the power spectrum shape come mostly from the temperature data; in practice for each data set we also use the polarization data, which is crucial in lifting degeneracies with the cosmological parameters. The light blue line is a concordance LCDM model.
• Figure 2: Large-scale structure power spectrum in units of ($h$/Mpc)$^3$, showing knot placement (triangles, arbitrary normalization) and cross-validation set-up. CV$_A$ is red and CV$_B$ is blue. Red points represent the LRG power spectrum from Ref. Reidetal09. The Lyman alpha measurement is represented by a filled box encompassing the constraints on the observed flux power spectrum from Ref. McDonaldLya06. The light blue line is a concordance LCDM model.
• Figure 3: Reconstructed spectral index $n_s(k)$ for various data combinations: WMAP5 with optimal penalty $\lambda^{\rm WMAP5}_{\rm opt}$ (top left), WMAP5$+$QUaD$+$ACBAR with optimal penalty $\lambda^{\rm CMB}_{\rm opt}$ (top right), and WMAP5$+$LRG$+$Ly$\alpha$ for two values of the penalty (bottom, left and right). Dark and light blue regions correspond to the best 95% and 68% reconstructions. The solid black line is the maximum likelihood fit. For comparison, the dashed line corresponds to a scale-invariant $P(k)$. See text for details.
• Figure 4: Reconstructed spectral index $n_s(k)$ from WMAP5, ACBAR, QUaD and SDSS DR7 LRG data with optimal penalty determined from cross-validation excluding Ly$\alpha$. The orange-red band shows the 95% and 68% $n_s$ constraints Reidetal09 for WMAP5$+$LRG data with a power-law prior.