Inflation model constraints from the Wilkinson Microwave Anisotropy Probe three-year data

TL;DR

This work leverages the WMAP three-year CMB data, augmented by SDSS, to systematically constrain single-field inflation by mapping observables to slow-roll parameters and exploring the inflationary potential via Monte Carlo reconstruction. It classifies inflationary models, quantifies constraints in the r–n–dn/dln k plane, and demonstrates that simple Harrison-Zel'dovich spectra remain viable within 95% confidence while ruling out certain hybrid and quartic potentials. A comprehensive Monte Carlo reconstruction reveals a clustering of viable models toward red spectra with negligible running and shows that late-time attractor hybrids are strongly disfavored, with no lower bound on gravitational waves unless a tensor detection occurs. These results refine the inflationary model space and place meaningful limits on the energy scale of inflation and the associated tensor modes.

Abstract

We extract parameters relevant for distinguishing among single-field inflation models from the Wilkinson Microwave Anisotropy Probe (WMAP) three-year data set, and also from WMAP in combination with the Sloan Digital Sky Survey (SDSS) galaxy power spectrum. Our analysis leads to the following conclusions: 1) the Harrison--Zel'dovich model is consistent with both data sets at a 95% confidence level; 2) there is no strong evidence for running of the spectral index of scalar perturbations; 3) Potentials of the form V \propto φ^p are consistent with the data for p = 2, and are marginally consistent with the WMAP data considered alone for p = 4, but ruled out by WMAP combined with SDSS. We perform a "Monte Carlo reconstruction" of the inflationary potential, and find that: 1) there is no evidence to support an observational lower bound on the amplitude of gravitational waves produced during inflation; 2) models such as simple hybrid potentials which evolve toward an inflationary late-time attractor in the space of flow parameters are strongly disfavored by the data, 3) models selected with even a weak slow-roll prior strongly cluster in the region favoring a "red" power spectrum and no running of the spectral index, consistent with simple single-field inflation models.

Figures (5)

• Figure 1: Constraints on the $n$---$r$ plane for different choices of experimental datasets. The analyses in the top panels include a running spectral index, while the analyses in the bottom panels are without running. The shaded regions indicate 68% and 95% C.L.
• Figure 2: Constraints on the $n$---$dn/d\ln k$ plane for different choices of experimental datasets. The shaded regions indicate 68% and 95% C.L.
• Figure 4: The $n$,$r$ parameter space WMAP3 alone (open contours) and WMAP3 + SDSS (filled contours), with a prior of $dn/d\ln{k} = 0$. The line segments show the predictions for $V(\phi) = m^2 \phi^2$ and $V(\phi) = \lambda \phi^4$ for $N = [46,60]$. The dashed lines show the 68% C.L. and 95% C.L. contours from the chains made public by the WMAP team, which do not include an HST prior on $H_0$ or an age prior. The scale of the plot is chosen to allow direct comparison with Fig. $14$ of Spergel et al.wmap3cosm. The shaded regions indicate 68% and 95% C.L.
• Figure 5: Inflationary models plotted against the 68% and 95% WMAP3 + SDSS error contours. Top panel: the predictions of various specific inflationary potentials (solid bands) plotted against the error bars from WMAP3 + SDSS with a prior of $dn / d\ln{k} = 0$. Bottom panel: 10,000 models generated by flow Monte Carlo consistent with the WMAP3 + SDSS data sets including running as a parameter, indicated by the larger error contours. The contours with a $dn / d\ln{k} = 0$ prior are plotted as a reference, and were not used in the Monte Carlo reconstruction. (Some data points fall outside the error contours plotted because likelihoods for the models were calculated using the full three-dimensional likelihood function ${\cal L}(n,r,dn/d\ln{k})$, and the contours were obtained by marginalizing over $dn/d\ln{k}$).
• Figure 6: Potentials generated by Monte Carlo Reconstruction consistent with WMAP3 + SDSS to 68% C.L. in the light shaded (yellow) region and 95% C.L. in the darker shaded (cyan) region. The WMAP3 data place an upper limit of about $2 \times 10^{16}\ {\rm GeV}$ on the energy scale of inflation. No lower limit is possible without a detection of a tensor mode signal. The concave-up line at the bottom of the figure is the single model in $10^{7}$ models generated which converged to an inflationary fixed point at late time.