Constraints on the Inflationary Expansion from Three Year WMAP, small scale CMB anisotropies and Large Scale Structure Data Sets

TL;DR

This paper constrains inflationary expansion by combining WMAP3 data with smaller-scale CMB observations and LSS surveys, using horizon flow function (HFF) parametrizations up to second order within a flat $\Lambda$CDM framework. The authors implement an MCMC analysis with analytic inflationary spectra derived from both the Green’s function method and the method of comparison equations, allowing both standard single-field inflation and NEC-violating scenarios (and relaxing the tensor–scalar consistency relation). They find $n_S\approx0.96$ and tight limits on the tensor-to-scalar ratio under standard inflation (e.g., $r_{0.01}<0.26$ at $2\sigma$), with small-scale data pushing $n_S$ toward red values and disfavouring $n_S=1$; allowing running via second-order HFFs yields a negative scalar running $\alpha_S$ around $-0.08$ (2$\sigma$) when $\epsilon_3$ priors are broad. NEC-violating models can fit the data comparably well, producing blue tensor spectra, while tensor and scalar runnings remain small; the results are sensitive to the LSS data set used, underscoring the importance of data choice and higher-order inflationary parametrizations for robust constraints.

Abstract

We present the constraints on inflationary parameters in a flat $Λ$CDM universe obtained by WMAP three year data release, plus smaller scale CMB and two LSS data sets, 2dF and SDSS (treated separately). We use a Markov Chain Monte Carlo (MCMC) technique combined with an analytic description of the inflationary spectra in terms of the horizon flow functions (HFF). By imposing a consistency condition for the tensor-to-scalar ratio, we study the constraints both on single field standard inflation and on inflation with the violation of the null energy condition, which leads to a blue spectrum for gravitational waves. For standard inflation, the constraint on the tensor-to-scalar ratio we obtain from CMB data and 2dF05 is: $r_{0.01} < 0.26$ at 2 $σ$ cl. Without the consistency condition between the tensor-to-scalar ratio and the tensor slope, the constraints on the tensor amplitude is not significantly changed, but the constraints on the HFFs are significantly relaxed. We then show that when the third HFF $ε_3$ is allowed to be non-zero and to be of order unity, a large negative (at $2 σ$) value for the running of the scalar spectral index in standard inflation is found in any set of data we consider.

Figures (11)

• Figure 1: One dimensional marginalized probabilities for cosmological parameters obtained by WMAP1+CMBsmall+2dF02 (black), WMAP3+CMBsmall plus 2dF02 (red) or plus 2dF05 (green).
• Figure 2: One dimensional marginalized probabilities for cosmological parameters obtained by using WMAP3+CMBsmall (black), WMAP3+CMBsmall plus 2dF02 (red) or plus SDSS (green).
• Figure 3: Constraints by using WMAP3 (red) vs WMAP1 (black) plus CMBsmall+2dF on ($\epsilon_2 \,, \epsilon_1$) and ($n_s - 1 \,, R_{10}$) planes at $1 \sigma$ and $2 \sigma$ level. On the right panel, the inflationary predictions for $V(\phi) = \lambda \phi^4/4$ (green) and for $V(\phi) = m^2 \phi^2/2$ (red) with $45 < \Delta N < 55$ are shown for comparison.
• Figure 4: Constraints by using WMAP3+SDSS (black) vs WMAP3+CMBsmall+SDSS (red) on ($\epsilon_2 \,, \epsilon_1$) and ($n_s - 1 \,, R_{10}$) planes at $1 \sigma$ and $2 \sigma$ level. The right panel shows how the inclusion of small scale CMB data drives the spectral scalar index towards red values, disfavouring at just 2$\sigma$ cl the $n_s=1$ line. The constraints on $n_S$ we obtain from WMAP3+SDSS are in complete agreemente with KKMRPE.
• Figure 5: Two dimensional contours at $1 \sigma$ and $2 \sigma$ level obtained by WMAP3+CMBsmall+2dF02. The plots reported from left to right are $(\epsilon_2\, \epsilon_1)$, $(n_S - 1 \,, R_{10})$, $(16 \, \epsilon_1 \,, r)$, respectively. In the plane $(16 \, \epsilon_1 \,, r)$, the red line describing the consistency condition $r=16 \epsilon_1$ is also drawn for comparison.