Natural Inflation: status after WMAP 3-year data

TL;DR

The paper tests Natural Inflation against WMAP3 constraints, showing consistency for f ≳ 0.7 m_Pl with Λ ∼ m_GUT. It analyzes scalar and tensor perturbations, their amplitudes, spectral indices, and the running, finding a small negative running (∼10^-3) and tensor amplitudes below current detectability. It also examines where observable 60 e-folds lie on the PNGB potential, concluding that for f ≳ 5 m_Pl the dynamics resemble a quadratic potential, while NI remains well-motivated against fine-tuning. Depending on f, NI can be categorized as small-field or large-field within the standard inflationary taxonomy, and its tensor predictions offer a potential avenue for future tests.

Abstract

The model of Natural Inflation is examined in light of recent 3-year data from the Wilkinson Microwave Anisotropy Probe and shown to provide a good fit. The inflaton potential is naturally flat due to shift symmetries, and in the simplest version takes the form $V(φ) = Λ^4 [1 \pm \cos(Nφ/f)]$. The model agrees with WMAP3 measurements as long as $f > 0.7 m_{Pl}$ (where $m_{Pl} = 1.22 \times 10^{19}$GeV) and $Λ\sim m_{GUT}$. The running of the scalar spectral index is shown to be small -- an order of magnitude below the sensitivity of WMAP3. The location of the field in the potential when perturbations on observable scales are produced is examined; for $f > 5 m_{Pl}$, the relevant part of the potential is indistinguishable from a quadratic, yet has the advantage that the required flatness is well-motivated. Depending on the value of $f$, the model falls into the large field ($f \ge 1.5 m_{Pl}$) or small field ($f < 1.5 m_{Pl}$) classification scheme that has been applied to inflation models. Natural inflation provides a good fit to WMAP3 data.

Figures (8)

• Figure 1: Natural inflation predictions and WMAP3 constraints in the $r$-$n_s$ plane. (Solid/blue) lines running from approximately the lower left to upper right are predictions for constant $N$ and varying $f$, where $N$ is the number of e-foldings prior to the end of inflation at which current modes of scale $k = 0.002$ Mpc$^{-1}$ were generated and $f$ is the width of the potential. The remaining (dashed/red) lines are for constant $f$ and varying $N$. The (light blue) band corresponds to the values of $N$ for standard post-inflation cosmology with $(\textrm{1 GeV})^4 < \rho_\textrm{RH} < V_\textrm{end}$. Filled (nearly vertical) regions are the parameter spaces allowed by WMAP3 at 68% and 95% C.L.'s (error contours taken from Ref. Kinney:2006qm). Natural inflation is consistent with the WMAP3 data for $f \hbox{$\, \buildrel { >}\over { \sim}\,$} 0.7m_\textrm{Pl}$ and essentially all likely values of $N$.
• Figure 2: The slow roll parameter $\epsilon$ is shown as a function of the potential width $f$ for various numbers of e-foldings $N$ before the end of inflation. The (light blue) band corresponds to the values of $N$ consistent with the standard post-inflation cosmology, as given by Eqn. (\ref{['eqn:Nk']}), for an end of reheating energy density $(\textrm{1 GeV})^4 < \rho_\textrm{RH} < V_\textrm{end}$, where the lower bound is a result of nucleosynthesis constraints.
• Figure 3: The potential height scale $\Lambda$ corresponding to $P_\mathcal{R}^{1/2} = 10^{-5}$ is shown as a function of the potential width $f$ for various numbers of e-foldingss $N$ before the end of inflation. The (light blue) band corresponds to the values of $N$ consistent with the standard post-inflation cosmology for $\rho_\textrm{RH} > (\textrm{1 GeV})^4$.
• Figure 4: The spectral index $n_s$ is shown as a function of the potential width $f$ for various numbers of e-foldingss $N$ before the end of inflation. The (light blue) band corresponds to the values of $N$ consistent with the standard post-inflation cosmology for $\rho_\textrm{RH} > (\textrm{1 GeV})^4$.
• Figure 5: The tensor to scalar ratio $r \equiv \frac{P_T}{P_\mathcal{R}}$ is shown as a function of the potential width $f$ for various numbers of e-foldingss $N$ before the end of inflation. The (light blue) band corresponds to the values of $N$ consistent with the standard post-inflation cosmology for $\rho_\textrm{RH} > (\textrm{1 GeV})^4$.