Inflation after WMAP3: Confronting the Slow-Roll and Exact Power Spectra with CMB Data

TL;DR

This paper tests single-field inflation against WMAP3 data by combining slow-roll theory (up to second order) with exact mode-by-mode computations of the inflationary power spectra, including a phenomenological reheating description. It systematically constrains slow-roll parameters and maps them onto large-field, small-field, hybrid, and running-mass potentials, showing that $\epsilon_1<0.022$, $-0.07<\epsilon_2<0.07$, and $r_{10}<0.21$, while large-field models with $p>3.1$ are excluded. The exact spectral analyses largely support the viability of small-field and running-mass models, though some parameter regions and reheating scenarios remain weakly constrained; hybrid models are disfavoured by the blue tilt. The study also tests trans-Planckian oscillations, finding no compelling evidence for them, though certain oscillatory templates can modestly improve fit at the cost of potential backreaction concerns. Overall, the vanilla slow-roll picture remains the most probable, with Planck-level data expected to tighten these constraints further.

Abstract

The implications of the WMAP (Wilkinson Microwave Anisotropy Probe) third year data for inflation are investigated using both the slow-roll approximation and an exact numerical integration of the inflationary power spectra including a phenomenological modelling of the reheating era. At slow-roll leading order, the constraints epsilon_1 < 0.022 and -0.07 < epsilon_2 < 0.07 are obtained at 95% CL (Confidence Level) implying a tensor-to-scalar ratio r_10 < 0.21 and a Hubble parameter during inflation H/Mpl < 1.3 x 10^(-5). At next-to-leading order, a tendency for epsilon_3 > 0 is observed. With regards to the exact numerical integration, large field models, V(phi) proportional to phi^p, with p > 3.1 are now excluded at 95% CL. Small field models, V(phi) proportional to 1-(phi/mu)^p, are still compatible with the data for all values of p. However, if mu/Mpl < 10 is assumed, then the case p = 2 is slightly disfavoured. In addition, mild constraints on the reheating temperature for an extreme equation of state w_reh = -1/3 are found, namely T_reh > 2 TeV at 95% CL. Hybrid models are disfavoured by the data, the best fit model having Delta chi^2 = +5 with two extra parameters in comparison with large field models. Running mass models remain compatible, but no prior independent constraints can be obtained. Finally, superimposed oscillations of trans-Planckian origin are studied. The vanilla slow-roll model is still the most probable one. However, the overall statistical weight in favour of superimposed oscillations has increased in comparison with the WMAP first year data, the amplitude of the oscillations satisfying 2|x|sigma_0 < 0.76 at 95% CL. The best fit model leads to an improvement of Delta chi^2 = -12 for three extra parameters. Moreover, compared to other oscillatory patterns, the logarithmic shape is favoured.

Figures (36)

• Figure 1: The quantity $\rho_{\mathrm{rad}}/\rho_{\mathrm{end}}$ for $w_{\mathrm{reh}}=2$ or $p=2$, $t_{\mathrm{end}}=1$ and $\Gamma =0.1$. The solid black line is the exact expression obtained from equation (\ref{['rhorad']}) while the dotted red line is the approximation of (\ref{['rhoappro']}). At $t/t_{\mathrm{end}}\gg 1$, the difference between the two curves is approximately a factor two. Here, the time $t$ and $\Gamma$ are measured in units of $m_{{\mathrm{Pl}}}$.
• Figure 2: Marginalised posterior probability distributions for the base $\Lambda$CDM cosmological parameters together with the cosmological constant and the Hubble parameter, obtained at first order in slow-roll expansion. The solid black lines correspond to an uniform prior choice on $\epsilon_1$ while the dashed blue ones to an uniform prior on $\log(\epsilon_1)$. The dotted black lines is the mean likelihood for the former prior.
• Figure 3: $68\%$ and $95\%$ confidence intervals of the two-dimensional marginalised posteriors in the slow-roll parameters plane, obtained at leading order in slow-roll expansion. The shading is the mean likelihood and the left plot is derived under an uniform prior on $\epsilon_1$ while the right panel corresponds to an uniform prior on $\log(\epsilon_1)$.
• Figure 4: Marginalised posterior probability distributions for the primordial parameters in the first order slow-roll expansion. As in figure \ref{['sr1st_cosmo_1D']}, the black solid curves are derived under an uniform prior on $\epsilon_1$ whereas the dashed blue curves correspond to a flat prior on $\log(\epsilon_1)$. At two-sigma level of confidence one has the marginalised upper bound $\epsilon_1 < 0.022$.
• Figure 5: One and two-sigma confidence levels of the two-dimensional marginalised posteriors in the Hubble-flow parameter planes (solid black contours). The dotted red contours correspond to the same confidence intervals derived at first order in slow-roll expansion (see figure \ref{['sr1st_eps12_2D']}). The left panels are associated with an uniform prior on $\epsilon_1$ whereas the right ones correspond to an uniform prior on $\log(\epsilon_1)$. As can be seen in the bottom right panel, positive values of $\epsilon_3$ are slightly favoured, but in a prior dependent way while $\epsilon_3=0$ remains within the one-sigma contour.