Updated constraints on large field hybrid inflation

We revisit the status of hybrid inflation in the light of Planck and recent BICEP2 results, taking care of possible transient violations of the slow-roll conditions as the field passes from the large field to the vacuum dominated phase. The usual regime where observable scales exit the Hubble radius in the vacuum dominated phase predicts a blue scalar spectrum, which is ruled out. But whereas assuming slow-roll one expects this regime to be generic, by solving the exact dynamics we identify the parameter space for which the small field phase is naturally avoided due to slow-roll violations at the end of the large field phase. When the number of $e$-folds generated at small field is negligible, the model predictions are degenerated with those of a quadratic potential. There exists also a transitory case for which the small field phase is sufficiently long to affect importantly the observable predictions. Interestingly, in this case the spectral index and the tensor to scalar ratio agree respectively with the best fit of Planck and BICEP2. This results in a $\mathrm{\Delta }{\chi }^2\simeq 5.0$ in favor of hybrid inflation for $\text{Planck}+\mathrm{BICEP}2$ ($\mathrm{\Delta }{\chi }^2\simeq 0.9$ for Planck only). The last considered regime is when the critical point at which inflation ends is located in the large field phase. It is constrained to be lower than about ten times the reduced Planck mass. The analysis has been conducted with the use of Markov-chain-Monte-Carlo Bayesian method, in a reheating consistent way, and we present the posterior probability distributions for all the model parameters.

Figure 1

Evolution of the first Hubble-flow parameter ${\epsilon }_1$ as a function of $\varphi /\mu$, computed by solving the full dynamics or using the slow-roll approximation. The full dynamics solution differs greatly from the slow-roll solution at small values of $\mu$. The kinetic energy acquired at the transition between the large field and the vacuum dominated phase prevents inflation to take place at small field values.

Figure 2

Evolution of the second Hubble-flow parameter ${\epsilon }_2$ as a function of $\varphi /\mu$, computed by solving the full dynamics or using the slow-roll approximation. As in Fig. 1 the full dynamics differs from the slow-roll approximation for small values of $\mu$, leading to different observable predictions.

Figure 3

Number of $e$-fold produced after reaching the maximum of ${\epsilon }_1$ as a function of $\mu$. Below the threshold value ${\mu }_{\mathrm{thr}}\sim 1.6M_{\text{pl}}$, only a few $e$-folds are realized at small field values in contradiction with slow-roll predictions.

Figure 4

${\varphi }_{*}$ (top) and corresponding $n_s$ (central) and $r$ (bottom) plotted as a function of ${\varphi }_{\text{end}}={\varphi }_{\mathrm{c}}$ using Eq. (33), for $\mu =1M_{\text{pl}}$ (solid blue), $\mu =5M_{\text{pl}}$ (dashed red), $\mu =10M_{\text{pl}}$ (dotted yellow) and $\mu =15M_{\text{pl}}$ (dash-dotted green), assuming $N_{*}=60$. The horizontal dotted lines in the central and bottom panels represent respectively the $2\sigma$ regions of Planck and BICEP2. The top black dotted line in the upper panel is obtained by using the approximation of Eq. (34).

Figure 5

Contour plot of the spectral index $n_s$ in the plane (${\log }_{10}{\varphi }_{\mathrm{c}},{\log }_{10}\mu )$, using the exact background dynamics and assuming $N_{*}=60$. The red dashed contours represent the $2\sigma$ confidence interval for Planck.

Figure 6

Contour plot of the tensor to scalar ratio $r$ in the plane (${\log }_{10}{\varphi }_{\mathrm{c}},{\log }_{10}\mu )$, using the exact background dynamics and assuming $N_{*}=60$. The red dashed contours represent the $2\sigma$ confidence interval for BICEP2.

Figure 7

Marginalized one-dimensional and two-dimensional posterior probabilities for the hybrid model parameters (in reduced Planck mass units) in the large field regime, for $\text{Planck}+\mathrm{BICEP}2$. The red contours are the $1\sigma$ and $2\sigma$ regions of confidence. The black contours are the $1\sigma$ and $2\sigma$ regions for Planck only. In the 1D plots, the black/red solid lines show the marginalized posterior distributions of the parameters respectively for Planck and $\text{Planck}+\mathrm{BICEP}2$. The dotted lines represent the mean likelihoods.

Figure 8

Two-dimensional marginalized posterior probabilities for the hybrid model parameters (in reduced Planck mass units) as well as the derived parameters $n_s$, $r$ and ${\rho }_{\text{end}}$ for $\text{Planck}+\mathrm{BICEP}2$. The red contours are the $1\sigma$ and $2\sigma$ regions of confidence. The black contours are the $1\sigma$ and $2\sigma$ regions for Planck only.

Figure 9

Distribution of 3000 points within the Markov chains in the plane (${\log }_{10}\mu ,{\log }_{10}{\varphi }_{\mathrm{c}}$). In the upper panel, the color scale represents the corresponding spectral index value, in the lower panel it represents the corresponding tensor to scalar ratio.