Graceful exit from Higgs $G$ inflation

Higgs $G$ inflation is a Higgs inflation model with a generalized Galileon term added to the standard model Higgs field, which realizes inflation compatible with observations. Recently, it was claimed that the generalized Galileon term induces instabilities during the oscillation phase and that the simplest Higgs $G$-inflation model inevitably suffers from this problem. In this paper, we extend the original Higgs $G$-inflation Lagrangian to a more general form, namely introducing a higher-order kinetic term and generalizing the form of the Galileon term, so that the Higgs field can oscillate after inflation without encountering instabilities. Moreover, it accommodates a large region of the $n_s\mathrm{\text{−}}r$ plane, most of which is consistent with current observations, leading us to expect the detection of B-mode polarization in the cosmic microwave background in the near future.

Figure 1

The relation between $M$ and $\tilde{M}$ that reproduces the correct amplitude of the power spectrum of the primordial scalar perturbation. For $M<1.6\times {10}^{-5}{M}_P$, the Galileon term induces instability at small scales during the oscillating stage, and the present universe will not be realized.

Figure 2

Analytical predictions of the Galileon term dominating models and the higher-order kinetic term dominating models, compared with the Planck results [27]. The double line shows the case where the higher-order kinetic term is dominant, with varying $\ell$, while the other lines show the predictions of the Galileon term dominating case, fixing $n$ and varying $m$ or vice versa.

Figure 3

Numerical results for $\ell =2$ and $m=1$ with several values of $n$. The continuous lines show the predictions for different combinations of $M$ and $\tilde{M}$ that lead to ${\mathcal{P}}_{\zeta }=2.2\times {10}^{-9}$ at $N=60$ while avoiding instabilities. The black line is the analytical prediction of the Galileon term dominant case.

Figure 4

Results for $\ell =2$, $m=2$ with several values of $n$, varying the value of $M$.

Figure 5

Results for $\ell =3$, $m=1$, and $\ell =5$, $m=5$ for $N=60$ (top), and $\ell =3$, $m=1$, $n=5$ with different numbers of $e$-folds (bottom). The value of $M$ for each point is shown in the lower figure.