Hybrid Higgs inflation: The use of disformal transformation

We propose a hybrid type of the conventional Higgs inflation and new Higgs inflation models. We perform a disformal transformation into the Einstein frame and analyze the background dynamics and the cosmological perturbations in the truncated model, in which we ignore the higher-derivative terms of the Higgs field. From the observed power spectrum of the density perturbations, we obtain the constraint on the nonminimal coupling constant $\xi$ and the mass parameter $M$ in the derivative coupling. Although the primordial tilt $n_s$ in the hybrid model barely changes, the tensor-to-scalar ratio $r$ moves from the value in the new Higgs inflationary model to that in the conventional Higgs inflationary model as $|\xi |$ increases. We confirm our results by numerical analysis by ADM formalism of the full theory in the Jordan frame.

Figure 1

The effective potential $U(\mathrm{\Phi })$ in the Einstein frame for the coupling parameters $M={10}^{-7},\xi =-{10}^4$ and $\lambda =0.1$.

Figure 2

The background dynamics for the coupling parameters $M={10}^{-7},\xi =-{10}^4$. We show the evolution of the Hubble expansion parameter $H$ with respect to the $e$-folding number $N$ in the Jordan frame. The orange curve denotes the evolution of the full theory in the Jordan frame, while the blue dotted curve shows the results obtained by the truncated model in the Einstein frame. We transform its solution into the Jordan frame variables. We also show the reference $e$-folding number of $N=50–60$ by the dotted light-gray lines.

Figure 3

The evolution of the Higgs field $\varphi$ with respect to the $e$-folding number $N$. The coupling parameters, the colored curves, and the dotted light-gray lines are the same setting as those in Fig. 2.

Figure 4

The phase-space evolution of the Higgs field ($\varphi -\dot{\varphi }$). The coupling parameters, the colored curves, and the dotted light-gray lines are the same setting as those in Fig. 2.

Figure 5

The constraint on the coupling parameters $M$ and $\xi$. The color dots with numbers or letters correspond to the coupling parameters given in Table. 1.

Figure 6

The colored lines show the numerical results of the hybrid Higgs inflation with $N=50–60$. The numbers or letters of lines correspond to the colored dots in Fig. 5 and the coupling parameters in Table 1. Ⓝ and Ⓒ correspond to new Higgs inflation model with $M=1.41\times {10}^{-8}$ and the conventional Higgs inflation with $\xi =-{10}^4$, respectively. Ⓟ shows the case with positive nonminimal coupling ($\xi =1/6$), which line is overlapping with the line Ⓝ. Ⓒ and ④ are almost the same. The observational constraints on $n_s$ and $r$ by Planck 2015 [57] are also shown.