Subcritical hybrid inflation in a generalized superconformal model

We study a generalized superconformal model that gives rise to a subcritical regime of $D$-term hybrid inflation. Exhibiting the model both in a Jordan frame and in the Einstein frame, the effective potential of the subcritical regime is derived in the Einstein frame. It turns out that the inflaton-waterfall field dynamics leads to various types of inflaton potential. Consequently, the tensor-to-scalar ratio is found to range from ${10}^{-4}$ (${10}^{-3}$) to 0.1 for getting 60 (50) $e$-folds before the end of inflation.

Figure 1

Allowed region for $\alpha$ and $\chi$ for $N_{*}=60$ (top) and 50 (bottom) by imposing $m_{\tau }^2>0$ and Eqs. (1)–(3). The right panels are the same as the left ones, but the range of $-1.5\le \chi \le -0.5$ is magnified. The color map indicates the minimum value of the predicted tensor-to-scalar ratio, $r_{\min }$.

Figure 2

Left: Predicted scalar spectral index and tensor-to-scalar ratio for fixed $e$-folds, $N_{*}=60$ (solid curves) and 50 (dotted curves), and various values of $\alpha$. In the top and bottom panels, $\chi$ is taken to $-5$ and 5, respectively. We impose $m_{\tau }^2>0$, and $1\sigma$ (dark shaded) and $2\sigma$ (light shaded) regions from the Planck Collaboration [2] are also shown. Right: Allowed regions for $\lambda$ and $\sqrt{\xi }$ for $\chi =-5$ (top) and 5 (bottom), by imposing Eqs. (1)–(3).

Figure 3

Normalized potential $\widehat{V}$ as a function of the canonically normalized inflaton field $\widehat{\varphi }$. Only the subcritical regime is shown, and the field values of $\widehat{\varphi }$ at the end of inflation and at 60 $e$-folds are indicated as ${\widehat{\varphi }}_{\mathrm{end}}$ and ${\widehat{\varphi }}_{*}$, respectively. Color codes are the same as in Fig. 2, and $\lambda$ and $\sqrt{\xi }$ are chosen to give the best-fit value of the observed $n_s$.

Figure 4

Trajectory of the waterfall field (red dotted curve) as a function of the canonically normalized inflaton field $\widehat{\varphi }$. As a reference, the normalized potential $\widehat{V}$ (blue solid curve) is also plotted. The field values of $\widehat{\varphi }$ at the end of inflation and at 60 $e$-folds are indicated as ${\widehat{\varphi }}_{\mathrm{end}}$ and ${\widehat{\varphi }}_{*}$, respectively. In the top (bottom) panels, $\chi$ is taken to $-5$ (5). The other parameters are $\alpha =1$, $\lambda =3.1\times {10}^{-3}$, $\sqrt{\xi }=6.5\times {10}^{15}\mathrm{GeV}$ (top-left); $\alpha =2.8/3$, $\lambda =3.3\times {10}^{-3}$, $\sqrt{\xi }=2.0\times {10}^{16}\mathrm{GeV}$ (top-right); $\alpha =2/3$, $\lambda =2.2\times {10}^{-3}$, $\sqrt{\xi }=5.5\times {10}^{15}\mathrm{GeV}$ (bottom-left); and $\alpha =2.04/3$, $\lambda =2.4\times {10}^{-3}$, $\sqrt{\xi }=6.6\times {10}^{15}\mathrm{GeV}$ (bottom-right). With these parameters, the median value of the observed $n_s$ is obtained, and the tensor-to-scalar ratio is predicted to be $7.9\times {10}^{-4}$ (top-left), $6.4\times {10}^{-2}$ (top-right), $4.2\times {10}^{-4}$ (bottom-left), and $8.5\times {10}^{-4}$ (bottom-right).

Figure 5

Same as Fig. 2, but taking $\chi =-1.03$ (top) and $\chi =-0.97$ (bottom) and different values of $\alpha$ accordingly. In the right panels, we take $N_{*}=60$ (solid curves) and $N_{*}=55$ (dot-dashed curves).

Figure 6

Same as Fig. 3, but with $\chi =-1.03$ and $-0.97$. Color codes are the same as in Fig. 5, and $\lambda$ and $\sqrt{\xi }$ are chosen to give the best-fit value for $n_s$.

Figure 7

Same as Fig. 4, but taking $\chi =-1.03$ (top) and $\chi =-0.97$ (bottom). The other parameters in each panel are $\alpha =1$, $\lambda =7.6\times {10}^{-4}$, $\sqrt{\xi }=1.8\times {10}^{16}\mathrm{GeV}$ (top-left); $\alpha =2.6/3$, $\lambda =5.5\times {10}^{-4}$, $\sqrt{\xi }=2.1\times {10}^{16}\mathrm{GeV}$ (top-right); $\alpha =2.1/3$, $\lambda =4.1\times {10}^{-4}$, $\sqrt{\xi }=1.9\times {10}^{16}\mathrm{GeV}$ (bottom-left); and $\alpha =2.3/3$, $\lambda =4.3\times {10}^{-4}$, $\sqrt{\xi }=2.3\times {10}^{16}\mathrm{GeV}$ (bottom-right). The tensor-to-scalar ratio is found to be $3.5\times {10}^{-2}$ (top-left), $5.9\times {10}^{-2}$ (top-right), $4.5\times {10}^{-2}$ (bottom-left), and $r=9.5\times {10}^{-2}$ (bottom-right).

Figure 8

Left: Predicted values of $n_s$ and $r$ for $N_{*}=60$ (solid curves) and 50 (dotted curves) and various values of $\chi$. $\alpha$ is taken to be 1 (top) and $2/3$ (bottom). Right: Allowed parameters for $N_{*}=60$. Here, $\alpha =1$ (top) and $2/3$ (bottom), and various values of $\chi$ are taken.

Figure 9

Normalized potential as a function of the canonically normalized inflaton field. Color codes are the same as in Fig. 8, and $\lambda$ and $\sqrt{\xi }$ are chosen to give the best-fit value for $n_s$.

Figure 10

Same as Fig. 2, but taking $\chi =0$ and different values of $\alpha$ accordingly. In the right panels, we take $N_{*}=60$ (solid curves) and $N_{*}=55$ (dot-dashed curves).

Figure 11

Same as Fig. 3, but with $\chi =0$. Color codes are the same as in Fig. 10, and $\lambda$ and $\sqrt{\xi }$ are chosen to give the best-fit value for $n_s$.

Figure 12

Same as Fig. 4, but taking $\chi =0$ and $\alpha =2/3$ (left) and $2.16/3$ (right). In the left panel, $\lambda =9.6\times {10}^{-4}$ and $\sqrt{\xi }=6.0\times {10}^{15}\mathrm{GeV}$. In the right panel, $\lambda =1.3\times {10}^{-3}$ and $\sqrt{\xi }=2.1\times {10}^{16}\mathrm{GeV}$. The tensor-to-scalar ratio is $6.1\times {10}^{-4}$ (left) and $7.0\times {10}^{-2}$ (right).