Nonlinear preheating with nonminimally coupled scalar fields in the Starobinsky model

We study the preheating after inflation in the Starobinsky model with a nonminimally coupled scalar field $\chi$. Using the lattice simulation, we analyze the rescattering between the $\chi$ particles and the inflaton condensate, and the backreaction effect of the scalar metric perturbations. We find that the rescattering is an efficient mechanism promoting the growth of the $\chi$ field variance. Meanwhile, copious inflaton particles can be knocked out of the inflaton condensate by rescattering. As a result, the inflaton field can become a non-negligible gravitational wave source, even comparable with the $\chi$ field in some parameter regions. For the scalar metric perturbations, which are on the sub-Hubble scale in our analysis, our results show that they have negligible effects on the evolution of scalar fields and the production of gravitational waves in the model considered in the present paper.

Figure 1

The evolution of $m_{\chi ,\mathrm{eff}}^2/(|\xi |{\mu }^2)$ as a function of $a(t)$ for the case of $m_{\chi }=0$.

Figure 2

The variance of the $\chi$ field versus $a(t)$ for $m_{\chi }=0$.

Figure 3

The maximum of the $\chi$ field variance as a function of $\xi$ for $m_{\chi }=0$.

Figure 4

(a) Left-hand plot: The variances of the $\varphi$ field (red line) and the $\chi$ field (blue line) versus $a(t)$ for $\xi =3$ and $m_{\chi }=0$. (b) Right-hand plot: The spatial averages of the inflaton field $⟨\varphi ⟩$ (red line) and $⟨m_{\chi ,\mathrm{eff}}^2⟩$ (blue line) versus $a(t)$ for $\xi =3$ and $m_{\chi }=0$.

Figure 5

The variances of the $\varphi$ field (red) and the $\chi$ field (blue) versus $a(t)$ for $\xi =5$ and $m_{\chi }=0$.

Figure 6

(a) Left-hand plot: The evolutions of $⟨\varphi ⟩$ (red line) and $-5\sqrt{\frac{3}{2}}⟨{\chi }^2⟩$ (blue line) as functions of $a(t)$ for $\xi =5$ and $m_{\chi }=0$. (b) Right-hand plot: The evolutions of $⟨m_{\chi ,\mathrm{eff}}^2⟩$ (red line) and $6\times 5^2⟨{\chi }^2⟩$ (blue line) as functions of $a(t)$ for $\xi =5$ and $m_{\chi }=0$.

Figure 7

The evolutions of the average gradient energy density of the $\varphi$ field (red line), the $\chi$ field (blue line), and their sum (green line) with $a(t)$ for $\xi =5$ and $m_{\chi }=0$.

Figure 8

(a) Left-hand plot: The variances of the $\varphi$ field (red) and the $\chi$ field (blue) versus $a(t)$ for $\xi =10$ and $m_{\chi }=0$. (b) Right-hand plot: The evolutions of the average gradient energy density of the $\varphi$ field (red line), the $\chi$ field (blue line), and their sum (green line) with $a(t)$ for $\xi =10$ and $m_{\chi }=0$.

Figure 9

(a) Left-hand plot: The variances of the $\varphi$ field (red) and the $\chi$ field (blue) versus $a(t)$ for $\xi =70$ and $m_{\chi }=0$. (b) Right-hand plot: The evolutions of the average gradient energy density of the $\varphi$ field (red line), the $\chi$ field (blue line), and their sum (green line) with $a(t)$ for $\xi =70$ and $m_{\chi }=0$.

Figure 10

(a) Left-hand plot: The variances of the $\varphi$ field (red) and the $\chi$ field (blue) versus $a(t)$ for $\xi =-3$ and $m_{\chi }=0$. (b) Right-hand plot: The evolutions of the average gradient energy density of the $\varphi$ field (red line), the $\chi$ field (blue line), and their sum (green line) with $a(t)$ for $\xi =-3$ and $m_{\chi }=0$.

Figure 11

(a) Left-hand plot: The variances of the $\varphi$ field (red) and the $\chi$ field (blue) versus $a(t)$ for $\xi =-5$ and $m_{\chi }=0$. (b) Right-hand plot: The evolutions of the average gradient energy density of the $\varphi$ field (red line), the $\chi$ field (blue line), and their sum (green line) with $a(t)$ for $\xi =-5$ and $m_{\chi }=0$.

Figure 12

(a) Left-hand plot: The variances of the $\varphi$ field (red) and the $\chi$ field (blue) versus $a(t)$ for $\xi =-30$ and $m_{\chi }=0$. (b) Right-hand plot: The evolutions of the average gradient energy density of the $\varphi$ field (red line), the $\chi$ field (blue line), and their sum (green line) with $a(t)$ for $\xi =-30$ and $m_{\chi }=0$.

Figure 13

The evolution of $(e^{\sqrt{2/3}\kappa \varphi }-1)$ as a function of $a(t)$.

Figure 14

(a) Left-hand plot: The variances of the $\chi$ field versus $a(t)$ for $m_{\chi }=0$, $m_{\chi }=0.5\mu$, $m_{\chi }=\mu$, $m_{\chi }=1.5\mu$, and $m_{\chi }=2\mu$ in the $\xi =10$ case. (b) Right-hand plot: The variances of the $\chi$ field versus $a(t)$ for $m_{\chi }=0$, $m_{\chi }=\mu$, $m_{\chi }=2\mu$, $m_{\chi }=3\mu$, $m_{\chi }=4\mu$, and $m_{\chi }=6\mu$ in the $\xi =-10$ case.

Figure 15

(a) Left-hand plot: The variances of the $\chi$ field versus $a(t)$ for $m_{\chi }=0$, $m_{\chi }=2\mu$, $m_{\chi }=3\mu$, $m_{\chi }=4\mu$, and $m_{\chi }=6\mu$ in the $\xi =-50$ case. (b) Right-hand plot: The variances of the $\varphi$ field versus $a(t)$ for $m_{\chi }=0$, $m_{\chi }=2\mu$, $m_{\chi }=3\mu$, $m_{\chi }=4\mu$, and $m_{\chi }=6\mu$ in the $\xi =-50$ case.

Figure 16

(a) Left-hand plot: The spatially averaged equation-of-state parameter $w$ versus $a(t)$ for $\xi =3$ and $m_{\chi }=0$. (b) Right-hand plot: The spatially averaged equation-of-state parameter $w$ versus $a(t)$ for $\xi =10$ and $m_{\chi }=0$.

Figure 17

The spatially averaged equation-of-state parameter $w$ versus $a(t)$ for $m_{\chi }=0$, $m_{\chi }=\mu$, $m_{\chi }=2\mu$, and $m_{\chi }=3\mu$ in the $\xi =-50$ case.

Figure 18

The total variance of the scalar fields (red line) and the variance of the scalar metric perturbations (blue line) versus $a(t)$ for $\xi =10$ and $m_{\chi }=0$.

Figure 19

The variances of the inflaton field (solid red line) and $\chi$ field (solid blue line) versus $a(t)$ for $\xi =10$ and $m_{\chi }=0$. The variances of the inflaton field (dashed purple line) and $\chi$ field (dashed green line) in a FRW metric are also shown for comparison.

Figure 20

The average total gradient energy density of the scalar fields ${\mathcal{G}}_{\mathrm{tot}}$ (red line) and the average gradient contribution of the scalar metric fluctuations ${\mathcal{G}}_{\beta }$ (blue line) versus $a(t)$ for $\xi =10$ and $m_{\chi }=0$.