Study of the preheating phase of chaotic inflation

Particle production and its effects on the inflaton field are investigated during the preheating phase of chaotic inflation using a model consisting of a massive scalar inflaton field coupled to $N$ massless quantum scalar fields. The effects of spacetime curvature and interactions between the quantum fields are ignored. A large $N$ expansion is used to obtain a coupled set of equations including a backreaction equation for the classical inflaton field. Previous studies of preheating using these equations have been done. Here the first numerical solutions to the full set of equations are obtained for various values of the coupling constant and the initial amplitude of the inflaton field. States are chosen so that initially the backreaction effects on the inflaton field are small and the mode equations for the quantum fields take the form of Mathieu equations. Potential problems relating to the parametric amplification of certain modes of the quantum fields are identified and resolved. A detailed study of the damping of the inflaton field is undertaken. Some predictions of previous studies are verified and some new results are obtained.

Figure 1

The evolution of the inflaton field for $g={10}^{-3}$. From left to right the top plots are for $g^2{\varphi }_0^2=35$ and 10. The bottom plot is for $g^2{\varphi }_0^2=1$.

Figure 2

The evolution of the inflaton field for $g^2{\varphi }_0^2=10$. The plot on the left is for $g={10}^{-2}$ and the one on the right is for ${10}^{-3}$.

Figure 3

The evolution of the energy density of the inflaton field, ${\rho }_{\varphi }$. Both plots are for $g={10}^{-3}$. The one on the left is for $g^2{\varphi }_0^2=1$ and the one on the right is for $g^2{\varphi }_0^2=10$.

Figure 4

This figure shows differences in $\varphi$ and $g^2⟨{\psi }^2{⟩}_{\mathrm{ren}}$ which occur for $g={10}^{-3}$ and $g^2{\varphi }_0^2=10$ and $g^2{\varphi }_0^2=10(1+{10}^{-5})$. In each plot the dashed curve corresponds to $g^2{\varphi }_0^2=10(1+{10}^{-5})$.

Figure 5

This figure shows the boundaries of the five lowest instability bands for the Mathieu equation using (3.5a, 3.5b). The lower bound of a given band is the dashed curve and the upper bound is the solid curve immediately above it. Note that there are no modes of the quantum field with $k^2<0$.

Figure 6

These plots show $g^2⟨{\psi }^2{⟩}_{\mathrm{ren}}$ for $g={10}^{-3}$ and $g^2{\varphi }_0^2=1$. The plots are identical except that the one on the left is plotted on a logarithmic scale and the one on the right is plotted on a linear scale.

Figure 7

This figure shows the integrand in Eq. (2.12a) for the case $g={10}^{-3}$, $g^2{\varphi }_0^2=1$, for the times $t=150$ (left panel) and $t=250$ (right panel). Note that at the later time the band of modes that contributes significantly to the integral includes modes with larger values of $k$ than the band at the earlier time.

Figure 8

The time evolution is shown for the case in which $g={10}^{-3}$ and $g^2{\varphi }_0^2=10$ for one solution and $10(1+{10}^{-7})$ for the other. The top plots show the evolution of $|\delta \varphi /\varphi |$ and $|\delta ⟨{\psi }^2{⟩}_{\mathrm{ren}}/⟨{\psi }^2{⟩}_{\mathrm{ren}}|$. The bottom plot shows the evolution of $|\delta \mathrm{Re}{f}_k/(\mathrm{Re}{f}_k)|$ for $k=0.5$. The dashed curve corresponds to $|\mathrm{Re}{f}_k|$ and is not a relative difference. Note that the growth of the relative difference for each quantity shuts off before its average becomes of order unity. The regions of the plots where the average evolves as a straight line with a nonzero slope correspond to times when the average relative differences are growing exponentially.

Figure 9

The time evolution is shown for the case in which $g={10}^{-3}$ and $g^2{\varphi }_0^2=10$ for one solution and $10(1+{10}^{-5})$ for the other. The top plots show the evolution of $|\delta \varphi /\varphi |$, $|\delta ⟨{\psi }^2{⟩}_{\mathrm{ren}}/⟨{\psi }^2{⟩}_{\mathrm{ren}}|$. The bottom plot shows the evolution of $|\delta \mathrm{Re}{f}_k/(\mathrm{Re}{f}_k)|$ for $k=0.5$. The dashed curve corresponds to $|\mathrm{Re}{f}_k|$ and is not a relative difference. The regions of the plots where the average evolves as a straight line with a nonzero slope correspond to times when the average relative differences are growing exponentially.

Figure 10

Shown is the real part of two mode functions $f_k$ for the case $g={10}^{-3}$, $g^2{\varphi }_0^2=10$. For the left plot $k={10}^{-4}$ and for the right plot $k=0.5$.