No quasistable scalaron lump forms after $R^2$ inflation

In the Einstein frame picture of Starobinky’s $R^2$ inflation model, cosmic inflation is driven by a slowly rolling inflaton field, called a scalaron, and followed by a coherently oscillating scalaron phase. Since the scalaron oscillates excessively many times in its potential, which has a quadratic minimum and is a little shallower than quadratic on the positive side, it may fragment into long-living localized objects, called oscillons or I balls, due to nonlinear growth of fluctuations before reheating of the Universe. We show that, while parametric self-resonances amplify scalaron fluctuations in the Minkowski background, the growth cannot overcome the decay due to expansion in the Friedmann background after $R^2$ inflation. By taking into account the backreaction from the metric of spacetime, modes that are larger than a critical scale are indeed amplified and become nondecaying. However, those nondecaying modes are not growing enough to form spatially localized lumps of the scalaron. Thus, reheating processes are unaltered by oscillons/I balls, and they proceed through perturbative decay of the scalaron as studied in the original work.

Figure 1

Time evolution of the fluctuations from $t=0[1/M]$ for each initial amplitude of the background field $\mathrm{\Phi }/M_p=2,0.5,0.1,0.01$. Each line is the snapshot of the spectrum of the fluctuations at several times. The vertical axis $\delta {\varphi }_k$ is the Fourier mode of the fluctuation $\delta \varphi (x)$, and the horizontal axis $k$ is the corresponding momentum.

Figure 2

Instability modes of the broad resonance for $\alpha <1$ and initial amplitudes $\mathrm{\Phi }/M_p=2,1,0.5,0.1$. The vertical axis is the right-hand side of the resonance condition (51). The horizontal axis is the time variable, which is defined as $\theta \equiv Mt(\text{modulo}2\pi )$. The enhancement of the fluctuations occurs in the $k^2/M^2>0$ region.

Figure 3

Instability modes of the broad resonance for $\alpha >1$ and initial amplitudes $\mathrm{\Phi }/M_p=10,2,1$. The vertical axis is the right-hand side of the resonance condition (59). The horizontal axis is the time variable, which is defined as $\theta \equiv 2Mt(\text{modulo}2\pi )$. The enhancement of the fluctuations occurs in $k^2/M^2>0$ region.

Figure 4

Time evolution of the fluctuations from $t=0[1/M]$ to $t={10}^3[1/M]$ for $\mathrm{\Phi }/M_p=10$. Each line is the snapshot of the spectrum of the fluctuations at several times. The vertical axis $\delta {\varphi }_k$ is the Fourier mode of the fluctuation $\delta \varphi (x)$, and the horizontal axis $k$ is the corresponding momentum.

Figure 5

Time evolution of the background from $t={10}^{-1}[1/M]$ to $t={10}^4[1/M]$. The vertical axis ${\varphi }_{\text{bg}}={\varphi }_0$ is the amplitude of the background field, and the horizontal axis $t$ is cosmic time in units of $M^{-1}$.

Figure 6

Time evolution of the fluctuations from $t=0[1/M]$ to $t={10}^4[1/M]$. Each line is the snapshot of the spectrum of the fluctuations at $t=0,10,{10}^2,{10}^3,{10}^4[1/M]$. The vertical axis $\delta {\varphi }_k$ is the Fourier mode of the fluctuation $\delta \varphi (x)$, and the horizontal axis $k$ is the corresponding momentum.

Figure 7

Time evolution of the fluctuations from $t=0[1/M]$ to $t={10}^4[1/M]$ with the Mukhanov–Sasaki equation. Each line is the snapshot of the spectrum of the fluctuations at $t=0,10,{10}^2,{10}^3,{10}^4[1/M]$. The vertical axis $\delta {\varphi }_k$ is the Fourier mode of the fluctuation $\delta \varphi (x)$, and the horizontal axis $k$ is the corresponding momentum.

Figure 8

Time evolution of $n_k$ from $t=0[1/M]$ to $t={10}^4[1/M]$. Each line is the snapshot of the spectrum of $n_k$ at $t=0,10,{10}^2,{10}^3,{10}^4[1/M]$. The horizontal axis $k$ is the comoving momentum corresponding to each $n_k$.