Nonperturbative analysis of the evolution of cosmological perturbations through a nonsingular bounce

In bouncing cosmology, the primordial fluctuations are generated in a cosmic contraction phase before the bounce into the current expansion phase. For a nonsingular bounce, curvature and anisotropy grow rapidly during the bouncing phase, raising questions about the reliability of perturbative analysis. In this paper, we study the evolution of adiabatic perturbations in a nonsingular bounce by nonperturbative methods including numerical simulations of the nonsingular bounce and the covariant formalism for calculating nonlinear perturbations. We show that the bounce is disrupted in regions of the universe with significant inhomogeneity and anisotropy over the background energy density but is achieved in regions that are relatively homogeneous and isotropic. Sufficiently small perturbations, consistent with observational constraints, can pass through the nonsingular bounce with negligible alteration from nonlinearity. We follow scale invariant perturbations generated in a matterlike contraction phase through the bounce. Their amplitude in the expansion phase is determined by the growing mode in the contraction phase, and the scale invariance is well preserved across the bounce.

Figure 1

The background solution of the scale factor $a$. The harmonic time $t$ is shifted so that the bounce occurs at $t=0$ and rescaled in units of $1/|H_0|$ where $H_0$ is the initial value of the Hubble parameter.

Figure 2

The ratio between the energy density of the scalar fields $\varphi$ and $\chi$. The $\chi$ field energy density is significant only near the bounce and otherwise negligible. The time coordinate is scaled in the same way as in Fig. 1.

Figure 3

Local expansion $\theta$ as a function of the coordinate $x$ at select times, computed with parameters in Eq. (67). The spatial coordinate $x$ is scaled in units of $1/|H_0|$, which is the size of a Hubble length at the initial time, and $t$ is scaled in the same way as in Fig. 1 so that $t=0$ corresponds to the bounce in the homogeneous solution. The expansion $\theta$ is scaled in units of $3|H_0|$. In this inhomogeneous case, the nonsingular bounce occurs when $\theta$ crosses zero from below, which happens at different times for different spatial points.

Figure 4

$|E_{\varphi }/E_{\chi }|$ (black continuous), $|\frac{1}{2}{}^{(3)}R/E_{\chi }|$ (red dashed), and $|{\sigma }^2/E_{\chi }|$ (blue dotted) as a function of the coordinate $x$ at select times, computed with parameters in Eq. (67). $x$ and $t$ coordinates are scaled in the same way as in Fig. 3. In this example, curvature and anisotropy are negligible compared to the energy density of the scalar fields.

Figure 5

Local expansion $\theta$ as a function of the coordinate $x$ at select times, computed with parameters in Eq. (68). $x$ and $t$ coordinates and the expansion $\theta$ are scaled in the same way as in Fig. 3. The nonsingular bounce does not occur in the shaded region (shown here in the middle of the $x$ range; though recall that $x$ is periodic, and each panel covers a single period) where the ratio $|{\sigma }^2/E_{\chi }|$ is greater than 1; see Fig. 6.

Figure 6

$|E_{\varphi }/E_{\chi }|$ (black continuous), $|\frac{1}{2}{}^{(3)}R/E_{\chi }|$ (red dashed), and $|{\sigma }^2/E_{\chi }|$ (blue dotted) as a function of the coordinate $x$ at select times, computed with parameters in Eq. (68); see Fig. 5 for the expansion at corresponding times. $x$ and $t$ coordinates are scaled in the same way as in Fig. 3. In this example, $|{\sigma }^2/E_{\chi }|>1$ in the shaded region (shown here in the middle of the $x$ range), preventing a nonsingular bounce.

Figure 7

Local expansion $\theta$ as a function of the coordinate $x$ at select times, computed with parameters in Eq. (69). $x$ and $t$ coordinates and the expansion $\theta$ are scaled in the same way as in Fig. 3. Part of the space (shown here in the middle region of the range of the periodic $x$ coordinate) bounces at a much later time compared to the other regions, causing inhomogeneity at late times.

Figure 8

$|E_{\varphi }/E_{\chi }|$ (black continuous), $|\frac{1}{2}{}^{(3)}R/E_{\chi }|$ (red dashed), and $|{\sigma }^2/E_{\chi }|$ (blue dotted) as a function of the coordinate $x$ at select times, computed with parameters in Eq. (69). $x$ and $t$ coordinates are scaled in the same way as in Fig. 3. In this marginal case, the ratio $|{\sigma }^2/E_{\chi }|$ between the anisotropy and the $\chi$ field energy density reaches as high as ${10}^{-2}$, causing significant nonlinearity.

Figure 9

Amplitude of the first few Fourier modes of the integrated expansion $\mathcal{N}$, computed with parameters in Eq. (69): $|{\mathcal{N}}^{(0)}|$ (black thick), $|{\mathcal{N}}^{(1)}|$ (red dashed), $|{\mathcal{N}}^{(2)}|$ (yellow dash-dotted), $|{\mathcal{N}}^{(3)}|$ (green dotted), and $|{\mathcal{N}}^{(4)}|$ (blue thin). $t$ is scaled in the same way as in Fig. 1, so that $t=0$ corresponds to the bounce in the homogeneous case. In this example, the amplitude of the higher Fourier modes, especially ${\mathcal{N}}^{(2)}$, is separated by less than 1 order of magnitude from the linear mode ${\mathcal{N}}^{(1)}$, indicating that nonlinearity becomes significant during the bouncing phase.

Figure 10

Comparison of the Fourier modes ${\mathcal{N}}^{(1)}$ (red dashed) and ${\mathcal{N}}^{(2)}$ (yellow dash-dotted). $t$ is scaled in the same way as in Fig. 9. The amplitude of ${\mathcal{N}}^{(2)}$ becomes substantial during the bounce as compared to ${\mathcal{N}}^{(1)}$.

Figure 11

The homogeneous part of the integrated expansion, ${\mathcal{N}}^{(0)}$ (continuous), as compared to the background solution $N$ (dashed). $t$ is scaled in the same way as in Fig. 9. The bouncing process in the inhomogeneous case deviates from the background solution due to the substantial anisotropy as compared to the scalar field energy density.

Figure 12

The first Fourier mode of the integrated expansion, ${\mathcal{N}}^{(1)}$ (dashed), as compared to the curvature perturbation $-{\psi }_{\mathrm{h}}$ (dotted) and lapse perturbation $\frac{1}{3}{A}_{\mathrm{h}}$ (continuous) calculated by linear perturbation theory. $t$ is scaled in the same way as in Fig. 9. The clear disagreement between the curves indicates the inaccuracy of linear perturbative calculations due to the anisotropy.

Figure 13

Amplitude of the first few Fourier modes of the integrated expansion $\mathcal{N}$, computed with parameters in Eq. (71): $|{\mathcal{N}}^{(0)}|$ (black thick), $|{\mathcal{N}}^{(1)}|$ (red dashed), $|{\mathcal{N}}^{(2)}|$ (yellow dash-dotted), $|{\mathcal{N}}^{(3)}|$ (green dotted), and $|{\mathcal{N}}^{(4)}|$ (blue thin). $t$ is scaled in the same way as in Fig. 1, so that $t=0$ corresponds to the bounce in the homogeneous case. In this example, the amplitude of higher Fourier modes are clearly suppressed with respect to the linear mode, indicating that nonlinearity is negligible.

Figure 14

Comparison of the Fourier modes ${\mathcal{N}}^{(1)}$ (red dashed) and ${\mathcal{N}}^{(2)}$ (yellow dash-dotted). $t$ is scaled in the same way as in Fig. 13. The amplitude of ${\mathcal{N}}^{(2)}$ stays negligible as compared to ${\mathcal{N}}^{(1)}$.

Figure 15

The homogeneous part of the integrated expansion, ${\mathcal{N}}^{(0)}$ (continuous), as compared to the background solution $N$ (dashed). $t$ is scaled in the same way as in Fig. 13. The perfect agreement shows that the background solution is a good approximation.

Figure 16

The first Fourier mode of the integrated expansion, ${\mathcal{N}}^{(1)}$ (dashed), as compared to the curvature perturbation $-{\psi }_{\mathrm{h}}$ (dotted) and lapse perturbation $\frac{1}{3}{A}_{\mathrm{h}}$ (continuous) calculated by linear perturbation theory. $t$ is scaled in the same way as in Fig. 13. All three curves agree to good approximation, showing that linear perturbation theory works well in this case.

Figure 17

Amplitude of the first few Fourier modes of the integrated expansion $\mathcal{N}$, computed with parameters in Eq. (72): $|{\mathcal{N}}^{(0)}|$ (black thick), $|{\mathcal{N}}^{(1)}|$ (red dashed), $|{\mathcal{N}}^{(2)}|$ (yellow dash-dotted), $|{\mathcal{N}}^{(3)}|$ (green dotted), and $|{\mathcal{N}}^{(4)}|$ (blue thin). $t$ is scaled in the same way as in Fig. 1, so that $t=0$ corresponds to the bounce in the homogeneous case. In this example, the amplitude of the two input modes ${\mathcal{N}}^{(1)}$ and ${\mathcal{N}}^{(3)}$ are comparable to each other, whereas the mixed modes ${\mathcal{N}}^{(2)}$ and ${\mathcal{N}}^{(4)}$ are suppressed, indicating that mode mixing is negligible.

Figure 18

$k=m$ mode in the two-mode computation (red dashed) and the single-mode computation (red continuous) and $k=3m$ mode in the two-mode computation (green dotted) and the single-mode computation (green continuous). $t$ is scaled in the same way as in Fig. 17. The agreement of two-mode and single-mode computations in each case shows that different Fourier modes evolve independently.

Figure 19

Amplitude of the growing mode (green continuous) and the constant mode (red dotted) with Bunch–Davies initial values for $m=0.01$. The growing mode gives the dominant contribution to the amplitude after the bounce.

Figure 20

Amplitude of the comoving curvature perturbation ${\mathcal{R}}_{\varphi }$ for wave numbers $m_1=0.01$ (green continuous) and $m_2=0.03$ (blue dashed). Their relative amplitude, given by the vertical distance, stays approximately the same after the bounce, showing that the scale dependence of the evolution of the perturbation amplitudes during the bounce is weak.

Figure 21

The ratio between the amplitude of the Fourier modes with $k=m$ and $k=3m$. The increase of the ratio after the bounce implies a small red tilt (though negligible for observable modes, see Fig. 22).

Figure 22

The power spectrum of the comoving curvature perturbation ${\mathcal{R}}_{\varphi }$ after the bounce. The change in the amplitude over $\sim 10$ $e$-folds of wave numbers is as small as ${10}^{-3}$ and becomes negligible for even smaller $k$ that corresponds to observable modes.