Cosmological perturbations through a simple bounce

We present a detailed study of a simple scalar field model that yields nonsingular cosmological solutions. We study both the qualitative dynamics of the homogeneous and isotropic background and the evolution of inhomogeneous linear perturbations. We calculate the spectrum of perturbations generated on super-Hubble scales during the collapse phase from initial vacuum fluctuations on small scales and then evolve these numerically through the bounce. We show there is a gauge in which perturbations remain well-defined and small throughout the bounce, even though perturbations in other commonly used gauges become large or ill-defined. We show that the comoving curvature perturbation calculated during the collapse phase provides a good estimate of the resulting large-scale adiabatic perturbation in the expanding phase while the Bardeen metric potential is dominated by what becomes a decaying mode after the bounce. We show that a power-law collapse phase with scale factor proportional to $(-t{)}^{2/3}$ can yield a scale-invariant spectrum of adiabatic scalar perturbations in the expanding phase, but the amplitude of tensor perturbations places important constraints on the allowed initial conditions.

Figure 1

The phase-trajectories in the $(x,y)$ and $(x,z)$ phase-planes for $\lambda =2$.

Figure 2

Phase-space trajectories in the $(\alpha ,\gamma )$ plane for $\lambda =2$. Everywhere within the dashed lines the universe will have a phantom equation of state.

Figure 3

Classification of phase-space trajectories in the $(\alpha ,\gamma )$ plane for $\lambda =2$.

Figure 4

Phase-space trajectories in the $(\alpha ,\gamma )$ plane for $\lambda =3$. Everywhere within the dashed lines the universe will have a phantom equation of state.

Figure 5

Classification of phase-space trajectories in the $(\alpha ,\gamma )$ plane for $\lambda =3$

Figure 6

Evolution of $\psi {|}_{\chi }$ shown as a function of expansion, $N=\pm \ln (a/a_0)$, for bounce solution with $\lambda =2$ model for each of the four independent modes: ${\phi }_Y$ (solid line), ${\phi }_J$ (dotted line), ${\chi }_Y$ (dashed line), and ${\chi }_J$ (dot-dashed line).

Figure 7

Comoving curvature perturbation $\mathcal{R}$ calculated from numerical solution for each of the four modes shown in Fig. 6.

Figure 8

Bardeen metric potential $\Phi$ calculated from numerical solution for each of the four modes shown in Fig. 6.

Figure 9

Evolution of $\psi {|}_{\chi }$ shown as a function of expansion, $N=\pm \ln (a/a_0)$, for bounce solution with $\lambda =\sqrt{3}$ model for each of the four independent modes: ${\phi }_Y$ (solid line), ${\phi }_J$ (dotted line), ${\chi }_Y$ (dashed line), and ${\chi }_J$ (dot-dashed line).

Figure 10

Comoving curvature perturbation $\mathcal{R}$ calculated from numerical solution for each of the four modes shown in Fig. 9.

Figure 11

Bardeen metric potential $\Phi$ calculated from numerical solution for each of the four modes shown in Fig. 9.

Figure 12

Evolution of $\delta g$ shown as a function of expansion, $N=\pm \ln (a)$, for a bounce solution in the $\lambda =\sqrt{3}$ model for each of the two independent tensor modes: $g_Y$ (solid line) and $g_J$ (dashed line).

Figure 13

Tensor-scalar ratio ${\mathcal{P}}_g/{\mathcal{P}}_{\mathcal{R}}$ after the bounce (solid line) shown against dimensionless parameter $C$ defined in Eq. (27). The dashed line shows the observational limit.

Figure 14

The violation of constraint Eqs. (60, 61) shown as a function of expansion, $N=\pm \ln (a)$, for a bounce solution in the $\lambda =2$ model for each of the four independent scalar modes: ${\phi }_Y$ (solid line), ${\phi }_J$ (dotted line), ${\chi }_Y$ (dashed line), ${\chi }_J$ (dot-dashed line). The $y$-axis of the left hand graph shows the logarithm of the absolute value of the left hand side of Eq. (60) divided by the sum of the absolute values of each term in the equation. The $y$-axis of the right hand graph shows the same thing for Eq. (61).

Figure 15

The violation of constraint Eqs. (60, 61) shown as a function of expansion, $N=\pm \ln (a)$, for a bounce solution in the $\lambda =\sqrt{3}$ model for each of the four independent scalar modes: ${\phi }_Y$ (solid line), ${\phi }_J$ (dotted line), ${\chi }_Y$ (dashed line), ${\chi }_J$ (dot-dashed line). The $y$-axis of the left hand graph shows the logarithm of the absolute value of the left hand side of Eq. (60) divided by the sum of the absolute values of each term in the equation. The $y$-axis of the right hand graph shows the same thing for Eq. (61).