Covariant perturbations through a simple nonsingular bounce

In this paper we study the evolution of cosmological perturbations through a nonsingular bouncing universe using covariant perturbation theory and examine the validity of linear perturbation theory. The bounce is modeled by a two component perfect fluid. Gauge invariant perturbations are defined in terms of the comoving observers. The scalar and vector perturbations become singular at the turning point, which is the boundary of the spacetime region where the null energy condition is violated. Nonadiabatic modes of comoving curvature perturbation diverge at the turning point. The gravitational waves oscillate around the bounce and the turning point. By computing the growth of linearity parameters, it has been shown that the perturbations do not remain linear at the turning point. We also study the matching conditions on scalar perturbations and $\mathcal{V}$, related to the spatial curvature perturbation, is found to be the appropriate variable to be matched across transition surface.

Figure 1

Plot of Hubble parameter as a function of conformal time $\eta$.

Figure 2

Plot of $\alpha$ as a function of $x$. $\alpha$ has a minimum value ${\alpha }_{\min }=-2$ at the bounce and two maxima ${\alpha }_{\max }=\frac{9}{8}$ at $x=\pm \sqrt{\frac{5}{3}}$. For $q^2<\frac{9}{8}$, there are two regions where $q^2-\alpha <0$. But for $q^2>\frac{9}{8}$, $q^2-\alpha$ is always positive.

Figure 3

Time evolution of $\mathcal{V}$ with $x$ with different values of wave number $q$.

Figure 4

Time evolution of $\mathcal{X}$ with $x$ with different values of wave number $q$.

Figure 5

Spectral distribution of $\mathcal{X}$ and $\mathcal{V}$ at a fixed time $x=100$.

Figure 6

Evolution of perturbations near the turning point for different wave numbers.

Figure 7

Evolution of $Z$ very close to the turning point for $q=0.01$.