Unstable Growth of Curvature Perturbations in Nonsingular Bouncing Cosmologies

Bouncing cosmologies require an ekpyrotic contracting phase ($w\gg 1$) in order to achieve flatness, homogeneity, and isotropy. Models with a nonsingular bounce further require a bouncing phase that violates the null energy condition ($w<-1$). We show that the transition from the ekpyrotic phase to the bouncing phase creates problems for cosmological perturbations. A component of the adiabatic curvature perturbations, though decaying and negligible during the ekpyrotic phase, is exponentially amplified just before $w$ approaches $-1$, enough to spoil the scale-invariant perturbation spectrum.

Figure 1

The kinetic term $P(X)$ versus $X\equiv \frac{1}{2}(\partial \varphi {)}^2$ for the ghost-condensate scalar field $\varphi$. As indicated by the dotted lines with arrows, during the ekpyrotic phase ($X>X_c$), the approximately canonical kinetic term $P(X)\approx X$ grows exponentially by a factor of $e^{2N_{\mathrm{tot}}}$; after the ekpyrotic phase, $X$ decreases by an even greater factor to reach $X=X_c$. This exponential decrease in $X$ is directly related to the problem with nonsingular bounces.

Figure 2

The effective potential $V(\varphi )$ in the ekpyrotic model with a nonsingular bounce. The evolution along the potential during the ekpyrotic and bounce phases is from right to left: the ekpyrotic phase refers to the exponential decline from $V_{\mathrm{ek}\mathrm{\text{−}}\mathrm{beg}}$ to $V_{\mathrm{ek}\mathrm{\text{−}}\mathrm{end}}$ near the minimum of the potential, then there is a brief kinetic energy dominated phase as $V$ quickly rises, and finally the nonsingular bouncing phase occurs as $V$ climbs sufficiently above zero.