Classically Stable Nonsingular Cosmological Bounces

One of the fundamental questions of theoretical cosmology is whether the Universe can undergo a nonsingular bounce, i.e., smoothly transit from a period of contraction to a period of expansion through violation of the null energy condition (NEC) at energies well below the Planck scale and at finite values of the scale factor such that the entire evolution remains classical. A common claim has been that a nonsingular bounce either leads to ghost or gradient instabilities or a cosmological singularity. In this Letter, we consider a well-motivated class of theories based on the cubic Galileon action and present a procedure for explicitly constructing examples of a nonsingular cosmological bounce without encountering any pathologies and maintaining a subluminal sound speed for comoving curvature modes throughout the NEC violating phase. We also discuss the relation between our procedure and earlier work.

Figure 1

A plot of the sound speed $c_S^2$ (solid blue curve) for comoving curvature perturbations as a function of time $t$. The time coordinate is given in Planck units and the value of $c_S^2$ is given in units where the speed of light is unity. Superimposed for illustration purposes are the shapes of the background solutions for $H(t)$ (dashed green curve; also shown in Figs. 2 and 3) and $\dot{H}(t)$ (dotted red curve). More specifically, the results correspond to $H(t)=H_{0}t{e}^{-F(t-t^{*}{)}^2}$ and $\gamma (t)={\gamma }_0{e}^{3\mathrm{\Theta }t}+H(t)$ with the parameter values $H_0=3\times {10}^{-5}$, $t^{*}=0.5$, $F=9\times {10}^{-5}$, ${\gamma }_0=-0.0044$, $\mathrm{\Theta }=0.0046$. Notably, throughout, the sound speed is real [$A(t)$, $B(t)>0$] and subluminal, with $0<c_S^2<1$. The characteristic energy scale $\sim H^2$ is well below the Planck scale, and the NEC violating phase lasts $\sim 150$ Planck times; it starts when $\dot{H}$ becomes positive at $t_{\mathrm{beg}}\simeq -74M_{\mathrm{Pl}}^{-1}$ and ends when $\dot{H}$ becomes negative at $t_{\mathrm{end}}\simeq 75M_{\mathrm{Pl}}^{-1}$; the bounce [$H(t)=0$] occurs at $t=0$. Note that the bounce stage occurs well within the classical regime.

Figure 2

A plot of the dimensionless couplings $k$ (solid blue curve), $q$ (dotted orange curve), and $b$ (dot-dashed red line) as a function of $\varphi$ for the example given in Fig. 1. The $x$ axis has Planck units and the $y$ axis has dimensionless units. As indicated by the labeling, we have rescaled the quadratic and quartic couplings for the purpose of illustration.

Figure 3

A plot of the sound speed $c_S^2$ (solid blue curve) for comoving curvature modes as a function of time $t$ during the bounce stage corresponding to the background given by $H(t)=H_{0}t{e}^{-F(t-t^{*}{)}^2}$ (dashed green curve; rescaled to show shape) and $\gamma (t)={\gamma }_0{e}^{3\mathrm{\Theta }t}+H(t)$ (dotted red curve) for the parameter values $H_0=3\times {10}^{-5}$, $t_{*}=0.5$, $F=7\times {10}^{-5}$, ${\gamma }_0=-0.0044$, and $\mathrm{\Theta }=4.6\times {10}^{-6}$. After the bounce and shortly before the end of NEC violation, $\gamma (t)$ goes from $-\infty$ to $+\infty$ while all fundamental physical quantities including $H(t)$ and $c_S^2$ remain finite and positive.