G-Bounce

TL;DR

The paper demonstrates that stable bouncing cosmologies arise naturally in General Relativity when a noncanonical scalar with Kinetic Gravity Braiding (Galileon-like) self-couplings is included. By deriving general background equations and explicit stability criteria (e.g., $D>0$ for ghost freedom and $c_s^2>0$ for gradient stability), the authors show that healthy bounces are generic within this framework, both with external matter and in the negligible-matter limit. A concrete hot G-bounce model yields a stable bounce for a finite range of the coupling parameter $\beta$, followed by either a hot Big-Bang-like radiation era or a possible inflationary stage if a flat potential is added; however, most trajectories encounter gradient instabilities or singularities either before or after the bounce. A separate Conformal Galileon example confirms the existence of stable bounces in two-dimensional phase space and highlights the sensitivity of global history to stability constraints and strong-coupling effects near phase-space boundaries. Overall, the work provides a versatile framework for constructing NEC-violating yet stable bounces in GR and clarifies the limitations and future directions for achieving a fully well-behaved, complete cosmological evolution.

Abstract

We present a wide class of models which realise a bounce in a spatially flat Friedmann universe in standard General Relativity. The key ingredient of the theories we consider is a noncanonical, minimally coupled scalar field belonging to the class of theories with Kinetic Gravity Braiding / Galileon-like self-couplings. In these models, the universe smoothly evolves from contraction to expansion, suffering neither from ghosts nor gradient instabilities around the turning point. The end-point of the evolution can be a standard radiation-domination era or an inflationary phase. We formulate necessary restrictions for Lagrangians needed to obtain a healthy bounce and illustrate our results with phase portraits for simple systems including the recently proposed Galilean Genesis scenario.

Figures (4)

• Figure 1: Plot of the phase space for the hot G-bounce model\ref{['hot:lagr']} in the presence of external radiation fluid ($w=1/3$), $\mu=\sqrt{\alpha}\dot\phi$, $h=\sqrt{\alpha}H$, $\beta=\varkappa/\alpha=1.5$. The solution is fully under control only in the blue region: pink regions are dynamically inaccessible, while the white regions suffer from instabilities (e.g. negative sound speed squared, $c_\text{s}^2 <0$, or ghosts, $D<0$.) In red, we have marked an example trajectory: it enters the healthy region from one where $c_\text{s}^2<0$, bounces, reaches a maximum Hubble parameter and then proceeds toward the origin. The equation of state of the scalar approaches $w_X=1$, so eventually it redshifts away, leaving only the accompanying radiation fluid. In blue, we have marked a nearby trajectory: here the Hubble parameter does not turn around, but the trajectory proceeds to a pressure singularity ($D=0$) where $\dot{H}$ diverges (but both the scale factor and Hubble parameter remain finite). Only a narrow range of bouncing trajectories arrive safely in the late-time radiation-domination era; the rest end in singularities.
• Figure 2: A selection of phase-space portraits for shift-symmetric G-bounce models with external matter. The classical evolution presented here is only under control in the blue regions: the pink colouring represents dynamically inaccessible regions and the white regions have unstable perturbations (either negative sound speed squared or ghosts). Top Panel:Hot G-bounce model with the $X^2$ term restored: $K=X+X^2/2 -X^3$, $\kappa=2\beta X$, $\beta=1.5$ in the presence of an accompanying radiation fluid. Restoring the $X^2$ term does not substantially change the phase space presented in figure \ref{['f:hot']}, provided its coefficient is somewhat smaller than one. Bottom Panel:Hot G-bounce model with an additional positive cosmological constant: $K=-\Lambda+X-X^3$, $\kappa = 2\beta X$, $\Lambda=0.05$, $\beta=1.5$, accompanying radiation fluid. Despite the addition of $\Lambda$, bouncing trajectories still exist, but now proceed to a late-time de Sitter attractor, providing an inflationary stage following the bounce.
• Figure 3: A selection of phase-space portraits for shift-symmetric G-bounce models with external matter. Color coding as in figure \ref{['shiftyfigs']}. Top Panel:G-bounce with destabilised Minkowski: $K=-X-X^3$, $\kappa=2\beta X$, $\beta=1.7$ in the presence of stiff matter, $w=1$. In this type of models, the trajectories generically begin in a pressure singularity and then cross $c_\text{s}^2=0$ (red) after bouncing, or vice-versa (blue). These models contain de Sitter attractors/repellers similar to those described in Ref. Deffayet:2010qz and trajectories end or begin there. Bottom Panel: bounce & recollapse model: $K= -\Lambda + X - X^3$, $\kappa=2\beta X + \gamma X^2$, negative $\Lambda = -0.05$, $\beta = 1.7$, $\gamma=-0.04$ with radiation, $w=1/3$. The red trajectory begins in a singularity at large $H$. The expansion slows down until the universe begins to recollapse. The crunching is prevented, however, and the universe bounces. The trajectory then crosses $c_\text{s}^2=0$ eventually evolving to an unstable de-Sitter attractor. The blue trajectory is the time reverse: first a bounce, then a recollapse.
• Figure 4: Phase portrait for the conformal Galileon model, Eq. \ref{['cg:action']} in rescaled coordinates $h=(f/\Lambda)^{3/2}H$, $\mu=(f/\Lambda)^{3/2}\dot \phi$. The main figure shows the stable region where $\mu>0$. The inset depicts the time-reversed region with $\mu<0$. The solution is under control fully only in the light blue regions: pink corresponds to dynamically inaccessible regions, white---to negative sound speed squared. Yellow and orange are regions where curvature is transplanckian for $(f/\Lambda)^3=1,2$, respectively. The blue line is a typical healthy bouncing trajectory (presented in Qiu:2011cy): it originates from a region where the theory is strongly coupled, but the background solution evolves as a collapsing radiation-dominated cosmology; the universe then bounces in a healthy region and then the trajectory very rapidly crosses into the region where $c_{\text{s}}^{2}<0$ and the classical solution should not be trusted. The red trajectory is the Galilean Genesis trajectory Creminelli:2010ba: it begins in the vicinity of the Minkowski origin; the universe is always expanding and eventually the trajectory crosses the line $c_{\text{s}}=0$ around $h=0.6$; depending on the choice of parameters this happens either before or after the curvatures become transplanckian. Both the trajectories merge to an attractor which evolves toward a Big Rip singularity. In the inset in green, we have marked a trajectory time-reversed with respect to the blue discussed above: this one begins in a (collapsing) Big Rip singularity, at some point crosses into a region of positive sound speed squared, bounces and then proceeds to expand in a radiation-domination-like phase which is also strongly coupled.