On Stability and Isotropization of Kasner Solution in $R^{2}$ Gravity

TL;DR

This work analyzes the stability of Kasner-like anisotropic expansion in vacuum $R^{2}$ gravity within Kantowski–Sachs geometry. By formulating a four-dimensional dynamical system in anisotropic variables and performing numerical evolutions of near-Kasner initial data, the authors uncover a bifurcation line that separates collapsing and isotropizing outcomes, with isotropization potentially accompanied by a Starobinsky-like slow-roll inflation. The findings show that forward-in-time Kasner expansion is generically unstable; sufficiently strong perturbations lead to isotropization and a finite inflationary period, while weaker perturbations yield a smooth dust-like expansion. These results imply that higher-curvature corrections can drive isotropization and, under suitable conditions, realize inflation in a vacuum cosmology, clarifying the role of initial anisotropy in early-universe dynamics.

Abstract

The fate of the Universe that initially expands anisotropically in the theory with $R^{2}$ quantum-gravitational term in the Lagrangian is investigated. The stability of Kasner-like expansion, specifically in the class of Kantowski-Sachs spacetimes, is analyzed. Kasner solutions are found to be unstable, with the bifurcation line between the initial conditions that lead to collapsing universes and the ones that set the universe for continuing expansion that becomes isotropic, established analytically. Under suitable conditions, the isotropized spacetime enters the intermediate slow-rolling inflationary stage similar to Starobinsky inflation.

Figures (4)

• Figure 1: The behaviour of the universe is determined by the values of $t_{0}$ and $p$ that are chosen. We found two clear situations, an isotropizing expanding universe and an anisotropically collapsing universe. The solid blue line indicates the bifurcation line between the regimes, where any point above that line would lead to a collapsing universe. Any point below that line would lead to an isotropizing universe, except along the line $p=0$ which gives the exact Kasner solution. The solid circles denote models that straddle the bifurcation line in our numerical tests. The red circles are just above it, and the green are just below. The dashed lines indicates the more quantitative assessment of isotropization roughly specifying the lines along each the amount of inflation is the same. There is no discernible stage of exponential behaviour above $p=-\frac{1}{4} t^2$ blue dashed line. To the left of the orange dotted line, the amount of inflation exceeds 65-efolds, the nominal value necessary for successful cosmological models. Overplotted are markers which indicate the initial conditions of other figures in this paper.
• Figure 2: A set of solutions with initial conditions that straddle the bifurcation line $p=\frac{1}{4} t_0^2$ corresponding to closed dots in Fig. \ref{['fig:BifurcationLine']}. Above the bifurcation line (top row) the space experiences fast accelerated anisotropic collapse, with reversal of the initial volume expansion to contraction. Below the bifurcation line (bottom row) initial anisotropic Kasner-like expansion transfers to isotropic one, following the "dust-like" behaviour $a,b \propto t^{2/3}$ with superimposed decaying oscillations. The dashed lines show $t^{2/3}$ asymptotics.
• Figure 3: A solution with the intermediate stage of near-exponential inflationary expansion. The left panel shows the scale factors $a(t)$ and $b(t)$ which approach isotropy $a(t) \propto b(t)$ at $t > 2$. The straight dashed lines corresponds to $\propto t^{2/3}$ "dust"-like asymptotitcs at $t \to \infty$. The rising dashed line is slow-roll Starobinsky inflationary regime of $R^2$ gravity $a(t) \sim e^{H_0 t - \frac{1}{72} m^2 t^2}$ that ends in this model at $m t \approx 30$. The right panel shows the behaviour of the Hubble parameter and its derivative, $H(t)=\frac{1}{3}y(t)$, $\dot{H}(t) = \frac{1}{3} z(t)$. Inflationary stage is clearly seen, corresponding to $\dot{H}(t) \approx -\frac{1}{36} m^2 = const$ and $H(t) = const -\frac{1}{36} m^2 t$, the behaviours shown by dashed lines. On subsequent "dust-like" stage at $m t > 30$, $H$ and $\dot{H}$ oscillate, passing through $(0,0)$ point at every oscillation KMP1987
• Figure 4: Development of the inflationary stage for the parameters below $p < -\frac{1}{4} t_0^2$ line. Left panel shows the example of a model starting on $p=-t_0^2$ line (grey dashed line). These models show already a discernible, but very short inflationary stage, giving only 3-4 efold expansion. The right panel shows example of the models that achieve cosmologically significant benchmark expansion of 65 e-folds. Such models lie, approximately, along the $p \approx -20 t_0^2$ line shown in Fig. \ref{['fig:BifurcationLine']}.