Chaotic universe in the $z=2$ Hořava-Lifshitz gravity

The deformed $z=2$ Hořava-Lifshitz gravity with coupling constant $\omega$ leads to a nonrelativistic mixmaster cosmological model. The potential of theory is given by the sum of IR and UV potentials in the ADM Hamiltonian formalism. It turns out that adding the UV potential cannot suppress chaotic behaviors existing in the IR potential.

Figure 1

3D Potential shapes $V({\beta }_{+},{\beta }_{-},\omega )$ with three canyon lines located at ${\beta }_{-}=0$ and ${\beta }_{-}=\pm \sqrt{3}{\beta }_{+}$: (a) for $\omega =100$, it corresponds to the GR mixmaster model ($V_{\mathrm{IR}}$ dominates). (b) For $\omega =0.01$, it corresponds to the potential of $z=2$ Hořava-Lifshitz gravity.

Figure 2

Potential graphs $V({\beta }_{+},{\beta }_{-},\omega )$ with ${\beta }_{-}=0$: the dashed curve is for $\omega =100$ (GR) and the solid curve for $\omega =0.01$ with a local maximum $V=V_{\mathrm{lm}}$.

Figure 3

Equipotential curves, developing triangles with corner: (a) for $\omega =100$ with $E=-6.5$, $-6.0$, $-5.0$, $-4.0$, respectively, (b) for $\omega =0.01$ with $E=-15.0$, $-13.0$, $-7.0$, $-4.0$, respectively. Three canyon lines (dashed lines) are developed along corners at ${\beta }_{-}=0$ and ${\beta }_{-}=\pm \sqrt{3}{\beta }_{+}$.

Figure 4

Poincaré sections for the $\omega =100$ case with (a) $E=-6.5$, (b) $E=-6.0$, (c) $E=-5.0$, and (d) $E=-4.0$.

Figure 5

Poincaré sections for the $\omega =0.01$ case with (a) $E=-15.0$, (b) $E=-13.0$, (c) $E=-7.0$, and (d) $E=-4.0$.