Solving the flatness problem with an anisotropic instanton in Hořava-Lifshitz gravity

In Hořava-Lifshitz gravity a scaling isotropic in space but anisotropic in spacetime, often called “anisotropic scaling,” with the dynamical critical exponent $z=3$, lies at the base of its renormalizability. This scaling also leads to a novel mechanism of generating scale-invariant cosmological perturbations, solving the horizon problem without inflation. In this paper we propose a possible solution to the flatness problem, in which we assume that the initial condition of the Universe is set by a small instanton respecting the same scaling. We argue that the mechanism may be more general than the concrete model presented here. We rely simply on the deformed dispersion relations of the theory, and on equipartition of the various forms of energy at the starting point.

Figure 1

Loglog plots of $a$ vs. $\tau /{\tau }_{\mathrm{in}}$ in solid blue with the analytic solution (26) superimposed in dashed red. We have $\lambda =2$, ${\alpha }_3=1$, ${\alpha }_2=0$ for both plots; however on the left we have $C=6$ while on the right $C=50$. This confirms the validity of the analytic solution in the large $C$ limit. The figures were obtained by solving Eq. (17) numerically from $\tau ={10}^{-4}$ using the small $\tau$ expansion until ${\partial }_{\tau }a{|}_{\tau ={\tau }_{\mathrm{in}}}\approx {10}^{-22}$.

Figure 2

The plot shows $a_{\mathrm{in}}^3/{\tau }_{\mathrm{in}}$ as a function of $C$ and confirms the expected analytic scaling behavior in the large $C$ limit shown in dashed red. To obtain the plot, we kept $\lambda =2$, ${\alpha }_3=1$, ${\alpha }_2=0$ and integrated the Euclidean equation of motion from $\tau =0$ to $\tau ={\tau }_{\mathrm{in}}$ for various values of the integration constant $C$.