Abstract
In Ho\v{r}ava-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, and rely simply on the deformed dispersion relations of the theory, and on equipartition of the various forms of energy at the starting point.
Highlights
In general relativity a homogeneous and isotropic Universe is described by the Friedmann equation 1⁄4 8πGρ −Λ; ð1Þ where H is the Hubble expansion rate; G is Newton’s constant; ρ is the energy density; K 1⁄4 0; 1; −1 is the curvature constant of a maximally symmetric 3-space; a is the scale factor; and Λ is the cosmological constant
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 have proposed a concrete framework for solving the flatness problem within HL gravity, the arguments presented are more general and may be valid on purely dimensional grounds for any UV complete theory with an anisotropic scaling of spacetime
Summary
Λ; ð1Þ where H is the Hubble expansion rate; G is Newton’s constant; ρ is the energy density; K 1⁄4 0; 1; −1 is the curvature constant of a maximally symmetric 3-space; a is the scale factor; and Λ is the cosmological constant. The ratio of the curvature term to 8πGρ grows but the initial value of the ratio at the end of inflation is so small that the Universe reaches the current epoch before the ratio becomes order unity This is how inflation solves the flatness problem. If the creation of the Universe is dominated by a small anisotropic instanton in the real time Universe after analytic continuation, the spatial curvature length scale will be much greater than the cosmological time scale. In this way the anisotropic instanton may solve the flatness problem without inflation.
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