Vector and tensor perturbations in Horava-Lifshitz cosmology

We study cosmological vector and tensor perturbations in Horava-Lifshitz gravity, adopting the most general Sotiriou-Visser-Weinfurtner generalization without the detailed balance but with projectability condition. After deriving the general formulas in a flat Friedmann-Robertson-Walker (FRW) background, we find that the vector perturbations are identical to those given in general relativity. This is true also in the nonflat cases. For the tensor perturbations, high order derivatives of the curvatures produce effectively an anisotropic stress, which could have significant efforts on the high-frequency modes of gravitational waves, while for the low-frequency modes, the efforts are negligible. The power spectrum is scale-invariant in the UV regime, because of the particular dispersion relations. But, due to lower-order corrections, it will eventually reduce to that given in general relativity (GR) in the IR limit. Applying the general formulas to the de Sitter and power-law backgrounds, we calculate the power spectrum and index, using the uniform approximations, and obtain their analytical expressions in both cases.

Figure 1

The three different regions: (a) the subhorizon region ($k\eta \to -\infty$), denoted by Sub-H; (b) the intermediate region ($\eta \simeq \overline{\eta }$), denoted by $II$; and (c) the superhorizon region ($k\eta \to 0^{-}$), denoted by Super-H. For well-defined ${\omega }_T$, the function $g(\eta )$ is positive for $\eta >\overline{\eta }$, and negative for $\eta <\overline{\eta }$, where $\overline{\eta }$ is called the turning point, and given by the negative root of $g(\overline{\eta })=0$.

Figure 2

The three different cases for the function $g(k,\eta )$ defined by Eq. (5.22): (a) $g(k,\eta )$ has one turning point; (b) $g(k,\eta )$ has two turning points; and (c) $g(k,\eta )$ has three turning points.