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Constraining the Detailed Balance Condition in Horava Gravity with Cosmic Accelerating Expansion

In 2009 Hořava proposed a power-counting renormalizable quantum gravity theory. Afterwards a term in the action that softly violates the detailed balance condition has been considered with the attempt of obtaining a more realistic theory in its IR-limit. This term is proportional to $ωR^{(3)}$, where $ω$ is a constant parameter and $R^{(3)}$ is the spatial Ricci scalar. In this paper we derive constraints on this IR-modified Hořava theory using the late-time cosmic accelerating expansion observations. We obtain a lower bound of $|ω|$ that is nontrivial and depends on $Λ_W$, the cosmological constant of the three dimensional spatial action in the Hořava gravity. We find that to preserve the detailed balance condition, one needs to fine-tune $Λ_W$ such that $- 2.29\times 10^{-4}< (c^2 Λ_W)/(H^2_0 \currentDE) - 2 < 0 $, where $H_0$ and $\currentDE$ are the Hubble parameter and dark energy density fraction in the present epoch, respectively. On the other hand, if we do not insist on the detailed balance condition, then the valid region for $Λ_W$ is much relaxed to $-0.39< (c^2 Λ_W)/(H^2_0 \currentDE) - 2 < 0.12$. We find that although the detailed balance condition cannot be ruled out, it is strongly disfavored.