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

In 2009 Ho\v{r}ava 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 $\omega R^{(3)}$, where $\omega$ is a constant parameter and $R^{(3)}$ is the spatial Ricci scalar. In this paper we derive constraints on this IR-modified Ho\v{r}ava theory using the late-time cosmic accelerating expansion observations. We obtain a lower bound of $|\omega|$ that is nontrivial and depends on $\Lambda_W$, the cosmological constant of the three dimensional spatial action in the Ho\v{r}ava gravity. We find that to preserve the detailed balance condition, one needs to fine-tune $\Lambda_W$ such that $- 2.29\times 10^{-4}< (c^2 \Lambda_W)/(H^2_0 \currentDE) - 2 < 0$, 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 $\Lambda_W$ is much relaxed to $-0.39< (c^2 \Lambda_W)/(H^2_0 \currentDE) - 2 < 0.12$. We find that although the detailed balance condition cannot be ruled out, it is strongly disfavored.