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In 2009 Ho\v{r}ava proposed a power-counting renormalizable quantum gravitytheory. Afterwards a term in the action that softly violates the detailedbalance condition has been considered with the attempt of obtaining a morerealistic theory in its IR-limit. This term is proportional to $\omegaR^{(3)}$, where $\omega$ is a constant parameter and $R^{(3)}$ is the spatialRicci scalar. In this paper we derive constraints on this IR-modifiedHo\v{r}ava theory using the late-time cosmic accelerating expansionobservations. We obtain a lower bound of $|\omega|$ that is nontrivial anddepends on $\Lambda_W$, the cosmological constant of the three dimensionalspatial action in the Ho\v{r}ava gravity. We find that to preserve the detailedbalance condition, one needs to fine-tune $\Lambda_W$ such that $- 2.29\times10^{-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 thepresent epoch, respectively. On the other hand, if we do not insist on thedetailed balance condition, then the valid region for $\Lambda_W$ is muchrelaxed to $-0.39< (c^2 \Lambda_W)/(H^2_0 \currentDE) - 2 < 0.12$. We find thatalthough the detailed balance condition cannot be ruled out, it is stronglydisfavored.

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