Gravity’s rainbow: a bridge between LQC and DSR

The doubly special relativity (DSR) theories are constructed in order to take into account an observer-independent length scale in special relativity framework. It is widely believed that any quan- tum theory of gravity would reduce to a DSR model at the flat limit when purely gravitational and quantum mechanical effects are negligible. Gravity’s rainbow is a simple generalization of DSR the- ories to incorporate gravity. In this paper, we show that the effective Friedmann equations that are suggested by loop quantum cosmology (LQC) can be exactly reobtained in rainbow cosmology setup. The deformed geometry of LQC then fixes the modified dispersion relation and results in a unique DSR model. In comparison with standard LQC scenario where only the geometry is modified, both geometry and matter parts get modified in our setup. In this respect, we show that the total number of microstates for the universe is finite which suggests the statistical origin of the energy and entropy density bounds. These results explicitly show that the DSR theories are appropriate candidates for the flat limit of loop quantum gravity..