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Inspired from the idea of minimally coupling of a real scalar field togeometry, we investigate the classical and quantum models of a flatenergy-dependent FRW cosmology coupled to a perfect fluid in the framework ofthe scalar-rainbow metric gravity. We use the standard Schutz' representationfor the perfect fluid and show that under a particular energy-dependent gaugefixing, it may lead to the identification of a time parameter for thecorresponding dynamical system. It is shown that, under some circumstances onthe minisuperspace prob energy, the classical evolution of the of the universerepresents a late time expansion coming from a bounce instead of the big-bangsingularity. Then we go forward by showing that this formalism gives rise to aSchr\"{o}dinger-Wheeler-DeWitt (SWD) equation for the quantum-mechanicaldescription of the model under consideration, the eigenfunctions of which canbe used to construct the wave function of the universe. We use the resultingwave function in order to investigate the possibility of the avoidance ofclassical singularities due to quantum effects by means of the many-worlds andBohmian interpretation of quantum cosmology.

Classical and quantum dynamics of a perfect fluid scalar-energy dependent metric cosmology