Power-law holographic dark energy and cosmology

We formulate power-law holographic dark energy, which is a modified holographic dark energy model based on the extended entropy relation arising from the consideration of state mixing between the ground and the excited ones in the calculation of the entanglement entropy. We construct two cases of the scenario, imposing the usual future event horizon choice, as well as the Hubble one. Thus, the former model is a one-parameter extension of standard holographic dark energy, recovering it in the limit where power-law extended entropy recovers Bekenstein-Hawking one, while the latter belongs to the class of running vacuum models, a feature that may reveal the connection between holography and the renormalization group running. For both models we extract the differential equation that determines the evolution of the dark-energy density parameter and we provide the expression for the corresponding equation-of-state parameter. We find that the scenario can describe the sequence of epochs in the Universe evolution, namely the domination of matter followed by the domination of dark energy. Moreover, the dark-energy equation of state presents a rich behavior, lying in the quintessence regime or passing into the phantom one too, depending on the values of the two model parameters, a behavior that is richer than the one of standard holographic dark energy.