Implications of Minimum and Maximum Length Scales in Cosmology

We investigate the cosmological implications of the generalized and extended uncertainty principle (GEUP), and whether it could provide an explanation for the dark energy. The consequence of the GEUP is the existence of a minimum and a maximum length, which can in turn modify the entropy area law and also modify the Friedmann equation. The cosmological consequences are studied by paying particular attention to the role of these lengths. We find that the theory allows a cosmological evolution where the radiation- and matter-dominated epochs are followed by a long period of virtually constant dark energy, that closely mimics the $\Lambda$CDM model. The main cause of the current acceleration arises from the maximum length scale $\beta$, governed by the relation $\Lambda\sim -\beta^{-1}W(-\beta^{-1})$. Using recent observational data (the Hubble parameters, type Ia supernovae, and baryon acoustic oscillations, together with the Planck or WMAP 9-year data of the cosmic microwave background radiation), we estimate constraints to the minimum length scale $\alpha \lesssim 10^{81}$ and the maximum length scale $\beta \sim -10^{-2}$.