Building regular cosmologies with an attractor quasi-de Sitter phase from quantum modified Friedmann equations

In this paper we obtain possible cosmologies without big bang singularity. We firstly get effective modifications of Friedmann equations by using a texture of the first law of thermodynamics with our proposed generalization of the Bekenstein-Hawking entropy. We get modified Friedmann equations leading to a universe without big bang singularity and with maximum allowed values for the Hubble flow, i.e. [Formula: see text], and for the energy density, i.e. [Formula: see text]. A general study of solutions of the aforementioned equations shows that an emergent universe filled with non-phantom matter arises starting with [Formula: see text] and with an exponential form for the scale factor [Formula: see text] corrected by a decreasing exponential together with regular expressions for [Formula: see text] everywhere. A further feature of these models is that the time derivative of [Formula: see text] ([Formula: see text]) when [Formula: see text] is reached is diverging, also by considering further quantum corrections. Finally, we show that a universe with finite [Formula: see text] can be easily obtained, for example, with a smooth transition from a phantom universe to a non-phantom one. In this way we obtain a regular universe with a quasi-de Sitter phase emerging naturally after Planckian times [Formula: see text] without fine tuning.