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.
S. Viaggiu (Fri,) studied this question.