Modified cosmology from quantum deformed entropy

In Ref. [S. Jalalzadeh, Phys. Lett. B 829 (2022) 137058], Jalalzadeh established that the thermodynamical entropy of a quantum-deformed black hole with horizon area $A$ can be written as $S_q=\pi \sin \left(\frac{A}{8G\mathcal{N}}\right)\mathrm{/}\sin \left(\frac{\pi }{2\mathcal{N}}\right)$, where $\mathcal{N}=L_q^2\mathrm{/}{L}_{\text{P}}^2$, $L_{\text{P}}$ being the Planck length and $L_q$ denoting, generically, the q-deformed cosmic event horizon distance $L_q$. Motivated by this, we now extend the framework constructed in [S. Jalalzadeh, Phys. Lett. B 829 (2022) 137058] towards the Friedmann and Raychaudhuri equations describing spatially homogeneous and isotropic universe dynamics. Our procedure in this paper involves a twofold assumption. On the one hand, we take the entropy associated with the apparent horizon of the Robertson-Walker universe in the form of the aforementioned expression. On the other hand, we assume that the unified first law of thermodynamics, $dE=TdS+WdV$, holds on the apparent horizon. Subsequently, we find a novel modified cosmological scenario characterized by quantum-deformed (q-deformed) Friedmann and Raychaudhuri equations containing additional components that generate an effective dark energy sector. Our results indicate an effective dark energy component, which can explain the Universe’s late-time acceleration. Moreover, the Universe follows the standard thermal history, with a transition redshift from deceleration to acceleration at $z_{\text{tran}}=0.5$. More precisely, according to our model, at a redshift of $z=0.377$, the effective dark energy dominates with a de Sitter universe in the long run. We include the evolution of luminosity distance, $\mu$, the Hubble parameter, $H(z)$, and the deceleration parameter, $q(z)$, versus redshift. Finally, we have conducted a comparative analysis of our proposed model with others involving non-extensive entropies.