Cosmological model from the holographic equipartition law with a modified R\'{e}nyi entropy

Cosmological equations were recently derived by Padmanabhan from the expansion of cosmic space due to the difference between the degrees of freedom on the surface and in the bulk in a region of space. In this study, a modified R\'{e}nyi entropy is applied to Padmanabhan's `holographic equipartition law', by regarding the Bekenstein--Hawking entropy as a nonextensive Tsallis entropy and using a logarithmic formula of the original R\'{e}nyi entropy. Consequently, the acceleration equation including an extra driving term (such as a time-varying cosmological term) can be derived in a homogeneous, isotropic, and spatially flat universe. When a specific condition is mathematically satisfied, the extra driving term is found to be constant-like as if it is a cosmological constant. Interestingly, the order of the constant-like term is naturally consistent with the order of the cosmological constant measured by observations, because the specific condition constrains the value of the constant-like term.Comment: Final version accepted for publication in EPJC. The titile is revised and references are added. [12 pages, 4 figures

General form of entropy on the horizon of the universe in entropic cosmology

Entropic cosmology assumes several forms of entropy on the horizon of the universe, where the entropy can be considered to behave as if it were related to the exchange (the transfer) of energy. To discuss this exchangeability, the consistency of the two continuity equations obtained from two different methods is examined, focusing on a homogeneous, isotropic, spatially flat, and matter-dominated universe. The first continuity equation is derived from the first law of thermodynamics, whereas the second equation is from the Friedmann and acceleration equations. To study the influence of forms of entropy on the consistency, a phenomenological entropic-force model is examined, using a general form of entropy proportional to the $n$-th power of the Hubble horizon. In this formulation, the Bekenstein entropy (an area entropy), the Tsallis--Cirto black-hole entropy (a volume entropy), and a quartic entropy are represented by $n=2$, $3$, and $4$, respectively. The two continuity equations for the present model are found to be consistent with each other, especially when $n=2$, i.e., the Bekenstein entropy. The exchange of energy between the bulk (the universe) and the boundary (the horizon of the universe) should be a viable scenario consistent with the holographic principle.Comment: Final version accepted for publication in PRD. Several pasragraphs and references are added and corrected. [10 pages

Cosmological model based on both holographic-like connection and Padmanabhan's holographic equipartition law

A cosmological model based on holographic scenarios is formulated in a flat Friedmann-Robertson-Walker universe. To formulate this model, the cosmological horizon is assumed to have a general entropy and a general temperature (including Bekenstein-Hawking entropy and Gibbons-Hawking temperature, respectively). In addition, a holographic-like connection [Eur. Phys. J. C 83, 690 (2023) (arXiv:2212.05822)] and Padmanabhan's holographic equipartition law are assumed for the entropy and temperature, and the Friedmann and acceleration equations are derived from these. The derived Friedmann and acceleration equations include both the entropy and the temperature and are slightly complicated, but can be combined into a single simple equation, corresponding to a similar equation that describes the background evolution of the universe in time-varying $\Lambda (t)$ cosmologies. The simple equation depends on the entropy but not on the temperature because the temperatures in the Friedmann and acceleration equations cancel each other. These results imply that the holographic-like connection should be consistent with Padmanabhan's holographic equipartition law through the present model and that the entropy plays a more important role. When the Gibbons-Hawking temperature is used as the temperature, the Friedmann and acceleration equations are found to be equivalent to those for a $\Lambda(t)$ model. A particular case of the present model is also examined, applying a power-law corrected entropy.Comment: 12 pages, 1 figur