An entropic-force scenario, i.e., entropic cosmology, assumes that the horizon of the universe has an entropy and a temperature. In the present study, in order to examine entropic cosmology, we derive entropic-force terms not only from the Bekenstein entropy but also from a generalized blackhole entropy proposed by C. Tsallis and L.J.L. Cirto [Eur. Phys. J. C 73, 2487(2013]. Unlike the Bekenstein entropy, which is proportional to area, the generalized entropy is proportional to volume because of appropriate nonadditive generalizations. The entropic-force term derived from the generalized entropy is found to behave as if it were an extra driving term for bulk viscous cosmology, in which a bulk viscosity of cosmological fluids is assumed. Using an effective description similar to bulk viscous cosmology, we formulate the modified Friedmann, acceleration, and continuity equations for entropic cosmology. Based on this formulation, we propose two entropic-force models derived from the Bekenstein and generalized entropies. In order to examine the properties of the two models, we consider a homogeneous, isotropic, and spatially flat universe, focusing on a single-fluid-dominated universe. The two entropic-force models agree well with the observed supernova data. Interestingly, the entropic-force model derived from the generalized entropy predicts a decelerating and accelerating universe, as for a fine-tuned standard ΛCDM (lambda cold dark matter) model, whereas the entropic-force model derived from the Bekenstein entropy predicts a uniformly accelerating universe.

We study two types of entropic-force models in a homogeneous, isotropic, spatially flat, matterdominated universe. The first type is a ΛðtÞ type similar to ΛðtÞCDM (varying-lambda cold dark matter) models in which both the Friedmann equation and the acceleration equation include an extra driving term. The second type is a BV type similar to bulk viscous models in which the acceleration equation includes an extra driving term but the Friedmann equation does not. In order to examine the two types systematically, we consider an extended entropic-force model that includes a Hubble parameter (H) term and a constant term in entropic-force terms. The H term is derived from a volume entropy, whereas the constant term is derived from an entropy proportional to the square of an area entropy. Based on the extended entropic-force model, we examine four models obtained from combining the H and constant terms with the ΛðtÞ and BV types. The four models agree well with the observed supernova data and describe the background evolution of the late universe properly. However, the evolution of first-order density perturbations is different in each of the four models, especially for low redshift, assuming that an effective sound speed is negligible. The ΛðtÞ type is found to be consistent with the observed growth rate of clustering, in contrast with the BV type examined in this study. A unified formulation is proposed as well, in order to systematically examine density perturbations of the two types.

In 'entropic cosmology', instead of a cosmological constant Λ, an extra driving term is added to the Friedmann equation and the acceleration equation, taking into account the entropy and the temperature on the horizon of the universe. By means of the modified Friedmann and acceleration equations, we examine a non-adiabatic-like accelerated expansion of the universe in entropic cosmology. In this study, we consider a homogeneous, isotropic, and spatially flat universe, focusing on the single-fluid (single-component) dominated universe at late-times. To examine the properties of the late universe, we solve the modified Friedmann and acceleration equations, neglecting high-order corrections for the early universe. We derive the continuity (conservation) equation from the first law of thermodynamics, assuming non-adiabatic expansion caused by the entropy and temperature on the horizon. Using the continuity equation, we formulate the generalized Friedmann and acceleration equations, and propose a simple model. Through the luminosity distance, it is demonstrated that the simple model agrees well with both the observed accelerated expansion of the universe and a fine-tuned standard ΛCDM (lambda cold dark matter) model. However, we find that the increase of the entropy for the simple model is likely uniform, while the increase of the entropy for the standard ΛCDM model tends to be gradually slow especially after the present time. In other words, the simple model predicts that the present time is not a special time, unlike for the prediction of the standard ΛCDM model.

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 Rényi 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 Rényi 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.

Thermodynamics on the horizon of a flat universe at late times is studied in holographic cosmological models that assume an associated entropy on the horizon. In such models, a Λ(t) model similar to a time-varying Λ(t) cosmology is favored because of the consistency of energy flows across the horizon. Based on this consistency, a Λ(t) model with a power-law term proportional to H α is formulated to systematically examine the evolution of the Bekenstein-Hawking entropy. Here, H is the Hubble parameter and α is a free parameter whose value is a real number. The present model always satisfies the second law of thermodynamics on the horizon. In particular, the universe for α < 2 tends to approach thermodynamic equilibrium-like states. Consequently, when α < 2, the maximization of the entropy should be satisfied as well, at least in the last stage of the evolution of an expanding universe. A relaxation-like process before the last stage is also examined from a thermodynamics viewpoint. 95.30.Tg, 98.80.Es

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 nth 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 ¼ 2, 3, and 4, respectively. The two continuity equations for the present model are found to be consistent with each other, especially when n ¼ 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.

The bulk viscosity of cosmological fluid and the creation of cold dark matter both result in the generation of irreversible entropy (related to dissipative processes) in a homogeneous and isotropic universe. To consider such effects, the general cosmological equations are reformulated, focusing on a spatially flat matter-dominated universe. A phenomenological entropic-force model is examined that includes constant terms as a function of the dissipation rate ranging fromμ ¼ 0, corresponding to a nondissipative ΛCDM (lambda cold dark matter) model, toμ ¼ 1, corresponding to a fully dissipative CCDM (creation of cold dark matter) model. A time-evolution equation is derived for the matter density contrast in order to characterize density perturbations in the present entropic-force model. It is found that the dissipation rate affects the density perturbations even if the background evolution of the late universe is equivalent to that of a fine-tuned pure ΛCDM model. With increasing dissipation rateμ, the calculated growth rate for the clustering gradually deviates from observations, especially at low redshifts. However, the growth rate for lowμ (less than 0.1) is found to agree well with measurements. A low-dissipation model predicts a smaller growth rate than does the pure ΛCDM model (for whichμ ¼ 0). More detailed data are needed to distinguish the low-dissipation model from the pure ΛCDM one.

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