Evolution of the universe in entropic cosmologies via different formulations

We study two types of entropic-force models in a homogeneous, isotropic, spatially flat, matter-dominated universe. The first type is a $\Lambda(t)$ type' similar to $\Lambda(t)$CDM (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 whereas 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 $\mathrm{\Lambda }(t)$ 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 $\mathrm{\Lambda }(t)$ 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 examine density perturbations of the two types systematically.