The logotropic dark fluid as an energy diffusion in unimodular gravity

Abstract

We show that the logotropic dark fluid, proposed as a single-fluid unification of dark matter and dark energy, is exactly dissipative unimodular gravity with pressureless matter and the logarithmic energy diffusion function $Q(a)=3A\ln a+Q_{c}$. The correspondence is exact at the level of the homogeneous background, rather than asymptotic, and introduces no parameters beyond those already present in either description. It provides a dynamical reinterpretation of the logotropic temperature $A$, whose origin was left open in the original proposal: up to the expansion rate, $A$ is the rate at which energy is diffused from matter into the geometric sector, $\dot Q=3AH$. The adiabatic sound speed of the effective fluid is $c_{s}^{2}=A/ρ$, where $ρ$ is the unimodular matter energy density, and reaches the speed of light at $a_{s}=a_{M}/2^{1/3}$; the matter energy density then vanishes and changes sign at $a_{M}$, which is precisely the scale factor at which the model is known to become phantom. Since $ρ$ coincides with the enthalpy density $ε+P$ of the logotropic fluid, the phantom branch is the branch on which the matter sector carries negative energy density and on which the apparent horizon entropy decreases. Finally, we show that a one-parameter saturating diffusion law removes these late-time pathologies while preserving the low-redshift phenomenology.