Chemical potential and the nature of dark energy: The case of a phantom field

Abstract

The influence of a possible nonzero chemical potential $\ensuremath{\mu}$ on the nature of dark energy is investigated by assuming that the dark energy is a relativistic perfect simple fluid obeying the equation of state, $p=\ensuremath{\omega}\ensuremath{\rho}$ ($\ensuremath{\omega}l0$, constant). The entropy condition, $S\ensuremath{\ge}0$, implies that the possible values of $\ensuremath{\omega}$ are heavily dependent on the magnitude, as well as on the sign of the chemical potential. For $\ensuremath{\mu}g0$, the $\ensuremath{\omega}$ parameter must be greater than $\ensuremath{-}1$ (vacuum is forbidden) while for $\ensuremath{\mu}l0$ not only the vacuum but even a phantomlike behavior ($\ensuremath{\omega}l\ensuremath{-}1$) is allowed. In any case, the ratio between the chemical potential and temperature remains constant, that is, $\ensuremath{\mu}/T={\ensuremath{\mu}}_{0}/{T}_{0}$. Assuming that the dark energy constituents have either a bosonic or fermionic nature, the general form of the spectrum is also proposed. For bosons $\ensuremath{\mu}$ is always negative and the extended Wien's law allows only a dark component with $\ensuremath{\omega}l\ensuremath{-}1/2$, which includes vacuum and the phantomlike cases. The same happens in the fermionic branch for $\ensuremath{\mu}l0$. However, fermionic particles with $\ensuremath{\mu}g0$ are permitted only if $\ensuremath{-}1l\ensuremath{\omega}l\ensuremath{-}1/2$. The thermodynamics and statistical arguments constrain the equation-of-state parameter to be $\ensuremath{\omega}l\ensuremath{-}1/2$, a result surprisingly close to the maximal value required to accelerate a Friedmann-Robertson-Walker--type universe dominated by matter and dark energy ($\ensuremath{\omega}\ensuremath{\lesssim}\ensuremath{-}10/21$).