Spectral energy distribution and generalized Wien's law for photons, cosmic string loops and related physical objects

Abstract

Physical objects with energy $u_w(l) \sim l^{-3w}$ with $l$ a characteristic length and $w$ a numerical constant ($-1 \leq w \leq 1$), lead to an equation of state $p=w\rho$, with $p$ the pressure and $\rho$ the energy density. Special objects with this property are, for instance, photons ($u = hc/l$, with $l$ the wavelength) with $w = 1/3$, and some models of cosmic string loops ($u = (c^4/aG)l$, with $l$ the length of the loop and $a$ a numerical constant), with $w = -1/3$, and maybe other kinds of objects as, for instance, hypothetical cosmic membranes with lateral size $l$ and energy proportional to the area, i.e. to $l^2$, for which $w = -2/3$, or the yet unknown constituents of dark energy, with $w = -1$. Here, we discuss the general features of the spectral energy distribution of these systems and the corresponding generalization of Wien's law, which has the form $Tl_{mp}^{3w}=constant$, being $l_{mp}$ the most probable size of the mentioned objects.