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What kind of "matter" is described by the following thermodynamic relation ? \begin{equation}\tag{1} p = -\, \frac{1}{3} \; \rho, \end{equation} Where $p$ and $\rho$ are the pressure and density respectively. The sign is important here. This isn't "radiation" ($w = +\, \frac{1}{3}$). I'm pretty sure this isn't "phantom matter" (negative kinetic term in the Lagrangian). And this $w = -\, \frac{1}{3}$ is larger than what we get for the "vacuum" ; $w = -\, 1$ (cosmological constant).

Of course, I know this is some kind of "exotic matter", but what is its usual name ? What else could we say about it from the relation (1) above ?

Mass
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Cham
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In "Adventures in Friedmann cosmology: A detailed expansion of the cosmological Friedmann equations" by Robert J. Nemiroff and Bijunath Patla in the American Journal of Physics volume 76, on page 265 (2008); http://dx.doi.org/10.1119/1.2830536 the authors call them "cosmic strings"

But this is in the context of cosmology, so its for a universe that on very large scales has energy that has an extent in just one direction (for the other directions it is quite small in comparison). And it is not made up of regular matter in that configuration, it has be immune to certain kinds of dilution that regular matter would not be immune to.

The authors cover the other cases too (at least within some kind of polynomial ansatz). But if $w=-1/3$ is indeed the case you are interested in then you can consider A. Vilenkin, “Cosmic strings,” Phys. Rev. D 24, 2082–2089 (1981).

Timaeus
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It is probably worth to point out that, in the context of the standard cosmological model where the Friedmann-Robertson-Walker metric is assumed as a background, the constant equation of state parameter $w=-1/3$ essentially gives the same contribution to the dynamics as a negative spatial curvature. It is possible to show this easily for FRW metrics, but I don't know if the result is more general.

  • let's consider first a flat FRW background metric and a fluid with barotropic equation of state $p=-1/3\, \rho$. Now implement such equation of state into the conservation of energy-momentum and get \begin{equation*} \dot{\rho} +2\, \frac{\dot{a}}{a}\, \rho = 0 \end{equation*} This straightforwardly gives an energy density of the form $\rho=\rho_0\, a^{-2}$ for some positive constant $\rho_0\equiv\rho(a=1)$. The Friedmann equations are then given by \begin{align}\label{flat} 3\left( \frac{\dot{a}}{a} \right)^2 &= 8\pi G\, \frac{\rho_0}{a^2}\\ 2 \frac{\ddot{a}}{a} + \left( \frac{\dot{a}}{a} \right)^2 &= \frac{8\pi G}{3}\, \frac{\rho_0}{a^2}\, . \end{align}
  • Now consider instead the vacuum Friedmann equations in a negatively curved FRW background ($k<0$): \begin{align}\label{curv} 3\left( \frac{\dot{a}}{a} \right)^2 - 3\, \frac{|k|}{a^2} &= 0\\ 2 \frac{\ddot{a}}{a} + \left( \frac{\dot{a}}{a} \right)^2 - \frac{|k|}{a^2} &= 0\, . \end{align} Note now that the curvature terms have the same power-law dependence $a^{-2}$ as the fluid in the previous example. If we bring these curvature terms on the right hand side -- that is, we pretend it to be a fluid source instead of a geometric spatial curvature -- we get a clear correspondence with the previous example, where now the curvature fluid has \begin{align*} \rho &= 3\, \frac{|k|}{8\pi G\, a^2}\\ p &= - \frac{|k|}{8\pi G\, a^2} \end{align*} Hence this fictitious fluid corresponding to negative spatial curvature has equation of state $p=-1/3\, \rho$.
Giovanni Acquaviva
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