Equation of state of cosmic strings and branes

I'm sure these are basic ideas covered in string cosmology or advanced GR, but I've done very little string theory, so I hope you will forgive some elementary questions. I'm just trying to fit some ideas together here. Following my answer to this question I began to wonder what sort of fluid would have a cosmological equation of state $w = -\frac{1}{3}$. Note the minus sign. I was interested because it seem like a critical case: the scale factor evolves as

$$a \propto t^{\frac{2}{3(1+w)}}$$

so for $w = -\frac{1}{3}$,

$$a \propto t$$

which is neither accelerating nor deccelerating. I was thinking that a fluid of cosmic strings fits the bill since the energy momentum tensor for a string directed along the 3-axis is (effective on scales $\gg$ than the transverse dimension of the string)

$$T^{\mu\nu}=T_{\text{str}}\left(\begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 \end{array}\right)\delta^{\left(2\right)}\left(x_{\perp}\right)$$

(c.f. Shifman Chapter 3) and $w$ is defined through the trace

$$T^\mu_\mu = \rho - 3 p = (1-3w) \rho = 2 \rho$$

the last equality using the explicit form of $T^{\mu\nu}$. This gives $w = -1/3$ as desired.

• Question 1: Does this continue to hold for a gas of non-interacting strings? I would expect so since the general case is described by the $T^{\mu\nu}$ above, suitably Lorentz transformed and convoluted with a distribution function. All the operations just mentioned being linear nothing should break, but is there a rigorous result?

• Question 2: Is the lack of acceleration/decceleration in any way related to the well known fact that cosmic strings do not gravitate? (There is a conical singularity on the string, yes, but no propagating curvature.) This is where I would really like some elaboration, because it seems intuitive. But I know that the gravitational effect of a slow rolling scalar is counter-intuitive so I don't want to jump to conclusions here.

Finally, the obvious extension of these ideas are to branes. Take 2-branes embedded in our ordinary 4D universe. The energy momentum tensor is

$$T^{\mu\nu}=T_{\text{w}}\left(\begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 0\\ 0 & 0 & 0 & -1 \end{array}\right)\delta^{\left(1\right)}\left(x_{\perp}\right)$$

for a brane (domain wall) oriented in the 2-3 plane (see Shifman again, chapter 2). This gives $w = -\frac{2}{3}$ and an expansion history $a \propto t^{2}$, an accelerating expansion. Is this connected to the fact that domain walls antigravitate?

EDIT: Partial answer: Kolb & Turner do the calculation I outlined above for a non-interacting gas of strings and domain walls. The result is a little more involved than I had envisioned. For strings:

$$w = \frac{2}{3} v^2 - \frac{1}{3}, (7.57)$$

where $v$ is the average velocity of the strings. For branes they find

$$w = v^2 - \frac{2}{3}, (7.45)$$

where again $v$ is the average velocity. The equation numbers refer to Kolb & Turner, The Early Universe, 1990 ed. So my speculations really only hold for the static case.

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