Equation of state of cosmic strings and branes

Question

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?

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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.

Answer 1

1. No, the simple result $w=-1/3$ ceases to hold when the strings start to interact. The cosmic strings must really be "long" for the classical estimates to be OK and for other portions of the energy to be negligible. Interactions add new terms to the energy and strings may also be short, tearing apart etc., and short and compact strings behave like particles (dust), and so on.

2. No, it is a coincidence that in $d=4$, the acceleration of the expansion vanishes for $w=-1/3$. It has to be true for some value. The value would be different in a different dimension, I think. Moreover, $d=4$ isn't the right spacetime dimension in string theory. On the other hand, the absence of gravitational waves holds because the codimension is just 2. That's why the geometry is flat almost everywhere (deficit angle only). In full superstring theory, the same property (codimension 2) is shared by 7-branes.