6 Peculiar velocities from cosmic strings

 As an example we consider the probability distribution of peculiar velocities within the cosmic string model of structure formation. Cosmic strings are one-dimensional topological defects possibly formed in a phase transition in the early Universe (Brandenberger 1994; Hindmarsh & Kibble 1995; Vilenkin & Shellard 1994). After the time of formation, the string network quickly evolves to a scaling solution with a constant number n_s of strings passing through a Hubble volume in one expansion time.

Since individual topological defects give rise to velocity perturbations of the dark matter through which they move up to a distance of a Hubble radius, one expects a non-Gaussian result, in contrast to inflationary theories which predict a Gaussian PDF. Therefore departures from a normal distribution may be a way to distinguish between the two main classes of theories of structure formation, inflation and topological defects. However, since many strings present between the time of equal matter and radiation contribute to the perturbations, a nearly Gaussian PDF can result due to the central limit theorem.

In order to estimate the deviation of the PDF from the normal distribution we use Petrov's formula (43) for the Edgeworth series in this section. We calculate the cumulants up to 12th order in the analytical model for the cosmic string network presented in Moessner (1995), where the cumulants are given up to 8th order.

For simplicity, let us write Eq. (43) schematically as  

$$q(x)=Z(x) \left\{ 1+ \sum_{s=1}^\infty t_{s} \right\} \; ,$$ (44)

i.e. denote the sth term in the sum by t_s. Let us further denote the sum up to s=N by $\Sigma_N \equiv \sum_{s=1}^N t_{s}$. In Fig. 9 we show the Edgeworth expansion of the PDF of peculiar velocities up to 10th order, for the case of n_s=1 string per Hubble volume. Expanding up to Nth order, the relative deviation of the PDF from a normal distribution is given by,

$$\frac{q(x)}{Z(x)}-1 = \Sigma_N \; .$$ (45)

It is only significant if the error of the asymptotic expansion, which is of the order of the last term included, is smaller than this deviation. In Fig. 10 we show the relative deviation $\Sigma_N$ and the error t_N associated with it. The wiggles or cusps in the graphs appear at zeros of $\Sigma_N$ and t_N which are both oscillating, changing their sign, and we have plotted the logarithms of the absolute values. So only the maxima of the curves give a true indication of the deviations and errors. For N=10 the error is clearly below the deviation and thus the latter is significant. For N=4, the error is still below the deviation for most values of x, but it is not as clear, especially since the error is not exactly equal to t_N but of that order only. In Fig. 11 we show the relative error $t_N/(1+\Sigma_N)$ in the expansion of q(x)/Z(x). It is smaller for an expansion to higher order for a given number of strings per Hubble volume. The error decreases more strongly with N for larger n_s. The relative error is also smaller, at fixed N, for a larger n_s, in which case the PDF is closer to a normal distribution.

It is interesting to compare these results with the general theory. Petrov (1972) proves that under certain conditions (which are fulfilled in most physically important cases)  

$$q(x) = Z(x)\left\{ 1+ \sum_{s=1}^N t_{s} \right\} +o(\sigma^N) \;$$ (46)

uniformly for $-\infty < x < +\infty$, when $\sigma \sim 1/\nu^{1/2}$and the PDF q(x) is for a sum of $\nu$ random variables. One should remember that each t_s is of the order of $\sigma^s$ in (43). In our case $\nu$ is just n_s, so the error of the truncated Edgeworth series scales as $\sim1/n_s^{N/2}$. From Fig. 11 we can see that the error for the case of n_s=10 strings is indeed N/2 orders less than for the case of n_s=1 string, i.e. 2 orders for N=4 and 5 orders for N=10 terms in the expansion. This is an illustration of the theory developed by Petrov (1972, 1987).