3. Cosmic field defects

The physics and cosmology of topological defects produced at phase transitions in the early universe are reviewed by Vilenkin and Shellard (1994). An example of dark energy is a tangled web of cosmic string, with fixed mass per unit length, which self-intersects without reconnection. ⁽⁴⁸⁾ In Vilenkin's (1984) analysis ⁽⁴⁹⁾ the mean mass density in strings scales as _string (ta(t))^-1. When ordinary matter is the dominant contribution to ² / a², the ratio of mass densities is _string / t^1/3. Thus at late times the string mass dominates. In this limit, _string a^-2, w_X = - 1/3 for the XCDM parametrization of Eq. (45), and the universe expands as a t. Davis (1987) and Kamionkowski and Toumbas (1996) propose the same behavior for a texture model. One can also imagine domain walls fill space densely enough not to be dangerous. If the domain walls are fixed in comoving coordinates the domain wall energy density scales as _X a^-1 (Zel'dovich, Kobzarev, and Okun, 1974; Battye, Bucher, and Spergel, 1999). The corresponding equation of state parameter is w_X = - 2/3, which is thought to be easier to reconcile with the supernova measurements than w_X = - 1/3 (Garnavich et al., 1998; Perlmutter et al., 1999a). The cosmological tests of defects models for the dark energy have not been very thoroughly explored, at least in part because an accurate treatment of the behavior of the dark energy is difficult (as seen, for example, in Spergel and Pen, 1997), but this class of models is worth bearing in mind.

⁴⁸ It would be helpful to have in hand a particle physics model which realises this scenario. As far as we are aware, strings in particle physics models reconnect. Back.

⁴⁹ The string flops at speeds comparable to light, making the coherence length comparable to the expansion time t. Suppose a string randomly walks across a region of physical size a(t)R in N steps, where aR ~ N^1/2t. The total length of this string within the region R is l ~ Nt. Thus the mean mass density of the string scales with time as _string l / a³ (ta(t))^-1. One randomly walking string does not fill space, but we can imagine many randomly placed strings produce a nearly smooth mass distribution. Spergel and Pen (1997) compute the 3 K cosmic microwave background radiation anisotropy in a related model, where the string network is fixed in comoving coordinates so the mean mass density scales as _string a^-2. Back.