Musings

Kibble Deficit

Topological defects are dangerous cosmologically. One of the original motivations for inflation was to solve the “monopole problem” – that monopoles created in the GUT phase transition would overclose the universe. The basic rule of thumb in the subject is the Kibble Mechanism — that, on average, you get one defect per horizon volume at the time of their formation. This is, obviously, a lower bound on the density of defects: the order-parameter cannot be correlated over length scales larger than the causal horizon. But, in almost all cases, one finds that the correlation volume is of the same order of magnitude as the horizon volume.

String cosmology provides a rich set of topological defects as well. A $D3$-$\overline{D3}$ pair rolling towards each other (a popular model for brane inflation) can annihilate in to (D- or F- or $(p,q)$-) strings. Non-BPS branes may be temporarily stabilized by finite temperature and decay into lower-dimensional branes when the universe cools below the transition temperature.

One of the striking things about these brane effects is that — in a systematic approximation — they are governed by a DBI action, which is nonpolynomial in derivatives. This can lead to effects dramatically different from naïve field theory expectations. As Alishahiha, Silverstein and Tong noted, the higher-derivative terms in the DBI action can produce slow-roll inflation, even for potentials that are relatively steep. The upper limit on $\dot \phi$ is easy to understand as a causal limit on the velocities of the branes in the extra dimensions.

But, apparently, the same DBI action, which prevents $\dot \phi$ from becoming arbitrarily large, can undermine the Kibble Mechanism. According to Barnaby, Berndsen, Cline and Stoica, because of effects of the DBI action, the correlation volume can be much smaller than the horizon volume. So one gets a density of cosmic strings (or other topological defects) much larger than one per horizon volume.

If true, this could be really important. It would provide really stringent bounds on whole classes of string models, lest they overproduce topological defects in the early universe.

Superficially, I see some plausibility in what they’re saying. First, there’s the aforementioned limit on $\dot\phi$, which prevents the order parameter from relaxing as efficiently as it would in ordinary field theory. Second, there’s the fact that the action of interest takes the form

where $V(\phi)$ vanishes at the minimum. This favours large fluctuations of the order parameter as you reach the endpoint of brane-antibrane annihilation. (Personally, I think this just signals the breakdown of the DBI approximation, due to the importance of higher string modes.) Anyway, between these two effects, they claim a much higher density of cosmic string formation (1 per $M_s^{-1}$, rather than one per $H^{-1}$).

Unfortunately, I don’t really understand many of the details of their calculation. Maybe someone can chime in with some explanations…