Caustic Formation in Tachyon Effective Field Theories

Certain configurations of D-branes, for example wrong dimensional branes orthe brane-antibrane system, are unstable to decay. This instability isdescribed by the appearance of a tachyonic mode in the spectrum of open stringsending on the brane(s). The decay of these unstable systems is described by therolling of the tachyon field from the unstable maximum to the minimum of itspotential. We analytically study the dynamics of the inhomogeneous tachyonfield as it rolls towards the true vacuum of the theory in the context ofseveral different tachyon effective actions. We find that the vacuum dynamicsof these theories is remarkably similar and in particular we show that in allcases the tachyon field forms caustics where second and higher derivatives ofthe field blow up. The formation of caustics signals a pathology in theevolution since each of the effective actions considered is not reliable in thevicinity of a caustic. We speculate that the formation of caustics is anartifact of truncating the tachyon action, which should contain all orders ofderivatives acting on the field, to a finite number of derivatives. Finally, weconsider inhomogeneous solutions in p-adic string theory, a toy model of thebosonic tachyon which contains derivatives of all orders acting on the field.For a large class of initial conditions we conclusively show that the evolutionis well behaved in this case. It is unclear if these caustics are a genuineprediction of string theory or not.Comment: 23 pages, 5 figures; accepted for publication in JHEP. Revised derivation of eikonal equation for the DBI action. Added comments concerning the relationship between p-adic string theory and tachyon matter. Added second example of inhomogeneous evolution in p-adic string theory. Misleading statements concerning caustic-free evolution removed, references adde