Caustics in Tachyon Matter and Other Born-Infeld Scalars

TL;DR

This work shows that scalar Born-Infeld theories with runaway potentials rapidly relax to a Hamilton-Jacobi free-streaming regime, leading to caustic formation and multi-valued field configurations beyond those caustics. By solving the free-streaming equation via characteristics, the authors derive explicit mappings from initial data to the evolving tachyon field and reveal the universal appearance of caustics across 1+1 and 3+1 dimensions, both in flat space and in expanding backgrounds. The cosmological analysis demonstrates that a rolling tachyon with sufficiently decreasing V(T) can behave as pressureless dark matter, but non-linear inhomogeneities inevitably generate caustics, challenging the interpretation of T in a single-valued effective field theory and highlighting the need to incorporate higher-derivative or interacting physics beyond caustics. The results bear on the use of tachyon BI models in cosmology and string-theory tachyon dynamics, clarifying when such effective theories remain valid and when they break down due to caustic formation.

Abstract

We consider scalar Born-Infeld type theories with arbitrary potentials V(T) of a scalar field T. We find that for models with runaway potentials V(T) the generic inhomogeneous solutions after a short transient stage can be very well approximated by the solutions of a Hamilton-Jacobi equation that describes free streaming wave front propagation. The analytic solution for this wave propagation shows the formation of caustics with multi-valued regions beyond them. We verified that these caustics appear in numerical solutions of the original scalar BI non-linear equations. Our results include the scalar BI model with an exponential potential, which was recently proposed as an effective action for the string theory tachyon in the approximation where high-order spacetime derivatives of T are truncated. Since the actual string tachyon dynamics contain derivatives of all orders, the tachyon BI model with an exponential potential becomes inadequate when the caustics develop because high order spatial derivatives of T become divergent. BI type tachyon theory with a potential decreasing at large T could have interesting cosmological applications because the tachyon field rolling towards its ground state at infinity acts as pressureless dark matter. We find that inhomogeneous cosmological tachyon fluctuations rapidly grow and develop multiple caustics. Any considerations of the role of the tachyon field in cosmology will have to involve finding a way to predict the behavior of the field at and beyond these caustics.