Caustics in Tachyon Matter and Other Born-Infeld Scalars

TL;DR

This paper analyzes scalar Born-Infeld theories with arbitrary potentials, demonstrating caustic formation in inhomogeneous solutions, and discusses implications for string theory tachyons and cosmological models involving dark matter.

Contribution

It provides an analytic description of caustic formation in Born-Infeld scalar models with runaway potentials, linking wave front propagation to inhomogeneous solution behavior.

Findings

Caustics form in solutions with runaway potentials.

Numerical solutions confirm caustic development.

Implications for string tachyon dynamics and cosmology.

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.