Gauge theories with time dependent couplings and their cosmological duals

We consider the N=4 super Yang-Mills theory in flat 3+1-dimensional space-time with a time dependent coupling constant which vanishes at t=0, like gYM2=tp. In an analogous quantum mechanics toy model we find that the response is singular. The energy diverges at t=0, for a generic state. In addition, if p>1 the phase of the wave function has a wildly oscillating behavior, which does not allow it to be continued past t=0. A similar effect would make the gauge theory singular as well, though nontrivial effects of renormalization could tame this singularity and allow a smooth continuation beyond t=0. The gravity dual in some cases is known to be a time dependent cosmology which exhibits a spacelike singularity at t=0. Our results, if applicable in the gauge theory for the case of the vanishing coupling, imply that the singularity is a genuine sickness and does not admit a meaningful continuation. When the coupling remains nonzero and becomes small at t=0, the curvature in the bulk becomes of order string scale. The gauge theory now admits a time evolution beyond this point. In this case, a finite amount of energy is produced which possibly thermalizes and leads to a black hole in the bulk.