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We consider time dependent backgrounds with a time-like linear dilaton whichleads to a weakly coupled theory at asymptotic future known as theSupercritical Linear Dilaton (SCLD) phase. Even after projecting out thetachyon there are naive divergences in the partition function due to thespectrum of relevant deformations. Although the partition function seemsdivergent, the background is actually well defined in the far future due tovanishing backreactions, provided we start from an appropriate initial statecalled the "capping" state. We use the technique of interpolating functions toconstruct this state by embedding the SCLD phase in a strong couplingcompletion of string theory. Although we are unable construct a specificexample, we do provide the general strategy in cases where S-duality mayprovide the completion.

Supercritical linear dilaton capping state using Interpolating functions