Quasiparticle Interference in STM As a Function of Temperature: Signatures of Coherence

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In this paper we explore the behavior of the quasi-particle interference pattern (QPI) of scanning tunneling microscopy as a function of temperature, T. After insuring a minimal consistency with photoemission, we find that the QPI pattern is profoundly sensitive to quasi-particle coherence and that it manifests two energy gap scales. The nearly dispersionless QPI pattern above Tc is consistent with data on moderately underdoped cuprates. To illustrate the important two energy scale physics we present predictions of the QPI--inferred energy gaps as a function of T for future experiments on moderately underdoped cuprates.