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In this paper we demonstrate how, using a natural generalization of BCStheory, superconducting phase coherence manifests itself in phase insensitivemeasurements, when there is a smooth evolution of the excitation gap \Deltafrom above to below Tc. In this context, we address the underdoped cuprates.Our premise is that just as Fermi liquid theory is failing above Tc, BCS theoryis failing below. The order parameter \Delta_{sc} is different from theexcitation gap \Delta. Equivalently there is a (pseudo)gap in the excitationspectrum above Tc which is also present in the underlying normal state of thesuperconducting phase, and can be directly inferred from specific heat andvortex core experiments. At the same time many features of BCS theory, e.g.,fermionic quasiparticles below Tc, are clearly present. These observations canbe reconciled by a natural extension of BCS theory, which includes finitecenter-of-mass momentum pair excitations, in addition to the usual fermionicquasiparticles. Applying this theory we find that the Bose condensation ofCooper pairs, which is reflected in \Delta_{sc}, leads to sharp peaks in thespectral function once $T \le T_c$. These are manifested in ARPES spectra aswell as in specific heat jumps, which become more like the behavior in a\lambda transition as the pseudogap develops. We end with a discussion oftunneling experiments and condensation energy issues. Comparison betweentheoretical and experimental plots of C_v, and of tunneling and vortex corespectroscopy measurements is good.Comment: 12 pages, 8 figures, ReVTeX 3.

Superconducting phase coherence in the presence of a pseudogap: Relation to specific heat, tunneling, and vortex core spectroscopies