Noninteracting Cooper pairs inside a pseudogap

I present a simple analytical model describing the normal state of a superconductor with a pseudogap in the density of states, such as in underdoped cuprates. In nearly-two-dimensional systems, where the superconducting transition temperature is reduced from the mean-field BCS value, Cooper pairs may be present as slow fluctuations of the BCS pairing field. Using the self-consistent $T$-matrix (fluctuation exchange) approach I find that the fermion spectral weight exhibits two BCS-like peaks, broadened by fluctuations of the pairing-field amplitude. The density of states becomes suppressed near the Fermi energy, which allows for long-lived low-energy Cooper pairs that propagate as a soundlike mode with a mass. A self-consistency requirement, linking the width of the pseudogap to the intensity of the pairing field, determines the pair condensation temperature. In nearly-two-dimensional systems, it is proportional to the degeneracy temperature of the fermions, with a small prefactor that vanishes in two dimensions.