Quantum cosmology: Exact solution for the Gowdy $T^3$ model

A complete set of orthonormal wave functions valid at all times including that of the classical singularity is obtained in closed form for each mode of the quantized Gowdy $T^3$ universe. These wave functions are superposed to yield that wave function for each mode which reduces to a given initial state near the classical singularity. The expected number of quanta in each mode at times far from the singularity is obtained and depends only on the constants which characterize the initial state. All expectation values agree with those obtained classically and semiclassically except for a smearing out of the essentially classical picture due to quantum fluctuations. The precise description of the initial state at the singularity yields a model with the size and shape parameter of the universe satisfying an initial free-particle-like equation and later captured in the $N$-quantum state of a rising harmonic-oscillator potential. This contrasts with the graviton creation from vacuum fluctuations description of an earlier treatment by Berger. Misner has shown that the Gowdy $T^3$ model universe may be described as a scattering process in minisuperspace. He obtains a Klein-Gordon equation for the wave function of the universe which is separable in Fourier components of the wave part of the gravitational field. It is shown here that exact solutions exist for the Klein-Gordon equation which reduces for each mode to the Schrödinger equation for a time-dependent-frequency harmonic oscillator. The methods of Salusti and Zirilli are used to obtain wave functions characterized by the quantum number $N$ with harmonic-oscillator spatial (in superspace) dependence and time-dependent coefficients. These $N$-quantum wave functions are fixed uniquely by requiring agreement with the known large-time-limit wave functions. Initial states are constructed for each mode near the time of the classical singularity which are wave packets characterized by an initial position and an initial momentum. These states form an overcomplete family of states. Their expectation values follow the classical equations of motion.