Solutions to the complete time-dependent scalar wave equation in the mixmaster universe are presented. Differential equations for the mode amplitudes are given corresponding to field expansions in terms of the asymmetric- and the symmetric-top wave functions and for both minimal and conformal couplings of the scalar field to gravity. The theory of quantized fields in the mixmaster space is summarized. Creation and annihilation operators in the Heisenberg picture develop in time under a general Bogolubov transformation which expresses mathematically the physical processes of particle creation and mode mixing. The problem of particle creation is discussed here in the light of the classical theory of wave propagation. In the adiabatic limit when the expansion for the universe is slow, little production takes place. Higher-order expressions for the production amount are derived by the method of successive WKB approximations. Finally, numerical solutions of the wave equation for the uncoupled modes are presented. The effects of geometric shape, level energy, and particle mass on production are studied for the characteristic "small oscillation" and "bounce" solutions of the mixmaster universe. The anisotropic dynamics of the background gives rise to strong directional effects in particle creation. Production of particles of higher mass or energy is less abundant and less sensitive to the shape or level. The process of mode mixing, which is a distinct feature of the mixmaster universe, will be studied in a later paper.

• Received 9 November 1973

DOI:https://doi.org/10.1103/PhysRevD.9.3263