Semiclassical cosmology with backreaction: The Friedmann-Schrödinger equation and inflation

We propose and study a scalar field cosmological model with backreaction. Scalar field evolution in the model is determined by the time-dependent Schr\"odinger equation. The corresponding wave function self-consistently informs the evolution of the scale factor via the Friedmann equation. We compare this model with the usual semiclassical approximation. For various scalar potentials, we show that (i) inflation arises naturally, without fine-tuning, for arbitrary initial wave functions of the scalar field, (ii) zero-point quantum fluctuations of the scalar field dynamically produce the equation of state $P=\ensuremath{-}\ensuremath{\rho}$ at late times for all initial states and potentials, and (iii) the results of general relativity emerge as the Universe expands. Lastly, we give a general construction of backreaction models using the canonical formulation of general relativity coupled to matter.