ON THE HORIZON HYPOTHESIS IN QUANTUM COSMOLOGY

Abstract

The mini-superspace approximation in quantum cosmology, whereby the space—time is restricted to the Friedmann form ds2=dt2−a2(t)dx2, requires the integrated three-space ∫d3x to be finite, in order that the operator replacements p→−iħ∂/∂q are well defined for the canonically conjugate momenta and coordinates (p, q). We have previously agued that this procedure can be made exact by cutting off the physical space at the causal horizon, so that ∫d3x=1 and a(t0)=(4π/3)1/3ξ−1(t0), where ξ≡d(ln a)/dt is the Hubble parameter and t0 is the present time, assuming dx2 to be flat. A corollary of this horizon hypothesis is that all quantum field theoretical integrals are similarly cut off. It is pointed out that the analysis by Jing and Fang of the two-year results for the COBE DMR observations of the quadrupole anisotropy of the cosmic microwave background radiation substantiates this idea (although the alternative explanation, that there are smaller density fluctuations at larger scales, is not ruled out). Also, the cosmological Schrödinger equation (Wheeler-DeWitt equation) for the wave function of the Universe Ψ, obtained from the heterotic superstring theory of Gross et al., is shown to be separable only when the physical space is isotropic and conformally flat, and in the approximation that quartic and higher-order terms in ξ are ignorable, and the solution is then expressed in terms of parabolic cylinder functions (Weber-Hermite functions). Finally, the occurrence of large density fluctuations via the indeterminacy relation ΔπξΔξ~ħ is further discussed.