Higher-derivative Quantum Cosmology

The quantum cosmology of a higher-derivative gravity theory arising from the heterotic string effective action is reviewed. A new type of Wheeler-DeWitt equation is obtained when the dilaton is coupled to the quadratic curvature terms. Techniques for solving the Wheeler-DeWitt equation with appropriate boundary conditions shall be described, and implications for semiclassical theories of inflationary cosmology will be outlined. Current version of the text of a talk given at ACGRG2, 9 July 1998 AMS Classification: 83-06, 83C45, 83E30, 83F05 1. Higher-Derivative Gravity Theories Since the renormalization of the Einstein-Hilbert action for general relativity introduces higher-order curvature terms, it has long been expected that the development of quantum cosmology would include higher-derivative gravity theories. Further support for this approach arises from the presence of quadratic and higher-order curvature terms in string effective actions, where modifications to the Ricci scalar are required for maintaining conformal invariance of the sigma model, describing string propogation in a given background, even after inclusion of quantum corrections to the β-function. Typically, one may consider actions of the form I = ∫ dx √ −g (R+ c1(gμν , φ, χγ, ...)(RμνρσR + αRμνR + β R) +O(R)) (1) and field-dependence of the coefficients c1, ... provides couplings between matter fields and the curvature. There are several types of higher-derivative gravity theories selected by physical considerations: (i) Lovelock gravity required for unitarity and elimination of ghosts Amongst the curvature invariants that arise in superstring effective actions are the dimensionally continued Lovelock invariants L(n) = (2n)! 2n R[i1i2 iiRi3i4 i3i4 ...Ri2n−1i2n] i2n−1i2n (2) which vanish in ten dimensions when n ≥ 6. As generalizations of the Gauss-Bonnet invariant, they are the only combinations of Riemann tensors that give rise to secondorder field equations for the metric. In a sigma-model with expansion parameter α, these curvature terms would occur at order O(α) in the effective action. The elimination of ghost fields in higher-order gravitational actions is achieved by eliminating terms of the type h∇h, n ≥ 2, where hμν is the perturbation about the flat-space metric, through metric redefinitions. The linearized equation for the graviton field is then second-order, possessing physically consistent stable classical solutions, in contrast to equations with third and higher-order derivatives of the metric, and at the quantum level, the ghost poles will cancel in the graviton propagator. The only non-zero Lovelock terms that could appear in ten-dimensional superstring effective actions are L(0), L(1), L(2), L(3), L(4), L(5) and functions of these tensors. While L(5) is a topological invariant in ten dimensions, the other three curvature combination L(2), L(3), L(4) contain dynamical higher-derivative terms.