Inflation, gravitational waves and the cosmic microwave background: reconciling CDM with COBE?

We examine the production of tensor (gravitational wave) perturbations in a universe where inflation is driven by a scalar field interacting via an exponential potential. In such a scenario, the Universe undergoes a power-law rather than exponential expansion, and density perturbations have a power-law but non-scale-invariant spectrum. We show that models which lead to only slight departures from scale-invariant density perturbations also produce significant gravitational wave perturbations that lead to significant anisotropics in the cosmic microwave background. For a spectral index n < 0.83, more than half the quadrupole anisotropy detected by COBE could be due to tensor, rather than scalar, perturbations. This result has profound implications for the cold dark matter model of galaxy and large-scale structure formation. If the standard version of CDM, in which the scalar perturbations are scale-invariant and tensor perturbation production is negligible, is normalized to produce the correct level of CMB anisotropy, it fails to account for the small-scale velocities and clustering properties of galaxies. On the other hand, if non-scale-invariant fluctuations are generated by the inflationary mechanism we suggest, a CDM model can simultaneously account for these properties and the COBE quadrupole anisotropy. The model may still have problems with large-scale clustering and bulk streaming motions, but observations on such scales are less well established and interpretation is made difficult by any scale-dependent bias.