Density Perturbations from Two-field Inflation

We discuss metric perturbations produced during a period of inflation in the early universe where two scalar fields evolve. The final scalar perturbation spectrum can be calculated in terms of the perturbed expansion along neighbouring trajectories in field-space. In the usual single field case this is fixed by the values of the fields at horizon-crossing, but in the presence of more than one field there is no longer a unique slow-roll trajectory. The presence of entropy as well as adiabatic fluctuations means that the super-horizon-sized metric perturbation $\zeta$ may no longer be conserved and the evolution must be integrated along the whole of the subsequent trajectory. In general there is an inequality between the ratio of tensor to scalar perturbations and the tilt of the gravitational wave spectrum, which becomes an equality when only adiabatic perturbations are possible and $\zeta$ is conserved.