Graviton Production in Inflationary Cosmology

We provide a completely quantum-mechanical derivation of the spectrum of gravitational waves produced in any inflationary cosmology. The gravitational waves result from a sequence of Bogoliubov transformations between creation and annihilation operators defined in de Sitter space and in radiation-and matter-dominated Robertson-Walker spacetimes. We discuss how the results depend on the initial state at the beginning of the inflationary period. The production of gravitational waves in an inflationary cosmology provides an important constraint on any model of inflation [1-4]. The Hubble constant during the de Sitter phase must be less than 1015 GeV or else these gravitational waves will produce unacceptably large anisotropies in the microwave background radiation [1-4]. Similar constraints hold for power law or any other form of inflation [4]. The gravitational waves can also affect pulsar timing measurements [5]. The calculation of the spectrum of gravitational waves produced by inflation is similar to, although simpler than, the calculation for the scalar energy-density fluctuations [6] which give rise to large-scale structure in an inflationary universe. In both cases, the quantum mechanical two-point function is related to a two-point statistical average of an ensemble of classical fields. The evolution of the fields to the present time is then governed by the classical equations of motion. Since the use of such a mixed quantum mechanical-classical formalism raises some subtle issues [7], we present here an alternate derivation of the gravitational wave spectrum which is purely quantum mechanical. Our results are not new, agreeing with those of refs. [ 1-4]. However, we believe that our presentation will help clarify the physics behind these important results and remove any doubts about their validity. We also address the question of the generality of the results and of their dependence on the initial state of the Universe at the beginning of the inflationary epoch. In our approach, gravitational waves exist today because the de Sitter vacuum established during the inflationary period looks like a multiparticle state when the definition of particles relevant to the present matter-dominated universe is used. In our computation we will determine the Bogoliubov transformations relating creation