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Boltzmann or Bogoliubov? Approaches Compared in Gravitational Particle Production

Gravitational particle production is a minimal contribution to reheating the Universe after the end of inflation. To study this production channel, two different approaches have commonly been considered, one of which is based on the Boltzmann equation, and the other is based on the Bogoliubov transformation. Each of these has pros and cons in practice. The collision term in the Boltzmann equation can be computed based on quantum field theory in the Minkowski spacetime, and thus many techniques have been developed so far. On the other hand, the Bogoliubov approach may deal with the particle production beyond the perturbation theory and is able to take into account the effect of the curved spacetime, whereas in many cases one should rely on numerical methods, such as lattice computation. We show by explicit numerical and analytical computations of the purely gravitational production of a scalar that these two approaches give consistent results for particle production with large momenta during reheating, whereas the Boltzmann approach is not capable of computing particle production out of vacuum during inflation. We also provide analytic approximations of the spectrum of produced scalar with/without mass for the low momentum regime obtained from the Bogoliubov approach.