So we have to restrict our results to cases in which the effects of interest take place on scales $\gg l_{\rm P}$ and $\gg t_{\rm P}$. However, it seems that there is no reason to restrict the calculation to masses $M > m_{\rm P}$, as long as the mass shell starts to expand at a radius $\gg l_{\rm P}$. We set $G = c = \hbar = 1$ in the following.

This paper is organized as follows. First we will need to find an observable of particle density in curved space, which will be done in Section 2. After this we will introduce our model of the expanding mass shell and discuss its properties. The computation of the number density operator is presented in Section 4. The results are discussed in section 5, which contains plots of the spectral energy flux and total energy flux through a 2-dimensional surface. We conclude with a short summary.

Quantum fields in curved space-time
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Let us first recall some of the basics of quantum field theory in curved space-time [@BiDa; @Wald; @Fulling]. Physics in flat space-time is invariant under Poincaré transformations. This singles out a set of (Fourier) modes and a vacuum state, 