Energy Fluctuations Generated by Inflation
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The energy density correlation function $C(x,y)=<ρ(x)ρ(y)>-<ρ>^2$ and its Fourier transform generated by the gravitational tidal forces of the inflationary de Sitter expansion are derived for a massless or light ($m<<H$) neutral scalar field $Φ$ without self--interactions, minimally coupled to gravity. The field has no classical background component, $<Φ>=0$. Every observationally relevant mode (which has today $λ_{phys}<H_0^{-1}$) had at early times $R/k_{phys}^2\to0$, and is taken to be initially in the Minkowski vacuum state. Our computation of $C(x,y)$, which involves four field operators, is finite and unambiguous at each step, since we use the following two tools: (1) We use a normal ordered energy density operator N$[ρ]$ and show that any normal ordering N gives the same finite result. (2) Since $C(x,y)$ has the universal $|x-y|^{-8}$ short--distance behaviour, the Fourier transform can only be performed after smearing the energy density operator in space and time with a smearing scale $τ$. The resulting energy density fluctuations are non--Gaussian, but obey a $χ^2$--distribution. The power spectrum $\dk$ involves a smearing scale $τ$, and we choose $τ=k^{-1}$. For massless scalars we obtain $k^3\dk\sim H^4k^4$ on super--horizon scales $k<H$. For light scalars with masses $m\ll H$ a plateau appears on the largest scales $k<\sqrt{m^3/H}$, the result is $k^3\dk\sim m^6H^2(k/H)^{4m^2/3H^2}$. On sub--horizon scales $k>H$ we have the universal law $k^3\dk\sim k^8$.
14 pages, no figures, AMS TeX, ETH--TH/93--15
14 pages, no figures, AMS TeX, ETH--TH/93--15