Energy Fluctuations Generated by Inflation

The energy density correlation function $C(x,y)=<\rho (x)\rho (y)>-<\rho {>}^2$ 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 $\mathrm{\Phi }$ without self–interactions, minimally coupled to gravity. The field has no classical background component, $<\mathrm{\Phi }>=0$. Every observationally relevant mode (which has today ${\lambda }_{phys}<H_0^{-1}$) had at early times $R\mathrm{/}{k}_{phys}^2\to 0$, 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$[\rho ]$ and show that any normal ordering N gives the same finite result. (2) Since $C(x,y)$ has the universal $\mathrm{\mid }x-y{\mathrm{\mid }}^{-8}$ short–distance behaviour, the Fourier transform can only be performed after smearing the energy density operator in space and time with a smearing scale $\tau$. The resulting energy density fluctuations are non–Gaussian, but obey a ${\chi }^2$–distribution. The power spectrum $\text{dk}$ involves a smearing scale $\tau$, and we choose $\tau =k^{-1}$. For massless scalars we obtain $k^3\text{dk}\sim H^4{k}^4$ on super–horizon scales $k<H$. For light scalars with masses $m\ll H$ a plateau appears on the largest scales $k<\sqrt{m^3\mathrm{/}H}$, the result is $k^3\text{dk}\sim m^6{H}^2(k\mathrm{/}H{)}^{4m^2\mathrm{/}3H^2}$. On sub–horizon scales $k>H$ we have the universal law $k^3\text{dk}\sim k^8$.