Energy Density Fluctuations in Inflationary Cosmology

Abstract

We analyze the energy density fluctuations contributed by scalar fields $Φ$ with vanishing expectation values, $\langleΦ\rangle=0$, which are present in addition to the inflaton field. For simplicity we take $Φ$ to be non--interacting and minimally coupled to gravity. We use normal ordering to define the renormalized energy density operator $ρ$, and we show that any normal ordering gives the same result for correlation functions of $ρ$. We first consider massless fields and derive the energy fluctuations in a single mode $\vk$, the two--point correlation function of the energy density, the power spectrum, and the variance of the smeared energy density, $\ddR$. Mass effects are investigated for energy fluctuations in single modes. All quantities considered are scale invariant at the second horizon crossing (Harrison--Zel'dovich type) for massless and for unstable massive fields. The magnitude of the relative fluctuations $\deρ/\rt$ is of order $(\Hi/\Mp)^2$ in the massless case, where $\Hi$ is the Hubble constant during inflation. For an unstable field of mass $m_Φ\ll\Hi$ with a decay rate $Γ_Φ$ the magnitude is enhanced by a factor $\sqrt{m_Φ/Γ_Φ}$. Finally, the prediction for the cosmic variance of the average energy density in a sample is given in the massless case.