Energy Density Fluctuations in Inflationary Cosmology

We analyze the energy density fluctuations contributed by scalar fields Φ with vanishing expectation values, 〈Φ〉 = 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 ~k, the two– point correlation function of the energy density, the power spectrum, and the variance of the smeared energy density, 〈δ2 R〉. 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 δρ/ρtot is of order (HI/MPl) 2 in the massless case, where HI is the Hubble constant during inflation. For an unstable field of mass mΦ ≪ HI with a decay rate ΓΦ the magnitude is enhanced by a factor √ mΦ/ΓΦ. Finally, the prediction for the cosmic variance of the average energy density in a sample is given in the massless case. PACS numbers: 04.62.+v, 98.80.Cq, 98.65.Dx