Spontaneous creation of almost scale-free density perturbations in an inflationary universe

Abstract

The creation and evolution of energy-density perturbations are analyzed for the "new inflationary universe" scenario proposed by Linde, and Albrecht and Steinhardt. According to the scenario, the Universe underwent a strongly first-order phase transition and entered a "de Sitter phase" of exponential expansion during which all previously existing energy-density perturbations expanded to distance scales very large compared to the size of our observable Universe. The existence of an event horizon during the de Sitter phase gives rise to zero-point fluctuations in the scalar field $\ensuremath{\varphi}$, whose slowly growing expectation value signals the transition to the spontaneous-symmetry-breaking (SSB) phase of a grand unified theory (GUT). The fluctuations in $\ensuremath{\varphi}$ are created on small distance scales and expanded to large scales, eventually giving rise to an almost scale-free spectrum of adiabatic density perturbations (the so-called Zel'dovich spectrum). When a fluctuation reenters the horizon ($\mathrm{radius}\ensuremath{\simeq}{H}^{\ensuremath{-}1}$) during the Friedmann-Robertson-Walker (FRW) phase that follows the exponential expansion, it has a perturbation amplitude ${\frac{\ensuremath{\delta}\ensuremath{\rho}}{\ensuremath{\rho}}|}_{H}=(4 or \frac{2}{5})H\frac{\ensuremath{\Delta}\ensuremath{\varphi}}{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\varphi}}({t}_{1})}$, where $H$ is the Hubble constant during the de Sitter phase (${H}^{\ensuremath{-}1}$ is the radius of the event horizon), $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\varphi}}({t}_{1})$ is the mean value of $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\varphi}}$ at the time (${t}_{1}$) that the wavelength of the perturbation expanded beyond the Hubble radius during the de Sitter epoch, $\ensuremath{\Delta}\ensuremath{\varphi}$ is the fluctuation in $\ensuremath{\varphi}$ at time ${t}_{1}$ on the same scale, and $4(\frac{2}{5})$ applies if the Universe is radiation (matter) dominated when the scale in question reenters the horizon. Scales larger than about ${10}^{15}\ensuremath{-}{10}^{16}{M}_{\ensuremath{\bigodot}}$ reenter the horizon when the Universe is matter dominated. Owing to the Sachs-Wolfe effect, these density perturbations give rise to temperature fluctuations in the microwave background which, on all angular scales \ensuremath{\gg}1\ifmmode^\circ\else\textdegree\fi{}, are $\frac{\ensuremath{\delta}T}{T}\ensuremath{\simeq}(\frac{1}{5})H\frac{\ensuremath{\Delta}\ensuremath{\varphi}}{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\varphi}}({t}_{1})}$. The value of $\ensuremath{\Delta}\ensuremath{\varphi}$ expected from de Sitter fluctuations is $O(\frac{H}{2\ensuremath{\pi}})$. For the simplest model of "new inflation," that based on an SU(5) GUT with Coleman-Weinberg SSB, $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\varphi}}({t}_{1})\ensuremath{\ll}{H}^{2}$ so that $\frac{\ensuremath{\delta}T}{T}\ensuremath{\gg}1$---in obvious conflict with the large-scale isotropy of the microwave background. One remedy for this is a model in which the inflation occurs when $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\varphi}}({t}_{1})\ensuremath{\gg}{H}^{2}$. We analyze a supersymmetric model which has this feature, and show that a value of ${\frac{\ensuremath{\delta}\ensuremath{\rho}}{\ensuremath{\rho}}|}_{H}\ensuremath{\simeq}{10}^{\ensuremath{-}4}\ensuremath{-}{10}^{\ensuremath{-}3}$ on all observable scales is not implausible.