Designing density fluctuation spectra in inflation

Abstract

Scale-invariant (flat) fluctuation spectra are the most natural outcomes of inflation. Nonetheless current large-scale-structure observations seem to indicate more fluctuation power on large scales than flat spectra give. We consider a wide variety of models based on the chaotic inflation paradigm and sketch the effects that varying the expansion rate, structure of the potential surface, and the curvature coupling constants have on the quantum fluctuation spectra. We calculate in detail the quantum generation of fluctuation spectra by numerically solving the linearized perturbation equations for multiple scalar fields, the metric, and the radiation into which the scalars dissipate, following the evolution from inside the horizon through reheating.We conclude that (1) useful extended nonflat power laws are very difficult to realize in inflation, (2) double inflation leading to a mountain leveling off at a high-amplitude plateau at long wavelengths is generic, but to tune the cliff rising up to the plateau to lie in an interesting wavelength range, a special choice of initial conditions and/or scalar field potentials is required, and (3) small mountains (moguls) on the potential surface lead to mountains of extra power in the fluctuations added on top of an underlying flat spectrum. For quadratic and quartic couplings, the mountain fluctuations may obey Gaussian statistics but the spectral form will be very sensitive to initial conditions as well as potential parameters; non-Gaussian mountain fluctuations which depend upon potential parameters but not on initial field conditions will be the more likely outcome. However, adding cubic couplings can give mountains obeying Gaussian statistics independently of initial conditions.Since observations only probe a narrow patch of the potential surface, it is possible that it is littered with moguls, leading to arbitrarily complex ``mountain range'' spectra that can only be determined phenomenologically. We also construct an inflation model which houses the chaotic inflation picture within the grand unified theory (GUT) framework. The standard chaotic picture requires an unnaturally flat scalar field potential, \ensuremath{\lambda}\ensuremath{\approxeq}5\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}14}$, and a strong curvature coupling parameter bound, \ensuremath{\xi}<0.002. By allowing the Higgs field to be strongly coupled to gravity through a large negative curvature coupling strength, \ensuremath{\xi}\ensuremath{\sim}${\mathrm{\ensuremath{-}}10}^{4}$, so the Planck mass depends on the GUT Higgs field, the Higgs field can be strongly coupled to matter fields [with \ensuremath{\lambda}\ensuremath{\sim}(\ensuremath{\xi}${/10}^{5}$${)}^{2}$]. This leads to both a flat Zeldovich spectrum of the ``observed'' amplitude and a high reheating temperature (\ensuremath{\sim}${10}^{15}$ GeV), unlike the \ensuremath{\lambda}\ensuremath{\sim}${10}^{\mathrm{\ensuremath{-}}13}$ standard case. The large -\ensuremath{\xi} would be related to the ratio of the Planck scale to a typical GUT scale. Although a single dynamically important Higgs multiplet gives flat spectra, a richer Higgs sector could lead to broken scale invariance.