A spatially flat cosmological scalar field $\Phi$ model with the scalar field potential $\propto \exp (-\frac{\Phi }{\sqrt{p}})$, $p>\mathrm{}1\mathrm{}$, provides a simple class of inflationary cosmologies (which includes the usual exponential expansion inflation) that may be used as an analytical testing ground to help understand the predictions of the inflation model of the very early Universe. We divide the evolution of this model into three distinct epochs: scalar-field dominance and conventional radiation and baryon dominance; in each epoch we only account for irregularities in the dominant form of matter. We present closed-form solutions of the (synchronous gauge) relativistic linear perturbation equations that govern the evolution of inhomogeneities. These classical solutions, augmented with quantum-mechanically motivated initial conditions and joining conditions to match the expressions for the irregularities at the scalar-field-radiation and radiation-baryon transitions, are used to estimate the large-time form of the spectrum of energy-density irregularities, of the local departure velocity from homogeneous expansion, of large-scale fluctuations in the microwave background temperature, and of the gravitational-wave energy density. The inflation epoch results agree with those found from a purely quantum-mechanical analysis. Depending on the value of $p$ this model can have more large-scale power than the usual scale-invariant spectrum (at the expense of less small-scale power) and would seem to be marginally better at forming large-scale structure than the canonical model; however, the decrease in small-scale power serves to exacerbate the problem of late galaxy formation. As the model approaches the exponential expansion inflation limit, the power spectrum tends towards the scale-invariant form, although, in this limit the numerical prefactor diverges. We find that transverse peculiar velocity perturbations are not generated. Normalizing by fitting to the observed large-scale departure velocity, we find that models which stop inflating around ${10}^7$-${10}^{16}$ GeV are not obviously observationally inconsistent.

• Received 26 May 1989

DOI:https://doi.org/10.1103/PhysRevD.45.1913