An exactly solvable inflationary model

Abstract

We discuss a model of gravity coupled to a scalar field that admits exact cosmological solutions displaying an inflationary behavior at early times and a power-law expansion at late times. We study its general solutions and the effect of the inclusion of matter.

Appendix

For completeness, in this section we briefly discuss the results obtained when the potential (1.2) is used instead of (1.4). In this case the potential has a stable minimum for $\phi =0$ and goes to infinity for $\phi \rightarrow \infty$. The properties of the solutions are therefore completely different. However, the field Eqs. (2.10), (2.11) are still valid, with the substitution $\lambda ^2\rightarrow -\lambda ^2$, and their first integrals read

$$\begin{aligned} \dot{\psi }^2=-\lambda ^2e^{2\psi }+q_1^2,\qquad \dot{\chi }^2=-\lambda ^2e^{2\chi }+q_2^2, \end{aligned}$$

(5.1)

with positive integration constants satisfying $q^2_1=\beta ^2q^2_2$. Integrating (5.1), one obtains

$$\begin{aligned} e^{2a}={q^2\over \lambda ^2}\left( {\beta \cosh ^{\beta ^2}[q(\tau -\tau _2)]\over \cosh [\beta q(\tau -\tau _1)]}\right) ^{2/\gamma },\qquad e^{2\sqrt{3}\phi /\beta }=\left( {\,\cosh [\beta q(\tau -\tau _1)]\over \beta \cosh [q(\tau -\tau _2)]}\right) ^{2/\gamma }.\nonumber \\ \end{aligned}$$

(5.2)

Contrary to the solutions of Sect. 2, these solutions are regular everywhere, since the hyperbolic cosine has no zeroes.

They represent universes starting with a big bang at $t=0$ and recontracting after a finite time.