General solutions for a flat Friedmann universe filled with a perfect fluid and a scalar field with an exponential potential (Q1886182)

The paper under review studies the problem of integrability by quadratures of a spatially homogeneous and isotropic Friedman model of the universe [\textit{W. Rindler}, Relativity, Special, General and Cosmological. (Oxford University Press, Oxford, New York) (2001; Zbl 0981.83001); \textit{N. Straumann}, General Relativity with Applications to Astrophysics. (Springer-Verlag, Berlin, Heidelberg, New York) (2004; Zbl 1059.83001)] containing both a weakly coupled scalar field \(\varphi\) with an exponential potential \(V(\varphi)\), as in Kaluza-Klein models, supergravity, and superstring theories. Moreover, the model of the universe assumes to contain a perfect fluid which is studied for several choices of the parameters governing its linear equation of state. The paper obtains the Einstein-scalar field equations in the form of Lagrange-Euler equations following from a Lagrangian. The dynamical system described by a Lagrangian of this type belongs to the class of pseudo-Euclidean Toda like systems. The method used for integration is based on reducing the Euler-Lagrange equations to the generalized Emden-Fowler second order ordinary differential equation [\textit{V. F. Zaitsev} and \textit{A. D. Polyanin}, Discrete-groups methods for integrating equations of nonlinear mechanics. CRC Press, Boca Raton, Florida (1994)]. Finally, the paper provides a discussion of the physical properties of the obtained exact solutions. The asymptotic behaviour of these solutions at early and late times is analyzed. An example of an explicit solution in terms of cosmic time is presented.