On the Equation of State for Scalar Field

We consider Friedmann cosmologies with minimally coupled scalar field. Exact solutions are found, many of them elementary, for which the scalar field energy density, rho_f, and pressure, p_f, obey the equation of state (EOS) p_f=w_f\rho_f. For any constant |w_f|<1 there exists a two-parameter family of potentials allowing for such solutions; the range includes, in particular, the quintessence (-1<w_f<0) and `dust' (w_f=0). The potentials are monotonic and behave either as a power or as an exponent for large values of the field. For a class of potentials satisfying certain inequalities involving their first and second logarithmic derivatives, the EOS holds in which w_f=w_f(\f) varies with the field slowly, as compared to the potential.