Gravitational instability of regular model-universes in a modified theory of general relativity

Any regular model-universe must be non-singular for an infinitesimal perturbation of the metric, if it exists at all. To examine this problem, the gravitational instability of regular isotropic model-universes in a modified theory of general relativity is investigated, whose gravitational Lagrangian is of the form Lg=R-(1/6l2)(l2R)n, where l is a characteristic length such that l \lesssim107(2n-3)/(n-1) cm (due to consideration of the solar gravitational field) and n is a numerical constant larger than unity. Dynamical equations for three independent wave modes (i.e., gravitational, rotational and longitudinal waves) classified by Lifshitz are derived and some characteristic features of these wave modes are touched upon. In particular, the temporal behavior of rotational waves allowable in the regular model-universes found by Breizman et al. (p=ε/3) and Gurovich (p=0) in the case n=4/3 is studied in detail. It is shown that their amplitudes become singular at respective characteristic epochs, except in a special model-universe which cannot be applied to the real universe. Several possibilities to overcome such a difficulty are also discussed.