Exact cosmological solutions with nonminimal derivative coupling

We consider a gravitational theory of a scalar field phi with nonminimal derivative coupling to curvature. The coupling terms have the form kappa R-1 phi(,mu)phi(,mu) and kappa R-2(mu nu)phi(,mu)phi(,nu), where kappa(1) and kappa(2) are coupling parameters with dimensions of length squared. In general, field equations of the theory contain third derivatives of g(mu nu) and phi. However, in the case -2 kappa(1)=kappa(2)equivalent to kappa, the derivative coupling term reads kappa G(mu nu)phi(,mu)phi(,nu) and the order of corresponding field equations is reduced up to second one. Assuming -2 kappa(1)=kappa(2), we study the spatially-flat Friedman-Robertson-Walker model with a scale factor a(t) and find new exact cosmological solutions. It is shown that properties of the model at early stages crucially depend on the sign of kappa. For negative kappa, the model has an initial cosmological singularity, i.e., a(t)similar to(t-t(i))(2/3) in the limit t -> t(i); and for positive kappa, the Universe at early stages has the quasi-de Sitter behavior, i.e., a(t)similar to e(Ht) in the limit t ->-infinity, where H=(3 kappa)(-1). The corresponding scalar field phi is exponentially growing at t ->-infinity, i.e., phi(t)similar to e(-t/kappa). At late stages, the Universe evolution does not depend on kappa at all; namely, for any kappa one has a(t)similar to t(1/3) at t ->infinity. Summarizing, we conclude that a cosmological model with nonminimal derivative coupling of the form kappa G(mu nu)phi(,mu)phi(,nu) is able to explain in a unique manner both a quasi-de Sitter phase and an exit from it without any fine-tuned potential.