We investigate cosmological solutions of Brans-Dicke theory with both the vacuum energy density and the gravitational constant decaying linearly with the Hubble parameter. A particular class of them, with constant deceleration factor, sheds light on the cosmological constant problems, leading to a presently small vacuum term, and to a constant ratio between the vacuum and matter energy densities. By fixing the only free parameter of these solutions, we obtain cosmological parameters in accordance with observations of both the relative matter density and the universe age. In addition, we have three other solutions, with Brans-Dicke parameter ω = −1 and negative cosmological term, two of them with a future singularity of big-rip type. Although interesting from the theoretical point of view, two of them are not in agreement with the observed universe. The third one leads, in the limit of large times, to a constant relative matter density, being also a possible solution to the cosmic coincidence problem.

We present some cosmological solutions of Brans-Dicke theory, characterized by a decaying vacuum energy density and by a constant relative matter density. With these features, they shed light on the cosmological constant problems, leading to a presently small vacuum term, and to a constant ratio between the vacuum and matter energy densities. By fixing the only free parameter of our solutions, we obtain cosmological parameters in accordance with observations of the relative matter density, the universe age and redshift-distance relations.Several authors have been considering the possibility of a varying cosmological term in order to fit the observed non-decelerated expansion of the universe and, at the same time, to explain the small value of the cosmological constant observed at present [1]- [13]. Such a variation of the vacuum density has found support in some quantum field approaches (see, for example, [6,9]), in which context an induced variation of the gravitational coupling constant G may also be expected [5,10,11,13].Our goal in this contribution is to present some cosmological solutions with decaying vacuum in the realm of Brans-Dicke theory. For this purpose, we will consider an empirical variation law for G, given by the Weinberg relation G ≈ H/m 3 π , where H =ȧ/a is the Hubble parameter, and m π is the energy scale of the QCD chiral symmetry breaking, the latest cosmological vacuum phase transition. Such a relation, originally based on the Eddington-Dirac large number coincidence, can find some support on theoretical, holographic arguments [11,14]. Let us write it aswhere the constant λ is positive and has the order of m 3 π . With this variation law for G, we will form an empirical ansatz to be used in Brans-Dicke equations. It is fulfilled by the additional constraintwhere ρ = ρ m + ρ Λ is the total energy density, and α is a positive constant of the order of unity. As we know, in the context of scalar-tensor theories, even for zero spatial curvature, the total energy density is not necessarily equal to the critical density ρ c ≡ 3H 2 /(8πG). Therefore, the above equation should also be considered an empirical coincidence, suggested by observation. Our point is that, since (1) and (2) are valid nowadays, they may be valid for any time, or at least in the limit of late times. Let us find solutions for this limit, by considering a spatially flat FLRW space-time, with a cosmic fluid formed * Associate Member by dust matter (i.e., p m = 0) plus a vacuum term with equation of state p Λ = −ρ Λ . The Brans-Dicke equations are then given by [15,16] where p = p m + p Λ is the total pressure, ω is the Brans-Dicke parameter, and the Brans-Dicke scalar field φ is related to the gravitational constant by φ = G 0 /G, G 0 being a positive constant of the order of unity. From our ansatz (1)-(2), we obtainwhere q = −aä/ȧ 2 is the deceleration parameter. By using (6)-(8), we can rewrite equations (3)-(5) in the form (3 + 2ω)λG 0 [q + 3(1 + q)H] = 3αλH + 3ρ Λ , (9) ρ m = αλ(1 + q)H,Equation (11) shows that q is a cons...