We study how may vary the gravitational and the cosmological “constants,” ( and ) in several scalar-tensor theories with Bianchi III, , and symmetries. By working under the hypothesis of self-similarity we find exact solutions for two different theoretical models, which are the Jordan-Brans-Dicke (JBD) with and the usual JBD model with potential (that mimics the behaviour of . We compare both theoretical models, and some physical and geometrical properties of the solutions are also discussed putting special emphasis on the study of the isotropization of the solutions. 1. Introduction Current observations of the large scale Cosmic Microwave Background (CMB) suggest to us that our physical universe is expanding in an accelerated way. Such observations [1–3] indicate that the universe is dominated by an unidentified “dark energy” (DE) and suggest that this unidentified dark energy has a negative pressure [4–6]. This last characteristic of the dark energy points to the vacuum energy or cosmological constant , as a possible candidate for dark energy. From the theoretical point of view, it is convenient to consider the cosmological constant as a dynamical quantity in order to solve the so-called coincidence and fine tuning problems. In the same way other observations have pointed out a possible variation of the gravitational constant [7, 8]. For example, observations of Hulse-Taylor binary pulsar [9, 10], and type Ia supernova observations [11]. For an extensive review see Uzan [12]. We have several theoretical models that consider both constants as variable with respect to the cosmic time. Such theories are modified general relativity (MGR), modified scalar cosmological models (MST), and several scalar-tensor theories (STT). The MGR and MST have a drawback, since in them the variations of and are introduced in an ad hoc manner. Nevertheless we consider that the STT are the best models to study the variation of and , since they have been deduced form variational principles and where the time dependence can occur in a natural way, without any new assumption or modification of the theory. This class of models has received a renewed interest in recent times, for two main reasons. Firstly, the new inflationary scenario as the extended inflation has a scalar field that solves several problems present in the old theories. Secondly, string theories and other unified theories contain a scalar field which plays a similar role to the scalar field of the STT. The scalar-tensor theories started with the work of Jordan in 1950 [13]. A prototype of such models was proposed
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