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Anisotropic Cosmological Model in Modified Brans--Dicke Theory

It has been shown that four dimensional Brans-Dicke theory with effective matter field and self interacting potential can be achieved from vacuum 5D BD field equations, where we refer to as modified Brans-Dicke theory (MBDT). We investigate a generalized Bianchi type I anisotropic cosmology in 5D BD theory, and by employing obtained formalism, we derive induced-matter on any 4D hypersurface in context of the MBDT. We illustrate that if the usual spatial scale factors are functions of time while scale factor of extra dimension is constant, and scalar field depends on time and fifth coordinate, then in general, one will encounter inconsistencies in field equations. Then, we assume the scale factors and scalar field depend on time and extra coordinate as separated variables in power law forms. Hence, we find a few classes of solutions in 5D spacetime through which, we probe the one which leads to a generalized Kasner relations among Kasner parameters. The induced scalar potential is found to be in power law or in logarithmic form, however, for constant scaler field and even when scalar field only depends on fifth coordinate, it vanishes. The conservation law is indeed valid in this MBDT approach for derived induced energy momentum tensor (EMT). We proceed our investigations for a few cosmological quantities, where for simplicity we assume metric and scalar field are functions of time. Hence, the EMT satisfies barotropic equation of state, and the model indicates that constant mean Hubble parameter is not allowed. Thus, by appealing to variation of Hubble parameter, we assume a fixed deceleration parameter, and set evolution of quantities with respect to the fixed deceleration, BD coupling and state parameters. The WEC allows a shrinking extra dimension for decelerating expanding universe that, in the constant scalar field, evolves same as flat FRW spacetime in GR.