Gravitational Instability in the Brans-Dicke Cosmology

In view of a possible relevance of the Brans-Dicke cosmology to the resolution of the primeval helium problem and the problem of galaxy formation, the gravitational instability in their cosmology is studied by the use of our Lagrangian gauge. It is shown that the density contrast consists, in general, of four independent terms, i.e. two terms resembling their counterparts in the general relativistic cosmology and the other two due to the variation of the gravitation “constant”. At the later stage of the universe, the density contrast can grow in time as δε/ε∝t2{1+2/(4+3ω)}/3 (ω is a parameter characteristic of their cosmology such as ω\gtrsim5). It is also shown that Mach's principle in the sense of Dicke provides us with some close connection between the last two terms and the time-growing term.