I Is the weak field limit valid for the Brans-Dicke scalar field?

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The Brans-Dicke theory is a scalar-tensor modification of general relativity that has been tested against solar system experiments which show that the parameter ##\omega \gt 40000## implying that the theory is almost indistinguishable from GR.

These tests assume the weak field limit of the theory so that the scalar field ##\phi = \phi_0 + \xi##.

But is this valid if the scalar field is long range so that all the mass in the Universe at any distance contributes to ##\phi## at any given point in space?
 

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