In cosmological models with a varying gravitational constant, it is not clear whether primordial black holes preserve the value of G at their formation epoch. We investigate this question by using the Tolman-Bondi model to study the evolution of a background scalar field when a black hole forms from the collapse of dust in a flat Friedmann universe. Providing the back reaction of the scalar field on the metric can be neglected, we find that the value of the scalar field at the event horizon very quickly assumes the background cosmological value. This suggests that there is very little gravitational memory.

We discuss the various ways in which primordial black holes may have formed in the early Universe and how the effects of such black holes can be used to place constraints on cosmological models. We show that such constraints may be severely modified if the value of the gravitational "constant" G varies with cosmological epoch, a possibility which arises in many scenarios for the early Universe. The nature of the modification depends upon whether the value of G near a black hole maintains the value it had at its formation epoch (corresponding to gravitational memory) or whether it tracks the background cosmological value. This is still uncertain but we discuss various approaches which might help to resolve the issue.

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