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Has there been any studies done on the correlation between etf vs cash (i.e. GLD vs GD) for example and how they should theoretically move together, and what fundamental reasons could cause them to diverge.

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A quick Google search gives a few hints:

Another quick search on scholar.google.com gives this research.

So some work has been done on the topic. However, I'd warn you about the credit risk of the ETF, as countre-parties are various, often unknown and might be really unreliable, and you might end up with a portfolio of junk bonds rather than the underlying your were looking for in case of default.

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