What would you consider the best online resources for learning the 3+1 ADM formalism and gauge invariant perturbation theory in cosmology? (Assuming intermediate level GR and QFT familiarity)
Hah, I just studied this a while ago with James Bardeen, so I would say he is the best resource for learning this! Since you probably don't have access to the physical Bardeen, you can check out:
Physical Review D, Vol 22 no 8 (1980) "Gauge-invariant cosmological perturbations"
Physical Review D, Vol 40 no 6 (1989) "Designing density fluctuation spectra in inflation"
There is also a set of lecture notes I have sitting on my desk by him that claim they are "to be published in Particle Physics and Cosmology" which are dated 1988, so presumably they were published within the next year or so.
If you can find them, the talks are probably the easier of the three, and the first PrD article is the second easiest. The third paper is very nice, but more technically difficult.
The single best reference for learning about gauge-invariant cosmological perturbation theory is ch. 7 of Mukhanov's book Physical foundations of Cosmology. Ok, not an online resource, but still the very best if that's what you're looking for.
On doing a search for one of York's original papers I ran into this wonderful site by Luca Bombelli at Ole' Miss (University of Mississippi). This contains a very comprehensive bibliography on the initial-value problem. I would also recommend Robert Wald's GR book. It has very nice coverage of the IVP in chapters and in an appendix.
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- Ma & Bertschinger (1995): http://adsabs.harvard.edu/abs/1995ApJ...455....7M