Cosmology and GR are not my areas of specialty, but I've been thinking of your post and considering the historical example of Michael Faraday. Faraday applied the proverb, "A man’s gift makes room for him and brings him before great men." He attended lectures of Sir Humphry Davy, took meticulous notes, and eventually prepared and presented Davy with a carefully bound 300 page book based on his lecture notes. A short time later, Davy took him into his employ, and the rest is history. The question is how to apply the same principles to open doors to a 21st century research opportunity for an amateur.
Most research scientists have several unfunded side projects on a back burner waiting for a combination of available resources and circumstances to pursue them. Free labor on various programming or EE type tasks could be the "gift" that opens a door for you, especially if a scientist has a side project he's been unwilling to interest a grad student or post doc in where tasks could be outsourced to the amateur.
So your immediate choice might be whether you can get remotely involved in a group that "fits your interests" directly, or whether you need to view involvement with a more local research group as an intermediate stepping stone. Getting a co-authorship or recommendation from a more local research group may help getting the attention of the more remote groups. We've mentored research of amateurs both locally and remotely, and the amateurs in the remote situations really need to want it more to be able to be sufficiently disciplined and focused to contribute much of value. A remote research mentoring relationship seems less likely to succeed for amateurs who are unfamiliar with the needs, demands, and time line of real research. We've had more success with amateurs who have done some research before.
We're a bit wary of amateurs who offer nothing more than words about how badly they want it and how hard they'll work. We are now more likely to give them a challenging task that they should be able to complete relatively independently. For example, write a program to convert data from some NASA database or file format into something we can more easily input into our software. If they could get this done right in a couple weeks without more investment from us than a couple emails and a half-hour Skype or two, they'd probably have our attention. We'd then figure out how to parcel out another sub-project. At some point, we'd put them in the critical path of projects intended for publication, especially if they are willing to keep working for free. We have lots of unfunded side projects.
It is tough to know how many GR/Cosmology groups might be willing to work with an amateur or which ones might have tasks they could farm out to you. You just need to do the legwork of google searches, website reviews, and review of their papers to figure this out. If you have good programming skills, this is most likely your "gift", and you need to find groups who can make use of it and are willing to take the gamble of trying. I would focus first on groups that are close enough to you so your cold email could request an in-person meeting to make most of your sales pitch. But you need to STOP thinking about what they can do for you and START thinking about what you can do for them.