Reddit reviews Physical Foundations of Cosmology
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Used Book in Good Condition
Even though you want the full tensor treatment, I would first go through Ryden and make sure you understand the basics well. This is a great undergraduate standard written at the level for those who know "calculus, linear algebra and classical mechanics" and teaches the undergraduate level basics as well as anything.
After this, the standard modern graduate texts are Modern Cosmology by Dodelson and Physical Foundations of Cosmology by Mukhanov. Both use tensors and the full GR treatment with the former, in my opinion, being an easier text (which I think have some great initial chapters describing GR) but Mukonov going through some very advanced concepts like renormalization in quantum field theory, etc...
In addition to textbooks, Baumann's lecture notes on inflation are very good.
Good luck.
Cool closely related to a lot of my own interests! I assume you've already seen some Cosmology at uni but you couldn't ask for a better overview of modern cosmology than Mukhanov https://www.amazon.co.uk/Physical-Foundations-Cosmology-Viatcheslav-Mukhanov/dp/0521563984
Even better in my opinion is his book on quantized fields in a gravitational background: https://www.amazon.co.uk/Introduction-Quantum-Effects-Viatcheslav-Mukhanov/dp/0521868343 If you want to see just how far canonical quantization can do (unruh and hawking effect!) as well as a detailed look into back reaction calculations then look no further. In particular it has a very nice section about quantizing fields in de Sitter space (i.e. inflation).
You may have seen some of this before but I would certainly recommend them. I don't know that much about axions but if you did a lit review on them I'm sure you know what to read up on to find out more.
I would be mindful that there is very little left to be done that is pure "pen and paper" in cosmology (and a lot of theoretical physics tbh). This doesn't mean you should lose hope, just be prepared to take on a bit of coding. I know professors who have rejected candidates because they said things like "I wouldn't want to do any coding" in interview.