I have finished Carroll's general relativity book, Spacetime and Geometry. I am specifically interested in cosmology, so is there any book which goes more in depth into cosmology? I prefer a mathematically rigorous book, which goes beyond the cosmology chapter in Carroll's book. The Big Bang and inflation theory should be dealt with in depth.
In addition to Weinberg's fantastic book which does cover things pretty well, a good companion book is "Relativistic Cosmology" by Ellis, Maartens and Maccallum, which covers a lot of stuff Weinberg doesn't and is in general a good book on Cosmology.
Aside from the standard introduction to FLRW cosmology, inflation, thermodynamics etc, it covers an additional range of topics in cosmology such as the role of Modified Gravity, Models of Dark Energy, Anisotropic cosmologies and Inhomogenous models.
You will need a good grasp of GR, some differential geometry and statistical mechanics if you want to grasp all of the material properly but it is worth putting the time in.
As far as cosmology is concerned, the book which I consider to be THE best for a mathematical treatment of cosmology, is AK Raychaudhuri's "General relativity, astrophysics, and cosmology". It is excellently presented, Raychudhuri doesn't shy away from the math, and the old-school style makes it all the more elegant. So, I would STRONGLY recommend it.
I also suggest Narlikar's book on cosmology, it is beautifully written. As other users pointed out above me, Weinberg is also an excellent choice. In case you want an extremely mathematical book with astrophysics on the side, go for Straumann. But, for me, nothing beats Raychaudhuri's book. Good Luck!
Steven Weinberg - Cosmology: Great book, it covers a lot of topics in a very rigorous way, at least for physicists, it can be hard to follow sometimes, as usual when talking about Weinberg this is not an introduction to the subject
Coles and Lucchin - Cosmology; the origin and evolution of cosmic structures: It's not a new book but it's one of the easiest introduction I know of. It's not complete and it's not up to date but the standard parts are very well written.
Viatcheslav Mukhanov - Physical foundation of cosmology: another great book in my opinion, differnt from the two mentioned above, more pedagogical than Weinberg, more physical than Lucchin, probably less complete than both, but a comparison on completeness is flawed cause they treat different topics. Surely worth read. I especially liked the chapters on inflation.
Scott Dodelson - Modern Cosmology: Not as complete as Weinberg but better than any other book I've read in certain topics: basically all the second part, the part that is devoted to the non homogeneous universe is in my opinion the best place to start studying it.
There's another book which I think could be very good for an expert physicist who wants to approach cosmology and astrophysics, it's by Tanu Padmanbhan and it's called Cosmology and astrophysics through problems. It's not an easy book at all, it's basically a book of exercises (some of them very hard) that builds complete knowledge of astrophysics and cosmology at least at a starring PhD level. There is a detailed solution for every exercise. There may not be many advanced research topics, but if you're able to go through the book you will surely have a solid base to start research in cosmology, I'd say more solid than a guy with a maser degree in these topics.
There are other books I can't review cause i didn't read much of them but I've heard good things about, so taking a look at them and see if one likes them can be useful. These books are:
Kolb and Turner - The early Universe
John Peacock - Cosmological physics
- Tanu Padmanabhan - Structure formation in the universe: I know a bit more about this book. It focuses more on structure formation but there is a good introductory part on the basics of cosmology before dwelling in the interesting field of structure formation, often ignored by theorists.
Daniel Baumann - Cosmology: you can find them here. These are the only lecture notes I know, and I think they are the best place to start studying cosmology. They are rather complete for an introduction and they are easy to follow. I like the structure and the approach in these lectures.
One can find other lecture notes and videos of Baumann in this page