Stretchy space - useful books
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What if a science writer also had done some work as a scientist in the field he is writing about? This is the case for The birth of time by John Gribbin, which results in an excellent book. Unfortunately for me he goes further than most and separates out Cosmological from Doppler redshift for a galaxy in the Virgo cluster. I couldn't help thinking "Now wait a minute, aren't they both described by general relativity?"
General relativity for the general readerThe books mentioned above raise the question of how to view the expansion of the universe, but don't really provide enough detail to understand where the arguments are coming from. For this you need a book that explains something about general relativity. One possibility is Black Holes, Wormholes and Time Machines by Jim Al-Khalili which grew out of a series of lectures which the author gave to teenage schoolchildren, and so would be a good place to start.
Alternatively you might like How the universe got its spots, by Janna Levin, which is based on a series of letters to her mother, and so is non-technical. Levin is putting forward the view that the topology of the universe might mean that it is finite, even if its geometry doesn't force it to be so. Hence it's not strictly related to the subject in hand, but gives a good impression of the way a cosmologist thinks.
If you're looking for a more general book on cosmology you should look at Our evolving universe by Malcolm Longair, which gives the impression of being a "coffee table book", (it's a large format book, and has many colour pictures), but is in fact a comprehensive introduction to the subject. Foundations of modern cosmology by Hawley and Holcomb is a similar book, suitable for slightly more advanced readers.
Evolution of the Universe by Igor Novikov is readable by the general reader, being philosophical rather than technical, but I think that you will get more out of it if you have already been introduced to the subject.
Student BooksFor those who don't mind a few equations there are plenty of cosmology books which supply the required amount of general relativity. Principles of cosmology and gravitation by Michael Berry is a good introduction to GR, leading on to cosmology. An introduction to the science of cosmology by Raine & Thomas has less GR, but still enough to get a feeling for the subject, and goes on to explore more advanced cosmological topics.
If you're looking for a mathematical presentation of General relativity A short course in general relativity by Foster and Nightingale is a Longman mathematical text and so is well structured for the student (this refers to the first edition).
TextbooksFinally, a look at the books for those with more experience of the subject - advanced undergraduate and above. Cosmology by Michael Rowan-Robinson is concise yet comprehensive. The standard textbook of cosmology is probably Principles of physical cosmology by P. J. E. Peebles, while for General Relativity there is General Relativiy by Robert Wald.
Gravitation by Misner, Thorne and Wheeler also deserves a mention. This large book was the standard work of the 1970's, and although it is somewhat out of date, you may prefer its more informal style. I would also claim some support from this book, as the discussion on p739 (Chapter 27) argues against the idea of "creating space".