You certainly need the calculus because you will need to understand how predictions emerge from the various theoretical aspects of a model, and these predictions are often in terms of how certain quantities change in time, or their total magnitude as observed in today's universe. If you want to probe deeper, you need tensor analysis to understand how the cosmological models flow out of general relativity which is currently our best theory for how gravity operates on large scales. But, to actually UNDERSTAND articles in cosmology, you need to know physics, which takes a lot more than simply mathematical manipulation. You must develop an intuition about physics that lets you see how matter and radiation interact. You can get some of this in courses by working out hundreds of simple 'problems', but there are no shortcuts. It is possible to know where all the keys on a piano are, but that doesn't mean you can play a fugue!