The early acceleration, called cosmic inflation, happened when the universe was extremely young and extremely energetic. It generated the initial density fluctuations which eventually led to the formation of the structures in the Universe: galaxies, stars, planets, and ultimately, ourselves. The late-time acceleration has only begun fairly recently in the history of the universe, and is a great mystery since it happens on very large scales (at a very low energy scale). In fact, the energies at which these two epochs occurred are at least 15 orders of magnitude apart!
The precise physical mechanisms of inflation and the late-time acceleration are not understood. What we do know is that physics beyond the standard model of particle physics is necessary. Learning about the physics of inflation corresponds to studying particle physics at energy scales out of reach of any current or future accelerators on Earth. On the other hand, if Einstein's theory of relativity is correct, the late-time acceleration means that there exists an additional exotic constituent (Dark Energy) in the energy budget of the Universe. Alternatively, the acceleration could signal that the theory of relativity is not a good description of nature on very large scales.
We hope to gain insight into these difficult but fascinating questions by using data sets on the Cosmic Microwave Background (CMB, from the fairly early universe), and the large-scale distribution of matter, galaxies, and gas (from the late universe). We do this by calculating precise theoretical predictions linking models of inflation or dark energy with observations, and comparing them with the data.