It has been suggested that cold dark matter (CDM) has difficulties in explaining tentative evidence for noncuspy halo profiles in small galaxies, and the low velocity dispersions observed in the largest Milky Way satellites (too-big-to-fail problem). Strongly self-interacting dark matter has been noted as a robust solution to these problems. The elastic cross sections required are much larger than predicted by generic CDM models, but could naturally be of the right size if dark matter is composite. We explore in a general way the constraints on models where strongly interacting CDM is in the form of dark atoms or molecules, or bound states of a confining gauge interaction (hadrons). These constraints include considerations of relic density, direct detection, big bang nucleosynthesis, the cosmic microwave background, and LHC data.
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