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Non-trivial dynamics within the dark sector can give rise to a complicated, non-thermal dark-matter phase-space distribution, which in turn can significantly influence the growth of cosmic structure. In this talk, we explore the cosmological implications of the non-trivial dynamics which may arise within dark sectors containing many degrees of freedom. We show how the non-trivial features in the phase-space distribution can induce modifications to quantities of structure formation such as the matter power spectrum and the halo mass function. We then examine the extent to which one can address the archaeological "inverse" problem of deciphering the properties of the underlying dark sector from the matter power spectrum and the halo mass function. We present simple one-line conjectures which can be used to “reconstruct” the dark-matter phase-space distribution directly from the shape of the matter power spectrum and the halo-mass function, and show that salient features of the distribution can be successfully reproduced -- even for non-trivial distributions which are highly non-thermal and/or multi-modal. Our conjectures therefore provide an operational tool for probing the dark sector which does not rely on the existence of non-gravitational couplings between dark and visible states.