  Most of the mass content of dark matter haloes is expected to be in the form
of tidal debris. The density of debris is not constant, but rather can grow due
to formation of caustics at the apocenters and pericenters of the orbit, or
decay as a result of phase mixing. In the phase space, the debris assemble in a
hierarchy which is truncated by the primordial temperature of dark matter.
Understanding this phase structure can be of significant importance for the
interpretation of many astrophysical observations and in particular dark matter
detection experiments. With this purpose in mind, we develop a general
theoretical framework to describe the hierarchical structure of the phase space
of cold dark matter haloes. We do not make any assumption of spherical symmetry
and/or smooth and continuous accretion. Instead, working with correlation
functions in the action-angle space, we can fully account for the hierarchical
structure (predicting a two-point correlation function ~ (\Delta J)^{-1.6} in
the action space), as well as the primordial discreteness of the phase space.
As an application, we estimate the boost to the dark matter annihilation signal
due to the structure of the phase space within virial radius: the boost due to
the hierarchical tidal debris is of order unity, whereas the primordial
discreteness of the phase structure can boost the total annihilation signal by
up to an order of magnitude. The latter is dominated by the regions beyond 20%
of the virial radius, and is largest for the recently formed haloes with the
least degree of phase mixing.
