Hierarchical phase space structure of dark matter haloes: Tidal debris, caustics, and dark matter annihilation

Niayesh Afshordi, Roya Mohayaee, and Edmund Bertschinger

Phys. Rev. D 79, 083526 – Published 24 April 2009

Abstract

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 that 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 $\propto Δ J^{- 1.6}$ 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. Nevertheless, as we argue in a companion paper, the boost due to small gravitationally-bound substructure can dominate this effect at low redshifts.

Received 12 November 2008

DOI:https://doi.org/10.1103/PhysRevD.79.083526

Authors & Affiliations

Niayesh Afshordi^1,*, Roya Mohayaee^2,†, and Edmund Bertschinger^3,‡

¹Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, ON, N2L 2Y5,Canada
²Institut d’Astrophysique de Paris, CNRS, UPMC, 98 bis boulevard Arago, France
³Department of Physics and Kavli Institute for Astrophysics and Space Research, MIT, Room 37-602A, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA

^*nafshordi@perimeterinstitute.ca
^†roya@iap.fr
^‡edbert@mit.edu

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Issue

Vol. 79, Iss. 8 — 15 April 2009

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Images

Figure 1
The top horizontal panel shows the phase space of the merger of two dark matter haloes, each of which has its own hierarchy of phase structure. The times on the top panel refer to the crossing times. The zooming shows that each hierarchy contains a lower level and so on. The hierarchy is cut at the scale of the smallest dark matter halo that has been accreted to the final halo. However, the phase space is not smooth below this scale. Indeed, the phase space is intrinsically discrete due to the coldness of dark matter shown by the last zooming on the left. (Top panel: courtesy of Vlasov-Poisson simulation [33].)Reuse & Permissions
Figure 2
A one-dimensional cartoon of the evolution of tidal streams in both phase and action-angle spaces. As structure wraps around the phase space, more streams cross the same angle coordinate, which leads to a discrete latticelike structure in the action space.Reuse & Permissions
Figure 3
The action-space distribution of debris in a unit 2d torus with unit particle mass and no potential. The debris is originally within $0 < x$ , $y < 0.1$ , and $- 10 < v_{x}$ , $v_{y} < 10$ . The figures show a cut through the action space with $0.09 < x$ , $y < 0.1$ , which is characterized by ${\tilde{f}}_{p} (J, x, t_{a c c, p})$ [Eq. (14)] in our formalism.Reuse & Permissions
Figure 4
The local boost factor due to primordial discreteness of the phase structure for an NFW potential: The plots show how the boost in the annihilation measure of a DM halo changes as we go from the inner part of halo to outer parts, due to the discrete phase space structure of CDM. The local boost increases as we go toward the outskirts of the halo and also as we decrease the concentration. The dashed curves on the left panel show the boost if we include corrections due to a finite initial phase for the debris (68), which become important for nearly degenerate frequencies. The right panel shows that most of the boost comes from regions beyond 20% of the virial radius.Reuse & Permissions
Figure 5
The estimated total boost in the annihilation measure of a DM halo, due to the discrete distribution in the CDM phase space is shown for an NFW halo. The lower dashed curve shows the total boost when we include corrections due to a finite initial phase for the debris (68), which become important for nearly degenerate frequencies.Reuse & Permissions
Figure 6
The estimated local boost including contributions from the debris, discreteness and the subhaloes, given by (69) with the first term set at its lowest value of unity, is shown for different redshifts. At high redshifts (e.g. $z = 10$ ), the primordial caustics dominate over all other effects. However, at low redshifts the discreteness effect due to caustics is only important in the outskirts of the haloes [notice that the discreteness/debris contribution remains independent of redshift in the units used in this plot, while the subhalo contribution evolves as $\sim H^{- 2} (z)$ ].Reuse & Permissions

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