Self-consistent phase-space distribution function for the anisotropic dark matter halo of the Milky Way

Mattia Fornasa and Anne M. Green

Phys. Rev. D 89, 063531 – Published 27 March 2014

Abstract

Dark matter (DM) direct detection experiments usually assume the simplest possible “standard halo model” for the Milky Way (MW) halo in which the velocity distribution is Maxwellian. This model assumes that the MW halo is an isotropic, isothermal sphere, a hypothesis that is unlikely to be valid in reality. An alternative approach is to derive a self-consistent solution for a particular mass model of the MW (i.e. obtained from its gravitational potential) using the Eddington formalism, which assumes isotropy. In this paper we extend this approach to incorporate an anisotropic phase-space distribution function. We perform Bayesian scans over the parameters defining the mass model of the MW and parametrizing the phase-space density, implementing constraints from a wide range of astronomical observations. The scans allow us to estimate the precision reached in the reconstruction of the velocity distribution (for different DM halo profiles). As expected, allowing for an anisotropic velocity tensor increases the uncertainty in the reconstruction of $f (v)$ , but the distribution can still be determined with a precision of a factor of 4-5. The mean velocity distribution resembles the isotropic case; however, the amplitude of the high-velocity tail is up to a factor of 2 larger. Our results agree with the phenomenological parametrization proposed in Mao et al. (2013) as a good fit to $N$ -body simulations (with or without baryons), since their velocity distribution is contained in our 68% credible interval.

Received 29 November 2013

DOI:https://doi.org/10.1103/PhysRevD.89.063531

Authors & Affiliations

Mattia Fornasa and Anne M. Green

School of Physics and Astronomy, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom

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Issue

Vol. 89, Iss. 6 — 15 March 2014

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Images

Figure 1
The dark (light) blue band shows the 68% (95%) credible region for the reconstruction of the rotation curve $Θ (r)$ of the MW for, from left to right, the NFW, Einasto, and Burkert profiles for the DM halo. The solid black line shows the best-fit rotation curve, while the dashed (dotted) line indicates the contribution of the DM halo (disk).
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Figure 2
The energy-dependent part of the phase-space density (in arbitrary units). Different shades of blue indicate the dependence on $ρ_{s}$ for fixed $r_{s}$ . The rest of the parameters in the scan are fixed to their best-fit point values for the case of a NFW DM halo, while the parameters of the $L$ -dependent part of the phase-space density are fixed to $(β_{0}, β_{\infty}, k_{L_{0}}) = (0.0, 0.5, 0.25)$ . The yellow bands show how $F_{E} (E)$ changes when $ρ_{s}$ is kept constant and $r_{s}$ is varied.
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Figure 3
Upper left panel: the $f_{1} (v)$ distribution for various values of $ρ_{s}$ . The yellow bands show the effect of changing $β_{0}$ . Upper right panel: the same as the left panel but for the dependence on the scale radius (lines) and on $β_{\infty}$ (bands). Lower panel: the same as the upper panels but for the dependence on $L_{0}$ . Note that the lines corresponding to $L_{0} = 10 L_{s}$ and $L_{0} = 1000 L_{s}$ coincide. For the parameters that are kept fixed we chose the best-fit values for the case of a NFW DM halo and $(β_{0}, β_{\infty}, k_{L_{0}}) = (0.0, 0.5, 0.25)$ .
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Figure 4
The probability distribution of the speed distribution, $f_{1} (v)$ , resulting from the scan of the parameters of the MW mass model (see Sec. 3) for, from left to right, the NFW, Einasto, and Burkert DM halo profiles. In this case the speed distribution is obtained using the Eddington formalism, which assumes isotropy. The light (dark) blue band indicates the 68% (95%) credible region, while the solid black line corresponds to the mean.
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Figure 5
Same as Fig. 4 but for the speed distribution, $f_{1} (v)$ , obtained using the procedure described in Sec. 4 which allows the DM halo to be anisotropic. The light (dark) green band indicates the 68% (95%) credible region, while the solid black line corresponds to the mean. The solid blue line indicates the mean of the distribution of $f_{1} (v)$ obtained from the Eddington procedure (as in Fig. 4). In the left panel (for the NFW DM halo), the dotted line is the standard halo model Maxwell-Boltzmann distribution, Eq. (1), while the dashed line shows the Mao et al. parametrization from Ref. [122]. In both cases the parameters are set to their mean values from Table 2.
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Figure 6
The mean velocity distribution at large velocities for the NFW (black lines), Einasto (red lines), and Burkert (blue lines) DM profiles. Solid lines are for the isotropic case, calculated using the Eddington formalism, and dashed lines allowing anisotropy.
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