Two-integral distribution functions in axisymmetric galaxies: Implications for dark matter searches

We address the problem of reconstructing the phase-space distribution function for an extended collisionless system, with known density profile and in equilibrium within an axisymmetric gravitational potential. Assuming that it depends on only two integrals of motion, namely the energy and the component of the angular momentum along the axis of symmetry $L_z$, there is a one-to-one correspondence between the density profile and the component of the distribution function that is even in $L_z$, as well as between the weighted azimuthal velocity profile and the odd component. This inversion procedure was originally proposed by Lynden-Bell and later refined in its numerical implementation by Hunter and Qian; after overcoming a technical difficulty, we apply it here for the first time in presence of a strongly flattened component, as a novel approach of extracting the phase-space distribution function for dark matter particles in the halo of spiral galaxies. We compare results obtained for realistic axisymmetric models to those in the spherical symmetric limit as assumed in previous analyses, showing the rather severe shortcomings in the latter. We then apply the scheme to the Milky Way and discuss the implications for the direct dark matter searches. In particular, we reinterpret the null results of the Xenon1T experiment for spin-(in)dependent interactions and make predictions for the annual modulation of the signal for a set of axisymmetric models, including a self-consistently defined corotating halo.

Figure 1

Radial (left) and azimuthal (right) velocity distributions in the galactic plane for various fraction of the axisymmetric component, parametrized by $x_{\mathrm{axi}}$. In lower panels we show the relative difference with respect to the $x_{\mathrm{axi}}=0$ case, computed at different radii.

Figure 2

Total velocity dispersion (left) and velocity anisotropy (right) in the galactic plane as a function of radial distance for various fractions of axisymmetric component, parametrized by $x_{\mathrm{axi}}$. In the lower left panel we show the relative difference of the velocity dispersion with respect to the $x_{\mathrm{axi}}=0$ case.

Figure 3

Radial (left) and azimuthal (right) velocity distributions in the galactic plane for various fractions of the stellar disc component, parametrized by $x_{\mathrm{disc}}$. In the lower panels we show the relative difference with respect to the $x_{\mathrm{disc}}=0$ case, computed at different radii.

Figure 4

Radial (left) and azimuthal (right) velocity distributions in the galactic plane for various ratios of $r_s/a_d$. In the lower panels we show the relative difference with respect to the $r_s/a_d=8$ case, computed at different radii.

Figure 5

Total velocity dispersion (left) and velocity anisotropy (right) in the galactic plane as a function of radial distance for various ratios of $r_s/a_d$. In the lower left panel we show the relative difference of velocity dispersion with respect to the $r_s/a_d=8$ case.

Figure 6

Radial (left) and azimuthal (right) velocity distributions for various heights $z$ above the galactic disc, normalized to the disc height $b_d$. In the lower panels we show the relative difference respect to the $z=0$ case, computed at different radii.

Figure 7

Total velocity dispersion (left) and velocity anisotropy (right) as a function of radial distance for various heights $z$ above the galactic disc (normalized to the disc height $b_d$). In the lower left panel we show the relative difference of velocity dispersion with respect to the $z=0$ case.

Figure 8

Radial (left) and azimuthal (right) velocity distributions in the galactic plane for various halo shapes, parametrized by $q$. In the lower panels we show the relative difference with respect to the spherical halo, computed at different radii.

Figure 9

Total velocity dispersion (left) and velocity anisotropy (right) in the galactic plane as a function of radial distance for various halo shapes, parametrized by $q$. In the lower left panel we show the relative difference of velocity dispersion with respect to the spherical halo.

Figure 10

Azimuthal velocity probability distribution (left) and average azimuthal velocity as a function of $R$ (right) for PSDF with $f_{-}=\alpha f_{+}$ and $f_{-}$ computed from ${\overline{v}}_{\varphi }(R)$ as defined in (23), using $r_a=r_s$ and $\omega$ such that the spin parameter $\lambda (0.25r_{200})=0.04$.

Figure 11

Local radial (left) and azimuthal (right) DM velocity distributions for SHM, spherical modelling through Eddington’s inversion and axisymmetric modeling with $q=1$ and $q=0.9$, computed for the considered Milky Way sample. In the lower panels we show the relative differences with respect to the SHM.

Figure 12

Local DM velocity dispersion (left) and anisotropy (right) as function of galactocentric distance for SHM (the results for spherical modeling through Eddington’s inversion are equivalent to the SHM ones, since we compute the velocity dispersion for the latter according to the Eq. (26)) and axisymmetric modeling with $q=1$ and $q=0.9$, computed for the considered Milky Way sample. In the lower left panel we show the relative difference of velocity dispersion with respect to the SHM.

Figure 13

Astrophysical factors defined in Eqs. (29) and (30), which appear in the deferential event rate, as a function of minimal scattering velocity for SHM and axisymmetric model with $q=1$ (HQ), $q=0.9$ and spherical rotating halo with ${\overline{v}}_{\varphi }(R)$ as defined in Eq. (23), using $r_a=r_s$ and $\omega$ such that the spin parameter $\lambda (0.25r_{200})=0.04$.

Figure 14

Xenon1T cross-section—mass exclusion plots for SHM and axisymmetric modeling with $q=1$, $q=0.9$ and spherical rotating halo with ${\overline{v}}_{\varphi }(R)$ as defined in Eq. (23), using $r_a=r_s$ and $\omega$ such that the spin parameter $\lambda (0.25r_{200})=0.04$. Left plot shows the results for spin-independent and right plot for spin-dependent scattering.

Figure 15

Left: Annual modulation of the differential scattering rate as a function of recoil energy. Top panel shows the two extreme cases with respect to the time of the year, while the lower plot shows the absolute difference between the two. Right: Annual modulation of the event rate as a function of time. Upper panel shows the absolute value, while the lower panel shows the relative value with respect to the annual average for the given model. The results are computed for ${}^{131}\mathrm{Xe}$, assuming SI corss-section ${\sigma }_n^{\mathrm{SI}}={10}^{-46}{\mathrm{cm}}^2$ and $m_{\chi }=20\mathrm{GeV}$, while the considered PSDF models are the same as in Fig. 14.