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A self-consistent phase-space distribution function for the anisotropic Dark Matter halo of the Milky Way
Cosmology and Nongalactic Astrophysics · Submitted 21 Nov 2013
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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The abstract, sentence by sentence
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, hypotheses that are 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 parameterising 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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | background 1.00 | 0.04 | 0.02 | asserted 0.99 |
| s1 | This model assumes that the MW halo is an isotropic, isothermal sphere, hypotheses that are unlikely to be valid in reality. | background 0.87 | 0.05 | 0.02 | hedged possibility 0.70 |
| s2 | 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. | background 0.97 | 0.06 | 0.02 | asserted 0.94 |
| s3 | In this paper we extend this approach to incorporate an anisotropic phase-space distribution function. | contribution 0.99 | 0.89 | 0.06 | asserted 0.99 |
| s4 | We perform Bayesian scans over the parameters defining the mass model of the MW and parameterising the phase-space density, implementing constraints from a wide range of astronomical observations. | method 0.95 | 0.96 | 0.04 | asserted 0.99 |
| s5 | The scans allow us to estimate the precision reached in the reconstruction of the velocity distribution (for different DM halo profiles). | method 0.47 | 0.90 | 0.07 | asserted 0.93 |
| s6 | 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. | result 0.99 | 0.95 | 0.38 | asserted 0.99 |
| s7 | The mean velocity distribution resembles the isotropic case, however the amplitude of the high-velocity tail is up to a factor of 2 larger. | result 1.00 | 0.94 | 0.22 | asserted 0.99 |
| s8 | 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. | result 0.57 | 0.75 | 0.22 | asserted 0.93 |
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Original 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, hypotheses that are 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 parameterising 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.
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References
How the literature uses it
25 arXiv papers cite this in 57 sentences
Papers citing it · 27 sentences without a confident label
Contests it 1 paper
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However, simulations tend not to find radial anisotropy in the inner regions of halos ( [ref] ), and in practice we find that PSDFs with $β_0 > 0$ do not satisfy the necessary condition for stability.[This was not accounted for in Ref. [ref] but as discussed extensively in Ref. [ref] , it is not satisfactory to estimate astrophysical uncertainties by considering PSDFs that are potentially unstable solutions of the collisionless Boltzmann equation.] Therefore, a more sensible upper limit is given by $β_0 ≤0$.
Towards a more rigorous treatment of uncertainties on the velocity distribution of dark matter particles for capture in stars · Estimating the uncertainty on the speed DF from the unknown anisotropy · contest 0.80
Builds on it 2 papers
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The last two cases were not previously considered in Ref. [ref] .
How to calculate dark matter direct detection exclusion limits that are consistent with gamma rays from annihilation in the Milky Way halo · Mass models of the Milky Way · 0.89
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Ref. [ref] extended this approach to include anisotropy of the velocity distribution.
Astrophysical uncertainties on the local dark matter distribution and direct detection experiments · Modelling the Milky Way · 0.90
Uses its method or tool 4 papers
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In this work, we use the mass model defined in Ref. [ref] .
How to calculate dark matter direct detection exclusion limits that are consistent with gamma rays from annihilation in the Milky Way halo · Mass models of the Milky Way · 0.86
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These were computed modifying the distribution (DF) by including suitable angular momentum factors along the lines suggested by Wojtak et al [ref] and more recently by Fornasa and Green [ref] .
Asymmetric Velocity Distributions from Halo Density Profiles in the Eddington Approach · Introduction · 0.91
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More specifically we compute the median speed DF and the corresponding 68% and 95% confidence regions.[We sample uniformly in linear space for $β_0$ and $β_∞$, and uniformly in log space for $L_0$ as in Ref. [ref] .
Towards a more rigorous treatment of uncertainties on the velocity distribution of dark matter particles for capture in stars · Estimating the uncertainty on the speed DF from the unknown anisotropy · 0.99
Uses its data or values 4 papers
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For consistency with the analysis of gamma-ray lines of Ref. [ref] , the direct detection bounds have been extracted assuming the same local DM density, $0.4$ GeV$/$cm$^3$, and the same choices of DM halo density profiles[Note that the profiles (both Einasto and NFW) considered in Ref. [ref] are slightly different from those of Ref. [ref] , used to extract the speed distributions in Ref. [ref] .
Enhanced lines and box-shaped features in the gamma-ray spectrum from annihilating dark matter in the NMSSM · Details of the scan and experimental constraints · 0.88
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Figure [ref] shows the capture boost where the high-velocity cut is decreased to 498 km/s [ref] and increased to be infinity [ref] .
Impact of Dark Matter Velocity Distributions on Capture Rates in the Sun · Uncertainties on the capture rate · 0.98
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In contrast, curves with lighter colours correspond to upper limits derived using the mean values that the authors of Ref. [ref] obtained for each of the above mentioned profiles, namely, $ρ^NFW_0=0.41$, $ρ^Ein_0=0.42$ and $ρ^Bur_0=0.41$ GeV/cm$^3$.
On the importance of direct detection combined limits for spin independent and spin dependent dark matter interactions · Limits on the SI cross section · 0.99
Compares with it 1 paper
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Finally, our results for the velocity DFs are consistent with the results of Ref. [ref] .
Towards a more rigorous treatment of uncertainties on the velocity distribution of dark matter particles for capture in stars · Relaxing the assumption of isotropy · 0.99
Cites it as background 10 papers
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The phase-space distribution, $F(x,v)$, of a matter component is self-consistent if, together with the gravitational potential of the system, $Φ(x)$, it satisfies the Boltzmann equation (see, e.g., Refs. [ref] ).
How to calculate dark matter direct detection exclusion limits that are consistent with gamma rays from annihilation in the Milky Way halo · Self-consistent velocity distributions · 0.98
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They differ in terms of assumptions and complexity, and range from the Eddington inversion approach, in which one can predict the DM PSDF from the knowledge of the underlying DM density and total gravitational potential [ref] , to its anisotropic extensions [ref] —see Ref. [ref] for a critical review discussing theoretical issues—through more sophisticated methods such as the angle-action formalism [ref] .
Towards a more rigorous treatment of uncertainties on the velocity distribution of dark matter particles for capture in stars · Introduction · 0.91
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In the literature there have been extensive discussions about the impact of uncertainties related to WIMP phase space on direct detection [ref] and indirect detection for more specific scenarios [ref] .
Impact of Dark Matter Velocity Distributions on Capture Rates in the Sun · Introduction · 0.95
What it cites · 103 sentences without a confident label
Builds on 5 papers
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In this paper, we apply the formalism developed in Ref. [ref] in the context of the DM halos of galaxy clusters to the MW DM halo (see also recent work in Ref. [ref] for an alternative approach to anisotropy).
The distribution function of dark matter in massive haloes · Introduction · 0.81
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In this paper, we apply the formalism developed in Ref. [ref] in the context of the DM halos of galaxy clusters to the MW DM halo (see also recent work in Ref. [ref] for an alternative approach to anisotropy).
Anisotropic dark matter distribution functions and impact on WIMP direct detection · Introduction · 0.81
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Thus, reconstructing $F(E,L)$ by inverting Eq. ([ref]) can be considered as an extension of the Eddington formulism to the case of an anisotropic DM halo[Note that the original Eddington formalism, for a spherical isotropic system, does not require any additional parameters and the speed distribution is completely determined by the gravitational potential [ref] .].
Deriving the velocity distribution of Galactic Dark Matter particles from rotation curve data · Self-consistent solution for the phase-space density of Dark Matter in the Milky Way halo · 0.95
Uses the method or tool of 8 papers
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Ref. [ref] parametrized the $L$-dependent component in terms of three quantities ($β_0$, $β_∞$ and $L_0$) and showed that, once this is done, self-consistent solutions can be obtained for each given mass model simply by inverting a Volterra integral equation.
The distribution function of dark matter in massive haloes · Summary and conclusions · 0.85
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We implement the information from the motions of masers, following Ref. [ref] , by marginalizing over the peculiar velocities and parallaxes of the masers.
The uncertainty in Galactic parameters · Proper motion of masers in high-mass star-forming regions · 1.00
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Our mass model of the MW follows closely that proposed in Ref. [ref] , and has four components: * stellar disk: Following Ref. [ref] , we model the stellar disk with a single component thin disc with density profile (in cylindrical coordinates): [equation] which is in agreement with the fit to the COBE Diffuse Infrared Background Experiment data discussed in Ref. [ref] .
COBE's Galactic Bar and Disk · Milky-Way mass contributors · 0.80
Uses data or values from 18 papers
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In light of this, we constrain the circular velocity by imposing the measurement of the proper motion of Sgr A$^⋆$ by Ref. [ref] , assuming the value of $V_⊙^$ quoted before from Ref. [ref] (see also Sec. [ref]).
The Proper Motion of Sgr A*: II. The Mass of Sgr A* · Local circular velocity · 1.00
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In light of this, we constrain the circular velocity by imposing the measurement of the proper motion of Sgr A$^⋆$ by Ref. [ref] , assuming the value of $V_⊙^$ quoted before from Ref. [ref] (see also Sec. [ref]).
The Milky Way's circular velocity curve between 4 and 14 kpc from APOGEE data · Local circular velocity · 1.00
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The clusters studied in Ref. [ref] had an isotropic velocity distribution (i.e. $β≈0$) close to their center, $β∼0.2$ at the scale radius, increasing to a value as large as 0.4 at $r ∼10 r_s$.
The distribution function of dark matter in massive haloes · Self-consistent solution for the phase-space density of Dark Matter in the Milky Way halo · 0.83
Compares with 10 papers
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For high speeds the mean anisotropic speed distribution is larger (up to a factor of 2 for of a NFW halo) than the mean isotropic distribution derived with the Eddington formula (consequently the peak in the distribution is also lower). * The parametrization of $f_1(v)$ introduced in Refs. [ref] , as a good fit to simulated DM halos, lies inside the band of our 68% credible interval.
Connecting Direct Dark Matter Detection Experiments to Cosmologically Motivated Halo Models · Summary and conclusions · 0.90
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For high speeds the mean anisotropic speed distribution is larger (up to a factor of 2 for of a NFW halo) than the mean isotropic distribution derived with the Eddington formula (consequently the peak in the distribution is also lower). * The parametrization of $f_1(v)$ introduced in Refs. [ref] , as a good fit to simulated DM halos, lies inside the band of our 68% credible interval.
Halo-to-Halo Similarity and Scatter in the Velocity Distribution of Dark Matter · Summary and conclusions · 0.90
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Our results are similar to those of Ref. [ref] .
A novel determination of the local dark matter density · Parameter estimation for the Milky Way mass models · 1.00
Cites as support 2 papers
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This model is unlikely to be a good approximation to the real MW DM halo. $N$-body simulations produce halos with velocity distributions which deviate systematically from a Maxwellian [ref] .
Phase-space structure in the local dark matter distribution and its signature in direct detection experiments · Introduction · 0.98
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This model is unlikely to be a good approximation to the real MW DM halo. $N$-body simulations produce halos with velocity distributions which deviate systematically from a Maxwellian [ref] .
Dark Matter Direct Detection with Non-Maxwellian Velocity Structure · Introduction · 0.98
Cites as background 28 papers
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We consider three different spherically symmetric parametrizations for the DM density profile: a Navarro-Frenk-White (NFW) profile [ref] [equation] an Einasto profile [ref] [equation] and a Burkert profile [ref] [equation] The scale radius, $r_s$, is related to the radius at which the logarithmic derivative of the density profile is equal to $-2$, while $ρ_s$ fixes the normalization.
A Universal Density Profile from Hierarchical Clustering · Milky-Way mass contributors · 0.87
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We consider three different spherically symmetric parametrizations for the DM density profile: a Navarro-Frenk-White (NFW) profile [ref] [equation] an Einasto profile [ref] [equation] and a Burkert profile [ref] [equation] The scale radius, $r_s$, is related to the radius at which the logarithmic derivative of the density profile is equal to $-2$, while $ρ_s$ fixes the normalization.
The Structure of Dark Matter Haloes in Dwarf Galaxies · Milky-Way mass contributors · 0.87
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The impact of microlensing data on the reconstruction of the MW potential has been studied in Ref. [ref] .
Dark Matter distribution in the Milky Way: microlensing and dynamical constraints · Microlensing · 0.80
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