Dark Matter Distribution in the Milky Way

The distribution of dark matter in the Milky Way is of central importance for any indirect search for a WIMP annihilation signal in the Milky Way Dark Matter Halo.

Consider a small volume with dark matter density ρ. For a WIMP with mass M, there are on average N = ∆V ρ/M dark matter particles in the considered volume

∆V. The probability that an annihilation occurs in the volume is proportional to the number of possible combinations of two particles that can annihilate. For Majorana (’self annihilating’) WIMPs this is ^N₂= N(N −1)/2∼ N²/2 for large number of particles N in the volume. For Dirac WIMPs in contrast, the number of possible combinations is smaller than for Majorana WIMPs. Given thatN Dirac particles are in a volume, there are on average N/2 particles and N/2 antiparticles and the number of combinations is N²/4. As a consequence of a WIMP annihilation,N_γ γ-rays are assumed to be created isotropically. The fraction dΩ/4π of the created γ-rays is emitted in the solid angledΩ.

The expected flux of γ-rays created in WIMP annihilations and observed in the field of viewdΩ is obtained by integrating many of the considered small volumes of dark matter

densityρ(s) over the line of sightds,

where the proportionality constanthσviis called the ’velocity averaged annihilation cross section’. For Dirac WIMPs, eq. 4.1 has to be divided by an additional factor of two¹. Given a line of sight from an observer on earth towards a direction with angleα to the galactic center, the distancer(s) from a pointson the line of sight to the galactic center is given by

r(s) =^qr²_E+s²−2r_Escos(α).

Here, r_E ∼ 8.5 kpc is the distance between the earth and the galactic center. The dependence of the expected signal strength on the squared dark matter density along the line of sight motivates a discussion of the current knowledge of the dark matter distribution in the Milky Way. Apart from details, there are two different approaches to gain information on the dark matter distribution in the Milky Way. Unfortunately both methods lead to two different predictions. This is sometimes called the ’core/cusp problem’ in literature.

Computer Simulations

Computer simulations are used to solve the N-body problem for a large set of ’particles’

that interact gravitationally starting from a nearly flat matter distribution with some seed density fluctuations in a realistic cosmology at the recombination epoch. Here,

’particles’ is used as a general term that denotes a mass unit in the simulation. The mass unit used for a simulation depends in general on the resolution, i.e. spatial scale, that is investigated. The large scale structure of the universe with the dynamics of galaxy clusters is simulated with particles that are much more massive than for example the galaxy scale structure. To investigate the matter density distribution on galaxy scales, a full simulation of a universe with low spatial resolution is typically considered and the formation of galaxies of a given size is searched for. Selected seed galaxies are then simulated in more detail with a higher spatial resolution and lower ’particle’ masses. In the following, the current main results from the investigation of Milky Way size galaxies are itemized.

• The dark matter distribution in a Milky Way size galaxy is neither spherically symmetric nor smooth. In other words, a substantial amount of substructure on top of a smooth component is predicted by N-body simulations.

• The smooth component of the dark matter density can be well fitted by universal

1It appears as if a substantial amount of confusion about the factors of 2 in eq. 4.1 exist in the literature.

No reference for a justification argument like the one given above could be found.

4.1 Introduction

Figure 4.1: Einasto and NFW parametrizations of the dark matter density as a function of the distance to the galactic center for the Milky Way. Parameters are adapted from Pieri et al. [2011]. The colored regions indicate the signal and background regions of the rotated pixel (signal region in pink, background re-gion in green) and the On/Off method (signal rere-gion in blue and background region in yellow) discussed later in this text.

and simple functions. The Einasto profile with α= 0.17,

describes the result of the Aquarius simulation and the Navarro Frenk White (NFW) profile,

fits the Via Lactea II simulation results (see Pieri et al. [2011] and references therein).

In both cases,rsandρsare the scaling radius and scaling density which are fit parameters respectively andRis the distance to the center of the galaxy. It is obvious that the dark matter density predicted by the mentioned N-body simulations is increasing towards the center of a galaxy where a ’cusp’ is expected. In the vicinity of the center of a galaxy, the smooth component of the dark matter density predicted in N-body simulations is dominant and the subhalo component is negligible. Figure 4.1 compares the Einasto and the NFW parametrization of the dark matter density profile with parameters taken from² Pieri et al. [2011] that lead to the dark matter density in the vicinity of the sun to be∼0.3 GeV/cm³.

Observations

The presence of dark matter in galaxies is inferred from investigations of the dynamics of galaxy constituents. Especially the flatness of the velocity curves of spiral galaxies hints towards the presence of dark matter in the investigated spiral galaxies. For the velocity v(R) of a test mass around the mass M(< R) enclosed within the radius R it holds in general that

v(R)∼ s

M(< R)

R .

To obtain a constant velocity curvev(R), the density profile must followρ(R)∼1/R² as this leads toM(< R) = 4π^R₀^Rdr r²ρ(r)∼R. This means that the dark matter density distribution must followρ(R)∼1/R² far away from the center of the galaxy, i.e. outside of the luminous radius of a galaxy where most of the visible mass resides. The density distributions quoted above that fit the results of N-body simulations are compatible with this behavior. Unfortunately it is very difficult to obtain conclusive information on the dark matter density distribution within the optical radius of a spiral galaxy through observations. The measurement of the velocity curves of investigated spiral galaxies is not precise enough to distinguish between different proposed density profiles. Especially,

2The parameters from the given reference are also used in the analysis presented below. They are ρs= 8.1·10⁶MSun/kpc³andrs= 21 kpc for the NFW profile andρs= 2.8·10⁶MSun/kpc³,rs= 20.0 kpc andα= 0.17 for the Einasto profile.

4.1 Introduction the precision is not sufficient to distinguish between a cusped profile motivated by N-body simulations and a profile with constant or nearly constant density within the optical radius of a spiral galaxy. An example for a dark matter density distribution with nearly constant density within the optical radius r₀ and a ρ(R)∼1/R² dependence outside of the optical radius is the Burkert profile

ρ_Burkert(R) =ρ₀ r³₀

(R+r0)(R²+r²₀) .

Gentile et al. [2004] investigate five spiral galaxies which are similar to the Milky Way and conclude that the velocity curves are better fit by Burkert profiles but a cusped profile like NFW cannot be ruled out. As discussed in Strigari [2012], the situation is very similar for the investigation of Milky Way Dwarf galaxies where cusped profiles appear also less preferred than cored profiles. In particular for the Fornax and Sculptor Dwarf galaxies, a NFW dark matter density profile is disfavored at a confidence level of 96% and 99%, respectivly, and in both cases cored profiles give compatible descriptions (Walker and Penarrubia [2011]).

Obviously there is a hint for a conflict between the prediction for dark matter density distributions from N-body simulations and the observational results. Possible sources for this and other discrepancies such as the missing satellite problem, which is the mismatch between the small number of observed Milky Way dwarf galaxies and the large predicted number (see Strigari [2012] for a discussion), are

• the possible failure of the ΛCDM cosmology that is underlying the N-body simu-lations (see Strigari [2012] for a discussion) or on a less fundamental scale

• the disregard of baryons in N-body simulations. Most N-body simulations neglect the influence of baryonic matter and the complicated physics associated with f.i.

supernovae explosions and star formation with the argument that galaxies are dominated by dark matter. Most older attempts to include baryonic effects in the N-body simulation predicted that the central dark matter density becomes even steeper than for dark matter particles only (see Abazajian and Harding [2012] and references therein). However, very recently the opposite conclusion, i.e. that the inclusion of baryons is flattening the central dark matter density distribution, was drawn (see Governato et al. [2012]). Those results indicate an attractive solution for the ’core/cusp problem’, i.e. the hint for a discrepancy between the cusped dark matter density distribution predicted by N-body simulations of only gravitationally interacting particles and the observations that tend to prefer cored dark matter density distributions. Additionally, the abundance of substructure in form of dark matter subhalos is predicted to be reduced when baryons are included in N-body simulations which is also in better agreement with the observational data (see Governato et al. [2012]).

Conclusion

Currently, it appears impossible to draw a definite conclusion on the dark matter density distribution in general galaxies and in the Milky Way in particular. For the concrete case of the Milky Way, three different possible dark matter density profiles are conceivable.

• A steep dark matter density profile which is singular towards the galactic center.

Figure 4.1 shows the Einasto and NFW dark matter density distribution with parameters adopted from Pieri et al. [2011] for the Milky Way. For distances larger than ∼ 45 pc to the galactic center, both characterizations obtained from different N-body simulations agree within a factor of two.

• The dark matter density distribution follows in general a parametrization as ob-tained from N-body simulations. However, the impact of baryons lead in the vicinity of the galactic center to a flattening of the dark matter density distribu-tion. Governato et al. [2012] find that this leads to a nearly constant dark matter density within the inner ∼500 pc of a galaxy (corresponding to∼3−4^◦ angular distance to the galactic center for an observer on earth when the Milky Way is considered).

• The dark matter density distribution of the Milky Way follows a Burkert profile and is thus constant up to the optical radius of the Milky Way.

A search for WIMP annihilation in the Milky Way halo should consider the uncertainty in the dark matter density profile as far as possible. The methods that are discussed below are not necessarily sensitive to all of the listed dark matter distributions. In particular, no sensitivity at all can be achieved in the case of a constant dark matter density distribution up to the optical radius parametrized by the Burkert profile.