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Deriving the velocity distribution of Galactic dark matter particles from the rotation curve data
Pijushpani Bhattacharjee,1,2,* Soumini Chaudhury,1,† Susmita Kundu,1,‡ and Subhabrata Majumdar3,§
1Astroparticle Physics and Cosmology Division and Centre for Astroparticle Physics, Saha Institute of Nuclear Physics,1/AF Bidhannagar, Kolkata 700064, India
2McDonnell Center for the Space Sciences and Department of Physics, Washington University in St. Louis,Campus Box 1105, One Brookings Drive, St. Louis, Missouri 63130, USA
3Department of Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India(Received 10 October 2012; revised manuscript received 21 January 2013; published 26 April 2013)
The velocity distribution function (VDF) of the hypothetical weakly interacting massive particles
(WIMPs), currently the most favored candidate for the dark matter in the Galaxy, is determined directly
from the circular speed (‘‘rotation’’) curve data of the Galaxy assuming isotropic VDF. This is done by
‘‘inverting’’—using Eddington’s method—the Navarro-Frenk-White universal density profile of the dark
matter halo of the Galaxy, the parameters of which are determined by using the Markov chain
Monte Carlo technique from a recently compiled set of observational data on the Galaxy’s rotation curve
extended to distances well beyond the visible edge of the disk of the Galaxy. The derived most-likely local
isotropic VDF strongly differs from the Maxwellian form assumed in the ‘‘standard halo model’’
customarily used in the analysis of the results of WIMP direct-detection experiments. A parametrized
(non-Maxwellian) form of the derived most-likely local VDF is given. The astrophysical ‘‘g factor’’ that
determines the effect of the WIMP VDF on the expected event rate in a direct-detection experiment can be
lower for the derived most-likely VDF than that for the best Maxwellian fit to it by as much as 2 orders of
magnitude at the lowest WIMP mass threshold of a typical experiment.
DOI: 10.1103/PhysRevD.87.083525 PACS numbers: 95.35.+d, 98.35.Df, 98.35.Gi
Several experiments worldwide are currently trying todirectly detect the hypothetical weakly interacting massiveparticles (WIMPs), thought to constitute the dark matter(DM) halo of our Galaxy, by looking for nuclear recoilevents due to scattering of WIMPs off nuclei of suitablychosen detector materials in low background undergroundfacilities. The rate of nuclear recoil events depends cru-cially on the local (i.e., solar neighborhood) density andvelocity distribution of the WIMPs in the Galaxy [1],which are a priori unknown. Estimates based on a varietyof observational data typically yield values for the localdensity of DM, �DM;�, in the range 0:2–0:4 GeV cm�3
[ð0:527–1:0Þ � 10�2M� pc�3] [2]. In contrast, not muchknowledge directly based on observational data is availableon the likely form of the velocity distribution function(VDF) of the WIMPs in the Galaxy. The standard practiceis to use what is often referred to as the ‘‘standard halomodel’’ (SHM), in which the DM halo of the Galaxy isdescribed as a single-component isothermal sphere [3], forwhich the VDF is assumed to be isotropic and of Maxwell-Boltzmann (hereafter simply ‘‘Maxwellian’’) form, fðvÞ /exp ð�jvj2=v0
2Þ, with a truncation at an assumed value of
the local escape speed, and with v0 ¼ vc;�, the circular
rotation velocity at the location of the Sun. Apart fromseveral theoretical issues (see, e.g., Ref. [4]) concerning
the self-consistency of the SHM as a model of a finite-size,finite-mass DM halo of the Galaxy, high resolutioncosmological simulations of DM halos [5] give strongindications of significant departure of the VDF from theMaxwellian. On the other hand, these cosmological simu-lations do not yet satisfactorily include the gravitationaleffects of the visible matter (VM) components of the realGalaxy, namely, the central bulge and the disk, whichprovide the dominant gravitational potential in the innerregions of the Galaxy including the solar neighborhoodregion.The VDF of the DM particles at any location in the
Galaxy is self-consistently related to their spatial density aswell as to the total gravitational potential, �ðxÞ, at thatlocation. For a spherical system of collisionless particles(WIMPs, for example) with isotropic VDF satisfying thecollisionless Boltzmann equation, the Jeans theorem [3]ensures that the phase space distribution function F ðx; vÞdepends on the phase space coordinates ðx; vÞ only throughthe total energy (per unit mass), E ¼ 1
2v2 þ�ðrÞ, where
v ¼ jvj, r ¼ jxj. For such a system, given a isotropicspatial density distribution �ðrÞ � R
d3vF ðEÞ, one can
get a unique F by the Eddington formula [3,6]
F ðEÞ ¼ 1ffiffiffi8
p�2
�Z E
0
d�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiE ��
p d2�
d�2þ 1
ffiffiffiE
p�d�
d�
�
�¼0
�; (1)
where�ðrÞ � ��ðrÞ þ�ðr ¼ 1Þ is the relative potentialand E � �Eþ�ðr ¼ 1Þ ¼ �ðrÞ � 1
2v2 is the relative
energy, with F > 0 for E > 0, and F ¼ 0 for E � 0.
*[email protected]†[email protected]‡[email protected]§[email protected]
PHYSICAL REVIEW D 87, 083525 (2013)
1550-7998=2013=87(8)=083525(5) 083525-1 � 2013 American Physical Society
The latter condition implies that at any location r, the VDFfrðvÞ ¼ F =�ðrÞ has a natural truncation at a maximum
value of v, namely, vmax ðrÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi2�ðrÞp
.Thus, given a isotropic density profile of a set of
collisionless particles, we can calculate the VDF, frðvÞ,using Eq. (1) provided the total gravitational potential�ðrÞin which the particles move is known. A direct observa-tional probe of�ðrÞ is provided by the rotation curve (RC)of the Galaxy, the circular velocity of a test particle as afunction of the galactocentric distance. In this paper wereconstruct the total gravitational potential �ðrÞ in theGalaxy directly from the Galactic RC data and then useEq. (1) to obtain the VDF, frðvÞ, of the WIMPs at anylocation in the Galaxy [7].
We shall assume that the DM density profile to be usedon the right-hand side of Eq. (1) is of the universalNavarro-Frenk-White [9] form, which, when normalizedto DM density at solar location, can be written as
�DMðrÞ ¼ �DM;��R0
r
��rs þ R0
rs þ r
�2; (2)
where R0 is the distance of the Sun from the Galacticcenter. The profile given by Eq. (2) has two free parame-ters, namely, the density �DM;� and the scale radius rs.
The total gravitational potential seen by the DM particle,�, is given by � ¼ �DM þ�VM, where �DM is the DMpotential corresponding to the density distribution given byEq. (2) and �VM is the total potential due to the VMcomponent of the Galaxy. The latter can be effectivelymodeled [10] in terms of a spheroidal bulge superposedon an axisymmetric disk, with density distributions given,
respectively, by �b ¼ �b0ð1þ ðr=rbÞ2Þ�3=2, where �b0 andrb are the central density and scale radius of the bulge,
respectively, and �dðR; zÞ ¼ ��2zd
e�ðR�R0Þ=Rd e�jzj=zd , whereR and z are the axisymmetric cylindrical coordinates with
r ¼ ðR2 þ z2Þ1=2, Rd and zd are the scale length and scaleheight of the disk, respectively, and �� is its local surfacedensity. The corresponding gravitational potentials forthese density models, �bulge and �disk, can be easily ob-
tained by numerically solving the respective Poisson equa-tions, thus giving �VM ¼ �bulge þ�disk.
The density models specified above have a total of sevenfree parameters, namely, rs, �DM;�, �b0, rb, ��, Rd, and zd.
We determine the most-likely (ML) values and the68% C.L. upper and lower ranges of these parameters byperforming a Markov chain Monte Carlo (MCMC) analy-sis (see, e.g., Ref. [11]) using the observed RC data of theGalaxy. For a given set of the Galactic model parameters,the circular rotation speed, vcðRÞ, as a function of thegalactocentric distance R, is given by
v2cðRÞ ¼ R
@
@R½�DMðR; z ¼ 0Þ þ�VMðR; z ¼ 0Þ�: (3)
For the observational data, we use a recently compiled set ofRC data [12] that extends to galactocentric distances wellbeyond the visible edge of the Galaxy. This data setcorresponds to a choice of the local standard of rest(LSR) set to ðR0;vc;�Þ¼ð8:0kpc;200kms�1Þ [13]. For theMCMC analysis, we use the �2-test statistic defined as
�2 � Pi¼Ni¼1 ð
vic;obs
�vic;th
vic;error
Þ2, where vic;obs and v
ic;error are, respec-
tively, the observational value of the circular rotationspeed and its error at the ith value of the galactocentricdistance, and vi
c;th is the corresponding theoretically calcu-
lated circular rotation speed. For priors on the free parame-ters involved, we have taken the following ranges of therelevant parameters based on currently available observa-tional knowledge: For the VM parameters, �b0: ½0:1–2��4:2�102M� pc�3 [10]; rb: ½0:01–0:2� � 0:103 kpc [10];��: ½35–58�M� pc�2 [14]; Rd: ½1:7–3:5� kpc [10,15].The parameter zd has been fixed at 340 pc [16] since theresults are fairly insensitive to this parameter. Forthe DM parameters we took a wide enough prior range forrs: ½0:1–100� kpc and �DM;�: ½0:1–0:5� GeV cm�3 consis-
tent with values recently quoted in the literature [2].The results of our MCMC analysis are summarized in
Table I and Fig. 1. Figure 2 shows the theoretically calcu-lated rotation curve for the most-likely set of values of theGalactic model parameters obtained from the MCMCanalysis and listed in Table I, and its comparison with theobserved rotation curve data. In Table II, we display thevalues of some of the physical quantities of interest char-acterizing the Galaxy, derived from the Galactic parame-ters listed in Table I. The values in Table II are inreasonably good agreement with the values of these quan-tities quoted in recent literature [8,12,17]. The relativelylarge uncertainties in the values of some of the quantities
TABLE I. The most-likely values of the Galactic model parameters, as well as their 68% C.L. lower and upper ranges, means andstandard deviations, obtained from our MCMC analysis using the observed rotation curve data.
Parameter rs (kpc) �DM;� ðGeV=cm3Þ �b0 � 10�4 ðGeV=cm3Þ rb (kpc) �� ðM�=pc2Þ Rd (kpc)
Most-likely 30.36 0.19 1.83 0.092 57.9 3.2
Lower 14.27 0.17 1.68 0.083 55.51 2.99
Upper 53.37 0.23 2.0 0.102 58.0 3.27
Mean 41.35 0.20 1.84 0.092 54.30 3.14
Standard deviations 20.51 0.02 0.059 0.001 3.47 0.11
BHATTACHARJEE et al. PHYSICAL REVIEW D 87, 083525 (2013)
083525-2
that receive dominant contribution from the DM haloproperties at large galactocentric distances are simply areflection of the relatively large uncertainties of the rota-tion curve data at those distances.
The Galactic model parameters determined above allowus to reconstruct the total gravitational potential �ðxÞ atany location in the Galaxy. Because of the axisymmetricnature of the VM disk, this potential is nonspherical. To useEq. (1), which is valid only for a spherically symmetricsituation, we use the spherical approximation [8,18],�VMðrÞ ’ G
Rr0 MVMðr0Þ=r02dr0, where MVM is the total
VM mass contained within r [19].The resulting normalized speed distribution frðvÞ �
ð4�v2ÞfrðvÞ [withRfrðvÞdv ¼ 1] evaluated at the loca-
tion of the Sun, giving the most-likely f�ðvÞ, is shown inFig. 3. For comparison, we also show in the same figurethe best Maxwellian fit (BMF) [with fMaxwell� ðvÞ /v2 exp ð�v2=v2
0Þ] to the most-likely f�ðvÞ obtained from
MCMC analysis. We also compare our results with thosefrom four large N-body simulations [5].
As evident from Fig. 3, the speed distribution differssignificantly from the Maxwellian form. We find that thefollowing parametrized form, which goes over to the stan-dard Maxwellian form in the limit of the parameter k ! 0,
0.5
1
1.5
2
2.5
3
3.5
4
0.01 0.1 1 10 100
v c [1
00 k
m/s
ec]
R [kpc]
VM
DM
FIG. 2. Rotation curve of the Galaxy with the most-likely setof values of the Galactic model parameters listed in Table I. Thedata with error bars are from Ref. [12].
30 60 900.1
0.2
0.3
0.4
0.5
0
0.2
0.4
0.6
0.8
FIG. 1. The 2D posterior probability density function for darkmatter parameters (rs � �DM;�), marginalized over the visible
matter parameters.
TABLE II. The most-likely values of various relevant physicalparameters of the Milky Way and their upper and lower rangesderived from the most-likely and 68% C.L. upper and lowerranges of values of the Galactic model parameters listed inTable I.
Derived quantities Unit Values
Bulge mass (Mb) 1010M� 3:53þ1:81�1:29
Disk mass (Md) 1010M� 4:55þ0:2�0:22
Total VM mass (MVM ¼ Mb þMd) 1010M� 8:07þ2:01�1:51
DM halo virial radius (rvir) kpc 199:0þ75�53:5
Concentration parameter ðrvirrsÞ � � � 6:55þ5:01
�2:05
DM halo virial mass (Mh) 1011M� 8:61þ14:01�5:22
Total mass of Galaxy (MVM þMh) 1011M� 9:42þ14:21�5:37
DM mass within R0 1010M� 1:89þ0:72�0:3
Total mass within R0 1010M� 7:09þ1:9�1:15
Total surface density at
R0ðjzj � 1:1 kpcÞM� pc�2 69:21þ2:52
�3:55
Total mass within 60 kpc 1011M� 3:93þ2:15�1:41
Total mass within 100 kpc 1011M� 5:92þ4:35�2:56
Local circular velocity ðvc;�Þ km s�1 206:47þ24:67�16:3
Local maximum velocity ðvmax ;�Þ km s�1 516:02þ120:85�97:58
0
0.1
0.2
0.3
0.4
0.5
0.6
0 1 2 3 4 5 6
f O· (v)
[(10
0 km
/sec
)−1 ]
v [100 km/sec]
Via Lactea
BNMF
BMF
ML
0 1 2 3 4 5 6v [100 km/sec]
Ling et al.
BNMF
BMF
ML
0
0.1
0.2
0.3
0.4
0.5
0.6
f O· (v)
[(10
0 km
/sec
)−1 ]
Aquarius
BNMF
BMF
ML
GHALO
BNMF
BMF
ML
FIG. 3. Normalized local speed distribution f�ðvÞ correspond-ing to the ML set of values of the Galactic model parametersgiven in Table I (solid curve) and its uncertainty band (shaded)corresponding to the 68% C.L. upper and lower ranges of theGalactic model parameters. The four panels show comparison ofour results with those from four different N-body simulations[5], as indicated. In each panel, the best non-Maxwellian fit(BNMF) [Eq. (4)—almost indistinguishable from the ML curve]as well as the BMF, the latter with the form fMaxwell� ðvÞ/v2expð�v2=v2
0Þ truncated at vmax ;�¼516 kms�1 (see Table II)
and with the free parameter v0 determined to be 206 km s�1, arealso shown.
DERIVING THE VELOCITY DISTRIBUTION OF . . . PHYSICAL REVIEW D 87, 083525 (2013)
083525-3
gives a good fit to our numerically obtained most-likelylocal speed distribution shown in Fig. 3:
f�ðvÞ � 4�v2ð�ð�Þ � �ð�max ÞÞ; (4)
where �ðxÞ ¼ ð1þ xÞke�xð1�kÞ, � ¼ v2=v2
0, �max ¼v2max ;�=v2
0, v0 ¼ 339 km s�1, and k ¼ �1:47. As a quan-titative measure of the deviation of a model form of thelocal speed distribution, fmodel, from the numericallyobtained ML form fML shown in Fig. 3, the quantity �2
f �ð1=NÞPN
i¼1½fMLðviÞ � fmodelðviÞ�2 has a value of 7:2�10�5 for the parametrized form given by Eq. (4) comparedto a value 1:7� 10�3 for the best Maxwellian fit shownin Fig. 3. Note also that our results differ significantly fromthose obtained from the N-body simulations.
In Fig. 4 we show the most-likely frðvÞ’s at severaldifferent values of the galactocentric distance r. Noticehow the peak of the distribution shifts towards smallervalues of v and the width of the distribution shrinks, aswe go to larger r, with the distribution eventually becom-ing a delta function at zero speed at asymptotically largedistances, as expected. The non-Maxwellian nature ofthe distribution at all locations is also clearly seen, withthe Maxwellian approximation always overestimating thenumber of particles at both low as well as extreme highvelocities. The inset in Fig. 4 shows our results for the
pseudophase space density, Q � �=hv2i3=2, as a functionof r, and its comparison with the power-law behaviorpredicted from simulation results [20]. Note the agreementwith the power-law behavior at large distances but strongdeviation from it at smaller galactocentric radii, whichwe attribute to the effect of the visible matter: For a givenDM density profile, the additional gravitational potentialprovided by the VM supports higher velocity dispersion
of the DM particles, making Q smaller than that for theDM-only case.We now discuss the implications of our results for the
analysis of direct-detection experiments. The differentialrate of nuclear recoil events per unit detector mass(typically measured in counts/day/kg/keV), in which aWIMP (hereafter generically denoted by � with massm�) elastically scatters off a target nucleus of mass mN
leaving the recoiling nucleus with a kinetic energy ER, canbe written as [1]
dRdER
ðER; tÞ ¼ �ðq2 ¼ 2mNERÞ2m��
2��gðER; tÞ; (5)
where �� � �DM;� is the local mass density of WIMPs,
�ðq2Þ is the momentum transfer dependent effectiveWIMP-nucleus elastic cross section, � ¼ m�mN=ðm� þmNÞ is the reduced mass of the WIMP-nucleus system, and
gðER;tÞ¼Z umax ðtÞ
u>umin ðERÞd3u
uf�ðuþvEðtÞÞ�ðumax�uminÞ; (6)
is the crucial ‘‘g factor’’ that contains all information aboutthe local VDF of the WIMPs [21]. In Eq. (6) the variable u(with u ¼ juj) represents the relative velocity of theWIMPwith respect to the detector at rest on Earth, and vEðtÞ is the(time-dependent) velocity of Earth relative to the Galactic
rest frame. The quantity umin ðERÞ ¼ ðmNER=2�2Þ1=2 is the
minimum WIMP speed required for giving a recoil energyER to the nucleus, and umax ðtÞ is the (time-dependent)maximumWIMP speed [4] corresponding to the maximumspeed vmax (defined in the Galactic rest frame) for the VDFunder consideration.
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6
f r(v)
[(10
0 km
/sec
)-1]
v [100 km/sec]
200 kpc
50 kpc
8 kpc
2 kpc
10-14
10-13
10-12
10-11
10-10
10-9
0.1 1 10
Q [G
eV m
-6 s
ec3 ]
r/ rs
(r/rs)-1.875
This Work
FIG. 4. Normalized speed distribution of the DM particles atvarious galactocentric radii (solid curves), corresponding to themost-likely set of values of the Galactic model parameters givenin Table I. The curves (dotted) for the corresponding bestMaxwellian fit are also shown for comparison. The inset showsthe pseudophase space density of DM, Q � �=hv2i3=2, as afunction of r.
10-6
10-4
10-2
100
102
104
1 10
ζ
mχ [GeV]
SodiumXenon
FIG. 5. The ratio (solid curves), � � gMLðEthÞ=gMaxwellðEthÞ,of the g factor calculated with our ML form of f�ðvÞ shown inFig. 3 to that for the best Maxwellian fit to it also shown in Fig. 3,as a function of the WIMP mass m�, for two different target
nuclei, namely, sodium and xenon, both with Eth ¼ 2 keV. Theshaded bands correspond to the uncertainty bands of f�ðvÞshown in Fig. 3. The calculations are for June 2, when Earth’svelocity in the Galactic rest frame is maximum.
BHATTACHARJEE et al. PHYSICAL REVIEW D 87, 083525 (2013)
083525-4
Note that the quantity gðER; tÞ takes its largest value atER ¼ Eth, the threshold energy for the experiment underconsideration. To illustrate the effect of the non-Maxwellian nature of the VDF and its uncertainty, wedefine the quantity � � gMLðEthÞ=gMaxwellðEthÞ, the ratioof the g factor calculated with our ML form of f�ðvÞ shownin Fig. 3 to that for the best Maxwellian fit to it also shownin Fig. 3, both evaluated at ER ¼ Eth. A plot of � as afunction of the WIMP mass m�, for two different target
nuclei, viz. sodium and xenon, in both case with Eth ¼2 keV, is shown in Fig. 5.
The lowest WIMP mass that can be probed by a givenexperiment is given by m�;min ¼mN½ð2mNðvmax ;�þvEÞ2=EthÞ1=2�1��1. As seen from Fig. 5, the effect of thedeparture from Maxwellian distribution is most significant
at the lowest WIMP mass where the difference can be asmuch as 2 orders of magnitude.To summarize, a first attempt has been made to derive
the velocity distribution (assumed isotropic) of the darkmatter particles in the Galaxy directly using the rotationcurve data. The distribution is found to be significantlynon-Maxwellian in nature, the implication of which is asizable deviation of the expected direct-detection eventrates from those calculated with the usual Maxwellianform.
We thank Satej Khedekar and Subir Sarkar for usefuldiscussions. P. B. thanks Ramanath Cowsik for discussionsand for support under the Clark Way Harrison VisitingProfessorship program at the McDonnell Center for theSpace Sciences at Washington University in St. Louis.
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