How cold is cold dark matter? - Institute of Astronomy€¦ · How cold is cold dark matter? Mark...

How cold is cold dark matter?

Mark Wilkinson

with Matt Walker, Justin Read, Walter Dehnen,

Gerry Gilmore,Ugur Ural, David Cole, David Maurin, etc.

Friday, 6 August 2010

Dark halo profiles of dSph galaxies

Homework assignment: How cold is cold dark matter?

Mark Wilkinson

with Justin Read, Walter Dehnen, Gerry Gilmore,Ugur Ural, David Cole, David Maurin, etc.


Hercules and the constant halo mass scale

Geha et al. (2008)

• Mass estimates of ultra-faint dSphs should be treated with caution:• stellar density profiles poorly determined • possible contamination of velocity samples• strong modelling assumptions required


Cleaned Hercules CMD

Aden et al., 2009, A&AFriday, 6 August 2010

Stromgren photometry in the Hercules dSph (Aden et al., 2009)

INT 14012

c1 = (u− v)− (v − b)Friday, 6 August 2010

Revising Hercules membership

Aden et al., 2009, ApJL

Velocity dispersion drops from to > 7 km s−1 3.7 km s−1



Geha et al. (2008)

• Hercules drops below common mass halo relation

• Mass estimates of ultra-faint dSphs should be treated with caution



Geha et al. (2008)

• Hercules drops below common mass halo relation

• Mass estimates of ultra-faint dSphs should be treated with caution

Aden et al., 2009


What’s the matter with Jeans Modelling?

2002MNRAS.330..778W

Wilkinson et al.,(2002)

Not all line profiles are well

fitted by Gaussians

Non-Gaussianity is the signature of

velocity anisotropy

Leads to mass-anisotropy degeneracy


What’s the matter with Jeans Modelling?

Wilkinson et al.,(2002)

Not all line profiles are well

fitted by Gaussians

Non-Gaussianity is the signature of

velocity anisotropy

Leads to mass-anisotropy degeneracy

2002MNRAS.330..778W


log[

!(r)

/M!

kpc"

3]

log[r/kpc]

"(r

)

r / kpc

#lo

s(R

)/km

s"1

R / kpc

LP(R

,vlo

s)

vlos / kms"1

$ = 0$ = 1/2$ = 1

0 10 200 0.1 0.2

0 0.5 1-2 -1 0

0

1

2

3

0

2

4

6

8

10

-1

-0.5

0

0.5

1

5

6

7

8

9

10

Dispersion profiles are full of degeneracies......


Fitting dSph dispersion profiles: Leo I

• Assume either NFW halo (1 free halo parameter) or generalised Hernquist profile (4 free halo parameters)

• Cored and cusped halo profiles fit almost equally well

! !"# !"$ !"% !"& '!

(

'!

'(

)*+,-./

!**+,0*1*2/

34)5

"*6*!!"(

"*6*!

"*6*!"(

"*6*'"789):

!

(

'!

'(

;<=

"*6*!!"(

"*6*!

"*6*!"(

"*6*'

Koch et al. (2007)Friday, 6 August 2010

Dark matter annihilation in dwarf galaxies: perspectives for a future !-ray observatory 9

4 DETECTABILITY OF MILKYWAY DSPHS

4.1 dSph Kinematics with the Spherical Jeans Equation

dSphs have negligible rotational support and in equilibrium their in-ternal gravitational potentials balance the random motions of theirstars. In order to estimate dSph masses, here we consider the be-havior of dSph stellar velocity dispersion as a function of distancefrom the dSph center (analogous to rotation curves of spiral galax-ies). Specifically, we use the stellar kinematic data of Walker et al.(2009) for the dSphs Carina, Fornax, Sculptor and Sextans, the dataof Mateo et al. (2008) for the Leo I dSph, and data from Mateoet al. (in preparation) for the Draco, LeoII and Ursa Minor dSphs.Walker et al. (2009, W09 hereafter) have calculated velocity disper-sion profiles from these same data under the assumption that line-of-sight velocity distributions are Gaussian. Here we re-calculatethese profiles without adopting any particular form for the velocitydistributions. Specifically, for a given dSph we parse the velocitysample into circular bins containing approximately equal numbersof member stars4, and within each bin we estimate the second ve-locity moment (squared velocity dispersion) as

!V 2" =1

N # 1

NX

i=1

[(Vi # !V ")2 # !2i ], (23)

where N is the number of member stars in the bin. We hold !V "fixed for all bins at the median velocity over the entire sample. Foreach bin we use a standard bootstrap resampling to estimate the as-sociated error distribution for !V 2", which is approximately Gaus-sian. Figure 11 displays the resulting velocity dispersion profiles,!V 2"1/2(R), which closely resemble the previously published pro-files.

In order to relate these velocity dispersion profiles to dSphmasses, we follow W09 in assuming that the data sample in eachdSph a single, pressure-supported stellar population that is in dy-namic equilibrium and traces an underlying gravitational potentialdominated by dark matter. Implicit is the assumption that the orbitalmotions of stellar binary systems contribute negligibly to the mea-sured velocity dispersions5. Further assuming spherical symmetry,the mass profile,M(r), of the dark matter halo relates to (momentsof) the stellar distribution function via the Jeans equation:

1"

ddr

("v2r) + 2

#(r)v2r

r= #GM(r)

r2, (24)

where "(r), v2r (r), and #r $ #(r) $ 1 # v2

!/v2r describe

the 3-dimensional density, radial velocity dispersion, and orbitalanisotropy, respectively, of the stellar component. Projecting alongthe line of sight, the mass profile relates to observable profiles, theprojected stellar density I(R) and velocity dispersion !p(R), ac-cording to (Binney & Tremaine 2008, BT08 hereafter)

!2p(R) =

2I(R)

Z !

R

„

1 # #rR2

r2

«

"v2rr%

r2 # R2dr. (25)

4 Kinematic samples typically are contaminated by interlopers from theMilky Way foreground. Following W09, we discard all stars for which thealgorithm described by Walker et al. (2009) returns a membership probabil-ity less than 0.95.5 Olszewski et al. (1996) and Hargreaves et al. (1996) conclude that this as-sumption is valid for the classical dSphs studied here, which have measuredvelocity dispersions of ! 10 km s"1. This conclusion does not necessarilyapply to recently-discovered “ultrafaint” Milky Way satellites, which havemeasured velocity dispersions as small as ! 3 km s"1 (McConnachie etal., in preparation).

Figure 11. Velocity dispersion profile data for the 8 classical dSphs, ob-tained as describe in the text. The solid lines correspond to the best-fit mod-els for the inner slope when ! is left free (dark), ! is fixed to 1 (blue), and! is fixed to 0 (red). (See next section for the list of free parameters in thefit.)

Notice that while we observe the projected velocity dispersion andstellar density profiles directly, the line-of-sight velocity dispersionprofiles provide no information about the anisotropy, #(r). There-fore we require an assumption about #(r). Here we assume # =constant, allowing for nonzero anisotropy in the simplest way. Forconstant anisotropy, the Jeans equation has the solution (e.g., Ma-mon & !okas 2005)

"v2r = Gr"2"r

Z !

r

s2"r"2"(s)M(s)ds. (26)

We shall adopt parametric models for I(R) andM(r) and then findvalues of the parameters ofM(r) that, via equations (25) and (26),best reproduce the observed velocity dispersion profiles.

4.1.1 Stellar Density

Stellar surface densities of dSphs are typically fit by Plummer(1911), King (1962) and/or Sersic (1968), profiles (e.g., Irwin &Hatzidimitriou 1995). For simplicity, here we adopt the Plummerprofile,

I(R) =L

$r2half

1[1 + R2/r2

half ]2, (27)

which has just two free parameters: the total luminosity L and theprojected6 half-light radius rhalf . Given spherical symmetry, thePlummer profile implies a 3-dimensional stellar density of (BT08)

"(r) = # 1$

Z !

r

dIdR

dR%R2 # r2

=3L

4$r3half

1[1 + r2/r2

half ]5/2

.

(28)

6 We define rhalf as the two-dimensional half-light radius—i.e., the radiusof the circle that encloses half of the total luminosity as seen in projection.

c" Xxxx RAS, MNRAS 000, 1–18

WalkerFriday, 6 August 2010

Jeans equations give simple relation between kinematics, the light distribution and the underlying mass distribution

Jeans equations allow unphysical models

M(r) = −r2

G

1ν

d νσ2r

d r+ 2

βσ2r

r

β(r) = 1 − v2

t 2v2

r

Plummer profile for stars+

NFW profile for dark matter+

isotropic velocity distribution

=

unphysical distribution function


Important discovery: Walker et al. observations of dSph dispersion profiles show that they are

generally flat to large radii

Using Jeans equations provides indication of mass scale of their haloes - not unreasonable procedure due

to above observation

However, something more sophisticated than Jeans modelling is needed to constrain the density profiles

of dSphs


• As ingredients for understanding dark matter/galaxy formation on small (sub-kpc) scales •To compare with simulations

• To interpret indirect detection data

Why might you care about dSph halo profiles?

dΦγ

dEγ(Eγ ,∆Ω) = Φpp(Eγ)× J(∆Ω)

J =

∆Ω

ρ2DM(l,Ω) dldΩ


Charbonnier et al., 2010

°int-210 -110 1

)-5

kpc

2lo

g10(

J/(M

8

9

10

11

12

13

14

15

16

FornaxMedian68% CL95% CL

DSphs and indirect detection


12 Charbonnier, Combet, Daniel, Hinton, Maurin, Power, Read, Sarkar, Walker, & Wilkinson

Table 2. Classical dSphs positions (Mateo 1998) sorted according to their distance. The remaing columns are the median, 68% and 95% CLs, respectively onthe inner slope !, and the J factor for "int = 0.01! and "int = 0.1! (from the six-parameter MCMC analysis).

dSph Position Median 68%95%

CLs

long. lat. d ! J(0.01!) J(0.1!) ! J(0.01!) J(0.1!)[deg] [deg] [kpc] [M2

" kpc#5] [M2" kpc#5] [M2

" kpc#5] [M2" kpc#5]

Ursa Minor 105.0 44.8 66 1.5 1.1 1013 1.2 1013 0.82#1.80.13#2.0

1.1 1011#3.7 1015

4.4 1009#4.2 1016

5.7 1011#3.7 1015

1.9 1011#3.9 1016

Sculptor 287.5 -83.2 79 1.4 8.4 1011 1.2 1012 0.53#1.80.06#2.0

8.2 1009#1.7 1015

1.4 1009#2.2 1016

1.8 1011#1.5 1015

9.6 1010#2.2 1016

Draco 86.4 +34.7 82 0.72 8.8 109 1.7 1011 0.21#1.40.02#1.8

1.3 1009#3.3 1011

5.7 1008#1.5 1014

9.1 1010#5.6 1011

5.7 1010#1.5 1014

Sextans 243.5 +42.3 86 0.94 2.1 1010 8.4 1010 0.30#1.60.03#1.9

2.5 1008#2.5 1013

3.7 1007#1.1 1016

5.2 1009#3.1 1013

1.7 1009#1.1 1016

Carina 260.1 -22.2 101 1.0 1.8 1010 4.6 1010 0.52#1.40.09#1.8

9.0 1008#1.7 1011

1.6 1008#1.4 1014

1.6 1010#2.2 1011

9.7 1009#1.4 1014

Fornax 237.1 -65.7 138 1.1 2.0 1010 1.2 1011 0.33#1.80.04#2.0

7.6 1008#3.1 1014

2.8 1008#1.7 1016

3.6 1010#3.2 1014

2.2 1010#1.6 1016

LeoII 220.2 +67.2 205 1.2 1.5 1011 2.6 1011 0.56#1.70.11#1.9

1.4 1010#7.5 1013

2.4 1009#4.3 1015

9.3 1010#8.4 1013

4.8 1010#4.5 1015

LeoI 226.0 +49.1 250 1.0 2.0 1010 6.0 1010 0.55#1.40.15#1.8

2.1 1009#1.1 1011

6.1 1008#3.9 1013

4.0 1010#1.3 1011

2.9 1010#4.1 1013

from GC [deg]!0 20 40 60 80 100

]-5

/kpc

2J

[M

910

1010

1110

1210

1310

1410

1510

1610

Galactic background (f=0.1)

=Einasto)sm" (smJ2(1-f)

=Einasto)cl"> (dP/dV=core, cl<J

>cl+<JsmJ2=(1-f)totJ

° =0.1int#

LeoISextans

LeoII

FornaxCarina

Draco

SculptorUrsa Minor68% and 95% CLs

Figure 15. J for the smooth (blue-dashed line), mean clump (red-dottedline) and sum (black-solid line) vs the angle from the Galactic centre. Sym-bols are J for the dSphs. The central point correspond to the median values,the solid bars to the 68% CLs, and the dotted bars to the 95% CLs. The in-tegration angle is 0.1!.

recover the standard result that the galactic centre outshines thedSphs, unless the latter have really cuspy profiles (the upper limitsin the dSphs correspond to large !).

The smaller 68% CLs for LeoI, Carina and Draco could seem,at a first glance, to be at odds with the fact that all the density profilelook alike (see Fig. 13). However, from Table 2, the upper value ofthe 68% CL on ! is respectively 1.4 for these 3 dSphs whereas itvaries between 1.6 and 1.8 for the others. The cuspier the profile,the brightest the signal, but this has to be weighted by the values of"s. As shown in Section 2.3, for profiles with given typical value ofM300 ! 107M", those with larger ! outshine those with smaller !.

clumps distributed in the range 10#6 ! 1010M" (see, e.g., Lavalle et al.2008, and references therein). The local DM distribution is fixed to the fidu-cial value #" = 0.3 GeV cm#3. The exact configuration is unimportanthere as this plot is mostly used for illustration purpose (a detailed and up-to-date study is left to a future analysis).

This is consistent with the fact that these 3 dSphs have smaller errorbars for the 68% CL. The same behavior is also found for the 95%CLs. From that we can conclude that the three most constraineddSphs (in terms of the J factor) are LeoI, Carina and Draco.

One may wonder whether or not a smaller integration anglemay increase the signal/background ratio (for this paragraph only,the signal is J from the dSph and the background is that of theDM galactic background), as the 68% lower limit for the dSph isclose to or even is below the DM background. The behaviour ofthe background (DM) and dSphs signal when changing the instru-ment resolution (i.e., #int) is understood as follows. Toward theGC, the integral is completely dominated by the central overden-sity and changing the integration angle has no impact on the re-sult. For any other position, the integrand appearing in Eqs. (14)and (15) does not vary much with #, so that Eq. (16) holds, givingan #2

int dependence. In a similar fashion, for dSphs having a cuspyprofile (! > 0), most of the flux is emitted in a small region encom-passed by the instrumental resolution, so that their J does not varywith the instrumental resolution. Conversely, if the profile is verysmooth and the integration angle smaller than #rs = rs/d (withinwhich the DM density is constant) J " #2

int (see also Section 3).For Draco, the decrease of the flux can be read off the 68% lowercurve (red-dashed line) in Fig. 14. Going from 0.1! to 0.01! de-creases J by a factor of # 100. It means that if the profile is core,increasing the angular resolution of the instrument has no effecton the signal/background of the dSphs. In the worst case scenario,dSphs are not even detectable against the DM smooth background.On the other hand, if the dSph profiles are steeper, having as smallas possible an angular resolution increase the signal/background ra-tio, which scales as #2

int. The extragalactic DM background is notconsidered in this work, but its J also scales as #2

int so that thesame reasoning holds true.

4.4 5-parameter MCMC analysis (inner slope ! fixed)

Form the !-free analysis of the previous section, the only robustprediction on the J factor that can be made at the 95% CL is thatit is at least above # 59M2

" kpc5. The constraint on ! is reallytoo loose, but on the other hand, profiles with value as high as! = 2 are not favored by any DM simulation. As increased resolu-

c" Xxxx RAS, MNRAS 000, 1–18

Charbonnier et al., 2010Friday, 6 August 2010

Distribution function modelling:breaking the mass-anisotropy degeneracy

• 2-parameter models

• Use un-binned spatial and velocity distribution of the stars

• Model convolved with observational errors and binary velocities

• Assumes equilibrium, spherical symmetry, cored inner halo

• N=160 stellar velocitiesKleyna et al. (2002); Wilkinson et al. (2002)

Draco


Assumptions:

• Spherical symmetry

• Equilibrium

• 2-integral distribution function: F(E,L)

The next step: Halo profiles


The next step: Halo profiles

ρhalo(r) =ρ0

r

rs

γ

1 +

r

rs

1/αα(β−γ)

• 2-integral distribution functions F(E,L) constructed using scheme of Gerhard; Saha

• Data analysed star-by-star: no binning

• No assumptions of Gaussianity

• General halo profile:


2-Integral Distribution function

x(E,L) =L

L0 + Lcirc(E)

F (E,L) = w(E)g(E,L) Gerhard (1991)

g(E,L) =

1 + c

1− x2a − 1

radial

1 isotropic1 + c

1− x2

a tangential

w(E) is a free function of energy - obtained numerically


Constructing the line of sight velocity distributions

• Choose surface brightness profile and halo

• Invert integral equation for using algorithm by P. Saha

• Project to obtain LOS velocity distribution on a grid of and

• Spline to required radii for observed stars, and convolve with individual velocity errors

vlos

R

R

Lmax =

2(Φ− E)r

ρ(Φ) =4π

r2

Φ

0w(E) dE

Lmax

0

g(E,L)LdL2(Φ− E)− L2/r2

w(E)


Fitting to kinematic data

• Surface brightness profile determined from data

• Markov-Chain-Monte-Carlo used to scan 13 dimensional parameter space

• Parameters: 3 velocity distribution parameters ( ); 5 halo parameters ( ); 5 light parameters ( );

• Multiple starting points for MCMC used - chains run in parallel and combined once “converged”

• Error convolution included

a, c, L0 α, β, γ, ρ0, rs

α, β, γ, ρ0, rs


Fornax - surface brightness profile

!""

!"!

!"#

!"!#

!"!!

!""

!"!

!$%&

%'$()*+,-&Friday, 6 August 2010

Fornax - velocity data

NB: Dispersion data not used to constrain modelsFriday, 6 August 2010

Fornax - PRELIMINARY density profile

log10(N)


Fornax - anisotropy profile

β(r) = 1− v2t (r)

2v2r(r)

log10(N)


Tests with spherical modelsCusp Core

• Artificial data sets of similar size, radial coverage and velocity errors to observed data set in Fornax


Tests with spherical models - large r_sCusp Core

• Artificial data sets of similar size, radial coverage and velocity errors to observed data set in Fornax


What about velocity anisotropy?

• Reassuring, but need to test with models that do not satisfy our modelling assumptions.....


Can we hide a small core?100 pc core 200 pc core

100pc core can be hidden, but 200 pc core starts to broaden 95% confidence limits


Likelihood distributionsFornax 200 pc “hidden” core

Clear signature of “hidden” core in distribution of likelihoods


Tidal tails and dSph dispersions

SDSS, 2002Friday, 6 August 2010

What about tides?

Profile can still be constrained even in presence of strong tides


No method is perfect.......

• Input (anisotropic) model not contained in family of models searched • Statistical test to identify models which fail is under development


Fornax - PRELIMINARY density profile

log10(N)


Implications for direct detection

[deg]int-210 -110 1

)-5

kpc

2lo

g10(

J/(M

8

9

10

11

12

13

14

15

16

FornaxMedian68% CL95% CL


Next generation dSph modelling: MCMC + Nbody

Ural et al., in prep.

• All dSphs orbit Milky Way

• Tides expected to affect kinematics at some level

• Need to explore degeneracies between tidal effects and static mass model parameters


Carina: Observational constraints

Ural et al., in prep.Figure 1.1: Surface brightness profile of Carina. Data from [72]

Rmax[kpc ] ! [M!/kpc2] d! [M!/kpc

2]

0.14835 1.64884 " 106 7.97466 " 104

0.2967 7.74082 " 105 3.39396 " 104

0.44505 2.36569 " 105 1.63247 " 104

0.5934 6.58713 " 104 1.04294 " 104

0.74175 2.47437 " 104 8.27153 " 103

0.8901 8.25756 " 103 6.8045 " 103

1.03845 9.03123 " 103 6.18768 " 103

1.1868 1.05641 " 103 5.70715 " 103

1.33515 5.16045 " 103 4.21236 " 103

1.4835 1.20763 " 103 2.5083 " 103

1.63185 3.0993 " 102 1.3362 " 103

Table 1.1: Surface brightness data for Carina [72]

3

Figure 1.2: Projected velocity dispersion profile. Data from [72]

R[kpc ] ! [km/s] d! [km/s]

0.0564 5.9 1.9

0.1051 6.1 2.0

0.1389 5.6 1.9

0.1697 6.4 1.9

0.1999 6.5 2.1

0.229 7.6 2.2

0.2622 6.8 2.0

0.3214 5.5 1.7

0.4261 6.0 1.9

0.8288 8.4 2.4

Table 1.2: Dispersion profile for Carina [72]

4


0.511.522.50

500

1000

‘!10

/!9

Nr p

0.511.522.5

50

100

e

0.511.522.50

0.5

1

Mh

alo

0.511.522.5

24

x 108

r ha

lo

0.511.522.50123

50 1000

500

rp

N

50 1000

0.5

1

50 100

24

x 108

50 100

0123

0 0.5 10

100

200

e

N

0 0.5 1

24

x 108

0 0.5 1

0123

2 4

x 108

0

200

400

Mhalo

N

2 4

x 108

123

1 2 30

200

400

rhalo

N

Figure 1.5: The correlations obtained from 960 in the MC1 set as in Figure 1.4. The x axes of each

column and the y axes of each row are fixed to one parameter. However, the two proper motion

values and the stellar mass are instead replaced by calculated orbital and dispersion parameters.

10

0.511.522.50

500

1000

‘!10

/!9

Nr p

0.511.522.5

50

100

e

0.511.522.50

0.5

1

Mh

alo

0.511.522.5

24

x 108

r ha

lo

0.511.522.50123

50 1000

500

rp

N

50 1000

0.5

1

50 100

24

x 108

50 100

0123

0 0.5 10

100

200

e

N

0 0.5 1

24

x 108

0 0.5 1

0123

2 4

x 108

0

200

400

Mhalo

N

2 4

x 108

123

1 2 30

200

400

rhalo

N

Figure 1.5: The correlations obtained from 960 in the MC1 set as in Figure 1.4. The x axes of each

column and the y axes of each row are fixed to one parameter. However, the two proper motion

values and the stellar mass are instead replaced by calculated orbital and dispersion parameters.

10

Ural et al., in prep

Pericentre

halo

sca

le le

ngth

Carina - preliminary

results


• Un-binned radial velocity data can be used to measure density profiles of dSph dark matter haloes

• Modelling of Fornax dSph suggests halo is cusped on scales 100pc - additional work underway to confirm performance of models.

• More data inside 100pc needed to probe innermost Fornax halo - data for this region in Carina in hand

• Current uncertainties on J for dSphs are very large - difficulty for indirect detection constraints

• Models which include tidal disturbance provide next step towards fully consistent modelling of dSphs (Ural et al. 2010)

Conclusions


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