Astronomy 541 NFW dark matter halo profile

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Astronomy 541 NFW dark matter halo profile

• Presented by
• Shirley Ho

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NFW profile(prelude)

• We have a lot of different analytic formula that
  try to describe the density distribution of a
  halo.
• For Example, we have the following
• - Softened power law ellipsoid
• a 1 - softened isothermal model
• a 0 -Hubble Model
• a -2 - Plummer model (see Binney Tremanine
  1987)

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More models

• - Pseudo-Jaffe ellipsoid (from Jaffe
  model(1993) where
• - King Model
• - De Vaucouleurs model
• - Hernquist model

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Finally, we have NFW

• An analytic formula that tells us how halos look
  like
• (Navarro, Frenk White 1996)
• This scales as 1/r at small radii, and 1/r3 at
  large radii.

So this is NFW? Where does it come from ?
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Outline

• What is a dark matter halo ?
• A crash course on spherical collapse
• Numerical determination of a fitting formula for
  halos
• What are the possible going-wrongs?
• What do people say about NFW?
• Recent developments
• Acknowledgments

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What is dark matter halo?
Not spherical!
Thanks to Gus Evrard and White et al. (1993)
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A crash course on spherical collapse
For simplicity, we are neglecting radiation and
Lambda, since structure formation (probably)
kicks in mainly after epoch of radiation and
matter equality and before Lambda comes into play
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The parametric solutions tells us a turn-around
that matter collapses. Note that the collapse
does happen in finite amount of time if we plot
the actual r instead of the development angle.
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After a simple calculation (which we will skip
here), we have
Then, we relate time to the scale factor (a), and
Relate the overdensity to the cosmological
parameters.
Then, we realize that A and B are INITIAL
conditions
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• Here we are , with the spherical collapse
  relation
• At the collapse, we have

But spherical collapse does NOT collapse to a
point in the end, Halo virializes !
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Schematically, after a halo virializes
Now, we have a SIMPLE spherical HALO. And
remember 178 is the magic number
(approximately 200) often used to define
collapsed objects .
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Now, instead of determining the halo profile
through calculation, we have a numerical
determination of a fitting formula for a halo
NFW profile

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More on the definitions
Concentration c r??? / rs
r200 is the radius at which the density is 200
times of the critical density.
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More definitions circular velocity

• V200 is the circular velocity define at r200
• Vc( r ) (GM/r)1/2
• And they are related by
• and x r/r200

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Why is NFW so cool ?

• A simple analytic form for halos of various mass
  scale !

Scaled density profile of the most massive and
least massive halos look very similar except
at the core. The less massive halo has higher
density at the center possibly due to the fact
that they form earlier, and density perturbation
at earlier times of the universe is more
concentrated.
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How does real data fit NFW?

• Projected radial profile from 2MASS

radial profile fits projected NFW form with
c3.00.3
Note that they are not fitting the whole range of
r, so this does not show the problem of the
inner slope (later).
Lin, Mohr, Stanford 2004
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Here are the questions

• Is there a universal profiles for dark matter
  halos?
• Why would there be a universal profile (if there
  is one) - hierarchical clustering (not to be
  discussed here today)
• Would NFW be the one profile (if there is a
  universal one) ?
• These are questions we would like to
  understand, hopefully this talk would stimulate
  some discussions or new ideas ) But before
  that, lets take a look at the issues that NFW
  has been facing

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Issues that NFW has to address

• -Numerical Limitations
• Gravitational softening Initial
  redshifts Number of particles Size of
  simulation box Number of time steps Boundary
  conditions, Size of the mesh

-Observational issues observations of
flatter inner slopes of halos using
LSB and Lensing -Theoretical side With
better simulations, one finds halos with
steeper inner slopes!
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Numerical issues Gravitational softening

• Due to the fact that gravitational force in small
  scale becomes very strong, thus it is hard for
  the equations to handle such big numbers,
  therefore, at twice the softening length, they
  will use the Newtonian value between 2 particles.
• The halo structure is resolved at radii larger
  than the softening length (0.01 r200)

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But NFW (the authors) have tested using different
parameters

• Tested different box size ranging from 3 to 58
  Mpc
• Tested different initial redshifts from 4.5 to 43
  (to start the simulations from) -- initial
  redshifts matter quite a bit to the inner slopes,
  since the density fluctuation are more
  concentrated at earlier times.
• Tested different gravitational softening lengths.
• They decided to use time steps of 10-4 to 10-5 of
  Hubble time, softening lengths of 10-2 r200,
  boxes of 360 and 30Mpc on the side. (I wont go
  on with this list for this talk ) )

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After the numerical issues, we have the
observational controversies

• - Rotational curves of LSB (low surface
  brightness) galaxies found an extended region in
  the center of the halo which has constant dark
  matter density (aka cusps) (work by Flores
  Primack 1994, Moore 1994)
• - Strong gravitation lensing found real
  clusters fitting various models (eg. Allen 1998
  Clowe et al. 2000 Clowe Schneider 2001
  Gavazzi et al. 2003, Gentile et al. 2004)
• Examples of these issues will follow

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Observational LSB rotation curves

• Flores Primack 1994 pointed out that at small
  radii, the halos are not going to be singular
  (aka the density would not increase monotonically
  with radius). It is ruled out by their analysis
  of the flat rotation curves of the low surface
  brightness (LSB) galaxies.
• LSBs are used because there are not much baryonic
  matter inside LSB systems, therefore, there are
  not much baryonic infall that could have modified
  the dark matter halo profile.

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Observations strong gravitational lensing

• Gavazzi et al 2003 analyzed the radial mass
  profile of MS2137.3-2353 and found that
  Isothermal sphere fits the system better than
  generalized NFW (where inner slope is allowed to
  vary).

How to decide the mass profile using
gravitational lensing ? One usually model the
system using some profiles (such as isothermal
sphere or NFW) and then try to generate arcs
that matches what are observed.
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Strong gravitational lensing..

• Gentile et al. 2004 have decomposed the
  rotational curves for 5 spiral galaxies into
  their stellar, gaseous and dark matter components
  and fit the inferred density distribution with
  various models and found that models with a
  constant density core is preferred.

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Difficulties in mass modeling using strong
lensing

• There are also other examples of strong lensing
  that I didnt bring up, but would like to talk
  about the difficulties of mass modeling and why
  should we be cautious about trusting them.
• Since strong lensing systems tend to be massive
• - they most probably contain a lot of
  substructures and even huge galaxies (eg. cD
  galaxies) and plus the fact that most of the
  time, halos are not spherical, there could be a
  huge parameter space to search through if we try
  to model all of these, so most of the time,
  people will deal with one or two of the issues,
  but not all of them, thus making their
  conclusions not as strong as it looks like.

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Then after the observations, we have the
theoretical issues

• - New and better simulations come up with higher
  resolution, which resolves more of the inner
  region of the halos, found steeper inner slopes .
• We have a lot of different ideas that comes in
  recently, such as work by Fukushige Makino
  2001 Moore et al. 1998,1999 Power et al. 2003,
  Navarro et al 2003, Hayashi et al. 2003, Navarro
  2004, Bullock 2001 and many many others)
• note I apologize in advance here to people who
  may be in this room and worked on similar things,
  but I didnt cite their papers here since there
  are just so many of good papers out there.

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Steeper inner slope?...
The dashed line shows the generalized NFW Profile
with inner slope -1.5

• Fukushige Makino (2001) and Moore et al. 1998,
  1999 both independently used better N-body
  simulations and reported that NFW fits to their
  simulated halos underestimate the dark matter
  density in the inner regions (r

Fukushige Makino 2001
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What could have gone wrong?
Hayashi et al. 2003

• Power et al. , Hayashi et al. and Navarro et al.
  2003 presented us a series of paper on inner
  structure of halos, suggesting ways to reconcile
  with the observations. Here is what they said
• With better simulations, spherically averaged
  cluster sized halos, the halos become
  progressively shallow from the virial radius
  inwards and show no sign of turning into a
  power-law.
• Most halo profiles become shallower more
  gradually than predicted by NFW, NFW turns over
  too sharply.

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More issues of halos One thing we should not
forget is that halos are not spherical!

• Jing Suto (2002)

There are a lot of issues that have been caused
by spherically the halos in either simulations
(spherically average over the halos, may cause
artificial clustering) or observations
(eg.modeling of mass systems without taking the
asphericity of the halos). However, as Jing
Suto (2002) showed us, halos are far from
spherical and this is important to bear in our
minds.
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New Models! (by Navarro !)
Shown below Logarithmic slope of the
density profile. Strong dashed line is the newly
suggested model, fitting (from left to right)
dwarf, galaxy- sized, cluster-sized halos.

• Navarro 2003 proposed a new analytic form
• Where r-2 is defined as the radius at which

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Conclusions (YEAH!)
What can we conclude from now on? - Is NFW a
universal profile for dark matter halos? -
Um.. Probably yes to a certain extent, since it
fits clusters pretty well, but when it
comes to the inner slope, then it is
questionable. It does however give us a useful
and simple model for halo mass profile -
Is there a really universal profile ? - It
seems like the simulations have fit NFW pretty
well over quite a scale (from dwarf to
cluster-sized halos), but again, we do not
see NFW fitting perfectly at the inner region. -
Since NFW is such a simple model, should one be
able to derive it from the power spectrum
??? - Halo mass profile is one of the frontiers
of Large scale structure!

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Acknowledgments

• I would like to thank Scott Tremaine for his
  suggestions and discussions. I would also like to
  thank Gus Evrard and Martin White who have first
  taught me about halos and clusters.
• Finally , Thanks to all of you for coming to this
  seminar )

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References ( a brief list)

• Navarro, Frenk White 1996, 1997
• Navarro et al. 2004
• Power et al. 2003
• Hayashi et al. 2003
• Fukushige Makino 2001
• Moore et al 1998, 1999
• Flores Primack 1994
• And many more .