Halos and Halo Substructure

Halos and Halo
Substructure:
operational definitions
and
properties
Importance of substructure
‰ Hierarchical models of large scale structure
– Halos: the small scale structure of LSS
– Subhalos: the small scale structure of halos
– Dynamical processes: tides, friction, interactions
‰ Galaxies as halo substructure
– The luminosity dependence of clustering
– Weak gravitational lensing
‰ Substructure of galaxies
– Strong gravitational lensing
‰ Substructure of our Galaxy
Halos and subhalos
As sites of galaxy formation
Vmax = the maximum of
circular velocity curve
= (GM(<r)/r)1/2
Galaxy formation
‰ Gas cools in virialized
dark matter ‘halos’.
Physics of halos is
nonlinear, but primarily
gravitational.
‰ Complicated
gastrophysics (star
formation, supernovae
enrichment, etc.) mainly
determined by local
environment (i.e., by
parent halo), not by
surrounding halos.
Halos and subhalos
As sites of galaxy formation
Also…
it is probably prudent to understand formation of
dark matter halos first before we dive into the messy
gastrophysics of galaxy formation…
"You've got to be very careful if you don't know where you are going
because you might not get there.” –Yogi Berra
Philosophical Foundations
We shall see that this is very relevant for (sub)halo
properties (especially mass)
So…
if it looks like substructure and if
It behaves like substructure, it’s substructure
Seriously though, following the above,
operational definition of substructure is
the method of how it is identified
This will be the main subject of this talk1
1Disclaimer:
“I'm not going to buy my kids an encyclopedia. Let them walk to school like I did.” – Yogi Berra
What’s a Halo?
time
density
Before defining halo substructure, define what is a halo…
What’s a Halo?
A halo is a nonlinear peak of the matter density field
With its boundary defined by a certain density contrast
The Clumpy Universe
Nonlinear peaks stand out visually
ART code:
LCDM 60 h-1 Mpc
σ8=0.9;
mp=109h-1 Msun;
ε = 0.5h-1 kpc
Halo Finders
‰ a fairly long history, but only recently halo finders were
challenged to identify both halos and their substructure
‰ each halo finder algorithm assumes a specific
(operational) definition of what a halo is and what are its
boundary and mass
‰ each particular definition is somewhat arbitrary and
may or may not suit a particular purpose.
‰ it’s not always obvious how to compare halos to
observed universe
“In theory, there is no difference between theory and practice.
In practice, there is.” – Yogi Berra
Traditional Halo Finders
originally not designed to identify substructure
Friends-Of-Friends (FOF)
Spherical Overdensity (SO)
Density Maxima (DenMax) algorithm
+ DenMax’s offsprings: SKID, HOP
“Halo identification is 90% halo definition. The other half is algorithmical”
– after Yogi Berra
Friends-Of-Friends
Einasto et al. 1984, Davis et al. 1985
http://www-hpcc.astro.washington.edu/tools/fof.html
All particles within a linking length, b, from each other are
linked into a single group. Therefore defines a halo to be a
connected region bounded by a density isosurface.
+ conceptually simple, b, is the only one free parameter
- can join nearby distinct halos into a single “virialized” group
- uses density field only, no physical criteria for checking
whether a halo is gravitationally bound
- mass of an FOF halo is not something that can be easily
related to masses measured in observations
Friends-Of-Friends
a cluster-sized halo identified by FOF in a
high-resolution cosmological simulation
Adopted from Fig. 4
Springel et al. 2001
Friends-Of-Friends
1 Mpc
the case of the Local Group
The Local Group
Local Group-like system in LCDM
(dark matter only)
Spherical
Overdensity
Lacey & Cole (1994)
Defines a halo as a matter within a sphere, centered on a density
peak and enclosing certain overdensity, δvir~200
+ conceptually simple, δvir, is the only one free parameter
± forces a halo to be spherical, although extension to ellipsoidal
overdensity is easy (e.g., M. Gross, PhD thesis,1998)
- uses density field only, no physical criteria for checking whether a
halo is gravitationally bound
Spherical
Overdensity
easy to code:
‰ calculate particle density (e.g., using SPH kernel with a smooth
routine: http://www-hpcc.astro.washington.edu/tools/smooth.html
‰ sort particles by density, start with the highest density and find
a sphere around it enclosing Dvir. Go to next densest particle, not
enclosed in the previous sphere – do the same for all particles with
local density > Dvir. Use linked list to search for particles within
a given R.
‰ I have a simple SO code in C, which I can give you.
Spherical Overdensity
a Milky Way-sized halo, circle indicates Rvir
Rvir
ART code sim:
LCDM, σ8=0.9;
Mvir=3x 1012h-1 Msun
Rvir=293h-1 kpc
~5x106 particles
within Rvir
ε = 0.1h-1 kpc
A note on the
Virial Radius/Overdensity
as a physical boundary
applies to all halo finders
‰ Motivated by the spherical collapse model and a reasonable choice,
but…
bound, virialized matter exists well beyond Rvir
(e.g., Gill et al. 2004; Kazantzidis et al. 2006)
equilibrium density profile extends beyond Rvir, without a clear infall
region for galaxy-sized halos (Prada et al. 2005)
virial shocks around halos in gasdynamics simulations are often found
at R~ few Rvir, not at the virial radius (Frenk et al. 1999; Nagai & Kravtsov
2003; Birnboim et al. 2006)
Density profiles of halos at r>Rvir
nothing special is happening at r=Rvir
Prada et al. (2005)
Mean radial velocity of dark matter
Halo density profile x radius2
cluster-sized halos
group-sized halos
MW-sized halos
dwarf-sized halos
Distance to the center of host halo in units of its
virial radius
“Virial” accretion shocks are
typically found at r>Rvir
gas temperature
gas density
“The virial radius is actually at three virial radii.” – Anatoly Klypin
A sample of galaxies simulated with ART code;
circles show the Rvir using the usual definition of Δvir
note that accetion shocks are typically at r>Rvir (Birnboim et al. 2006)
This means that usual mass
definitions are quite arbitrary
from the physical standpoint
‰ This is partly why so many different definitions are used in the
literature
However, all definitions define mass within some overdensity, so
different definitions differ in the overdensity they use
Some commonly used “virial” overdensity values, defined either wrt
critical density ρcrit or wrt mean density of the universe:
2500c, 500c, 200c, Dvir (=337m for LCDM with Ω=0.3), 200m, 180m
see an accurate fitting formula in Bryan & Norman (1998) for Δvir
for different cosmologies
My favorite definition of
virial Overdensity:
is the one that is prominently stated
in the text of a paper!
Failure to spell out the mass definition may cause a lot of
confusion later on…
And generates quotes, rivaling those of Yogi Berra…
“Let’s define virial radius as the virial radius.” – James Bullock
DenMax
Bertschinger & Gelb (1991)
Defines halos as connected regions above certain
overdensity, breaking neighboring peaks at the saddle points
‰ density is smoothed on a grid with a gaussian filter of a fixed width
‰ particles are moved along density gradients until they reach a
local density peak. Particles in the same peak are grouped into a
halo with the FOF algorithm with a possible check for E<0
+ fixes the “bridge” problem of the FOF
- uses a fixed smoothing of the density field -> fixed spatial resolution
SKID
Weinberg, Katz & Hernquist (1997)
http://www-hpcc.astro.washington.edu/tools/skid.html
Defines halos as connected regions above certain
overdensity, breaking neighboring peaks at the saddle points
‰ Similar to the DenMax in spirit, but uses Lagrangian estimator of the
local density, using Nneigh neighboring particles and SPH kernel
+ fixes the “bridge” problem of the FOF
+ adaptive spatial resolution
- imposes a mass resolution
HOP
Eisenstein & Hut (1998)
http://cmb.as.arizona.edu/~eisenste/hop/hop.html
Defines halos as connected regions above certain
overdensity, applying a set of rules to break neighboring peaks
into separate halos using isodensity contours
‰ Like SKID, uses Lagrangian density estimator
‰ Particles ‘hop’ to a neighbor with the maximum density, until they
reach a local maximum
+ more flexibility than previous method
- at the expense of more parameters
- no check for E<0
Halo statistics
halos are expected to be associated with galaxies and we can
study their, abundance, properties, clustering – important input
and test for models (e.g., semi-analytic models, halo models)
but…if accuracy is required (precision cosmology!), must
remember
‰ the different definitions of halos by different methods
‰ the meaning of mass in a particular halo finder and in
the analytic models (e.g., EPS) may differ
The Halo Mass Function
Reed et al. 2003
Can be rescaled to a “universal”
Form for all redshifts and
different cosmologies
(current parameterizations
by Sheth & Tormen 1999;
Jenkins etal. 2001;
Reed et al. 2005)
Halo mass function
Frac. deviation
Halo number density
Cosmology
Warren et al. 2005
Halo mass (FOF)
Sheth & Tormen 2001, 2002;
Jenkins et al. 2001;
Reed et al. 2003, 2005;
Warren et al. 2005
Halo mass (FOF)
Spatial distribution of halos
(the bias)
Seljak & Warren (2004)
Halo bias
b(M/M*) is approximately
independent of cosmology,
but M* depends on
cosmological parameters
⇒ can use this to provide
cosmology information in
addition to the mass function
Halo mass (in units of M*)
Cosmology
Halo mass
Merger histories of halos
Halo conecntration
time
Wechsler et al. 2002
Halo formation time
(PINpointing Orbit-Crossing Collapsed HIerarchical Objects)
http://adlibitum.oats.inaf.it/monaco/Homepage/Pinocchio/
a new fast code to generate catalogues of
cosmological dark matter halos with their
merger history.
It is able to reproduce, with very good
accuracy, the hierarchical formation of
dark matter halos from a realization of an
initial (linear) density perturbation field,
given on a 3D grid.
It works similarly to a conventional N-body
simulation, but it is based on a powerful
analytic perturbative approach, called
Lagrangian Perturbation Theory and runs
in just a small fraction of the computing
time, producing promptly the merging
histories of all halos in the catalogue.
Accurately reproduces merger
Histories from N-body simulations,
Not very accurate in spatial
distribution of halos though…
Li et al. 2005 (astroph/0510372)
Formation time dependence of halo bias
30 ChiMpc
youngest 20%
of halos
oldest 20% of
halos
dark matter
Gao, Springel & White 2005
Formation time dependence of halo bias
quantitatively
oldest 10%
Halo bias
oldest 20%
youngest 20%
youngest 10%
mean
Halo mass (in units of M*)
Gao, Springel & White 2005
Halo bias
Concentration time dependence of halo bias
Halo mass (in units of M*)
Wechsler, Zentner et al. 2006
Spherical or
non-spherical?
Should we define the mass within a spherical region (SO) or
non-spherical region (FOF, SKID, HOP)?
‰ given that there is no well-defined physical boundary of
halos, the answer depends on what you want to do with the
halo masses
‰ if you are studying halo formation and mass assembly
theoretically – you may want to use non-spherical definition
if, however, you are comparing halo masses to masses of say
clusters measured within a spherical aperture, you should use
spherical radius to define the mass..
Cumulative abundance (>M)
in this comparison of theory and data
of the cluster mass function, the halo masses
are defined within spherical radius as is done
for the observed cluster masses
evolution of
theoretical
halo mass function
compared
to the
cluster abundance
evolution in the
400d X-ray survey
Vikhlinin et al. 2006
In preparation
M500 - mass within radius enclosing overdensity
of 500 x ρcrit
Halo Substructure
Virialized regions of halos are not smooth
"Nobody goes there anymore because it's too crowded.” – Yogi Berra
Halo Substructure
Depends on numerical resolution
and was not observed in dissipationless cosmological simulations
prior to ~1996-1997
phase space density
“by no stretch of the imagination does one form
‘galaxies’ in the current cosmological simulations: the
physical model and dynamic range are inadequate to
follow any but the crudest details...” Summers et al. 1995
about dissipationless N-body simulations
space density
Credit: Ben Moore http://www.nbody.net
Halo Substructure
Depends on how you look at it
The same MW-sized halo viewed
in a random direction
along its major axis and filament
Origin of anisotropic subhalo distribution:
anisotropic accretion along filaments
[Knebe et al. 04; Zentner et al. 05; Liebeskind et al. 05]
+ dynamical evolution of orbits [Zentner et al. 05]
accreting
subhalo
tracks
vertical is
direction
of host halo
major axis
Formation of a Milky Way-sized halo
“You can observe a lot just by watching.” – Yogi Berra
z = 10
z=7
z=5
z=3
time
z=2
z=1
z = 0.5
z=0
ART code simulation:
standard LCDM, σ8=0.9; mp=6x105h-1 Msun; ε = 0.1h-1 kpc
Mvir=3x1012h-1 Msun; Rvir=293h-1 kpc; ~5x106 particles within Rvir
Halo Subhalos
were themselves isolated, distinct halos
in the past
Halo Substructure
Approx. (but not exactly) self-similar
Halo Substructure
Approx. (but not exactly) self-similar
A cluster or a galaxy?
Halo Substructure
Depends on how cold the dark matter is
Credit: Ben Moore http://www.nbody.net
Substructure savvy
Halo Finders
"He hits from both sides of the plate. He's amphibious.“ - Y.B.
‰ almost all halo finders identify substructure with density
peaks in configuration space, but use velocity information
later
‰ virial radius is not meaningful for subhalos. Various
definitions of truncation radius, rt, loosely related to the tidal
radius, are adopted. Subhalo mass is m(<rt)
‰ Vmax is often used as a property of subhalos, in lieu of m
‰ every halo finder checks for E<0 and removes unbound
particles. Halo properties are computed using bound
particles
Substructure savvy
Halo Finders
“It’s déjà vu all over again” – Yogi Berra
FOF -> Hierarchical FOF (HFOF; Gottloeber, Klypin & Kravtsov 1998)
1) run halo finder with an hierarchy of linking lenghts
(smoothing scales, or mass resolution)
2) combine resulting catalogs, removing duplicate
objects and identifying which halos are substructure of
another halo
3) remove ‘fake’ halos by checking whether they are selfbound (E<0)
similarly, hierarchical DenMax (Neyrick et al. 2004)
SKID (Ghigna et al. 1998; recently, Diemand et al. 2004), HOP (???)
Substructure savvy
Halo Finders
"You can't compare me to my father. Our similarities are different." – Dale Berra
SO -> Bound Density Maxima (BDM; Klypin et al 1999)
1a) tracer spheres of radius rsearch, around a subsample of dm
particles, move along density gradients to a nearest density
peak (Klypin et al. 1999)
1b) use SPH kernel to estimate local density for each particle.
Sort particles by density. Identify peaks, as spheres of radius
rsearch, centered on the currently densest particle in the list.
(Kravtsov et al. 2004)
2) for each peak, iteratively remove unbound particles
3) compute halo/subhalo properties: Mvir, Rvir, Mt, rt, Vmax, etc.
using only gravitationally bound particles
BDM in operation
Kravtsov et al. 2004
"I wish everybody had the drive he (Joe DiMaggio) had. He never did anything wrong
on the field. I'd never seen him dive for a ball, everything was a chest-high catch,
and he never walked off the field.” – Y.B.
BDM vs SKID
cumulative circ velocity function
average for several MW-sized halos
Vmax = the max
of circular velocity
curve
=(GM(<r)/r)1/2
SKID
BDM
comparisons
in collaboration
with Juerg Diemand
and Andrew Zentner
BDM vs SKID
cumulative mass function
average for several MW-sized halos
SKID
BDM
comparisons
in collaboration
with Juerg Diemand
and Andrew Zentner
BDM vs SKID
differences in the truncation radius
(and hence mass) definition
Substructure savvy
Halo Finders
Subhalo Finder (SubFind; Springel et al. 2001)
1) identify parent groups (e.g., with the
standard FOF)
2) compute local density for particles using
Nneigh neighbors with SPH kernel
3) identify substructure as regions enclosed
by isodensity surfaces traversing a saddle
point.
4) check whether subhalos are bound,
remove unbound particles
SubFind in operation
original FOF halo
subhalos
diffuse particles
Springel et al. 2001
Substructure savvy
Halo Finders
MLAPM Halo Finder (MHF; Knebe, Gill & Gibson 2004)
+ MLAPM Halo Tracker (MHT)
1) identifies density peaks with centroids of the
deepest refinement levels of MLAMP
2) find a halo radius: minimum of Rvir or rt
3) iteratively remove unbound particles
4) compute properties of halos
5) Halo Tracker uses an earlier epoch output to
correlate particles of halos existing then with
particles at the current epoch -> track particles
that used to belong to a halo, check whether they
(or a fraction of them) still form a bound halo
Radial distribution of subhalos:
MHT vs the rest of the world
“I can understand how he won twenty five games.
What I don’t understand is how he lost five” – Yogi Berra
Subhalos:
M > 2x1010h-1 Msun
Gill, Knebe & Gibson 2004
Distance to the center of host halo in units of its
virial radius
Substructure savvy
Halo Finders
Voronoi Bound Zones (VOBOZ; Neyrinck, Gnedin & Hamilton 2005)
VD
cells
1) estimate local density for each particle using V
Voronoi diagram (VD)
2) particles hop to the their VD neighbors with
largest density, until they reach a density peak.
This defines zones – VD volume which contributes
particles to the same peak
3) particles in the zone are grouped similarly to SubFind.
VD
zones
4) check whether subhalos are bound, iteratively remove
unbound particles
5) use VD to estimate probability for a halo to be real
VOBOZ in operation
Different curves correspond to samples of halos with
different significance thresholds
Halo correlation function
Subhalo mass function in a
Cluster-sized halo
dotted
line:
BDM
Neyrinck, Gnedin & Hamilton 2005
Velocity and spatial biases
radial distribution of subhalos
Surface density profile
Colin et al. 99, 00; Ghigna et al. 00; Diemand et al. 04; Gao et al. 04; Nagai & Kravtsov 05
factor of two difference in
abundance of galaxies
and subhalos?
projected cluster-centric radius in units of the virial radius
But…
radial distribution of subhalo selected by
Vmax depends on σ8
Radial (number) density profile
"I never blame myself when I'm not hitting. I just blame the bat and if it keeps up, I change bats.”
dark matter
S8=0.9 (red
and blue)
S8=0.7
comparisons
in collaboration
with Juerg Diemand
and Andrew Zentner
halo-centric radius in units of the virial radius
radial distribution of subhalo depends on
how they are selected (i.e., on their operational definition!) …
[Nagai & Kravtsov 2005, using N-body + gasdynamics simulations]
Radial distribution of subhalos for the same cluster with different selection
subhalos
selected
using
Vmax today
subhalos
selected
using
bound
mass today
subhalos
selected
using
Vmax they
had at the
accretion
epoch
subhalos
selected
using
the mass they
had at the
accretion
epoch
cluster-centric distance in units of the virial radius
because present day subhalo mass and Vmax
are affected by tidal forces from the host
fraction of initial mass, Vmax
lost due to tides
and the average effect depends on radius
[Nagai & Kravtsov 2005]
Selection based on a weakly evolving property,
such as stellar mass or subhalo M/Vmax before it
is accreted, results in
much reduced spatial and velocity bias
surface
density
profile
Nagai & Kravtsov 05;
Faltenbacher et al. 05
Maccio et al. 2006
projected cluster-centric radius in units of R200
Substructure in
Phase space
Motivation
Methods
Results
Phase space density
of halos
Is close to a simple power law, simpler than the density
Structure of halos -> probably hinting at the processes that
determine halo structure
Navarro & Taylor:
f NT
ρ (r )
= 3
∝ r −1.875
σ v (r )
V(f)df = volume of phase space occupied by f in
the range (f,f+df)
Halo Phase-Space Density
Real Density
Phase-Space Density
Halo Phase-Space Density
Spatial Density
Phase-Space Density
Computing phase
space density of halos
Is tricky…
several methods have been developed in the last couple of years
Arad, Dekel & Klypin 04
Arad & Dekel 05
Ascasibar & Binney 05
Sharma & Steinmetz 05
Delaunay Tesselation
Delaunay 1934; Tanemura, Ogawa & Ogita 1983; van de Weygaert 1994
used to measure phase space density by
Arad et al. 2004; Arad & Dekel 2005
Measuring f(x,v): Delaunay Tesselation
Field Estimator (DTFE)
SHESHDEL package: available by request from Itai Arad
www.phys.huji.ac.il/~itaia
particle i
m
f i = (d + 1)
Vi
Binary Tree (FiEstAS)
Ascasibar & Binney 2005
Sharma & Steinmetz 2005
Phase space density distribution
is nearly power law
Arad, Dekel & Klypin 04;Ascasibar & Binney 05;Sharma & Steinmetz 05
v(f)
f
PDF of Phase-Space Density: Power law
V(f)
109Mʘ
1012Mʘ
1015Mʘ
Arad, Dekel & Klypin
f
Sharma & Steinmetz 2005
Is v(f)∝f-2.5 determined by substructure?
ΛCDM
No short waves
Real-Space Density
Moore et al.
Phase-Space density
ΛCDM
No short waves
Without small-scale
structure f(v) is steeper
ΛCDM
v∝f-2.5
v(f)
no short waves
v∝f-3.2
f
Conclusions
‰ halo finders, and hence halo and substructure
definitions, have matured
‰ however, we now need more detailed comparisons of
how different halo identification algorithms define halo
properties, because conclusions will often depend on this
‰ subhalos provide a wealth of information useful for
understanding galactic satellites, galaxy clustering, ++
No, thank you. “I usually take a two hour nap, from 1 to 4”
“I want to thank you all for making this day necessary” – Yogi Berra
Who is Yogi Berra?
"They say he's funny. Well, he has a lovely wife and family, a beautiful
home, money in the bank, and he plays golf with millionaires.
What's funny about that?" – Casey Stengel