Transcript Document

Effects of correlation
between halo merging steps
J. Pan
Purple Mountain Obs.
outline
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Introduction to the excursion set theory of
halo model
The fractional Brownian motion
Modified excursion set theory based on FBM
summary
The halo model of large scale
structure
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To understand LSS we need knowledge of
dark matter distribution and corresponding
evolution
The halo model approximates the LSS by
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Dark matter groups into halos gravitationally
Baryons trapped in halo form galaxies
How matter is partitioned into halos?
LSS
What is the status of matter inside a halo?
Starting points: dark matter’s distribution
as a random surface
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Spatial density fluctuation of dark matter is
stochastic: a random surface defined on 3D
space
Standard description of the random
surface: random walks
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For a general random surface:
2
2
2
h
~
r

x

y

z
local height v.s. distance
For a cosmic matter field:
R
Smoothed density fluctuation  ( R)   W ( R)
2
S


( R)
v.s. the variance over the smoothed field
the random walk:
 ( R) ~ S ( R)
Halos: as peaks of the random surface
PEAK
HALO
Dark matter distribution (evolution)
= halo distribution (evolution) + matter distribution (evolution) in halos
Excursion set theory
Excursion set theory (1) the frame
The random walk is a Brownian Motion
Q: the number density of trajectories
within dS and d
If no boundary
R  , S  0,   0
Excursion set theory (2) halo formation as
a single barrier first-upcrossing problem
If  > c, the matter within R will collapse
to form a halo
c serves as an absorbing barrier
on the random walk
No. of halos of mass M
S  S (R), R  M halo
1/ 3
n( M )dM  n( S )dS
Connection to halo mass
S  S (R), R  M
1/ 3
halo
No. of trajectories which
firstly crosses barrier at S
Excursion set theory (3) merging as a two
barriers first-upcrossing problem
M1 ,  C ( z1 )
M 2 ,  C ( z2 )
z1  z2
 C ( z1 )   C ( z2 )
M1  M 2
No. of progenitors of mass M1 at z1
given a parent halo of mass M2 at z2
n(M 2 , z2 | M1 , z1 )dM2
 n[S2 ,  C ( z2 ) | S1 ,  C ( z1 )]dS2
No. of trajectories has first-upcrossing over c(z1)
at S1 given its first upcrossing at S2 over c(z2)
Halo formation/merging processes:
Fokker-Planck equation
No matter 1-barrier or 2-barriers jumping
 C
Q
1  2Q
Q



S
2  2
S

If the collapsing is spherical,
there is analytical solution
If ellipsoidal collapsing,
no analytical solution
numerical
 C  const.
 C  B(S )
Success of the excursion set theory
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Agreement with numerical simulations
Conditional mass func.
Halo mass function
Halo bias
more application …
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The central engine of semi-analytical models
of galaxies (halos are warm beds of galaxies)
the 21cm emission during re-ionization
Critical problems : halo formation time
distribution
Critical problems: age dependence of halo
bias
Old halos are more strongly clustered than young halos of the same mass
Dense environment induces more old halos
The missing link: correlation between
random walk steps
Progenitor
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Brownian motion
contains no memory of
its past history
Environment impact is
null in standard
excursion set theory
Halo forms
Halo
forms
Environment:
The overdensity
at large scale
P(2|1)=P(2, 1)/P(1)=P(2)xP(1)/P(1)=P(2) !
2: merging
1
So…
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What if there is correlation, i.e. the random walk
performed by cosmic density field is not a random
Brownian motion?
How to model this correlation?
What will the correlation bring up to halo growth
history, exactly?
No one knows.
We need a class of random walk of which Brownian
motion is just a special case.
Fractional Brownian motion: definition
FBM:
the simplest random walks of
sub-diffusion, proposed by
Mandelbrot, a fractal concept.
X (t   )
X (t )
t
t 
Widely used in modeling
random surface in geology,
self-organization growth of
structures in solid physics,
even the stock price fluctuation…
If the hurst exponent a = ½, it is the random Brownian motion
FBM: properties
0.2
X (t   )
X (t )
t
t 
0.5
0.8
a < 0.5 anti-persistent negatively correlated with past
a > 0.5 persistent
positively correlated with past
a = 0.5 stable
no memory of past
back to excursion set …correlation!
2
1
0
modified excursion set theory: FokkerPlanck equation
2
 C
Q

Q
Q
 aS 2a 1


S
 2
S

By substitution S 2a
~
S
It is the equation for random Brownian motion!
The standard excursion set theory results can be easily converted
for solutions.
modified excursion set theory: halo mass
function
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the correlation between
merging steps can
change halo mass
function dramatically
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positive correlation
reduces number of large
mass halos
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No way to work out the
ellipsoidal collapse
(moving barrier problem)
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spherical collapse
modified excursion set theory: conditional
mass function – weak positive correlation?
modified excursion set theory: halo
formation time distribution
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effects of positive
correlation a > 0.5
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small mass halos
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less young, more old
large mass halos
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more young, less old
summary
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With FBM, the excursion set theory can be modified to include
correlation between merging steps with minimal efforts:
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It seems there is weakly positive correlation shown in simulations.
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easily transplanted from known results
easy implementation to SAM, Monte-Carlo merging tree algorithm
conditional mass function
halo formation time distribution
Troubles: no analytical to solve the Fokker-Planck equation for FBM
with moving barriers, we are struggling to have accurate
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mass function
halo bias
environmental effects