effects of correlation between halo merging steps j. pan purple mountain obs
TRANSCRIPT
Effects of correlation between halo merging steps
J. Pan
Purple Mountain Obs.
outline
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 To understand LSS we need knowledge of
dark matter distribution and corresponding evolution
The halo model approximates the LSS by Dark matter groups into halos gravitationally Baryons trapped in halo form galaxies
LSSHow matter is partitioned into halos?
What is the status of matter inside a halo?
Starting points: dark matter’s distribution as a random surface Spatial density fluctuation of dark matter is
stochastic: a random surface defined on 3D space
Standard description of the random surface: random walks For a general random surface:
local height v.s. distance For a cosmic matter field:
222~ zyxrh
R
Smoothed density fluctuation )()( RWR v.s. the variance over the smoothed field )(2 RS
the random walk: )(~)( RSR
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
0,0, SR
Excursion set theory (2) halo formation as a single barrier first-upcrossing problem
3/1),( haloMRRSS
3/1),( haloMRRSS Connection to halo mass
If > c, the matter within R will collapseto form a halo
c serves as an absorbing barrier on the random walk
dSSndMMn )()(
No. of halos of mass M
No. of trajectories whichfirstly crosses barrier at S
Excursion set theory (3) merging as a two barriers first-upcrossing problem )(, 11 zM C
)(, 22 zM C
21
21
21
)()(
MM
zz
zz
CC
21122
21122
)](,|)(,[
),|,(
dSzSzSn
dMzMzMn
CC
No. of progenitors of mass M1 at z1 given a parent halo of mass M2 at z2
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
const.C If the collapsing is spherical,there is analytical solution
Q
S
Q
S
Q C2
2
2
1
No matter 1-barrier or 2-barriers jumping
If ellipsoidal collapsing, no analytical solutionnumerical
)(SBC
Success of the excursion set theory Agreement with numerical simulations
Halo mass function
Conditional mass func. Halo bias
more application …
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 Brownian motion
contains no memory of its past history
Environment impact is null in standard excursion set theory
1
2: merging
Environment: The overdensityat large scale
Haloforms
ProgenitorHalo forms
P(2|1)=P(2, 1)/P(1)=P(2)xP(1)/P(1)=P(2) !
So…
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
)(tX
t
)( tX
t
If the hurst exponent = ½, it is the random Brownian motion
FBM: the simplest random walks of sub-diffusion, proposed byMandelbrot, a fractal concept.
Widely used in modeling random surface in geology, self-organization growth of structures in solid physics,even the stock price fluctuation…
FBM: properties
)(tX
t
)( tX
t
< 0.5 anti-persistent negatively correlated with past > 0.5 persistent positively correlated with past = 0.5 stable no memory of past
0.8
0.5
0.2
back to excursion set …correlation!
1
2
0
modified excursion set theory: Fokker-Planck equation
Q
S
QS
S
Q C2
212
By substitution SS~2
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 the correlation between
merging steps can change halo mass function dramatically
positive correlation reduces number of large mass halos
No way to work out the ellipsoidal collapse (moving barrier problem)
spherical collapse
modified excursion set theory: conditional mass function – weak positive correlation?
modified excursion set theory: halo formation time distribution
effects of positive correlation > 0.5 small mass halos
less young, more old large mass halos
more young, less old
summary
With FBM, the excursion set theory can be modified to include correlation between merging steps with minimal efforts: easily transplanted from known results easy implementation to SAM, Monte-Carlo merging tree algorithm
It seems there is weakly positive correlation shown in simulations. 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 mass function halo bias environmental effects