Effects of correlation between halo merging steps J. Pan Purple Mountain Obs.

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