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

2. outline Introduction to the excursion set theory of halo model The fractional Brownian motion Modified excursion set theory based on FBM summary

3. 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 LSS How matter is partitioned into halos? What is the status of matter inside a halo?

4. 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

5. Standard description of the random surface: random walks For a general random surface: local height v.s. distance For a cosmic matter field: R Smoothed density fluctuation v.s. the variance over the smoothed field the random walk:

6. Halos: as peaks of the random surface PEAK HALO

7. Dark matter distribution (evolution) = halo distribution (evolution) + matter distribution (evolution) in halos Excursion set theory

8. Excursion set theory (1) the frame The random walk is a Brownian Motion Q: the number density of trajectories within d S and d  If no boundary

9. Excursion set theory (2) halo formation as a single barrier first-upcrossing problem Connection to halo mass 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 No. of trajectories which firstly crosses barrier at S

10. Excursion set theory (3) merging as a two barriers first-upcrossing problem No. of progenitors of mass M 1 at z 1 given a parent halo of mass M 2 at z 2 No. of trajectories has first-upcrossing over  c(z 1 ) at S 1 given its first upcrossing at S 2 over  c(z 2 )

11. Halo formation/merging processes: Fokker-Planck equation If the collapsing is spherical, there is analytical solution No matter 1-barrier or 2-barriers jumping If ellipsoidal collapsing, no analytical solution  numerical

12. Success of the excursion set theory Agreement with numerical simulations Halo mass function Conditional mass func.Halo bias

13. more application … The central engine of semi-analytical models of galaxies (halos are warm beds of galaxies) the 21cm emission during re-ionization

14. Critical problems : halo formation time distribution

15. 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

16. 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 overdensity at large scale Halo forms Progenitor Halo forms P(2|1)=P(2, 1)/P(1)=P(2)xP(1)/P(1)=P(2) !

17. 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.

18. Fractional Brownian motion: definition If the hurst exponent  = ½, it is the random Brownian motion FBM: the simplest random walks of sub-diffusion, proposed by Mandelbrot, a fractal concept. Widely used in modeling random surface in geology, self-organization growth of structures in solid physics, even the stock price fluctuation…

19. FBM: properties  < 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

20. back to excursion set … correlation! 1 2 0

21. modified excursion set theory: Fokker- Planck equation By substitution It is the equation for random Brownian motion! The standard excursion set theory results can be easily converted for solutions.

22. 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

23. modified excursion set theory: conditional mass function – weak positive correlation?

24. 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

25. 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