Effects of correlation between halo merging steps

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 How matter is partitioned into halos? LSS 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 dS and dd If no boundary

9. Excursion set theory (2) halo formation as a single barrier first-upcrossing problem If d > dc, the matter within R will collapse to form a halo dc serves as an absorbing barrier on the random walk No. of halos of mass M Connection to halo mass 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 M1 at z1 given a parent halo of mass M2 at z2 No. of trajectories has first-upcrossing over dc(z1) at S1 given its first upcrossing at S2 over dc(z2)

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

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

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. Brownian motion contains no memory of its past history Environment impact is null in standard excursion set theory The missing link: correlation between random walk steps Progenitor Halo forms Halo forms 2: merging Environment: The overdensity at large scale 1 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 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… If the hurst exponent a = ½, it is the random Brownian motion

19. FBM: properties 0.2 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

20. back to excursion set …correlation! 2 1 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. 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: halo mass function

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

24. effects of positive correlation a > 0.5 small mass halos less young, more old large mass halos more young, less old modified excursion set theory: halo formation time distribution

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