Universal Mass Accretion Rates &

Concentrations of

Dark Matter HalosDonghai ZHAO

Shanghai Astronomical Observatory / MPA

Yipeng Jing (SHAO), Houjun Mo (UMASS), Gerhard Börner (MPA)

Montpellier 2010.07.27

ΩmΩΛhσ8 …

ray

Annihilatio

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ecayDM

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Dark matter distribution and its evolution

Galaxy formation model

Cosmology

Lensing

model

Formation and evolution of galaxies

Gravitational lensing

Dark matter halos connect theory with observation

Either for constraining galaxy formation model with galaxy observation, or for constraining cosmological model with gravitational lensing observation, one need a precise model for dark matter distribution and its evolution for any given cosmology.

nonlinear evolution of cosmic structure

initial density field

gravitational instability in an expanding universe

Initial tiny density fluctuation is amplified by gravitational instability;

Dark matter collapse into small quasi-equilibrium objects (dark matter halos) first due to higher fluctuation amplitude on those small scales;

Hierarchical formation through mergers

Simple picture: in the standard framework of structure formation, DM halos are very nonlinear objects formed from initial tiny density fluctuation by gravitational instability magnification.

1) How does total mass of a halo grow? mass growth history

2) How does matter settle down inside a halo?density profile and its evolution history

Merging history of DM halos

Wechsler et al. 2002

Jing & Suto 2000

In this talk, we focus on for any given cosmology and initial density field:

Introduction I

1. Navarro, J. F., Frenk, C. S., & White, S. D. M. 1997, ApJ, 490, 493

- Spherically averaged halo density profile can be well described by a function of two free parameters.

2. Wechsler, R. H., Bullock, J. S., Primack, J. R.,

Kravtsov, A. V., & Dekel, A. 2002, ApJ, 568, 52 - Halo Mass Accretion Histories can be well described by a

exponential function of one free parameter.

- Halo concentration increases as c a.

N-body cosmological simulation

• Assumption1: dark matter particles have NO strong self interaction• Assumption2: MAHs are well simulated with pure dark matter

Model ni Γ σ8 Ω0 Λ0 Np L [Mpc/h] NNQ η zi

SF1 1 K cut 1 1 0 5123 160 512 0.016 1481.3

SF2 0 1 1 0 5123 160 512 0.016 363.2

SF3 -1 1 1 0 5123 160 512 0.016 75.0

SF4 -2 1 1 0 5123 160 512 0.016 11.3

LCDM1 1 0.2 0.9 0.3 0.7 2563 25 512 0.0025 72

LCDM2 1 0.2 0.9 0.3 0.7 5123 100 512 0.01 72

LCDM3 1 0.2 0.9 0.3 0.7 5123 300 512 0.03 36

LCDM4 1 0.19,BO 0.85 0.268 0.732 10243 300 1024 0.01 72

LCDM5 1 0.19,BO 0.85 0.268 0.732 10243 1200 1024 0.072 72

LCDM6 1 0.19,BO 0.85 0.268 0.732 10243 1800 1024 0.144 72

OCDM1 1 0.2 1.0 0.3 0 2563 50 256 0.02 72

SCDM1 1 0.5 0.55 1 0 2563 25 512 0.0025 72

SCDM2 1 0.5 0.55 1 0 5123 100 256 0.02 72

SCDM3 1 0.5 0.55 1 0 5123 300 256 0.03 36

1. MARs depend on power index of the initial density fluctuation power spectrum — since structure forms form initial density fluctuation

MARs depend on halo mass (acctually on fluctuation amplitude)

Simulated halo mass accretion histories

SF1SF2SF3SF4

2. MARs depend on cosmology — since cosmology determines gravity in the expanding background universe

MARs depend on redshift

Simulated halo mass accretion histories

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power index represents relative power

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mass scale

redshift

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1. Different factors entangling together2. For LCDM universe, there are characteristic scales both in space and in time, and so each mass scale is special, each time is special and each piece of MAH... Analytical approach (EPS) : assumptions are too simple and can not produce median/average MAHs analytically Empirical approach (van den Bosch 2002; Wechsler 2002; …; McBride 2009): just a fit to numerical results of some specific cases, neither universal nor accurate, and hard to help understand structure formation

A

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power index represents relative power

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mass scale

1. Different factors entangling together2. For LCDM universe, there are characteristic scales both in space and in time, and so each mass scale is special, each time is special and each piece of MAH... Rather than seeking a universal functional form for Mass Accretion History, we turn to disentangle and model the dependences of Mass Accretion Rate on all relevant factors (in order to predict MAH piece by piece according to the local features of cosmology and power spectrum on the correspondent redshift and mass scale), and this is the reason why we explore a variety of cosmologies.

redshift

M

redshift

mass sca

le

universal mass accretion rate

85.5

),,(]10)(/[)()(lg/)(lg

|lg/lgobsobs

Mmddc

c

MzzpMzzdMd

initial density fluctuation amplitude in a given volume/scale

relation between different scales

nonlinear effects: environmental heating, stripping …

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the smooth

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universe a

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dshift

Zhao et al. 2009, ApJ, 707, 354 (astro-ph/0811.0828)

Integrating MAR to predict MAHs

• In this special parameter space, MAH also has a very simple form — straight line.

85.5

),,(]10)(/[)()(lg/)(lg

|lg/lgobsobs

Mmddc

c

MzzpMzzdMd

• For a halo of given mass at a given redshift, one can estimate MAR for it. With this MAR, one can trace its main progenitor to a little bit higher redshift, and then calculate the progenitor’s MAR. One can do this iteratively to trace the whole MAH.

Zhao et al. 2009, ApJ, 707, 354 (astro-ph/0811.0828)

• It can transform to various shapes in the traditional M-z space.

• M(z) has a uniform asymptotic behavior to high redshift.

• The most massive halos grow faster and faster for SF cases.

•From SF to LCDM/OCDM/SCDM case,the model needn’t further tuning

Red lines: Zhao et al. 2009 No free parametersGreen lines: van den Bosch 2002Blue lines: Wechsler 2002

Also work for truncated power spectrum (even sharper than HDM)• Useful for modeling simulation artifact due to cutoff of initial density fluctuation power and study spectrum features.

• MAHs indeed have no universal shape in M-z space.

Power been cut off, even more extreme than hot dark matter power spectrum

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mass scale

Independent testOur model predictions also match PINOCCHIO MAHs very well, which are output automatically by the semi-analytical code ‘PINOCCHIO’ (Monaco 2002) and so are much independentto our numerical simulations and to our algorithm to build halo merger trees.

Points:Red lines: Zhao 2009

Why all halos show two growth phases in LCDM and OCDM universe

Because both decreasing of density parameter and increasing of power index will slow down halo growth, all LCDM and OCDM halo MAHs show a initial fast and a late slow growth phase, as found in Zhao et al. 2003a,b.

Halo density profile is connected to it’s MAHs

• Statistically– NFW 1997– Bullock 2001– Eke 2001– Zhao 2003b– Salvador-Solé 2007– Neto 2007 (MS)– Maccio 2008– Gao 2008 (MS)

• Individually– NFW 1997– Wechsler 2002– Zhao 2003a

Fluctuation amplitude

• Concentration of a halo is strongly correlated to the universe age when its main progenitor first gain 4% of its current mass.

• This relation can be used to predict the mass- and redshift-dependence of halo concentration, more accurately than all previous models.

• It reproduced our earlier simulation results and model predictions (Zhao 2003ab), which are included in 《 Galactic Dynamics II 》 (J. Binney) and have been confirmed by the Millennium simulation, Bolshoi simulation and by many other authors.

Points: Zhao 2003bRed lines: Zhao 2009

Independent test

1) Points: Maccio et al 2008 z = 0

2) Points: Klypin et al 2010 with Bolshoi Simu.

Lines: Zhao et al 2009 with NO free parameter (parameters are the same for all cases)Maccio’s model needs free parameters to reproduce their own simulation data

3) J.C. Munoz-Cuartas1, A.V. Maccio, S. Gottlober, & A.A. Dutton 2010 (1007.0438): When compared with the model proposed in Zhao et al. (2009) our results show a very good agreement at the low mass end. For high masses we find a slightly higher difference but never exceeding few percent.

• It’s again very accurate, universal and simple: the same little set of parameters works well for various cosmological models, power spectra and for halos of different masses and different redshifts.

111* Bolshoi Vol.

Red lines: Zhao et al 2009 with the same parameters for all cosmologies and all redshifts

• It can also predict the evolution of halo structural properties along the main branch of merger trees, thus, one can plot its NFW density profile at any point of its history and then obtain density evolution at any fixed radius.

• It can also predict the evolution of halo structural properties along the main branch of merger trees, thus, one can plot its NFW density profile at any point of its history and then obtain density evolution at any fixed radius.

1) How does total mass of a halo grow? mass growth history

2) How does matter settle down inside a halo?density profile and its evolution history

Merging history of DM halos

We now have a simple but precise universal model which can be utilized to predict both mass growth and density profile evolution for halos of any mass at any redshift in any cosmology with any initial density field, with the same set of parameters.

Wechsler et al. 2002

Jing & Suto 2000

In this talk, we focus on for any given cosmology and initial density field:

http://www.shao.ac.cn/dhzhao/mandc.html (web calculator and code)

Two phase halo formation -We found that the mass assembly history of dark matter haloes consists of two phases, an early fast phase and a late slow phase, which are quite different in many characteristics. The concentration increases steadily with time during the slow phase because the newly-accreted matter is mostly added into the outskirt. In contrast, the concentration parameter of dark matter haloes remains a constant during the early fast phase as matter within haloes mixes rapidly through violent relaxation. This is also the reason why the concentration flattens at large masses and at high redshifts.

Accurate universal model for MARs & concentrations of dark matter halos -We disentangled and properly modeled all effects that are involved in the nonlinear process of halo formation (cosmology, initial density fluctuation spectrum, halo mass, redshift and nonliear effects) and connected halo interior mass concentrations with their MAHs in a simple way. These models can be used to predict the MAHs, the mass & redshift dependence of concentrations and the individual concentration evolution histories of dark matter halos. These models successfully predict nonlinear structure formation from initial density fluctuation spectrum and cosmology, and are simple, accurate and universal.

Conclusions I

Conclusions II

• We explained why this extremely simple model works universally and accurately: 1) we disentangled and properly modeled all effects that are involved in the nonlinear process of structure formation; 2) modeling mass growth rate rather than MAH helps to evade factors’ time- and scale-varying problem.

• We found that contrary to traditional understanding, halo mass grow faster and faster and concentration decrease with time for the most massive halos in SF models.

• We explained why halos on all scales show two distinct growth phases in LCDM and OCDM universes as found in Zhao et al. 2003a,b: both decreasing of density parameter and increasing of power index will slow down halo mass growth.

Thank you！

Universal density profile － NFW function

Zhao et al. 2003a

with two free parameters: rs & s

or Mh & c ≡ rh / rs

rs

rhs

Now the problems become to predict:1) Mh (z | Mh,0, z0), and2) c (z | Mh,0, z0) for given cosmology and initial density field.

– Van den Bosch 2001 argues that his formula is universal, but the formula is based on MAHs constructed from EPS formalism, which are not consistent with those from simulations as already noticed by himself. Furthermore, actually it’s not universal even for EPS MAHs.

Zhao et al. 2003a, MNRAS

Two distinct phases in the growth of DM halos

c ≡ rh / rs

The faster the mass grows, the slower the concentration increase.

r h halo radius

rs NFW scale

M s mass in

r s

M h halo mass

Two distinct phases in the growth of DM halos

1. an initial phase characterized by rapid halo growth through major mergers, which effectively reconfigure the gravitational potential wells and cause the collisionless DM particles to undergo dynamical violent relaxation and mix up sufficiently to form the core structure; c ≡ rh / rs ≈ 4, quite different from former assumption ( c 1/(1+z))

2. a late slower quiescent growth predominantly through accretion of material onto the halo outskirt, little affecting the inner structure and potential; c ≡ rh / rs grows, the slower the mass accretion, the faster the c increasing

Mass and redshift dependence of halo concentration

Zhao et al. 2003b, ApJ, 597, L9

Cook, M., Lapi, A., Granato, GL. 2009, MNRAS, “Two phase galaxy formation”Cook, M., Evoli, C., Barausse, E., Granato, GL., Lapi, A. astro-ph/0906.4115, “Two phase galaxy formation: The Gas Content of Normal Galaxies”Cook, M., Barausse, E., Evoli, C., Lapi, A., Granato, GL. Astro-ph/0910.3910, “Two phase galaxy formation: The Evolutionary Properties of Galaxies”

- partition each mass accretion history (MAH) into two phases: A ‘fast accretion’, merger dominated phase corresponding to spheroid-SMBH co-evolution, followed by a ‘slow accretion’, quiescent phase allowing for disk structure to form around the pre-processed halos (see Zhao et al. 2003); - compare their results on several properties of the local galaxy population with observations, finding an encouraging agreement.

Two phase galaxy formation

Why we still need an universal model when cosmology parameters are nearly

pin down? Even though cosmological model and power spectrum of initial density fluctuation have been pin down, cosmology parameters change with time and power index of power spectrum vary with mass scale dramatically. A universal model is needed for properly applied to different mass scales and different redshifts. Even though you only deal with a halo of a single mass at a given redshift, you still need a universal model, as its history spans different redshifts and different mass scales.

Why there is no universal integral function form for MAH in [M,z] space?

These factors keep changing in their own complicated ways, and so we may not expect that variations of all these factors compensate exactly to bring us a universal function form for MAH in terms of M and z. That’s why former efforts all failed at end.