Accurate universal models for the mass accretion histories and concentrations of dark matter halos

Accurate universal models for the mass accretion histories and concentrations of dark matter halos Donghai Zhao 趙東海 (SHAO) Co-workers: Yipeng Jing (SHAO) Houjun Mo (UMASS) Gerhard Boerner (MPA) 2009.2.24

Cosmology Dark matter distribution and its evolution DM model Galaxy formation model Precise cosmology era Either for constraining galaxy formation model with galaxy observation, or for constraining cosmological model with gravity lensing observation, a precise model for dark matter distribution and its evolution is needed. Formation and evolution of galaxies

Halo merger tree

Universal density profile－NFW function Two free parameters: M & c

Part I A universal model for the mass accretion histories of dark matter haloes, M(z|z0,M0)

Constructing Mass Accretion Histories, M(z|z0,M0) 1. Numerical Simulation: Wechsler 02, Zhao 03a,b 2. EPS: Kauffmann 93, Somerville 99, van den Bosch 02 3. ST: not appropriate for constructing MAHs • Analytic Model of MAHs 1. Based on Simulation: Wechsler 02, Zhao 03 (both CDM only, two free parameters) 2. Based on EPS: van den Bosch 02 (CDM only) This work is based on a large set of cosmological simulations with a wide variety of cosmologies and power spectra.

N-body cosmological simulation • Assumption: MAHs are well simulated with pure dark matter

Simulated halo mass accretion histories I 1.MAHs depend on initial density fluctuation power spectrum — sincestructure forms form initial density fluctuation

Simulated halo mass accretion histories II 2.MAHs depend on cosmology — since cosmology determines the expansion behavior of the background universe

Bad news for modeling MAHs 1. Different factors entangling together 2. Factors varing from point to point and from hitory to history Cosmological parameters Initial density power spectrum Nonlinear evolution of structures [CM2,n2] [CM1,n2] [CM1,n1] Mass of themselves a=1/(1+z)

Good news for modeling MAHs It seems that there is no more bad news.

Choosing quantities for modeling • Given cosmology and power spectrum, after extrapolated linearly to z=0, linear mass variance of given volume σ is determined by M, and linear critical collapse overdensity δc by z.

Growth rate as a function of scaled “mass”

In this special parameter space, mass accretion rate is determined by a combination of halo mass, redshift, parameters of cosmology and spectrum. A tight universal correlation for mass accretion rate • In this special parameter space, MAH also has a very simple form — straight line.

It can transform to various shapes in the traditional M-z space. From SF to LCDM/OCDM/SCDM case,the model needn’t further tuning • M(z) has a uniform asymptotic behavior to high redshift. • Most massive halos grow faster and faster.

Accurate and universal: With the same set of parameters, the model can make accurate prediction for different cosmology and for halos of different masses at different redshifts, because 1) it has disentangled all effects that are involved in the nonlinear process of structure formation; 2) the differential form helps to evade factors’ time- and scale-varying problem.

For truncatedP(k) • Valid even for truncated power spectrum (really universal) • Power been cut off, more extreme than hot dark matter power spectrum • Useful for modeling simulation artifact due to cutoff of initial density fluctuation power • MAHs indeed have no universal shape in M-z space.

Part IIAn universal model for the concentration evolution histories of dark matter haloes, c(z|z0,M0)

Statistically NFW 1997 Bullock 2001 Eke 2001 Zhao 2003b Salvador-Solé 2007 Individually NFW 1997 Wechsler 2002 Zhao 2003a Halo density profile is connected to it’s MAHs

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 (Zhao 2003b), which are included in《Galactic Dynamics II》(J. Binney) and have been confirmed by Gao et al. (2008) with the Millennium simulation and many other authors.

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

It can also predict the evolution of halo structural properties along the main branch of merger tress, i.e., one can plot its NFW density profile at any point of the MAH and then obtain density evolution at any fixed radius.

Website • Relevant paper: arXiv: 0811.0828 • A calculatorwhich allows one to interactively generate data for any given cosmological model is provided at http://www.shao.ac.cn/dhzhao/mandc.html • And there a user-friendly code to make the relevant calculations is also provided

Conclusion and discussion I • We found that halo mass accretion rate is determined by a combination of halo mass, redshift, parameters of cosmology and initial density field. • Using this correlation, we constructed a model to predict halo median MAHs and scatters, which is accurate even when mass has grown by several thousand times and is valid for halos of various mass, redshift, cosmology and power spectra (even truncated). This is because different effects have been disentangled. In [M,z] space, there is no universal integral function form for MAHs. • The model can naturally predict mass accretion rate, halo formation time, halo survival time, and can be used to model environment dependence of halo MAHs. • We found a characteristic mass scale, halos of which scale have a straight median MAH, and median MAHs of different final masses will all approach this scale at early regime. Halos massive than it grow faster and faster. • There is no universal integral function form in terms of M and z for MAHs.

Conclusion and discussion II • We found another strong universal correlation, which shows that median concentration c of halos of a given mass is tightly correlated with the universe age when their median main progenitor gained 4% of the current mass. • Combining this correlation with the above MAH model, we can predict accurately halo concentration as a function of M and z,c(z,M),for various cosmological models and power spectra (but failed somewhat in SF case with n=1). • This model can also be used to predict the evolution of c along main branches, c(z|z0,M0), and so mass assembling history at different radii.

Future work • Individual history and dispersion Application 4: halo formation time distribution (half mass redshift, or any ratio) Application 5: halo survival time (time when merging with a more massive halo) distribution (mass function needed) Application 6: modeling environment dependence of MAHs (???) • Universal mass function n(m,z)? from P(M,z) • Inner slope of halo density profile?

Thank you！

Wechsler et al. 2002assumed an exponential function form with one free parameter, based on numerical simulation. • There is no recipe to determine the free parameter; • Function form works bad in many cases. Zhao et al. 2003b

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.

Is there any accurate universal model? • Theoretically: disentangling different effects in the nonlinear halo formation process for a thorough understanding • Practically: usage for different masses and different redshifts; differentiating cosmological models

- TruncatedP(k) Power been cut off, more extreme than hot dark matter power spectrum Modeling artifact due to cutoff of initial density fluctuation power

Cosmological parameters change with time and power index of power spectrum vary with scale.

Accurate universal models for the mass accretion histories and concentrations of dark matter halos