margarita petkova- cosmological simulations of galaxy clusters

8/3/2019 Margarita Petkova- Cosmological Simulations of Galaxy Clusters

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Abstract

The entropy and temperature profiles and the M-T, S-T and Lx-T scaling relations

of two groups of clusters from two different simulations are studied. The first simulation

is non-radiative and the second one includes radiative cooling and star formation. The

results are then compared with observations and theory.The slopes of the scaling relations of the clusters from the non-radiative simulation

are within 5% to 1% of the slopes predicted from gravitational structure formation.

The scaled entropy and temperature profiles of clusters from the same simulation are

self-similar and in agreement with observations outside the cluster cores. The slope of

the Lx-T relation derived from clusters from the simulation with radiative cooling is

steeper than the one predicted by self-similarity and is consistent with observations.

The normalizations of the M-T and S-T relations from the simulation with radiative

cooling are larger than the ones from the non-radiative simulation.

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Contents

1 Introduction 4

1.1 Basic Properties of Galaxy Clusters . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Observing Galaxy Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Cluster Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.1 Scaling Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.2 Radial Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 Insights from Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Theoretical Aspects 11

2.1 Navarro-Frenk-White (NFW) profile . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Entropy Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Numerical Technique 12

3.1 ENZO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2 Parameters and Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . 13

3.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4 Results 14

4.1 Non-Radiative Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.1.1 Structure Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.1.2 Radial Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15


4.2 Simulation with Radiative Cooling . . . . . . . . . . . . . . . . . . . . . . . 18

4.2.1 Radial Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18


4.3 Comparison between Simulations . . . . . . . . . . . . . . . . . . . . . . . . 20

5 Discussion and Conclusions 23

6 Appendix A 27

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1 Introduction

Cosmological simulations of galaxy clusters are a powerfull tool for studying the formation

of large-scale structures in the Universe. However, the interplay of physics that drives

cluster formation is complicated. Processes such as radiative cooling, star formation andstellar feedback are already included into simulations. Other phenomena such as active

galactic nuclei (AGN) feedback, thermal conduction and magnetic fields are currently

being implemented. Therefore, it is important to study the properties of simulated galaxy

clusters and understand the roles of different physical processes.

1.1 Basic Properties of Galaxy Clusters

In the standard cold dark matter (CDM) scenario initial density fluctuations grow under

the influence of gravity. As a result at large scales the Universe has a web-like structure

called the cosmic web - large expanding voids are surrounded by collapsing gas filaments.

Consistent with the model of hierarchical structure formation, subclusters form first at

about redshift z = 2 at the nodes of those filaments. As they merge and mass accretes

onto them they become clusters (at about redshift z = 1) - the largest structures in the

Universe with total masses ranging from 1013M up to 1014M. Typical temperatures

are between 106 K and 104 K at which hydrogen is completely ionized. Densities are in

the range of 104cm3 in outer regions to 102cm3 in the core. The intracluster medium

(ICM) is enriched with metals and has a mean metalicity value of Z = 0.3Z. Dark matter

constitutes the bigger fraction of the mass of a cluster. The smaller fraction is the baryonic

content - mainly hot, x-ray emitting intracluster gas, whose spectrum fits to a thermalBremsstrahlung emission.

Clusters have various morphologies and structures, indicating that they are dynam-

ical systems. Evidence for recent mergers is consistent with the idea that clusters still

form today, as predicted by the hierarchical structure formation model. Clusters at high

redshifts have more structures - an indication of higher merger frequency. As subclusters

merge along the filaments of the cosmic web the induced shocks influence the temperature

and density of the ICM. This results in a rise of the virial temperature with increasing

mass, which is consistent with observations.

The fundamental model that describes galaxy clusters is the self-similar model. Itstates that clusters form through gravitational interactions only, their mass is dominated

by dark matter, major merger events are rare and the gas is in hydrostatic equilibrium. This

model predicts that clusters have self-similar profiles and scaling relations, i.e. the shapes

of the scaled profiles are similar. However, observations show that some clusters deviate

from this model. Most of them have cool cores - low temperature regions in the center

of the gravitational wells, that results from radiative cooling and other non-gravitational

processes such as star formation and AGN feedback.

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Galaxy clusters are important objects for cosmological studies. Their large scales

make them good probes for cosmology. After the supernovae and the cosmic microwave

backround, they are the major source of information about the density parameters. For

example, the baryonic density parameter b is determined from the x-ray luminosity and

the mass density parameter M - from the gas fraction of the cluster [1]. Moreover, theexistence of clusters at high redshift favors a low density (low M) Universe [2]. Further-

more, studying them reveals more about the processes that govern the formation of the

Universe.

1.2 Observing Galaxy Clusters

The ICM radiation fits to a thermal Bremsstrahlung emission which is located in the x-ray

region of the spectrum. Since x-rays do not penetrate the Earths atmosphere, space-based

x-ray satellites are best suited for galaxy clusters observations. Chandra and XMM-Newton

are two of the modern x-ray observatories. They use CCD cameras and can measure

spatially resolved spectra. The two instruments are complementary - Chandra has a good

spacial resolution ( = 0.5) and XMM-Newton is very sensitive and has an effective

area typically 3 to 5 times bigger than that of Chandra [2].

The data from the x-ray satellites provides a measure for the two basic properties

of the ICM - the temperature and the emission measure. The temperature is derived

from the shape of the spectrum and the emission measure EM =

n2edV - from the

normalization of the spectrum. More precisely, the temperature is obtained by fitting the

observed spectrum with a thermal emission model and correcting the wavelengths for the

redshift of the cluster. However, it is important to note that the temperature is affected

by the exponential cut-off in the spectrum of the Bremsstrahlung emission that scales as

f(E, T)T1/2exp(E/kBT). Hence the instruments used on the x-ray satellites should be

sensitive to energies greater than the cut-off. It should also be considered that the ICM

is not isothermal and thus any measured temperature is the mean value along the line of

sight.

The emission measure of the cluster can be obtained from the surface brightness

and the emissivity. The radial gas density profile is then derived by assuming spherical

symmetry and the isothermal -model

n(r) = no

1 + (r/rc)23/2

(1)

where rc is a scale radius. The -model fits well in the outer regions, but underestimates

the density in the centers of the clusters [2], where cooling takes place and isothermal

approximation is not valid.

The x-ray luminosity of the cluster can be further derived using the photon flux

detected by the measuring instrument. The temperature and the density profiles can be

used to derive the entropy profile. Moreover the total mass profile can be deduced from

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the equation for hydrostatic equilibrium

dp

dr=

GM(r)

r2(2)

and the equation of state for ideal gas p = nkBT, where p is the hydrostatic pressure,

= nmp is the density and is the average mass per proton. The mass profile

M(r) = kBT r

Gmp

d ln n

d ln r+

d ln T

d ln r

(3)

is then obtained by substituting for the pressure and the temperature in equation (2) and

rearranging the terms.

1.3 Cluster Scaling

Clusters are characterized by their mass and redshift. Since the mass is very difficult to

measure empirically, the temperature is considered instead. The basic properties of the

clusters, such as x-ray luminosity, mass and entropy can be determined as functions of

temperature and have a power law form Q T, where is the slope of the relation

and Q is the property of interest. A summary of the slopes of observed scaling relations is

shown in Table 1.

Relation Slope Reference Note

LX T 2.88 0.15 [3] a

2.64 0.16 [4] b

M T 1.88 0.27 [5]

1.71 0.09 [6] c1.56 0.16 [7] d

1.52 0.36 [8] e

1.49 0.15 [6] f

S T 0.65 0.05 [9]

Table 1: Slopes of the three basic scaling relations obtained from observations of nearby

clusters. Notes: a)clusters lack strong cooling flows; b)corrected for cooling flow; c)whole

sample is considered; d) = 2500; e)hot clusters (kBT > 5.5 keV); f) hot cluster subsample

(kBT > 3.5 keV)

1.3.1 Scaling Relations

Self-Similar Model

The self-similar model describes the properties of clusters assuming gravitational structure

formation only. The mass density distribution of an isothermal sphere is given by

(r) =2

2Gr2(4)

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where is the velocity dispersion of the cluster. If the virial radius rvir is defined as the

radius at which the density equals times the critical density of the Universe, then

M =4

3r3vircrit (5)

Using the relation between the gravitational and the kinetic energy of the cluster, the

velocity dispersion becomes

= M1/3

H(z)2G2/161/6

(6)

where H(z) is the redshift dependent Hubble constant in a CDM cosmology

H(z) =

100h

M(1 + z)3 +

1/2(7)

If the baryonic gas also follows an isothermal distribution, then kBT = mp2, where is

the average mass per proton ( = 0.59) and mp is the proton mass. Combining equations(4), (5) and (6) gives an expression for the mass of the cluster in terms of its temperature

M =

16G2

mp3

3/2H(z)1T3/2 (8)

Thus self-similar clusters obey the M-T relation M H(z)1T1.5.

The x-ray luminosity of the cluster can be computed using LX =

2gasdV(T),

where (T) T1/2 is the Bremsstrahlung emissivity. Thus LX M T1/2. If the density

is expressed in terms of the mass and the radius, then LX T1/2M2/r3. From equation

(5) holds that r

H(z)1

T1/2

. Subsituting the values for M and r, the x-ray luminos-ity becomes LX H(z)T

2. Using the same flow of arguments, the entropy-temperature

relation is derived as S H(z)4/3T.

M-T Relation

The M-T relation is important in constraining the scales of clusters. It gives a direct

measurement of the mass of the cluster if its x-ray temperature is known. Pointecouteau

et al. [7] show from a study of five hot galaxy clusters with XMM-Newton that the slope

of the M-T relation is 1.560.16 at a density contrast = 2500 with respect to the critical

density of the Universe. This measurement is consistent with the slope form gravitational

structure formation.Arnaud et al. [6] study the M-T relation of a sample of 10 nearby clusters (z < 0.15)

observed with XMM-Newton. The relations are computed for a set of density contrasts

and the mass is derived from a NFW profile (see Section 2.2). The slope of the relation

is constant for all considered density contrasts, implying that the mass-profiles are self-

similar. At r500 the slope of the M-T relation for the hottest (kBT > 3.5 keV) clusters is

1.49 0.15. It is consistent with the theoretical predictions from gravitational structure

formation. However, the slope steepens as the whole sample is considered - it becomes

1.71 0.09.

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Thus there are variations of the slope of the M-T relation with the temperature

of the system. The slope consistent with the theoretical predictions is true only for high

temperature clusters and steepens for low temperature ones. When clusters from a larger

temperature range are taken into account, the slope also gets steeper. Finoguenov et al.

[10] also confirm this deviation for two samples of galaxy clusters observed with ASCA.The slope for clusters with temperatures above 3 keV is 1.48 0.12 and the slope for

the sample including the low temperature clusters is steeper - 1.87 0.15. Allen et al. [8]

find a slope of 1.51 0.27 for a sample consisting of 5 hot (kBT > 5.5 keV) cluster with

measurements taken at 0.3r200, which is also consistent with the deviation of cluster

mass at high temperatures.

Lx-T Relation

The Lx-T relation is indicative of structural regularity in galaxy clusters. Since the lumi-

nosity depends on the baryon mass and the temperature - on the total mass, the Lx-T

relation can trace variations in the gas fraction. Markevitch et al. [4] find a slope of2.64 0.16 for a sample of 10 nearby hot (T > 3.5 keV) clusters. Arnaud & Evrard [3]

find a similar tight slope of 2.88 0.15 for a sample of 24 nearby clusters that lack strong

cooling cores. They also find that there is not a large scatter of the luminosity at low

temperature and thus the substructure of the clusters is very regular.

Fabian et al. [11] show that deviations from the theoretically predicted slope of the

Lx-T relation are related to the the strength of the emission determined by the cooling

flow in the cluster core. They also argue that the cooling flows are associated with the

instability of the ICM plasma and are periodically ceased and generated by merger events.

Periodic mergers are a natural phenomena in the hierarchical structure formation modeland numerical simulations [12] also suggest that cooling flow features are created and

destroyed by them.

S-T Relation

The S-T relation reflects the thermodynamic and accretion history of the ICM. Deviations

from the theoretically predicted slope are due to non-gravitational processes such as heat-

ing during a merger or radiative cooling. Entropy measured at 0.1r200 is bigger than the

value expected by gravitational heating alone. This is mostly evident in low mass clusters

[2]. Non-gravitational processes such as heating during collapse or radiative cooling have

been used to explain this. The slope of the S-T relation obtained for real clusters is con-siderably shallower than the theoretically predicted one. Pratt & Arnaud [13] find a slope

of 0.65 from observations with XMM-Newton. Another study with ASCA shows a slope

of 0.65 at 0.1r200 [9]. Pratt et al. [14] also find a slope of 0.64 0.11 for a sample of 10

nearby (z < 0.2) clusters.

All observations show that there is a large dispersion of data in the cluster core at

radii less than 0.1r200. This is due to the interplay of various non-gravitational processes.

In the center the gas density is high and the cooling time that scales as tcool T1/2/ne

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can be shorter than the dynamical time of the cluster (the time since the last merger

event). Thus the temperature in the core decreases due to radiative cooling and the baryon

density increases to maintain hydrostatic equilibrium. Recent observations with Chandra

[8] confirm that temperatures drop in the center of the cluster, forming a cooling core.

However, the lack of very cool gas is inconsistent with current cooling flow models [2]. Oneof the reasons could be AGN-cool gas interaction. For example, AGN heating can limit

cooling. Thermal conduction by electrons could also play a role in heating the core.

1.3.2 Radial Profiles

The radial profile of any physical quantity of a cluster can be expressed in scaled coordi-

nates. The radius is given in term of the virial radius and the other quantities - in terms

of their value at the virial radius.

Entropy Profiles

Entropy profiles are an imprint of the thermal history of the clusters. The accretion model

of Tozzi & Norman [15] present a power-law relation between entropy and cluster radius

of the form S(r) r, where = 1.1. Pratt et al. [14] find a slope = 1.08 0.04

for r 0.01rvir from a sample of 10 nearby (z < 0.2) morphologically relaxed clusters

observed with XMM-Newton. The profiles have been scaled with the virial entropy. They

obey self-similarity oustide the cluster core. Ponman et al. [9] find a slightly shallower

slope - = 0.94 0.14.

The entropy floor present in the central regions of the clusters is a result of the

formation of galaxies at larger redshifts [16]. Thus gas that had lower temperatures than

this floor is either heated up by non-gravitational processes or collapsed and formed stars

due to the short cooling time [17].

Mass Profiles

Pointecouteau et al. [18] present their results from a sample of 10 nearby (z < 1.5) clus-

ters from observations with XMM-Newton. The mass profiles are obtained by assuming

hydrostatic equilibrium and spherical symmetry. All ten profiles are self-similar and well

described by the NFW model (see Section 2.1) in the range 0.1 < r/r200 < 0.5.

Temperature Profiles

There is a similarity between the observed temperature profiles outside the cooling core.

There is a tendency for clusters with cooling core to form flatter temperature profiles at

larger radii. This suggests that the temperature profiles depend largely on the dynamical

state of the system [2].

Vikhlinin et. al [19] present a fit to the temperature profiles for a sample of 13

nearby, relaxed clusters. They find a relation T / < T >= 1.22 1.2r/r180 in the range

0.125 r/r180 0.6. The average temperature < T > excludes the values in central

regions. The profiles are self-similar outside the cooling core, which suggests that radiative

cooling does not have a strong effect on the regions outside the core.

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1.4 Insights from Simulations

Simulations have enabled astrophysicists to analyse and study the effects of gravitational

and non-gravitational processes in cluster formation and cluster properties.

Motl & Burns [20] argue that processes such as major merger events, radiative

cooling, supernovae (SN) feedback, AGN heating and thermal conduction by electrons

play a major role in shaping the temperature and density profiles of the clusters. Thus

all of them should be included into cosmological simulations in order to produce realistic

results. The different mechanism have different input into the system and different effect.

Burns et al. [21] propose a new, hierarchical model for the formation of cool cores in

galaxy clusters. When radiative cooling is introduced into a simulation, subcluster cores

often collide and merge. Thus in a standard CDM Universe cores are formed through the

merging. This scenario is consistent with the rich subcluster structure of galaxy clusters

that is found from observations with Chandra and XMM-Newton. The observed contact

discontinuities in the central regions of the clusters, the so called cold fronts, also arise

naturally from cluster mergers. Moreover, cluster cores have a relatively short cooling

time and high density, which breaks the self-similarity scaling in the centers. Burns et

al. [21] also find that clusters from non-radiative simulations obey the self-similar model

and that the most massive clusters are slightly overluminous and the less massive ones -

underluminous.

Mechanism Energy Contribution [1065erg]

Structure Formation 0.1

Radiative Cooling 103

SN Feedback 104

Thermal Conduction 104 to 105

AGN Feedback 106 to 105

Table 2: Energy contribution by different physical phenomena for a typical cluster. The

values are taken from [20].

Motl et al. [17] show that if there is only radiative cooling introduced into the

simulation, the cooling cores become much denser and more rigid than the realistic ones.They do not merge and act as rigid bodies. The process of star formation provides a natural

sink for the cooling gas - it is transformed into star particles and does not contribute to the

properties of the baryon gas. Thus when star formation is introduced into the simulation

the average temperature remains higher and the density - lower. However, the modeled

clusters still do not form realistic cool cores. Motl et al. [17] also find that introducing a

truncation of star formation at about redshift z 1.5 to z 2 provides a solution to the

problem - the cool cores become realistic. The truncation of astration is consistent with

observations since star formation rates of galaxies peak at about the same epoch.

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Bryan & Norman [22] show that cluster luminosity is dependent on the resolution

of the simulation as L r, where r is the spacial resolution and = 1.17. Thus

luminosity profiles from simulations with adaptive mesh refinement need to be corrected

for this dependence.

Motl & Burns [20] give an overview of the energy balance in galaxy clusters. Theystate that structure formation, radiative cooling, SN feedback, themal conduction and

energy feedback have major roles in shaping the properties of a cluster. They estimate

that the potential energy of a typical cluster is W 1065 erg and express the contribution

of each of these processes as a fraction of W. Table 2 summarizes these values.

2 Theoretical Aspects

2.1 Navarro-Frenk-White (NFW) profile

Navarro, Frenk & White [23] show that the equilibrium density profile of dark matter

haloes in a CDM Universe is given by

DM(r) =crit(z)c

(r/rs)(1 + r/rs)2(9)

where DM(r) is the radial density of the cluster, crit is the critical density of the Universe

given by crit = 3H2(z)/8G and c is the characteristic density given by

c =200

3

c3

[ln(1 + c) c/(1 c)](10)

where c = r200/rs is the concentration of the halo and rs is a scale radius. Thus for a

constant c the profiles of the clusters are self-similar.

2.2 Entropy Normalization

The specific entropy of the clusters is given by S = TXn2/3e , where ne is the electron

density, and is related to the real entropy by a logarithm and an additive constant. It

is closely related to the thermal history of the cluster and traces radiative cooling and

heating.

The specific entropy at a given radius is given by S(r) = TX(r)ne(r)2/3. The

electron density inside a given radius is computed as ne = gas(r)/emH, where e = 1.14

is the electron mass per particle and mH is the Hydrogen mass. The gas density and

the x-ray weighted temperature are computed from base quantities such as mass, volume

and x-ray emissivity. The profiles are scaled with respect to the virial specific entropy

S200 = T200n2/3e Assuming an isothermal sphere and applying the Virial theorem (see

Appendix A), the temperature at the virial radius can be expressed as

T200 =3GmpM200

2kBr200(11)

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The average electron density ne inside r200 equals 200 times the average electron density

of the Universe.

ne =200bcrit

emH(12)

Thus ne = 4.42 105cm3 and T200 = 6.39 10

7M200/r200.

3 Numerical Technique

3.1 ENZO

ENZO 1 is a hybrid cosmological simulation code created at the Laboratory for Computa-

tional Astrophysics at the University of California, San Diego [24]. It implements N-body

dynamics for the dark matter (DM) particles and hydrodynamics for the baryon gas. It

uses the structured adaptive mesh refinement (AMR) of Berger & Collela [25]. This method

has no restrictions on the number of grids at a level of refinement or the number of levelsof refinement. It is similar to a tree data structure [26]. There are several criteria that are

used for refinement - mass, Jeans length, slope, shock and cooling time. For cosmological

simulations, refinement by mass and degree of refinemet two are recommended [24]. For a

refinement higher than two all DM particles become accretion centers.

The N-body dynamics method assumes collisionless DM particles. They obey the

Newton equations of motion in comoving coordinates

dx

dt=

1

av (13)

dv

dt=

a

av

1

a (14)

The trajectories are computed by solving the Poisson equation with fast Fourier transform.

In comoving coordinates it has the form

2 =

4G

a( ) (15)

where a is the scale factor, G is the gravitational constant, is the local comoving mass

density and is the average density of the simulation. This method is more effective than

solving the collisionless Boltzmann equation in three dimensions because it involves three

instead of six variables [26].

The baryon gas is described by the Euler and Navier-Stokes equations. In comoving

coordinates they have the following form

t+

1

av =

1

a v (16)

v

t+

1

a(v )v =

a

av

1

ap

1

a (17)

1http://cosmos.ucsd.edu/enzo/

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where p is the pressure. The equations are solved by the piecewise parabolic method

(PPM) of Woodward & Collela adapted by Bryan et al. [27], where the gravitational

potential includes also the contribution from the DM particles. This method is third order

accurate and conserves energy density, momentum and mass flux in comoving coordinates.

Thus it produces correct entropy jumps at shocks and accurate temperature and pressurein hypersonic flows. The time step of the hydrodynamics and the N-body dynamics is

the same [26]. It is constrained such that a particle does not move more than a certain

fraction of the grid size in one time step. ENZO also implements a dual energy formalism

- internal and kinetic energy are solved for each grid at each time step. The Eulerian

hydrodynamics used in ENZO is an alternative to the smoothed particle hydrodynamics

(SPH). It has, however some advantages over the SPH - better shock resolution and flexible

mass resolution [24].

ENZO provides several additional physical packages. One of them is star formation.

It has been developed by using a scheme similar to the one by Cen & Ostriker [28]. Thestar formation algorithm sweeps through the grids and identifies (i)if the overdensity is

above a certain threshold , (ii)if the flow in the grid is convergent ( v ) < 0, (iii)if

the free-fall time is longer than the cooling time and (iv)if the mass of the baryonic gas

exceeds the Jeans mass. If all of these conditions are met, then a collisionless star particle

is created. Its mass is given by m = mbt/tdyn, where is an efficiency coefficient, mb

is the mass of the baryon gas in the grid cell, t is the time step of the simulation and

tdyn is the local dynamical time. Motl et al. [17] have determined experimentally that the

overdensity threshold should be in the range 10 < < 104 and the star particle mass

efficiency - in the range 0.01 < < 1.Another additional physical package is radiative cooling. The simpler cooling model

assumes that all species of the baryon gas are in equilibrium. The cooling curve is cal-

culated for a metalicity Z = 0.3Z. A more complex cooling model assumes primordial

gas abundances for nine species of hydrogen and helium. It implements collisional and

radiative processes for the multispecies [24]. For both models at each time step energy is

subtracted from the collapsing regions.

3.2 Parameters and Initial Conditions

In my thesis I study two groups of clusters from two different simulations. Both simulations

model a CDM Universe with the parameters = 0.73, M = 0.27, b = 0.044,

CDM = 0.226, h = 0.71 and 8 = 0.9. The comoving length of the simulations box is

128h1Mpc and the maximum comoving resolution is 7.8h1kpc. There are 1283 particles

in the simulated box. The boundary conditions are parabolic. The maximum level of

refinement is set to 7 and the criterion for refinement is baryon and dark matter overdensity

of 8.0. Each cell is refined by 2. The first simulation (hereon non-radiative simulation) is

non-radiative. The second simulation (hereon simulation with radiative cooling) includes

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the simpler model of radiative cooling and star formation, without SN feedback. The

clusters used for analysis have mass bigger that 1013M with the most massive ones up

to 2.3 1014M.

3.3 Analysis

The DM halos of the clusters are localized by the hop algorithm of Eisenstein & Hut that

traces increasing density. An analysis package generates spherically averaged profiles for

various physical quantities such as x-ray weighted temperature, entropy, mass and density.

The measurements are logarithmically spaced and the virial radius r200 is computed for a

density contrast = 200. Several procedures self-written IDL are used to plot and further

analyze the obtained data.

Figure 1: A 2D projection of the gas den-

sity of the whole simulated volume at

redshift z = 0. The clusters are located

at the nodes of the filaments of the cos-

mic web.

Figure 2: A zoom into the temperature

slice of the most massive cluster from the

non-radiative simulation. The color map

shows the log of the temperature (in K).

The background image is the 2D density

projection of the whole simulated vol-

ume.

4 Results

4.1 Non-Radiative Simulation

4.1.1 Structure Formation

Figure 1 shows a two-dimensional projection of the density of the whole simulated volume

at redshift z = 0. The galaxy clusters are located at the nodes of the filaments of the

cosmic web. Figure 2 shows a zoom into the most massive cluster at redshift z = 0. The

color map shows the logarithm of the temperature.

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Figure 3: Scaled entropy profiles of the

15 most massive clusters. The squares

show the mean value of the entropy and

the error bars - the 1 standard devi-

ation. The dashed line represents the

power-law fit S/S200 = 1.28(r/r200)1.07

in the range 0.2 r/r200 1.0.

Figure 4: Scaled entropy profiles of 31

clusters [29] from a non-radiative simu-lation with ENZO. The squares show the

median profile and the error bars - the 1

deviation. The dashed line illustrates the

power-law fit S/S200 = 1.32(r/r200)1.1 in

the range 0.2 r/r200 1.0.


Entropy ProfilesThe entropy profiles are an important source of determining the ther-

modynamical history of the the clusters. Entropy is scaled with the virial entropy of

each individual cluster. A detailed derivation of the virial entropy is given is Section

2.2. Figure 3 shows the scaled entropy profiles of the 15 most massive clusters from the

non-radiative simulation. The profiles are self-similar and obey a power law in the range

0.2 r/r200 1.0. The flattening of the profles inside the cluster core (r < 0.2r200) is due

to gravitational softening, where the gravitational potential of the cluster deviates from

the inverse square law. The power law fit has the form

S/S200 = 1.28(r/r200)1.07 (18)

and is consistent with results from observations [14] and from simulations with ENZO [29].

The slope is also consistent with the value 1.1 derived from the accretion model of Tozzi

& Norman [15]. Figure 4 shows overplotted entropy profiles by Voit et al. [29] from a non-

radiative simulation with ENZO. The slope of the power law that they find is within 3%

of the one from equation (18). Their normalization is, however, larger due to differences

in the scaling of the entropy.

Temperature Profiles Figure 5 shows overplotted scaled temperature profiles of

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Figure 5: Overplotted scaled tempera-

ture profiles of the 15 most massive clus-

ters at z=0. The solid line represents the

mean profiles and the error bars - the

1 deviation. The dashed line is the best

linear fit in the range 0.3 r/r200 0.6.

Figure 6: Overplotted scaled tempera-

ture profiles of 13 nearby galaxy clusters

[19]. The blue line represents the mean

profile and the red lines bound the width

of the scatter of the best fit.

the 15 most massive clusters at redshift z = 0. The profiles obey the following relation

T/ < T >= 1.17 0.61r/r200 (19)

in the range 0.3 r/r200 0.6, where < T > is the average x-ray weighted temperature

of the clusters inside the virial radius. The profiles are self-similar. The big temperature

fluctuations of one of the clusters are due to a merger. Even with this heat source present,

the profile is similar to the other ones. Figure 6 shows the overplotted scaled profiles

for 13 nearby real clusters [19]. The profiles of the simulated clusters are consistent with

the profile T / < T >= 1.22 1.2r/r180 from the real clusters, out of the cluster core

(r > 0.2rvir). The small deviations may be due to the different scaling of the radius.

However, the similarity suggests that radiative cooling does not have a strong effect on

the regions outside the core and that the results from the non-radiative simulation are

applicable to real clusters.


Figure 7 shows the relation of x-ray luminosity, gas mass and entropy with x-ray weighted

temperature of the 500 most massive cluster from the non-radiative simulation at redshift

z = 0 and redshift z = 1. A summary of the slopes of the relations found in Table 3. The

slopes for redshift z = 0 are within 5% (for Lx and M) to 1% (for S) within the slopes

predicted by gravitational cluster formation. Thus the clusters formed in the non-radiative

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(a) (b)

Figure 7: Scaling relations of x-ray luminosity, gas mass and entropy with x-ray weighted

temperature of the 500 most massive clusters at (a) redshift z=0 and (b) redshift z=1. Each

point represents a single cluster measurement at the virial radius. The solid line shows

the best linear fit - LX T2.10, Mgas T

1.59 and S T0.99 for z=0 and LX T1.72,

Mgas T1.15 and S T0.86 for z=1.

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relation slope (z=0) slope (z=1) theoretical slope

Lx-T 2.10 0.08 1.72 0.06 2.00

M-T 1.59 0.05 1.15 0.04 1.50

S-T 0.99 0.03 0.86 0.02 1.00

Table 3: Slopes of the relation of virial x-ray weighted luminosity, gas mass and entropy

with x-ray weighted temperature. The slopes at redshift z = 0 are consistent with the

theoretical slopes. The slopes for redshift z = 1 are shallower indicating that the clusters

are not yet virialized.

relation normalization (z=0) normalization (z=1)

Lx-T 3.24 1018 2.04 1015

M-T 0.13 1.89

S-T 125.2 5.87 104

Table 4: Normalization of the relation of virial x-ray weighted luminosity, gas mass and

entropy with x-ray weighted temperature.

simulation obey the self-similar model and are virialized. The data from redshift z = 1

is more dispersed and the slopes of the scaling relations are shallower. The normalization

for the M-T relation (see Table 4) is considerably lower than the one from redshift z = 0.

Thus not all mass has infallen into the gravitational well and the clusters have not formed

fully yet at redshift z = 1, as predicted by the hierarchical structure formation model.

4.2 Simulation with Radiative Cooling


The entropy profiles of the 15 most massive clusters from the simulation with radiative

cooling at redshift z = 0.1 are shown in Figure 8. The entropy profiles are very spiky -

entropy drops drastically in regions of progressive cooling. There is no flattening of entropy

observed in the cluster core. Figure 9 show the temperature profile of the most massive

cluster from the simulation with radiative cooling. Temperature falls drastically to 104

Kin the core - this is the threshold of the employed cooling scheme. Such cooling is not

observed in real clusters. Thus other non-gravitational processes such as AGN feedback

limit the radiative cooling.

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Figure 8: Overplotted scaled entropy

profiles of the 15 most massive clusters

at z = 0.1 from the simulation with ra-diative cooling. There are drastic drops

of entropy values.

Figure 9: Temperature profile of the

most massive cluster from the simulation

with radiative cooling. Temperature inthe core drops to 104 K - the threshold

value of the radiative cooling model.


Figure 10 shows the relations of x-ray luminosity, gas mass and entropy with x-ray weighted

temperature of the 450 most massive clusters at redshift z = 0.1 from the simulation

with radiative cooling. The values of the slopes are summarized in Table 5 together with

the ones obtained from observations. The slope of the Lx-T relation is consistent with

observations, indicating that radiative cooling in the form of Bremsstrahlung emission is

the primary source of x-ray luminosity in clusters. The slopes for the other two relations

are inconsistent with observations and thus other non-gravitational processes such as AGN

feedback are important in shaping the clusters mass and entropy.

relation slope (z=0.1) slope from observations

Lx-T 2.83 0.10 2.88 0.15 [3]

M-T 1.55 0.10 1.88 0.27 [5]

1.49 0.15 [6]S-T 0.88 0.10 0.65 0.05 [9]


with x-ray weighted temperature. The slope for the x-ray luminosity is consistent with

observations.

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Figure 10: Scaling relations of x-ray lu-

minosity, gas mass and entropy x-ray

weighted with temperature of the 450

most massive clusters at redshift z=0.1.

Each point represents a single cluster

measurements at the virial radius. The

solid line shows the best linear fit - LX

T2.83, Mgas T1.55 and S T0.88.

4.3 Comparison between Simulations

The slopes of the scaling relations Lx-T, M-T and S-T obtained from observations, from the

non-radiative simulation, from the simulation with radiative cooling and from gravitational

structure formation are summarized in Table 6. The slopes from the non-radiative simula-

tion are consistent with gravitational structure formation. Thus clusters from non-radiative

simulations are self-similar. The slope of the Lx-T relation from the simulation with ra-

diative cooling is consistent with observation, indicating that thermal Bremsstrahlung is

the major luminosity source. The Lx-T slope from the simulation with radiative cool-ing is bigger than the one from the non-radiative simulation. Thus x-ray luminosity is

boosted by the radiative cooling. The slopes for the M-T and S-T relations from both

simulations are similar. Thus not only radiative cooling and star formation, but also other

non-gravitational processes influence cluster formation. The slope of the M-T relation from

both simulations is consistent with the slope from observations of hot, massive clusters.

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Figure 11: Overplotted scaled mass pro-

files of the 15 most massive clusters atredshift z = 0 from the non-radiative

simulation.

Figure 12: Overplotted scaled mass pro-

files of the 15 most massive clusters atz = 0.1 from the simulation with radia-

tive cooling.

Figure 13: Scaling relations of x-ray lu-

minosity, gas mass and entropy with

x-ray weighted temperature of the 450most massive clusters at redshift z = 0

(non-radiative simulation) and redshift

z = 0.1 (simulation with radiative cool-

ing). The solid line and the diamond

points represent the data from the non-

radiative simulation and the dashed line

and the square points - the data from the

simulation with radiative cooling.

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slope from slope from slope from

slope from non-radiative simulation with gravitational

relation observations simulation radiative cooling structure

(z = 0) (z = 0.1) formation

Lx-T 2.88 0.15 [3] 2.10 0.08 2.83 0.10 2.00M-T 1.88 0.27 [5] 1.59 0.05 1.55 0.10 1.50

1.49 0.15 [6]

S-T 0.65 0.05 [9] 0.99 0.03 0.88 0.10 1.00


with x-ray weighted temperature obtained from observations, from the non-radiative sim-

ulation, from the simulation with radiative cooling and from gravitational structure for-

mation.

Figure 13 shows the overplotted scaling relations from both simulations. The nor-

malization of the M-T and S-T relations are larger for the simulation with radiative cooling.

This is due to the increase in density with the drop of temperature. Figures 11 and 12

show the overplotted mass profiles of the 15 most massive clusters from the non-radiative

simulations and from the simulation with radiative cooling. The normalization of the mass

inside the clusters cores from the latter simulation is much higher. The reason for this rise

is the increase of the density due to the cooling in the central regions. Figure 12 also shows

that there is a deviation from the self-similarity inside the clusters core, as predicted by

Fabian et al. [11].

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6 Appendix A

The density of an isothermal sphere is given by

(r) =2

2Gr2(20)

where is the velocity dispersion. In a virialized cluster the temperature is given by

T =mp

2

kB(21)

and the mass by

M =4

3r3(r) (22)

Thus the temperature can be expressed as

T200 =3GmpM200

2kBr200 (23)

27

margarita petkova- cosmological simulations of galaxy clusters

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