2. What are the largest objects that could have formed so far? 3. How do the cosmological parameters influence structure formation?

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1 Einführung in die beobachtungsorientierte Kosmologie I / Introduction to observational Cosmology I LMU WS 2009/10 Rene Fassbender, MPE Tel: , rfassben@mpe.mpg.de 1. Cosmological Principles, Newtonian Cosmology 2. Metric and Measurements in the Cosmos 3. Evolution of the Large-Scale-Structure 4. Observations of the Large-Scale-Structure 5. Relativistic Cosmology and World Models 6. Distance and Structure Measurements 7. Inflation 8. Cosmic Microwave Background 9. Dark Energy 10. Observational Evidence for Dark Energy 11. Formation of Galaxy Clusters 12. Structure and Evolution of Galaxy Clusters 13. Summary of the Cosmological Standard Model 1 Questions of structure formation for today 1. How many collapsed objects are there at any given epoch of the Universe and what is their mass distribution? 2. What are the largest objects that could have formed so far? 3. How do the cosmological parameters influence structure formation? 4. What is the internal structure of the formed objects? z=1.4 z=0 Source: R. Fassbender: Introduction to Observational Cosmology I WS09/10 2

2 Agenda I. The Dark Matter Halo Mass Function and its Evolution II. III. The Formation of Galaxy Clusters Universal Dark Matter Halo Profiles R. Fassbender: Introduction to Observational Cosmology I WS09/10 3 I. The Dark Matter Halo Mass Function and its Evolution

3 Summary of the Spherical Collapse Model (L3) Linear Regime: δ<<1 the density contrast δof overdense regions grows linear to the scale factor a(t) as long as the matter density is (close) to the critical density δ a 1/(1 + z) perturbations do not grow in an empty Universes (Ω m 0), i.e. perturbation growth slows down once the matter density drops significantly below the critical density Non-Linear Regime: the spherical collapse model provides a simplified but analytic solution in the non-linear regime for a homogeneous overdense spherical region in the EdS Ω m =1 case, an object detaches from the Hubble flow and recollapses once the critical density contrast is reached: equivalent to a real (non-linear) density: the collapse stops once a virial equilibrium configuration is reached at time, radius, and a final density contrast of the collapsed object of: or R. Fassbender: Introduction to Observational Cosmology I WS09/10 5 Schematic View on Structure Formation critical collapse density contrast δ c Source: H.Böhringer R. Fassbender: Introduction to Observational Cosmology I WS09/10 6

4 Starting Point: Gaussian Fluctuation Field a normalized Gaussian is fully characterized by a mean value μ (1. moment) and a width σ(2. moment) a Gaussian fluctuation density field is (at time t) statistically fully characterized by the mean density Ω m (1. moment) and the second moment as a function of length scale L, i.e. the power spectrum P(k) R. Fassbender: Introduction to Observational Cosmology I WS09/10 7 Filtering and Mass Scale spatial filtering on length scale R of a density field corresponds mathematically to a convolution with a filter function W R (x) in real space, or a multiplication with the Fourier transform of the filter function in Fourier space: the simplest filter in 3D space is the top hat filter: = const and W R =0 for r>r the average mass contained within the filter is then: the variance of an isotropic smoothed field with smoothing length R or mass scale M is then easiest expressed in Fourier space: R. Fassbender: Introduction to Observational Cosmology I WS09/10 8

5 The cosmological parameter σ 8 (1) the smoothing scale R is directly related to the mass scale M and the mass variance: the transition from the linear to the non-linear regime happens at σ M ~1 we still need a single number that characterizes the amplitudes of the density fluctuation field and fixes the normalization of the power spectrum σ 8 scales >8 h -1 Mpc non-linear power spec. linear PS choose a filter scale R in the transition regime, i.e. larger scales (smaller k) are still linear: R = 8 h -1 Mpc (=11.4 h 70-1 Mpc) k = h Mpc -1 the filtering/smoothing supresses scales of L<8 h -1 Mpc or k> h Mpc -1 R. Fassbender: Introduction to Observational Cosmology I WS09/10 9 The cosmological parameter σ 8 (2) σ 8 is the average (rms) fluctuation amplitude of the linearly evolved, present day (z=0) dark matter density contrast field smoothed with a top-hat filter of radius R=8 h -1 Mpc, or the power spectrum normalization (k-space volume) on large scales: WMAP5 (temperature) fluctuation map at z~1100 linear evolution to z=0 smoothing with R=8 h -1 Mpc average rms amplitude Source: R. Fassbender: Introduction to Observational Cosmology I WS09/10 10

6 The cosmological parameter σ 8 (3) σ 8 is the least intuitive cosmological parameter and is also the least well constrained one of the standard parameters σ 8 can be considered as the characteristic amplitude of the present day linear density contrast field on 8 h -1 Mpc scales or as the corresponding power spectrum normalization together with the primordial power spectrum slope n s ~1, which determines the initial spatial distribution of fluctuations, σ 8 is a measure of the characteristic amplitude of the fluctuations and is the second cosmological parameter that characterizes the inhomogeneous Universe present constraints: σ 8 = Source: Komatsu et al R. Fassbender: Introduction to Observational Cosmology I WS09/10 11 Goal: Press-Schechter Mass Function (1) dark matter halo mass function n(m,z) as a function of redshift z Idea of the Press-Schechter Approach (1974): 1. take primordial Gaussian density contrast field and evolve it forward in time according to the linear theory of structure growth 2. a space region is identified with a collapsed object once the critical spherical collapse threshold is reached: [in EdS] 3. objects that formed at z>0, i.e. reached equilibrium at redshift z form, have a present day z=0 density that is higher by the inverse of the linear structure growth function:, in an Einstein-de Sitter Universe this is given by: [δ c is slightly cosmology dependent, D + follows the form of L10] Source: Press & Schechter 1974 R. Fassbender: Introduction to Observational Cosmology I WS09/10 12

7 Procedure: Press-Schechter Mass Function (2) 1. determine the smoothing length scale R corresponding to the mass scale M 2. filter the present day (z=0) fluctuation field with the window function W R to arrive at the filtered density field 3. count the smoothed density peaks above the critical density contrast δ c at redshift z; i.e. peaks with a present minimum amplitude of: these peaks are interpreted as collapsed objects of mass M at redshift z the probability p(δ 0,c ) to find amplitudes above the critical collapse threshold δ 0,c follows from Gaussian statistics using the error function erfc: Source: Press & Schechter 1974 with R. Fassbender: Introduction to Observational Cosmology I WS09/10 13 Press-Schechter Mass Function (3) is the mean rms amplitude of the smoothed fluctation field with mass scale M the probability can be interpreted as a spatial filling factor or a mass fraction of collapsed objects above the threshold mass the final Press-Schechter mass function n(m,z) for the redshift evolution of the dark matter halo number density can be derived from this filling factor of collapsed objects at redshift z in the mass interval [M, M+dM]: for very large R or M: general case: the larger the mass M, the larger the smoothing radius R, the smaller the resulting rms amplitude σ M, i.e. the mass scale M is a monotonic function of: σ -1 M or ln(σ-1 M ) Source: Press & Schechter 1974 R. Fassbender: Introduction to Observational Cosmology I WS09/10 14

8 Press-Schechter Mass Function (4) Remarks on the PS Mass Function: a. to go from a filling factor per unit volume to a cluster number density n, one has to divide by the volume of the object b. the additional factor 2 corrects for a shortcoming of the Press-Schechter theory since the integral over only accounts for half of the total mass, and leaves out the initially underdense regions (related to the cloud-in-cloud problem ) c. the mass dependence is implicitly contained in the smoothed rms amplitude σ M, which also contains the dependence on the cosmological parameters [together with δ c ] through the redshift evolution of the linear growth factor D + and the linear power spectrum P(k) d. very massive objects are exponentially supressed / rare; the number density of the high-mass end of the mass function is thus exponentially sensitive to changes of cosmological parameters e. the PS mass function is based on the assumption of spherical collapse without substructure f. the choice of a top-hat filter function is ad-hoc R. Fassbender: Introduction to Observational Cosmology I WS09/10 15 Improved Mass Functions more rigorous derivations of the mass function exist, that can also account for the ad hoc factor 2 (Bond et al. 1991, Bower 1991, Lacey & Cole 1993) the more realistic case of an ellipsoidal collapse has also been derived (Sheth & Torman 2001) a particularly well tested and widely used universal fitting formula is the Jenkins mass function (Jenkins et al. 2001) which is based on large numerical simulations: it turns out that the Press-Schechter mass function underestimates the number if massive halos and overestimates the abundance of low-mass objects the unusual form of the mass variable lnσ -1 (M,z) ensures a universal (i.e. cosmology and redshift independent) form of the mass function R. Fassbender: Introduction to Observational Cosmology I WS09/10 16

9 The Universal Form of the Mass Function 1) power law form at low-mass end 2) characteristic mass scale M* at the knee 3) exponential decline at M>M* Source: Warren et al R. Fassbender: Introduction to Observational Cosmology I WS09/10 17 Mass Functions for Different Cosmologies Ω m =1.0 ΩΛ=0 σ 8 =0.5 fast evolution (EdS) Ω m =0.3 ΩΛ=0 σ 8 =0.9 slow evolution (open) solid line CCM Ω m =0.3 Ω Λ =0.7 w=-1 σ 8 =0.9 (bit too high) Ω m =1.0 ΩΛ=0 σ 8 =0.5 h=0.46 Ω m =0.3 ΩΛ=0.7 σ 8 =0.9 w=-0.8 Source: Voit 2005 R. Fassbender: Introduction to Observational Cosmology I WS09/10 18

10 The faster the structure growth, the fewer massive collapsed objects at higher redshift Concordance Model Ω m =0.3 Ω Λ =0.7 w=-1 Ω m =1 Einsteinde Sitter Universe Source: Borgani & Guzzo 2001 R. Fassbender: Introduction to Observational Cosmology I WS09/10 19 Evolution of the Mass Function from Simulations in Concordance Cosmology Press-Schechter Jenkins multiplicity function: mass fraction carried by objects per logarithmic unit mass bin hierarchical structure growth Evolution z=1.5 z=0 M=10 15 h -1 M sun : x10 3 M=10 14 h -1 M sun : x10 1 Source: Springel et al R. Fassbender: Introduction to Observational Cosmology I WS09/10 20

11 II. The Formation of Galaxy Clusters The Observed Galaxy Cluster Mass Function in cumulative form N(>M) Source: Vikhlinin et al. 2009; the observed evolution of the galaxy cluster mass function was the basis for the Dark Energy constraints from clusters discussed in L10 R. Fassbender: Introduction to Observational Cosmology I WS09/10 22

12 From a Dark Matter Halo to a Galaxy Cluster Galaxy Clusters are Dark Matter Halos filled with a hot X-ray-emitting Intracluster Medium (ICM) and Galaxies Simulated Dark Matter Halo X-rays: hot ICM optical: Galaxies Dark Matter 80-87% of total mass dominates gravitational potential and dynamics skeleton of the cluster structure Intracluster Medium (L12) 11-14% mass fraction dominates baryonic matter fills the cluster potential very hot and dilute plasma (~ K) emits in X-ray Galaxies (L12) 2-5% mass fraction 100 s-1000 s galaxies move like test-particles in DM-potential optical & infrared observations R. Fassbender: Introduction to Observational Cosmology I WS09/10 23 z=6 z=2 Simulated mass evolution of a single massive galaxy cluster The Press-Schechter approach only provides statistical information on the mass spectrum, but not on the growth evolution of individual objects. This hierarchical growth of massive structures can now be followed with large DM N-body simulations. z=1 z=0 infalling subhalos main halo the total mass of the main halo has grown by more than a factor of 10 since z=2 z=1 2 Source: Boylan-Kolchin et al., 2009 R. Fassbender: Introduction to Observational Cosmology I WS09/10 24

13 Dark Matter ICM Density Temperature Simulated Formation of a Galaxy Cluster since z=2 now with the baryonic ICM component the most massive M~10 15 M sun objects form last, i.e. at relatively low redshifts z<1-2 strong evolution effects of the DM and ICM component over last 10Gyr (z~2) the formation of the DM halo and the ICM is to first order driven by gravity only; more complex gas physics (e.g heating & cooling) can lead to deviations from the expected gravity only expectations of the ICM properties Source: galform/data_vis/index.shtml R. Fassbender: Introduction to Observational Cosmology I WS09/10 25 Why are there no objects with M>10 16 sun in the Universe? 16 M sun Answer 1: Press-Schechter theory a mass of M sun corresponds to a smoothing scale of R~40 Mpc according to: the local linear density contrast fields smoothed on this scale does not have any peaks above the collapse threshold of δ c ~1.69 Answer 2: Virialization Time a mass of M sun corresponds to a galaxy cluster radius of R cl ~4.5 Mpc according to: approximate real (nonlinear) overdensity of an object at collapse time the crossing time of a particle/galaxy with typical velocities of ~1000 km/s approaches the Hubble time, i.e. the object cannot be in virial equilibrium yet R. Fassbender: Introduction to Observational Cosmology I WS09/10 26

14 Why is the Intracluster Medium so hot? Answer: Virial Theorem the depth of the gravitational potential of clusters is so deep that the kinetic particle velocities in virial equilibrium correspond to temperatures of ~1-10 kev or K initially cold gas (from outside) is gravitationally shock heated to the virial temperature during infall into the potential well of the cluster R. Fassbender: Introduction to Observational Cosmology I WS09/10 27 Schematic View of Cosmic Gravitational Potentials Virialized (collapsed) Objects Non-Virialized Superstructures Velocity Dispersion Virial Gas Temperature Source: H.Böhringer R. Fassbender: Introduction to Observational Cosmology I WS09/10 28

15 III. Universal Dark Matter Halo Profiles Galaxy Clusters as Dark Matter Halos massive DM halo from the Millennium Simulation galaxy clusters are dominated by the structure of the Dark Matter halo the (radially symmetric) Dark Matter density profile ρ(r) determines the mass profile M(<R) and the gravitational potential Φ(R), and hence the interaction with the surrounding largescale structure the Dark Matter dominated gravitational potential binds the hot ICM gas and determines the dynamics of the galaxies the mass-to-light ratio of galaxy clusters is M/L ~ , i.e. they are the objects with the highest Dark Matter fraction / domination R. Fassbender: Introduction to Observational Cosmology I WS09/10 30

16 The NFW Profile for Dark Matter Halos Navarro,Frenk & White 1996 (NFW) proposed a universal Cold Dark Matter halo density profile the NFW profile is an empiric approximation of the virialized equilibrium configuration of Dark Matter halos form simulations the NFW profile is in principle valid for the DM halos of galaxies and galaxy clusters, i.e. it is (almost) self-similar several alternatives or extensions to the NFW DM density profiles exist (e.g. Hernquist-, Moore-, or Burkert-profile), with different radial characteristics in the center and/or at large R the NFW profile is up to date the most popular model for today: closer look at NFW profiles as the base model for galaxy clusters R. Fassbender: Introduction to Observational Cosmology I WS09/10 31 The DM Density Profile of (ideal) Galaxy Clusters ρs ρdm ( r) = 2 r r 1 + rs rs NFW profile: central cusp: outer region: ρ ρ DM DM ( r << ( r >> r ) r s r ) r s 1 3 two free parameters: r s determines the size (alternatively: compactness c=r 200 /r s ) ρ s determines the characteristic density (at r s ) a central cusp (r -1 ) is a prediction of CDM simulations the characteristic density ρ s is related to the formation epoch of the halo: smaller halos form earlier and are thus denser and more compact R. Fassbender: Introduction to Observational Cosmology I WS09/10 32

17 Definition of Cluster Radii galaxy clusters are not isolated objects with a sharp edge, i.e. the size and the edge of a cluster are not well defined the cluster radius R xxx is defined as the radius within which the average mass density of the total enclosed mass M(<R) is equal to xxx* the critical density ρ cr (z) the cluster radius is defined with respect to the evolving background density ρ cr (z), i.e. the radius itself is evolving towards smaller values with increasing redshift since the background density is increasing R 200 can be considered as an approximation for the total virialized cluster region other widely used definitions are R 500 as (typical) accessible radius for X-ray observations and R 2500 for the core region R. Fassbender: Introduction to Observational Cosmology I WS09/10 33 The NFW Mass Density Profile ρ NFW (r) with c = R 200 /r s = 5 in units of ρ cr = M sun sun /Mpc 3 ρ NFW (r). M(<R) / V(<R) V in units of [ρ cr ] R. Fassbender: Introduction to Observational Cosmology I WS09/10 34

18 The NFW Mass Density Profile ρ NFW (r) is (slightly) divergent at r 0, r but enclosed mass is finite R 200 ~1 Mpc for M 200 =10 14 M R 200 ~2 Mpc for M 200 =10 15 M R. Fassbender: Introduction to Observational Cosmology I WS09/10 35 NFW Mass Profile M(<R) of Galaxy Clusters M ( R) = 4 π ρ ( r) r R 0 2 dr R. Fassbender: Introduction to Observational Cosmology I WS09/10 36

19 from: NFW Gravitational Potential Φ(R) d Φ( R) F( R) G M ( < = = 2 dr m R R) normalized and scaled to Φ(R 200 )=0 R. Fassbender: Introduction to Observational Cosmology I WS09/10 37 Velocity of a radially infalling test particle with v(r200)=0 and 0.5*v 2 (R)= Φ(R) in [km/sec] typical (randomized) particle velocities in galaxy clusters are of order: v~1000 km/sec R. Fassbender: Introduction to Observational Cosmology I WS09/10 38

20 Results from X-ray X Observations are consistent with NFW Mass Profiles for Clusters mass profiles for 10 clusters with M sun physical scale scaled units to R 200 Source: Pointecouteau et al., 2005 R. Fassbender: Introduction to Observational Cosmology I WS09/10 39 Summary 1. the Press-Schechter approach for counting peaks in smoothed density fields provides a framework for an analytic solution of the mass spectrum of collapsed Dark Matter halos 2. the mass function has a characteristic universal form following a power law at the low mass end and an exponential cut-off above a characteristic mass M* 3. the cosmological parameter σ 8 characterizes the average rms amplitude of the present linear DM fluctuation field on large scales 4. the most massive objects with M>10 15 M sun appear last on the cosmic stage end show strong evolution effects with increasing redshift 5. galaxy clusters can be considered as Dark Matter halos filled with a hot intracluster gas and a bound galaxy population 6. the density profile of Dark Matter halos / galaxy clusters is well described by a NFW profile with a universal self-similar form R. Fassbender: Introduction to Observational Cosmology I WS09/10 40

21 Appendix The Fourier Transform of the Top-Hat Function Source: /Useful_Tables/Maths/fourier /Maths_Fourier_transforms.html#Top_hat R. Fassbender: Introduction to Observational Cosmology I WS09/10 42