Introduction to Observational Cosmology II

Transcription

1 Introduction to Observational Cosmology II Summer-semester 2010 in collaboration with Dr. Stefanie Phelps and Dr. Rene Fassbender Hans Böhringer Max-Planck Institut für extraterrestrische Physik, Garching H. Böhringer LMU Lecture: Observational Cosmology II ( 1) SS

2 Content of the Lecture Review of Cosmological Framework 3.5. Thermodynamik Evolution and Nucleosynthesis First Objects and Reionization Advanced Topics on the Cosmological Mikrowave Background I Advanced Topics on the Cosmological Mikrowave Background II 7.6. High Redshift Galaxies Strong Gravitational Lensing Distant Galaxy Clusters and Galaxy Evolution in Clusters Galaxy and AGN (Co-)Evolution 5.7. Weak Gravitational Lensing and Cosmic Shear The Sunyaev-Zeldovich Effect Cosmological Backgrounds Homework for the lecture certificate from 5.7 to H. Böhringer LMU Lecture: Observational Cosmology II ( 1) SS

3 Literature I P.J.E.Peebles Physical Cosmology 1993,Princeton Series in Physics J.A. Peacock Cosmological Physics 1999, Cambridge University Press S. Weinberg Gravitation and Cosmology 1972, John Wiley & Sons E.W. Kolb & M.S. Turner The Early Universe 1990, Addison-Wesley Co. P. Coles & F. Lucchin Cosmology, The origin and Evolution of Cosmic Structure 1996, John Wiley & Sons; 2nd edition 2002 G. Börner The Early Universe, Facts and Finction 1998, Springer Verlag T. Padmanabhan Structure Formation in the Universe 1993, Cambridge University Press T. Padmanabhan Cosmology and astrophysics through problems 1996, Cambridge University Press E.R. Harrison Cosmology, the Science of the Universe 1981, Cambridge University Press M.S. Longair Galaxy Formation 1998, A&A Library, Springer Verlag H. Böhringer 3

4 Literature II V. Mukhanov, Physical Foundations of Cosmology, 2005, Cambridge University Press Barbara Ryden, Introduction to Cosmology, 2003, Addison & Wesley D.-E. Liebscher, Cosmology, 2005, Springer Tracts in Modern Physics Scott Dodelson, Modern Cosmology, 2003, Academic Press, Elsevier Very solid introdunction to GR: B. Schutz, A first course in general relativity, 1985, Cambridge University Press H. Böhringer 4

5 Lecture 1 Repetition of the Basic Framework for Cosmological Studies 1. Basic cosmological principles 2. Friedmann-LeMaitre World Models 3. Robertson-Walker Curved Space-Time Metric 4. Distance Measurements over Cosmological Distances 5. Origin and Evolution of Structure in the Universe 6. Description and Observation of the Large-Scale Structure H. Böhringer 5

6 Distance Scales in the Universe z =1 Most distant Quasars, z = 6 Deep redshift surveys bis z ~ 0.2 Horizon logd [pc] Coma Galaxy Cluster, z = Virgo Galaxy Cluster, D = 17 Mpc Andromeda Galaxy, M31, D = 0.7 Mpc Large Magellanic Cloud, D = 55 kpc Radius of the Milkey Way, R ~ 20 kpc Star cluster Plejadies, D ~ 130 pc α Centauri, D ~ 1.3 pc Sun, D = 150 Mill. km 1 pc = 3.26 light years H. Böhringer 6

7 Prerequisite for a Reasonably Simple and Understandable Model of the Universe The physical laws are the same in the whole Universe Mater has the same nature in the whole Universe The Universe is approximately homogeneous and istropic Cosmological principle (very generally formulated): we are not. living in a special place in the Universe Assumption of a scenario where the current order in the Universe. Originates from arbitrary ( chaotic ) initial conditions This can probably not complete be carried through, since in the current cosmological models we experience some fine tuning of physical constants (see e.g. P.C.W. Davies The Accidental Universe Cambridge University Press 1982) H. Böhringer 7

8 Fundamental Principles Cosmology can be formulated as an excat science based on a set of principles: 1. Cosmological Principle: on large scales we can consider the Universe to be homogeneous and isotropic Robertson-Walker Metric for Curved Space Time 2. Equivalence Principle: the equivalence of gravitational and inertial mass is the basis of general relativity Einstein Field Eq. of GR Friedmann-Lemaitre-Equations 3. Action Principle: is the source to formulate field theories and derive the corresponding field equations Klein Gordon Equation for Inflaton Fields and Dark Energy H. Böhringer 8

9 Friedmann LeMaitre Equations modified with Λ The following equations are extended by the Λ-term introduced by Einstein and now used to describe an accelerated Universe: H. Böhringer 9

10 World Models Following from FL-Equations We have: Def.: Index 0 is for today constant 1. for k = 0 for 2. for k>0 a has a turn around point at: a(t) k<0 k=0 3. for k<0 remains always positive (infinite expansion) k>0 t H. Böhringer 10

11 Introducing the Parameters H 0 and Ω 0 =Ω m FL-Equations without Λ Definitions for present time: Then we can eliminate k/a 2 by: We can choose the length scale such that a 0 =1 H. Böhringer 11

12 General World Models with Parameters H 0, Ω m, Ω Λ 1. dynamical equation with : we substitute k with a 0 = 1: H. Böhringer 12

13 Differential Equation for the World Model 13

14 Free Expansion and Einstein-DeSitter Universe 1. Case Ω 0 = 0 free expansion 2. Case Ω 0 = 1 : ansatz for the solution:! H. Böhringer 14

15 Models in which the Λ-term dominates When the scaling parameter, a, gets very large the Λ-term dominates: a(t) H. Böhringer 15 t

16 Evolution of the Density Parameter Ω(t) for a 0 = 1 Substituted: Ω(t) Ω>1 1 Ω<1 t H. Böhringer 16

17 The Matter and Radiation Dominated Universe Density evolution : Matter Radiation log ρ(t) ρ m t eq ρ γ log t H. Böhringer 17

18 Equipartition and Recombination Time Equipartition time (matter and radiation have the same mass density): Example: H 0 = 70 km s -1 Mpc -1, Ω m = 0.3 Recombination time (the Universe becomes transparaent) : H. Böhringer 18

19 Age of the Universe H. Böhringer 19

20 Age of the Universe in Different Models Λ=0.9 Λ=0.7 Λ=0.5 Λ=0 H. Böhringer 20

21 Age Determination For Ω 0 = 1 : for H 0 =70 km s -1 Mpc -1 t = 9.3 Gyr For Ω 0 = 0.3, Ω Λ = 0.7, H 0 = 70 km s -1 Mpc -1 : t = 13.6 Gyr H. Böhringer 21

22 Curved Space-Time Metric Possible test of the geometry of a person living in the hyper-surface: angular sum in a triangle. H. Böhringer 22

23 Robertson-Walker Metric Tensor spatial part Where R relates to the curvature radius and the sign of k gives positive (convex) or negative (concave) curvature. k = 0 flat space k = +1 closed space (hypersphere) k = -1 open space The time dependent part is R(t) which can be separated concept of comoving coordinates! H. Böhringer 23

24 From Einstein s Field Equations to Freidmann-LeMaitre Formalism With the energy momentum tensor for a homogenous Universe given by: Inserting the Robertson-Walker-Metric for the geometry terms in the Einstein Equation leads to the Friedmann-Lemaitre-Equations (including Lambda). p. 9 of this lecture

25 Comoving Distance r as a Function of t For the measurement we use a null-geodesic : With an integration along the light path we obtain: Integral: for k=1 for k=0 for k=-1 with : for k=1 for k=0 for k=-1 H. Böhringer 25

26 Comoving Distance r as a Function of z Relation between z and t substituted: The function in the dynamical equation for a(t) coded by means of the function E(z) has to be derived by solving the cosmic field equations. with : H. Böhringer 26

27 Diameter Distance Physical extension of an object with the obsvered angular diameter θ at the redshift z : Definition of the diameter distance : This is the physical diameter of the object when the light was emitted and not the comoving size of the object (if it is comoving)! H. Böhringer 27

28 Luminosity Distance Ansatz: the source<is located at the origin, the observer is located on the sphere with radius = distance : bolometric flux! The surface of the sphere is calculated for the time t 0 : Definition of the luminosity distance: S 0 =4πR 02 r 2 Without the effect of the redshift and thinning of the photons : 1. Factor (1+z) due to the loss of energy by the redshift 2. Factor (1+z) due to time dilation from a(t) to a 0 H. Böhringer 28

29 Calculation of the Parameter R 0 The parameter R 0 appears in the formulae for the calculation of distances which only cancles in the case of k = 0. This parameter can be calculated as follows : H. Böhringer 29

30 Distance as a Function of Redshift H. Böhringer 30

31 Diameter as a Function of z for Different Cosmological Models H. Böhringer 31

32 Energy Flux as a Function of z for Different Cosmological Models Euklidian H. Böhringer 32

33 Results of Perlmutter et al. 99 I H. Böhringer IMPRS Lecture: Cosmology and Large-Scale Structure ( 2) SS

34 Results of Perlmutter et al. 99 III H. Böhringer IMPRS Lecture: Cosmology and Large-Scale Structure ( 2) SS

35 Brief Review of Structure Formation The Universe on large scale can be considered homogenous, but pronounced structure is seen in the Universe today at scales smaller than few 100 Mpc. We want to describe the origin and evolution of this structure.

36 Growth of Density Perturbations Sketch of the Program

37 Cosmological model H 0 Ω m Ω Λ Ω B (w) Generation of density fluctuations. σ 8 P 0 (k) Nature of the Dark Matter. CDM (HDM) Structure evolution (gravitational effect) Metric Test SN Ia Early structure: Cosmic microwave background Galaxies Galaxy clusters Large-scale galaxy/cluster distribution

38 Origin of Density Fluctuations in the Standard Model Early Universe Inflation Quantumfluctuations Streching Quantum fluctuations are suddenly magnified to macroscopic scales during an inflationary phase in the early Universe and they are preserved by this magnification. Their amplitude grows subsequently by gravitational instability processes. Gravitational growth ρ c ρ 0

39 Growth of Density Fluctuations one possible approach for the calculation Birkhoffʻs Theorem: the matter inside a homogeneous sphere in a surrounding homogeneous Universe is not subjected to outside forces, except for tidal forces, which we can neglect here as second order effects. The matter density inside the sphere is evolving like a mini-universe with the same density and expansion parameters. Each part of the Universe evolves approximately like a homogeneous Universe with the same mean density. Application to an inhomogeneous Universe : ρ 1 ρ 2 ρ 3 local homogeous approximation <ρ> ρ(x)

40 Gravitationsinstabilitäten VIII result of the perturbation analysis Differential equation for the evolution of density fluctuations (for pressureless fluid, applies after recombination): Only important for a fluid with pressure!

41 Repetition: Linear Growth of Density Fluctuations Mater dominated universe with critica density : With low density : Radiation dominated universe with critical dxensity : The density fluctuations grow in the radiation dominated universe only oiutside the horizon

42 Structure Growth in a Critical Density and Open (Empty) Universe For small negative deviations of Ω from 1 the parameter evolves increasingly rapidly to small values. When Ω becomes significantyl smaller then 1 the growth of fluctuations is stopped the fluctuation spectrum is frozen. Ω(t) δ(t) 1 a b t a b t

43 Growth of Density Fluctuations in Different Cosmological Models Friedman-Lemaitre-Models Λ-Models

44 Growth of Density Fluctuations in Cosmological Models with Dark Energy e.g. Wang & Steinhardt 1998, ApJ, 508, 483

45 Examples for the Growth of Structure in Different Cosmological Models including DE Density fluctuation growth: open Λ The more negative w, the stronger the evolution, that is, less clusters at high redshift for given normalization at present.

46 Fourier Description of the Density Fluctuation Field Description of an arbitrary function Δ(x) in a three-dimensional cube : With the power spectrum: Representation in the 3-dimensional continuum :

47 Fourier Decomposition II Power spectrum and variance of the density fluctuation field are (we assume here that the spatial distribution to be described is isotropic) : In the standard cosmological model the density fluctuation field is assumed (with good observational support) to be a Gaussian fluctuation field. A Gaussian field is defined such that P(k) is the only statistical description that is needed to characterize the field while all the rest is completely random. realization of a Gaussian random field as initial conditions for N-body simulations.

48 Power Spectra for Different Cosmologies

49 Relation Between P(k) and the Mean Amplitude of the Fluctuations as a Function of Scale (R) With window function : W(kR) ( Δ here is identical to the previously defined variance σ(r) )

50 Popular Filter Functions Gaussian filter set : Top hat filter set : Sharp k-space filter set:

51 Variance of the Density Fluctuations as Function of the Wavelength for Different Cosmologies

52 Power Spectra for Dark Matter Scenarios Harrison-Zeldovich Spectrum Horizon at the time of equipartition Power in arbitrary units Overdense regions collapse simultaneously at all scales

53 2dF-Galaxy Redshift Survey Redshifts of > 100,000 galaxies CDM-Simulations of this Survey H. Böhringer 53

54 Composite Observed Power Spectrum H. Böhringer 54