Observational Cosmology: 3.Structure Formation

Transcription

1 Observational Cosmology: 3.Structure Formation An ocean traveler has even more vividly the impression that the ocean is made of waves than that it is made of water. Arthur S. Eddington ( ) 1

2 Radiation: CMB - Isotropic to 1 part in 10 5, 0.003%, 2µK 3.1: Isotropy & Homogeneity on the Largest Scales Isotropy and Homogeneity on the largest scales Cosmological Principle: The Universe is Homogeneous and Isotropic True on the largest Scales Matter: Large scales > 100Mpc (Clusters / Superclusters) : Universe is smooth Radio Sources: isotropic to a few percent Small scales : Highly anisotropic 2

3 3.1: Isotropy & Homogeneity on the Largest Scales Isotropy and Homogeneity on the largest scales ~1000Mpc 200Mpc Walls Filaments Clusters Superclusters Voids 3

4 3.2: The Growth of Structure Primordial Density Fluctuations Origin of LSS today - primordial density fluctuations Density perturbation δ = ρ ρ ρ = Δρ ρ 1) Primordial Quantum Fluctuations Gaussian Fluctuations from inflation 2) Cosmic Defects Defects from phase transitions Cosmic String Domain Walls Textures Scale Free Harrison - Zeldovich spectrum model: Fluctuations have the same amplitude when they enter the horizon ~ δ ~ 10-4 Scale free Harrison-Zeldovich Spectrum of power P(k) = δ k 2 k n, n =1 4

5 3.2: The Growth of Structure Primordial Density Fluctuations δ = ρ ρ = Δρ ρ ρ 5

6 3.2: The Growth of Structure Primordial Density Fluctuations δ = ρ ρ = Δρ ρ ρ ρ ISOTHERMAL FLUCTUATIONS Fluctuations in matter ONLY No perturbations in the Temperature ρ ADIABATIC FLUCTUATIONS Fluctuations in matter and radiation (changes in volume in the early Universe change in number densities) ρ ISO-CURVATURE / ISENTROPIC FLUCTUATIONS No Perturbations in the density field Fluctuations in the matter relative to the radiation δ m δ γ Fluctuations in radiation field leave scar on CMB observed as deviations from 2.73K BB 6

7 The Jeans Length 3.2: The Growth of Structure Consider a homogeneous universe of average density M r ρ = ρ(1+δ) ρ = ρ Embed a sphere of mass, M = 4π ρ = ρ (1+ δ), δ = ρ ρ = Δρ 3 ρ (1+ δ)r3 with over density ρ ρ <<1 r = GΔM = 4πGρ r t Sphere collapses from rest equilibrium under self gravity δ 1 t 3 During Collapse δ <<1, M = 4π 3 ρ (1+ δ t ) r t 3 = constant r t = 3M 4πρ 1/ 3 (1+ δ t ) 1/ 3 = r to (1+ δ t ) 1/ 3 (1+ δ t ) 1/ 3 1 δ t 3, r t r to r r δ t 3 1 = 2 δ t = 4πGρ δ t Solutions of the form δ t = δ o 2 et /τ ff + δ o Where, τ ff = (4πGρ ) 1/ 2 is the dynamical free fall time t e t / t ff 2 r t 2 2 e t /τ ff 0 Only exponentially increasing term survives Conclusion: Density perturbations will grow exponential under the influence of self gravity ρ 7

8 The Jeans Length 3.2: The Growth of Structure Equation of State for an Ideal Gas P = ωρc 2 = kt µ ρ = v s 3 ρ (v s << c) v s = the sound speed ( fundamental cosmology 5.3) In absence of pressure, an overdense region collapses on order of the free fall time τ ff = (4πGρ ) 1/ 2 Compression Acoustic Oscillations Pressure Pressure Gradient - Resists Collapse if a pressure gradient can be created over a timescale given by τ J < τ ff GRAVITY Pressure Expansion τ J r v s Define a critical length over which density perturbation will be stable against collapse under self gravity r critical = λ J ~ v s τ ff ~ v s 2 Gρ λ J = π v 2 s Gρ 1/ 2 M J = 4π 3 ρ λ 3 J = 2π v s τ ff JEANS LENGTH JEANS MASS 8

9 Formal Jeans Theory 3.2: The Growth of Structure Continuity Equation Euler Equation Poisson Equation Entropy Equation 9

10 3.2: The Growth of Structure Jeans Mass, Silk Mass and the decoupling epoch Friedmann eqn. (k=0) expansion rate of Universe given by Hubble parameter H 2 = 8πGρ 3 Free Fall Time τ ff = (4πGρ ) 1/ 2 H 1 τ ff v λ J = 2π v s τ ff 2π(2 /3) 1/ 2 s Jeans Length v H s = c Photon sound speed (ω=1/3) 3 0.6c Before epoch of decoupling, photons and Baryons bound together as a single fluid At decoupling (z=1089) λ J,γ 3 c H λ J,γ (dec) 0.6Mpc c M J,baryon 36π ρ H 3 Super-horizon scales Sub horizon scales cannot grow M J,baryon (dec) M o This mass is larger than the largest Supercluster today! 10

11 3.2: The Growth of Structure Jeans Mass, Silk Mass and the decoupling epoch λ J,γ (dec) 0.6Mpc After epoch of decoupling, photons and Baryons behave as separate fluids M J,baryon (dec) M o Photon sound speed Baryon sound speed v s,γ = c 3 0.6c v s,baryon = kt mc 2 1/ 2 c c Jean s Length λ J = π v 2 s Gρ Jean s Mass after decoupling 1/ 2 After decoupling M J,baryon = λ J (dec) λ J λ J = v s,baryon v s,γ 3 λ J (dec) ~ 2x10 5 λ J (dec) M J,baryon (dec) ~ 10 5 M o This mass is approximately the same mass as Globular Cluster today! Until decoupling, structures over scales of globular clusters up to superclusters could not grow 11

12 3.2: The Growth of Structure Jeans Mass, Silk Mass and the decoupling epoch lg(m J ) {Mo} unstable Acoustic oscillation Radiation dominated Matter-radiation equality recombination Asymptotic value unstable Matter dominated λ J,γ (dec) 0.6Mpc M J,baryon (dec) M o λ J,γ 12pc M J,baryon 10 5 M o M silk 5x10 14 M o lg(t r ) {K} Close to decoupling / recombination : Baryon/photon fluid coupling becomes inefficient Photon mean free path increases diffuse / leak out from over dense regions Photons / baryons coupled smooth out baryon fluctuations Damp fluctuations below mass scale corresponding to distance traveled in one expansion timescale 3 / 2 Ω M silk 2x10 12 o THE SILK MASS ( Ω o h 2 ) 5 / 4 M o 5x10 14 M o Ω b 12

13 3.2: The Growth of Structure Growth of Perturbations in an expanding universe : The Hubble Friction Growth of structure - competition between 2 factors: Expanding Universe ρ = ρ (t) R(t) 3 Embed a sphere of density ρ, in a homogeneous universe Total Gravitational acceleration at surface of the sphere r = GM = G 4πρr 3 r 2 r 2 3 Free Fall Time τ ff = (4πGρ ) 1/ 2 Hubble Expansion M = const = 4π 3 ρ (1+ δ ) r 3 t t r t = 3M 4π ρ(t) = ρ (t)[ 1+ δ(t) ] H 1 = 8πGρ 3 r r = 4π 3 G ρ + ρ δ 1/ 3 ( ) 1 1/ 2 M H 1 τ ff r ρ = ρ(1+δ) ρ t 1/ 3 (1+ δ t ) 1/ 3 r t R(t)(1+ δ t ) 1/ 3 During Collapse/expansion 2 ρ d 2 2 = 1 dt 2 R R δ 3 2 R δ = 4π 3R 3 Gρ 4π 3 Gρ δ For homogeneous, isotropic universe, δ=0 R R = 4π 3 Gρ Subtracting the homogeneous component δ + 2H δ = 4π G ρ δ For a static universe, H=0 δ = 4π G ρ δ Extra term in expanding universe HUBBLE FRICTION slows the growth of the density pertubations 13

14 3.2: The Growth of Structure Growth of Perturbations in an expanding universe δ + 2H δ = 4π G ρ δ Rewrite in terms of density parameter δ + 2H δ 3 2 Ω H 2 m δ = 0 Radiation Era Ω m <<1 δ + 2H δ = δ + 1 t δ 0 Solution: δ(t) A + Bln( t) Fluctuations in matter (non-baryonic) can only grow logarithmically Lambda Era Η=Η Λ = Const δ + 2H δ = δ + H Λ δ 0 Solution: δ(t) A + Be 2 H Λ t F;uctuations in matter tend to a constant fractional amplitude. Matter Era (Ω m =1, H=2/3t) δ + 2H δ 3 2 Ω H 2 m δ = δ + 4 δ 2 3t 3t δ = 0 2 Solution: Growing mode and a decaying mode exist δ(t) A t 2 / 3 + B t 1 Density fluctuations in a flat, matter dominated Universe grow as δ A t 2 / 3 R(t) 1 (1+ z), δ <<1 14

15 3.3: Structure Formation in a Dark Matter Universe Growth of Perturbations in an expanding universe Density fluctuations in a flat, matter dominated Universe grow as δ A t 2 / 3 R(t) 1 (1+ z), δ <<1 δ<<1 linear regime δ~1 non-linear regime Require N-body simulations Baryonic Matter fluctuations can only have grown by a factor (1+z dec ) ~ 1000 by today for δ~1 today require δ~0.001 at recombination δ~0.001 δτ/τ ~0.001 at recombination But CMB δτ/τ 10 5!!! MATTER PERTURBATIONS DON T HAVE TIME TO GROW IN A BARYON DOMINATED UNIVERSE DARK MATTER Dark Matter Condenses at earlier time Matter then falls into DM gravitational wells 15

16 3.3: Structure Formation in a Dark Matter Universe Dark Matter To be born Dark, to become dark, to be made dark, to have darkness COLD DARK MATTER Non Relativistic at decoupling Heavy Neutrino SUSY Particles WIMPs Axions HOT DARK MATTER Relativistic at decoupling Light Neutrino Monopoles COSMIC RELICS Symmetry Defects Cosmic Strings Cosmic Textures 16

17 3.3: Structure Formation in a Dark Matter Universe Dark Matter Weakly interacting no photon damping Structure formation proceeds before epoch of decoupling Provides Gravitational sinks or potholes Baryons fall into potholes after epoch of decoupling Mode of formation depends on whether Dark Matter is HOT/COLD Hot /Cold DM Decouple at different times Different effects on Structure Formation Chandra website 17

18 3.3: Structure Formation in a Dark Matter Universe Dark Matter Actual picture of dark matter in the Universe!!! 18

19 3.3: Structure Formation in a Dark Matter Universe Dark Matter Actual picture of dark matter in the Universe!!! 19

20 3.3: Structure Formation in a Dark Matter Universe Hot Dark Matter Any massive particle that is relativistic when it decouples will be HOT Characteristic scale length / scale mass at decoupling given by Hubble Distance c/h(t) Radiation Dominates ρ R 4 (R = R(t)) R Friedmann eqn. H(t) 2 = Ω o r,o R Radiation dominated H(t) = 1 2t 4 2 H o For radiation (photons) 1+z ~ 3500 Epoch of Matter/Radiation Equality Matter Dominates ρ R 3 (R = R(t)) R Friedmann eqn. H(t) 2 = Ω o m,o R Radiation dominated H(t) = 2 3t 3 2 H o Substituting for (Ro/R), The Hubble Distance at t eq is Mass inside Hubble volume M eq = 4π 3 3 c H(t eq ) ρ(t eq ) = 4π 3 c H(t eq ) = c 3 / 2 Ω r,o 2ct 2 eq 30kpc H o c H o 3 3 / 2 Ωr,o 2 Ω m,o Ω m,o 3 Ω m,o ρ c,o R o R eq 3 = π 3 c H o 3 3 / 2 Ωr,o ρ 2 c,o ~ M o Ω m,o Other relativistic species >> M Supercluster 1+z ~ >3500, M H <10 17 M o Epoch of equality defined when k B T~mc 2 1/ 4 Recall Fundamental Cosmology 7.2 At a time given by T = 32πGa 3c 2 t 1/ x10 10 t 1/ 2 20

21 3.3: Structure Formation in a Dark Matter Universe Hot Dark Matter For a hot neutrino, mass m ν (ev/c 2 ) : T eq m ν k 11600m ν {K} t eq =1.7x1012 (m ν ) 2 {s} Before t eq, neutrinos are relativistic and move freely in random directions Absorbing energy in high density regions and depositing it in low density regions Like waves smoothing footprints on a beach! Effect smooth out any fluctuations on scales less than ~ ct eq λ eq c t eq ~ 17 m ν ( ) 2 kpc λ o = R o R eq λ eq ~ T eq λ eq m ν Mpc This Effect is known as FREE STREAMING Fluctuations suppressed on mass scales of M = 4π 3 λ 3 oω m,o ρ c,o ~ m ν M o Large Superstructures form first in a HDM Universe TOP-DOWN SCENARIO 21

22 3.3: Structure Formation in a Dark Matter Universe Cold Dark Matter For a CDM WIMP, mass m CDM ~1GeV : T eq m CDM k 10 9 {K} t eq = 5s c H = 2ct = 3x109 m λ o = R o R eq λ eq ~ T eq λ eq 0.04 kpc 2.73 M = 4π 3 λ 3 oω m,o ρ c,o << M o Fluctuations λ > λ ο will grow throughout radiation period Fluctuations λ < λ ο will remain frozen until matter domination when Hubble distance has grown to ~0.03Mpc corresponding to M o Scales > Hubble distance at matter domination retain original primordial spectrum Structure forms hierarchically in a CDM Universe BOTTOM-UP SCENARIO 22

23 3.3: Structure Formation in a Dark Matter Universe Structure Formation in a Dark Matter universe CDM - Bottom-Up Hierarchical Scenario HDM - Top-Down Pancake Scenario Simulation of CDM and HDM Structure formation seeded by cosmic strings ( 23

24 3.4: The Power Spectrum Quantifying the power in fluctuations on large scales We would like to quantify the power in the density fluctuations on different scales long wavelength (large scales) Density fluctuation field Fourier Transform of Density fluctuation field δ( r ) = ρ ρ = Δρ ρ ρ δ k = δ( r )e ik r Power of the density fluctuations P ( k) = δ k 2 High Power (large amplitude) Short wavelength (small scales) Low Power (small amplitude) 24

25 3.4: The Power Spectrum Quantifying the power in fluctuations on large scales Inflation Scale Free Harrison - Zeldovich spectrum model: P(k) = δ k 2 k n, n =1 Fluctuations have the same amplitude when they enter the horizon ~ δ ~ 10-4 Inflation field is isotropic, Homogeneous, Gaussian field (Fourier modes uncorrelated) All information contained within the Power Spectrum P(k) Value of δ(r) at any randomly selected point drawn from GPD (δ) = lg(p(k)) σ = Average mass contained with a sphere of radius λ (=2π/k) σ δ 2π e 2σ 2 1 V V P(k)d 3 k = (2π) 3 2π 2 M = 4π 3 P(k) k 2 dk 2π k 3 ρ small k large scales lg(k) large k small scales Mean squared mass density within sphere M M M 2 k 3 P(k) δm M 2 Instead of simply P(k) often plot (k 3 P(k)) 1/2 the root mean square mass fluctuations 25

26 The Transfer Function 3.4: The Power Spectrum Matter-Radiation Equality: Universe matter dominated but photon pressure baryonic acoustic oscillations Recombination Baryonic Perturbations can grow! Dark Matter free streaming & Photon Silk Damping erase structure (power) on smaller scales (high k) After Recombination Baryons fall into Dark Matter gravitational potential wells The transformation from the density fluctuations from the primordial spectrum through the radiation domination epoch through the epoch of recombination to the post recombination power spectrum, P(k,t) = T(k) 2 P(k,t primordial ) given by TRANSFER FUNCTION T(k), contains messy physics of evolution of density perturbations HDM T(k) =10 k k ν 1.5 k ν 0.4Ω o h 2 Mpc 1 (for a 30eV neutrino) supress all fluctuation modes λ < 2π k ν 120 m ν (ev ) Mpc [( ) ν ] 1/ν T(k) = f (Γ) = 1+ ((ak) + (bk) 3 / 2 + (ck) 2 ) CDM k 0, T(k) 2 1 P(k) k unchanged! k T(k) k 2 P(k) k 3 Small scale power! a = 6.4(Ω o h 2 ) 1 b = 3.0(Ω o h 2 ) 1 c =1.7(Ω o h 2 ) 1 ν =1.13 Γ = Shape Parameter 26

27 The Transfer Function 3.4: The Power Spectrum Primordial (P k) T(k) Baryons HDM MDM CDM P(k) HDM MDM CDM k {Mpc -1 } k {Mpc -1 } (k 3 P(k)) 1/ Primordial (P k) CDM (k 3 P(k)) 1/ CDM Primordial (P k) Mo 10-4 HDM MDM 10-4 MDM HDM Mo k {Mpc -1 } M {Mo} 27

28 The Transfer Function 3.4: The Power Spectrum Tegmark

29 The Power Spectrum 3.4: The Power Spectrum Vanilla Cosmology: Ω Λ =0.72, Ω m =0.28, Ω b =0.04, H=72, τ=0.17, b SDSS =0.92 Tegmark

30 The Power Spectrum 3.4: The Power Spectrum Tegmark

31 3.5: The Non-Linear Regime The non-linear Regime Primordial Fluctuations the seeds of structure formation Fluctuations enter horizon grow linearly until epoch of recombination Post recombination growth of structure depends on nature of Dark Matter Fluctuations become non-linear i.e. δ > 1 How can we model the non-linear regime? 31

32 3.5: The Non-Linear Regime (1) The Zeldovich Approximation (relates Eulerian and Lagragian co-ordinate frames) Eulerian coords (r) at time t related to Lagrangian coords (q) by initial velocity, s(q); r (q,t) = q + s(q)t In an expanding Universe: r (q,t) = R(t) q + δ(t) s(q) [ ] where δ + 2H δ 3 2 Ω H 2 m δ and s(q) = Φ o (q) Provides particle displacements with respect to initial Laplacian Mass Conservation: ρ(q,t) = ρ o r q = ρ 1 δ(t) r i q q r i q q = Deformation Tensor In 3-D, tensor eigenvectors define 3 orthogonal axes describing contraction/expansion: α(q), β(q), γ(q), α << β γ Pancake / Sheet α β << γ string / Filiament α β γ Knot / Sphere In the Zeldovich Approximation, the first structures to form are giant Pancakes (provides very good approximation to the non-linear regime until shell crossing) 32

33 (2) N-Body Simulations 3.5: The Non-Linear Regime PP Simulations: Direct integration of force acting on each particle PM Simulations: Particle Mesh Solve Poisson eqn. By assigning a mass to a discrete grid P3M: Particle-particle-particle-Mesh Long range forces calculated via a mesh, short range forces via particles ART: Adaptive Refinment Tree Codes Refine the grid on smaller and smaller scales PP Direct summation O(N 2 ) Practical for N<10 4 PM, P 3 M Particle mesh O(N logn) Use FFTs to invert Poisson equation. ART codes O(N logn) Multipole expansion. Strengths Self consistent treatment of LSS and galaxy evolution Weaknesses Limited resolution Computational overheads 33

34 3.5: The Non-Linear Regime (2) SAM - Semi Analytic Modelling Merger Trees; the skeleton of hierarchical formation Cooling, Star Formation & Feedback Mergers & Galaxy Morphology Chemical Evolution, Stellar Population Synthesis & Dust Hierarchical formation of DM haloes (Press Schecter) Baryons get shock heated to halo virial temperature Hot gas cools and settles in a disk in the center of the potential well. Cold gas in disk is transformed into stars (star formation) Energy output from stars (feedback) reheats some of cold gas After haloes merge, galaxies sink to center by dynamical friction Galaxies merge, resulting in morphological transformations. Strengths No limit to resolution Matched to local galaxy properties Weaknesses Clustering/galaxies not consistently modelled Arbitrary functions and parameters tweaked to fit local properties 34

35 3.5: The Non-Linear Regime N-Body Simulations - Virgo Consortium τ CDM Ω m =1, σ 8 =0.6, spectral shape parameter Γ=0.21 comoving size simulation 2/h Gpc (2000/h Mpc) cube diagonal looks back to epoch z = 4.6 cube edge looks back to epoch z = 1.25 half of cube edge looks back to epoch z = 0.44 simulation begun at redshift z = 29 force resolution is 0.1/h Mpc Λ CDM Ω m =0.3, Ω Λ =0.7, σ 8 =1, Γ =0.21 comoving size simulation 3/h Gpc(3000/h Mpc) cube diagonal looks back to epoch z = 4.8 cube edge looks back to epoch z = 1.46 half of cube edge looks back to epoch z = 0.58 simulation begun at redshift z = 37 force resolution is 0.15/h Mpc two simulations of different cosmological models : tcdm & LCDM one billion mass elements, or "particles" over one billion Fourier grid cells generates nearly 0.5 terabytes of raw output data (later compressed to about 200 gigabytes) requires roughly 70 hours of CPU on 512 processors (equivalent to four years of a single processor!) 35

36 3.5: The Non-Linear Regime N-Body Simulations - Virgo Consortium The "deep wedge" light cone survey from the τcdm model. The long piece of the "tie" extends from the present to a redshift z=4.6 Comoving length of image is 12 GLy (3.5/h Gpc), when universe was 8% of its present age. Dark matter density in a wedge of 11 deg angle and constant 40/h Mpc thickness, pixel size 0.77/h Mpc. Color represents the dark matter density in each pixel, with a range of 0 to 5 times the cosmic mean value. Growth of large-scale structure is seen as the character of the map turns from smooth at early epochs (the tie's end) to foamy at the present (the knot). The nearby portion of the wedge is widened and displayed reflected about the observer's position. The widened portion is truncated at a redshift z=0.2, roughly the depth of the upcoming Sloan Digital Sky Survey. The turquoise version contains adjacent tick marks denoting redshifts 0.5, 1, 2 and 3. 36

37 N-Body Simulations 3.5: The Non-Linear Regime 37

38 3.5: The Non-Linear Regime N-Body Simulations - formation of dark Matter Haloes R=0.02R o t=0.002t o 10Mpc The hierarchical evolution of a galaxy cluster in a universe dominated by cold dark matter. 38

39 N-Body Simulations 3.5: The Non-Linear Regime 39

40 N-Body Simulations 3.5: The Non-Linear Regime 40

41 N-Body Simulations 3.5: The Non-Linear Regime SPH Simulations Bevis & Oliver

42 Quantifying Clustering 3.6: Statistical Cosmology Underlying Dark Matter Density field will effect the clustering of Baryons Baryon clustering observed as bright clusters of galaxies Only the tip of the iceberg??? Baryons Dark Matter Baryons Dark Matter Baryons may be biased We would like to quantify the clustering on all scales from galaxies, clusters, superclusters 42

43 Quantifying Clustering 3.6: Statistical Cosmology 43

44 Quantifying Clustering 3.6: Statistical Cosmology Statistical Methods for quantifying clustering / topology The Spatial Correlation Function The Angular Correlation Function Counts in Cells Minimum Spanning Trees Genus Void Probability Functions Percolation Analysis Generally we want to measure how a distribution deviates from the Poisson case 44

45 The Correlation Function 3.6: Statistical Cosmology Angular Correlation Function w(θ) : Describes the clustering as projected on the sky (e.g. the angular distribution of galaxies, e.g. in a survey catalog) Spatial Correlation Function ξ(r) : Describes the clustering in real space For any random galaxy: Probability, δp, of finding another galaxy within a volume, V, at distance, r δp = ndv δp = n[ 1+ ξ(r) ]dv (ξ(r) 1, r 0 ξ(r) 0) (ξ(r) 1 r 0 ξ(r) 0) In a homogeneous Poisson distributed field (n = the number density) Since Probabilities are positive For a mean density to exist for the sample Assume ξ(r) is isotropic (only depends on distance not direction) ξ(r) = ξ(r) r V If the field is clustered In practice: the correlation function is calculated by counting the number of pairs around galaxies in a sample volume and comparing with a Poisson distribution 45

46 The Correlation Function 3.6: Statistical Cosmology Similarly the angular correlation function i given by δp = n[ 1+ w(θ) ]dω 1 dω 2 n = mean surface density For a catalog of n galaxies covering a solid angle, Ω Mean number of galaxies at θ±δθ from any randomly selected galaxy in mean solid angle <dω> ; ( n 1) ( 1+ w(θ) ) dω /Ω The total number of pairs with separations θ±δθ is therefore DD(θ) = n ( 2 n 1 )( 1+ w(θ) ) DD(θ) can be measured from the catalog Knowing DD(θ) and <dω>/ω w(θ) For All Sky Survey, Ω=4π, dω=2πsin(θ)δθ For more complicated geometries, easier to calculate <dω>/ω from the number of random-random pairs laid down over same area w(θ) = (DD/RR) 1 RR(θ) = n ( 2 n 1 ) Strictly require more random points than data points and need to correct for edge effects Use DR(θ) number of pairs with separations θ±δθ where one point is taken from random and real data set Standard Estimator : w(θ) = (2DD/DR) 1 dω Ω dω Ω 46

47 3.6: Statistical Cosmology The Correlation Function and the relation to the power spectrum Standard Estimator : Landy & Szalay- SL Estimator : Smaller uncertainties on large scales The angular correlation function is found to have the relation The spatial correlation function w(θ) = (2DD/DR) 1 w(θ) = (DD 2DR + RR) /RR Hamilton Estimator : w(θ) = 4(DDxDR) /(DR 2 1) ξ(r) = r r o γ w(θ) = A ω θ 1 γ Galaxies γ=1.8, r o =5h -1 Mpc Clusters γ=1.8, r o =12-25h -1 Mpc 1+ w(θ) Δw(θ) = DD The spatial correlation function is the Fourier transform of the Power Spectrum P(k) = ξ(r)e ik r d 3 r ξ(r) sin(kr) dr r The spatial correlation function is related to the mass density variation in spheres of radius,r σ R 2 = ΔM M 2 R = δ 2 = 1 R 3 ξ(rr)r2 dr σ R ~ unity on scales of 8Mpc normalize power spectrum at that scale b = σ 8,G σ 8,DM b is the bias parameter for galaxy biasing w.r.t. underlying Dark Matter Distribution 47

48 3.6: Statistical Cosmology The Correlation Function ELAIS APM w(θ) = A ω θ β A ω = ± β = 0.94 ± 0.09 w(θ) = A ω θ β β = 0.7 SDF z=4 LBG w(θ) = A ω θ β A ω = 0.71± 0.26 β = 0.8 ±

49 Limber Equation 3.6: Statistical Cosmology w(θ) = A ω θ 1 γ ξ(r) = r Limber Equation r o γ r o γ = A ω 1 C 0 1 γ F(z) D A (z) N(z) 2 g(z) dz ( N(z) dz) 2 0 Angular Diameter Distance Evolution of the spatial correlation function Cosmology term Numerical Contants Redshift Distribution (measured or predicted) D A F(z) (1+ z) ε g(z) = H o c Γ (γ 1) /2 C = π 1/ 2 Γ(γ /2) N(z) [(1+ z) 2 (1+ zω m + ((1+ z) 2 1)Ω ] Λ [ ] 49

50 Counts in Cells σ Scale (r) 2 ( z) = 0.2 Cosmic Variance too high 2 3.6: Statistical Cosmology Too Few boxes OK Boxes too sparse z Φ( L( z, s r min )) = f ( σ ( r)) / V ( z) ~ r N 2 = nv + (nv ) 2 + n 2 dv 1 dv 2 ξ Can derive divide the Universe into boxes of side r and count the number of galaxies, n i in each cell Divide into smaller and smaller boxes until n i =0 or 1 mean no. galaxies in volume N V = n i ndv = nv V Variance N 2 = 2 n i + n i n j i j either 0 or 1 n 2 dv 1 dv 2 ( 1+ ξ 12 ) N N N = ΔN 2 N = 1 N + 2 V Σ 2 V = variance of the density field smoothed over the cell 50

51 3.6: Statistical Cosmology Counts in Cells PROBING LARGE SCALE STRUCTURE 1000 ASTRO-F Herschel SCALE / h -1 Mpc PSC-z SWIRE Redshift 51

52 Minimum Spanning Trees 3.6: Statistical Cosmology 52

53 Genus 3.6: Statistical Cosmology 53

54 Void Probability Functions 3.6: Statistical Cosmology 54

55 Percolation Analysis 3.6: Statistical Cosmology 55

56 Large Scale Surveys 3.7: Large Scale Surveys ASTRO-F All Sky FIR µm = mJy ASTRO-F Team Pearson et al. (2004) 56

57 Summary 3.8: Summary Structure Formation in the Universe is determined by Initial Primordial Fluctuations Dark Matter (free streaming - Top Down / Bottom-Up Hierarchal) Acoustic Oscillations over the Jeans Length Photon Damping The epoch of decoupling and recombination Structure Formation in the Universe can be analysed by The Power Spectrum N-body Simulations Cosmological Statistics (e.g. correlation functions) Require large scale surveys and redshifts 57

58 Summary 3.8: Summary Observational Cosmology 3. Structure Formation Observational Cosmology 4. Cosmological Distance Scale 58