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Observational Cosmology: 3.Structure Formation

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Title: Observational Cosmology: 3.Structure Formation


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 (1882-1944)
2
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
Radiation CMB - Isotropic to 1 part in 105,
0.003, 2mK
Matter Large scales gt 100Mpc (Clusters /
Superclusters) Universe is smooth Radio
Sources isotropic to a few percent Small scales
Highly anisotropic
3
3.1 Isotropy Homogeneity on the Largest Scales
  • Isotropy and Homogeneity on the largest scales

200Mpc
4
3.2 The Growth of Structure
  • Primordial Density Fluctuations

Origin of LSS today - primordial density
fluctuations
5
3.2 The Growth of Structure
  • Primordial Density Fluctuations

6
3.2 The Growth of Structure
  • Primordial Density Fluctuations

Fluctuations in radiation field ? leave scar on
CMB ? observed as deviations from 2.73K BB
7
3.2 The Growth of Structure
  • The Jeans Length

Conclusion Density perturbations will grow
exponential under the influence of self gravity
8
3.2 The Growth of Structure
  • The Jeans Length

In absence of pressure, an overdense region
collapses on order of the free fall time
? Define a critical length over which density
perturbation will be stable against collapse
under self gravity
9
3.2 The Growth of Structure
  • Formal Jeans Theory

Continuity Equation
Euler Equation
Poisson Equation
Entropy Equation
10
3.2 The Growth of Structure
  • Jeans Mass, Silk Mass and the decoupling epoch

Before epoch of decoupling, photons and Baryons
bound together as a single fluid
This mass is larger than the largest Supercluster
today !
11
3.2 The Growth of Structure
  • Jeans Mass, Silk Mass and the decoupling epoch

After epoch of decoupling, photons and Baryons
behave as separate fluids
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
12
3.2 The Growth of Structure
  • Jeans Mass, Silk Mass and the decoupling epoch
  • 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

13
3.2 The Growth of Structure
  • Growth of Perturbations in an expanding universe
    The Hubble Friction

Growth of structure - competition between 2
factors
14
3.2 The Growth of Structure
  • Growth of Perturbations in an expanding universe

Rewrite in terms of density parameter
15
3.3 Structure Formation in a Dark Matter Universe
  • Growth of Perturbations in an expanding universe
  • dltlt1 ?? linear regime
  • d1 ?? non-linear regime ? Require N-body
    simulations
  • Baryonic Matter fluctuations can only have grown
    by a factor (1zdec) 1000 by today
  • for d1 today require d0.001 at recombination
  • d0.001 ? dT/T 0.001 at recombination
  • But CMB ? dT/T 10-5 !!!
  • MATTER PERTURBATIONS DONT HAVE TIME TO GROW IN
    A BARYON DOMINATED UNIVERSE

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

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
18
3.3 Structure Formation in a Dark Matter Universe
  • Dark Matter

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

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)

gtgt MSupercluster
21
3.3 Structure Formation in a Dark Matter Universe
  • Hot Dark Matter

For a hot neutrino, mass mn(eV/c2)
  • Before teq, 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 cteq

This Effect is known as FREE STREAMING
Fluctuations suppressed on mass scales of
Large Superstructures form first in a HDM
Universe ? TOP-DOWN SCENARIO
22
3.3 Structure Formation in a Dark Matter Universe
  • Cold Dark Matter

For a CDM WIMP, mass mCDM1GeV
Fluctuations l gt lo will grow throughout
radiation period
Fluctuations l lt lo will remain frozen until
matter domination when Hubble distance has grown
to 0.03Mpc corresponding to 1017Mo ? Scales gt
Hubble distance at matter domination retain
original primordial spectrum
Structure forms hierarchically in a CDM Universe
? BOTTOM-UP SCENARIO
23
3.3 Structure Formation in a Dark Matter Universe
  • Structure Formation in a Dark Matter universe

Simulation of CDM and HDM Structure formation
seeded by cosmic strings (http//www.damtp.cam.ac.
uk)
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

25
3.4 The Power Spectrum
  • Quantifying the power in fluctuations on large
    scales
  • Fluctuations have the same amplitude when they
    enter the horizon d 10-4
  • Inflation field is isotropic, Homogeneous,
    Gaussian field (Fourier modes uncorrelated)
  • ? All information contained within the Power
    Spectrum P(k)

Instead of simply P(k) ? often plot (k3P(k))1/2
the root mean square mass fluctuations
26
3.4 The Power Spectrum
  • The Transfer Function
  • 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,
  • given by TRANSFER FUNCTION T(k), contains messy
    physics of evolution of density perturbations

27
3.4 The Power Spectrum
  • The Transfer
  • Function

28
3.4 The Power Spectrum
  • The Transfer
  • Function

Tegmark 2003
29
3.4 The Power Spectrum
  • The Power Spectrum

Vanilla Cosmology WL0.72, Wm0.28, Wb0.04,
H72, t0.17, bSDSS0.92
Tegmark 2003
30
3.4 The Power Spectrum
  • The Power Spectrum

Tegmark 2003
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. d gt 1

How can we model the non-linear regime ?
32
3.5 The Non-Linear Regime
  • (1) The Zeldovich Approximation

(relates Eulerian and Lagragian co-ordinate
frames)
In the Zeldovich Approximation, the first
structures to form are giant Pancakes (provides
very good approximation to the non-linear regime
until shell crossing)
33
3.5 The Non-Linear Regime
  • (2) N-Body Simulations
  • 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(N2) Practical for Nlt104
PM, P3M 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

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

35
3.5 The Non-Linear Regime
  • N-Body Simulations - Virgo Consortium

t CDM Wm1, s80.6, spectral shape parameter
G0.21 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   L CDM Wm 0.3, WL
0.7, s8 1, G 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!)

36
3.5 The Non-Linear Regime
  • N-Body Simulations - Virgo Consortium
  • The "deep wedge" light cone survey from the tCDM
    model.
  • The long piece of the "tie" extends from the
    present to a redshift z4.6
  • 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 z0.2, 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.

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

38
3.5 The Non-Linear Regime
  • N-Body Simulations - formation of dark Matter
    Haloes
  • The hierarchical evolution of a galaxy cluster in
    a universe dominated by cold dark matter.
  • Small fluctuations in the mass distribution are
    barely visible at early epochs.
  • Growth by gravitational instability accretion ?
    collapse into virialized spherical dark matter
    halos
  • Gas cools and objects merge into the large
    galactic systems that we observe today

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

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

41
3.5 The Non-Linear Regime
  • N-Body Simulations

SPH Simulations
Bevis Oliver 2002
42
3.6 Statistical Cosmology
  • Quantifying Clustering
  • 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 may be biased
We would like to quantify the clustering on all
scales from galaxies, clusters, superclusters
43
3.6 Statistical Cosmology
  • Quantifying Clustering

44
3.6 Statistical Cosmology
  • Quantifying Clustering

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
45
3.6 Statistical Cosmology
  • The Correlation Function

Angular Correlation Function w(q) Describes
the clustering as projected on the sky (e.g. the
angular distribution of galaxies, e.g. in a
survey catalog)
Spatial Correlation Function x(r) Describes
the clustering in real space
For any random galaxy Probability , dP, of
finding another galaxy within a volume, V, at
distance , r
Assume x(r) is isotropic (only depends on
distance not direction) ? x(r) x(r)
? 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
46
3.6 Statistical Cosmology
  • The Correlation Function

Strictly require more random points than data
points and need to correct for edge effects
Use DR(q) number of pairs with separations qDq
where one point is taken from random and real
data set
47
3.6 Statistical Cosmology
  • The Correlation Function and the relation to the
    power spectrum

b is the bias parameter for galaxy biasing w.r.t.
underlying Dark Matter Distribution
48
3.6 Statistical Cosmology
  • The Correlation Function

49
3.6 Statistical Cosmology
  • Limber Equation

Limber Equation
50
3.6 Statistical Cosmology
divide the Universe into boxes of side r and
count the number of galaxies, ni in each cell
  • Counts in Cells

S2V variance of the density field smoothed over
the cell
51
3.6 Statistical Cosmology
  • Counts in Cells

52
3.6 Statistical Cosmology
  • Minimum Spanning Trees

53
3.6 Statistical Cosmology
  • Genus

54
3.6 Statistical Cosmology
  • Void Probability Functions

55
3.6 Statistical Cosmology
  • Percolation Analysis

56
3.7 Large Scale Surveys
  • Large Scale Surveys

57
3.8 Summary
  • 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

58
3.8 Summary
  • Summary

?
Observational Cosmology 3. Structure Formation
Observational Cosmology 4. Cosmological
Distance Scale
?
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