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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)

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

3.1 Isotropy Homogeneity on the Largest Scales

- Isotropy and Homogeneity on the largest scales

200Mpc

3.2 The Growth of Structure

- Primordial Density Fluctuations

Origin of LSS today - primordial density

fluctuations

3.2 The Growth of Structure

- Primordial Density Fluctuations

3.2 The Growth of Structure

- Primordial Density Fluctuations

Fluctuations in radiation field ? leave scar on

CMB ? observed as deviations from 2.73K BB

3.2 The Growth of Structure

- The Jeans Length

Conclusion Density perturbations will grow

exponential under the influence of self gravity

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

3.2 The Growth of Structure

- Formal Jeans Theory

Continuity Equation

Euler Equation

Poisson Equation

Entropy Equation

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 !

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

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

3.2 The Growth of Structure

- Growth of Perturbations in an expanding universe

The Hubble Friction

Growth of structure - competition between 2

factors

3.2 The Growth of Structure

- Growth of Perturbations in an expanding universe

Rewrite in terms of density parameter

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

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

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

3.3 Structure Formation in a Dark Matter Universe

- Dark Matter

Actual picture of dark matter in the Universe !!!

3.3 Structure Formation in a Dark Matter Universe

Actual picture of dark matter in the Universe !!!

- Dark Matter

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

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

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

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)

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

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

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

3.4 The Power Spectrum

- The Transfer
- Function

3.4 The Power Spectrum

- The Transfer
- Function

Tegmark 2003

3.4 The Power Spectrum

- The Power Spectrum

Vanilla Cosmology WL0.72, Wm0.28, Wb0.04,

H72, t0.17, bSDSS0.92

Tegmark 2003

3.4 The Power Spectrum

- The Power Spectrum

Tegmark 2003

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 ?

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)

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

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

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!)

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.

3.5 The Non-Linear Regime

- N-Body Simulations

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

3.5 The Non-Linear Regime

- N-Body Simulations

3.5 The Non-Linear Regime

- N-Body Simulations

3.5 The Non-Linear Regime

- N-Body Simulations

SPH Simulations

Bevis Oliver 2002

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

3.6 Statistical Cosmology

- Quantifying Clustering

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

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

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

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

3.6 Statistical Cosmology

- The Correlation Function

3.6 Statistical Cosmology

- Limber Equation

Limber Equation

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

3.6 Statistical Cosmology

- Counts in Cells

3.6 Statistical Cosmology

- Minimum Spanning Trees

3.6 Statistical Cosmology

- Genus

3.6 Statistical Cosmology

- Void Probability Functions

3.6 Statistical Cosmology

- Percolation Analysis

3.7 Large Scale Surveys

- Large Scale Surveys

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

3.8 Summary

- Summary

?

Observational Cosmology 3. Structure Formation

Observational Cosmology 4. Cosmological

Distance Scale

?