Title: Observational Cosmology: 3.Structure Formation
1Observational 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)
23.1 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
33.1 Isotropy Homogeneity on the Largest Scales
- Isotropy and Homogeneity on the largest scales
200Mpc
43.2 The Growth of Structure
- Primordial Density Fluctuations
Origin of LSS today - primordial density
fluctuations
53.2 The Growth of Structure
- Primordial Density Fluctuations
63.2 The Growth of Structure
- Primordial Density Fluctuations
Fluctuations in radiation field ? leave scar on
CMB ? observed as deviations from 2.73K BB
73.2 The Growth of Structure
Conclusion Density perturbations will grow
exponential under the influence of self gravity
83.2 The Growth of Structure
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
93.2 The Growth of Structure
Continuity Equation
Euler Equation
Poisson Equation
Entropy Equation
103.2 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 !
113.2 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
123.2 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
133.2 The Growth of Structure
- Growth of Perturbations in an expanding universe
The Hubble Friction
Growth of structure - competition between 2
factors
143.2 The Growth of Structure
- Growth of Perturbations in an expanding universe
Rewrite in terms of density parameter
153.3 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
163.3 Structure Formation in a Dark Matter Universe
- To be born Dark, to become dark, to be made dark,
to have darkness
173.3 Structure Formation in a Dark Matter Universe
- 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
183.3 Structure Formation in a Dark Matter Universe
Actual picture of dark matter in the Universe !!!
193.3 Structure Formation in a Dark Matter Universe
Actual picture of dark matter in the Universe !!!
203.3 Structure Formation in a Dark Matter Universe
- 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
213.3 Structure Formation in a Dark Matter Universe
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
223.3 Structure Formation in a Dark Matter Universe
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
233.3 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)
243.4 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
253.4 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
263.4 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,
- given by TRANSFER FUNCTION T(k), contains messy
physics of evolution of density perturbations
273.4 The Power Spectrum
283.4 The Power Spectrum
Tegmark 2003
293.4 The Power Spectrum
Vanilla Cosmology WL0.72, Wm0.28, Wb0.04,
H72, t0.17, bSDSS0.92
Tegmark 2003
303.4 The Power Spectrum
Tegmark 2003
313.5 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 ?
323.5 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)
333.5 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
- Strengths
- Self consistent treatment of LSS and galaxy
evolution - Weaknesses
- Limited resolution
- Computational overheads
343.5 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
353.5 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!)
363.5 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.
373.5 The Non-Linear Regime
383.5 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
393.5 The Non-Linear Regime
403.5 The Non-Linear Regime
413.5 The Non-Linear Regime
SPH Simulations
Bevis Oliver 2002
423.6 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 may be biased
We would like to quantify the clustering on all
scales from galaxies, clusters, superclusters
433.6 Statistical Cosmology
443.6 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
453.6 Statistical Cosmology
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
463.6 Statistical Cosmology
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
473.6 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
483.6 Statistical Cosmology
493.6 Statistical Cosmology
Limber Equation
503.6 Statistical Cosmology
divide the Universe into boxes of side r and
count the number of galaxies, ni in each cell
S2V variance of the density field smoothed over
the cell
513.6 Statistical Cosmology
523.6 Statistical Cosmology
533.6 Statistical Cosmology
543.6 Statistical Cosmology
- Void Probability Functions
553.6 Statistical Cosmology
563.7 Large Scale Surveys
573.8 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
583.8 Summary
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Observational Cosmology 3. Structure Formation
Observational Cosmology 4. Cosmological
Distance Scale
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