1. Growth of Structure
Notes based on Teaching Company lectures, and associated undergraduate text – with some additional material added.
For a more detailed discussion, see the article by Peacock taken from a 2002 winter school, which is a briefer version of his textbook:

2. Preliminaries
Inflation, we think, created very slight variations in density from place to place, which we describe using: δ(r) = δρ/<ρ>(r).
During expansion, although <ρ> decreases, δρ/<ρ> can increase.
When δρ/<ρ> reaches ~1 (i.e. ρ ~ 2<ρ>), a region breaks away from Hubble expansion, slows, turns around, and collapses – this is now “object formation” (stars, galaxies, clusters).
We describe δ(r) = δρ/<ρ>(r) using its Power spectrum, P(k):
Questions:
What was inflation’s P(k), and what is P(k) for today’s Universe?
Can we understand how P(k) evolves from one to the other?

3. Growth in roughness:   
Density: 
Roughness:
Density :    
Wavelength: 
Wavenumber: k = 2π 
Both co-moving.
Position
Position
Even though ρ is dropping due to expansion, δρ/ρ can increase.

4. Position (e.g light years)
Spatial Frequency: k
Long
waves
Short
P(k)
Density Variations
200
400
600
800
1000
100
300
500
strong
long 
medium
medium 
weak
short   
Sum all sine waves
All these type of plots are mine.
I may need to make cleaner and/or slightly different versions
Example 1-D power spectrum, P(k), and its density field, δρ/ρ.
In this, and all other cosmological examples, the phases are random

5. Today’s measured power spectrum, P(k)
Roughness P(k)
Spatial Frequency: k
Short waves
Long waves
P(k)  1/k2
Peak
 ≈ 200 Mly
wiggles
P(k)  k
WMAP website for the CMB image
2dF website for the galaxy pie diagram
Unfortunately, I can’t recall where I got the power spectrum from -- I think it was the 2dF site -- I’ll check. It’s rather important that we have this particular image.

6. Simple 1-D example of today’s P(k)
P(k)  1/k 2
Peak
P(k)  k
Short
waves
Long
Power P
Roughness 
Density variations
Position (light-years)
Spatial Frequency: k
Super-cluster
Void
Largest features (super-clusters/voids) have scale corresponding to peak of P(k). Larger scales are progressively smoother.

7. Super/Sub-Horizon Growth
P(k) for the density field contains both short and long λ modes.
In linear growth (when δρ/ρ << 1), the growth of each λ mode is independent of all the other modes.
However, the growth of a mode depends on whether the mode’s λ is smaller or larger than the horizon at that time.
Let’s look at super-horizon growth, and sub-horizon growth.
Initially, we are only interested in the dark matter component.
This ultimately dominates the matter components
It is pressureless, so only responds to gravity.

8. Example of two component density pattern.
Position (light years)
Horizon size
at 30 years
Roughness: 
200
400
600
800
1000
Small wavelength component ( ~ 10 ly)
Sub-horizon roughness
Actual roughness
Large wavelength component ( ~ 100 ly)
Super-horizon roughness
My diagram

9. Super-Horizon Growth Lower density Higher density C A B Average
Horizon Size

10. Different expansion rates amplify density variations
C: under-dense R
B: average
A: over-dense R R R
= 5%
Region Size, R R R R
= 3% R R V V   R
all increase
Big
Bang
Time
During radiation era: δρ/ρ t, so from inflation (10-36 s) to 1000 yrs δρ/ρ grows by factor ~1046 (on all scales). Incredible growth!

11. Sub-Horizon Growth Lower density Higher density C A B Average
Horizon Size
Local forces at work – gravity pulls material into denser regions away from less dense regions – density contrast grows.

12. Perturbation growth in an expanding medium
This process has a long history – the Newton-Bentley dialogue:
Newton realized infinite universe unstable to local collapse.
For static system, collapse proceeds exponentially: δρ/ρ ~ et .
However, expansion reduces the density, slowing the growth rate:
Static collapse time: tc ~ (Gρm)-1/2,
Cosmic expansion timescale: 1/H ~ (Gρc)-1/2
Richard Bentley
Isaac Newton
During the matter era:
ρm ≈ ρc and δρ/ρ a t2/3
During the radiation era:
ρm << ρc so δρ/ρ ~ frozen
In fact, super-horizon growth slowly redshifted away, so δρ/ρ ln a.
This is the Mésáros effect.
Both come from wikipedia
Both said the copyrights had expired.

13. Density Contrast Growth Rates.
Roughness Growth Rates
Radiation Era
Matter Era
Dark Energy Era
Super-horizon
a2
t
a
t2/3
= const
(frozen)
Sub-horizon
ln a

14. Position (light years)
200
400
600
800
1000
Actual roughness
Large wavelength component ( ~ 100 ly)
Roughness: 
Super-horizon roughness
Horizon size
at 30 years
My diagram
Small wavelength component ( ~ 10 ly)
Sub-horizon roughness
200
400
600
800
1000
Position (light years)

15. Rescaled to show y-range
Same y-scale
Rescaled to show y-range
Short : 1 Long : 1.5
Time
Short : 1 Long : 3
Roughness: 
Time
Short : 1 Long : 6
My diagram
Position
Because sub-horizon scales have suppressed growth, the overall nature of the density pattern changes over time.

16. Evolution of P(k)
Quantum fluctuations during (exponential) inflation provide an “Initial Power Spectrum” (IPS):
P(k) = A k n with n = 1
This is also called the Harrison-Zel’dovich spectrum, and/or the scale-invariant spectrum.
This latter term is used because: Δk2(Φ) = k3P(k) = constant. This means that the rms variation in gravitational potential, Φ, within regions of size r ~ 1/k, is independent of r. I.e the gravitational “wrinkliness” of the (initial) universe looks the same on all scales.
Let’s see how an IPS of P(k) k evolves over time.

17. Gravitational roughness is similar on all scales
0.01%  
1 Gly box
10 Gly
0.01%
0.01%  
1 Mly box
10 Mly
0.01%
I made these.
0.01%  
1 kly box
10 kly
0.01%

18. The Initial Power Spectrum and its corresponding density (1-D)
Position (light-years)
Density variations
Roughnenss 
The Initial Roughness spectrum: P(k)  k
Power P
P(k)  k
Weak
long waves
Strong
short waves
Spatial Frequency: k

19. Expansion of different sized waves
Radiation Era
Matter Era 5
long 4
Horizon size
c  t 3
medium
Log Size 2
expansion 1
short
long 5 4
wave
enters
horizon
medium 3 2
short 1
Equality
Now
Big Bang
Log Time

20. Growth of roughness for different sized waves
medium 3
Radiation Era
Matter Era 2 4
short 1
wave
enters
horizon
long
Roughness: P(k) 5
Final roughness
spectrum: peaked
short 1 2
medium 3
Initial roughness
spectrum: P(k)  k 4
long 5
Equality
Now
Big Bang
Time

21. CDM P(k) from CMBFAST models
Age/yr
today
1010
109
108
Matter
era
107
Roughness Log P(k):
106
105
provisional
Radiation
era
104
early universe
P(k)  k
1000
100
Long waves
Short waves
Spatial Frequency: Log k

22. advanced illustrations
Slightly more
advanced illustrations

27. Change in gradient by 4: n = 1 to -3
It is relatively easy to see why there is a change by 4 in the gradient of log P(k) vs log k due to suppressed growth in the radiation era:
Consider two waves, 1 dex apart in k. Since wavelengths grow ~ t1/2 in the radiation era and the horizon grows ~ t, then there is a two-dex delay in time between the horizon entry of the two waves. Since δρ/ρ grows ~ t in the radiation era, then there is a two-dex difference in growth of δρ/ρ, which corresponds to a four-dex difference in P(k) (since P(k) ~ (δρ/ρ)2). Hence, the high-k (small-λ) wave now has P(k) four-dex lower due to its extra suppression in the radiation era.
In practice, (i) the peak is so broad that it takes a few dex in k to reach the k -3 part, and (ii) non-linear effects make the high-k side less steep anyway.

28. Roughness from the Big Bang to today
Initial roughness spectrum
from inflation: P(k)  k
Growth of short waves suppressed during the radiation era
Roughness spectrum: P(k)
Horizon size near equality
provisional
Final roughness
spectrum
Long waves
Short waves
Spatial Frequency: k

29. Non-linear effects: high k side less steep
Spatial Frequency: k

30. The Initial Power Spectrum
Review using 1-D examples:
The Initial Power Spectrum
P(k)  k
Strong
short waves
Weak
long waves
Position (light-years)
Spatial Frequency: k
Power P
Roughnenss 
The Initial Roughness spectrum: P(k)  k
Density variations

31. Today’s Power Spectrum
P(k)  1/k 2
Peak
P(k)  k
Short
waves
Long
Power P
Roughness 
Final roughness spectrum
Density variations
Position (light-years)
Spatial Frequency: k
Super-cluster
Void

32. Primordial roughness Today’s roughness Density:  Position: x
Galaxy clusters
Density: 
Voids
Position: x
Position: x
Roughness
spectrum
Roughness
spectrum
time
Roughness: P(k)
Need to flip direction of x-axis, to have large on left and small on right.
Slope: n = 1
Primordial
long
waves
Spatial
frequency: k
short
waves
long
waves
Spatial
frequency: k
short
waves

33. Comparing P(k) for CDM & Baryons
The Baryons are tied to the photons via Thomson opacity, so they experience pressure.
On scales less than the Jean’s length, pressure dominates and the photon-baryon gas oscillates as sound waves.
Since λJ ≈ cs(π/Gρ)1/2 and cs ≈ c/√3, then λJ ≈ c/√(Gρ) ≈ c/H.
So essentially all scales within the horizon are dominated by pressure and undergo oscillatory (acoustic) motion, with no steady growth in δρ/ρ.
At the time of the CMB, therefore, δρ/ρ is much less in the baryons than in the dark matter.
Only when the radiation decouples and the pressure drops do the baryons fall into the DM pockets and inherit its density pattern.

34. Growth of roughness in dark & atomic matter
foggy
clear
Horizon entry
Dark Matter
Roughness: 
Atomic Matter
If no dark matter
My diagram
I will probably make a better version of this.
Note to self: This is P(k) so it’s (drho/rho)**2, so you may need to half the logarithmic scale.
102
103
104
105
106
107
108
Time (years)

35. Variations in density at time of CMB
1% variation in
dark matter density 1%
Dark matter
Density
0.01% variation in
atomic matter density
Before CMB: atomic matter oscillates in the DM potentials

36. Atomic matter’s pattern quickly matches dark matter
400,000 years
5 million years
10 million years
After CMB: atomic matter falls into the DM potentials – takes on its 3-D density pattern.

37. Rapid growth of baryon P(k)1 3 5 7

39. CDM & baryon evolution to z ~ 30

40. Turnaround and Collapse
Previous discussion was for linear growth regime, so up to δρ/ρ ~ 1.
After that, a region deviates significantly from the Hubble flow, and ultimately halts expansion, turns around and collapses.
Here are some figures illustrating the situation.

41. Global expansion & local collapse
rough
very
smooth
smooth
Global & Local
Global expansion
Expansion
Local expansion 1 2
very rough 3
First stars form inside these regions
My figure -- it may be possible (indeed, better) to do an animated version of this.
Global expansion
Local collapse 4

42. Collapse of denser regions
Average
Density
Region
Time
Higher
Density
Region
expansion
turnaround
collapse
Star forms
provisional

43. Dark matter’s inefficient collapse
C: lower density
B: average
200 density
difference
Region Size
A: higher density
“violent relaxation”
Rturn
1/2 Rturn
Big
Bang
Time

44. Collapse of dark matter region: “violent relaxation”
initial collapse
big crunch
first rebound
final state
Once collapse starts in the early Universe, it is a rapid process – early structure formation is fast.
The reason is simple: tcoll ~ 1/√Gρ, and since ρ is high at early times, collapse times are high (roughly, tage).

45. Hierarchy of Collapse
The first objects form where the density is highest.
Where are these peaks in the density field?
They occur where short wave peaks align with medium wave peaks which in turn align with long wave peaks (see figures).
Because of this, the first stars are born in groups, which then fall together in star-clusters, which fall together into infant galaxies, and so on, in a hierarchy.

46. Time Adult galaxy Infant galaxies Stars Full density pattern
collapse
& merge
Infant galaxies
collapse
& merge
Stars
collapse
Full density
pattern
Short “waves”
(Stars)
Medium “waves”
density
(Infant galaxies)
Long “waves”
(Adult galaxy)
Sum
Sum

47. This is often shown as a merger “tree”.
Time
Merger Tree
Reproduce this diagram (should be easy)
Note: I have blocked out some black lines which shouldn’t be included.

48. Collapse Time = Cosmic Age
Time to
turnaround
Time to
collapse
Overall average
expansion
Highest density collapses first
Then next highest
Region Size
Big
Bang
Time
Notice that the earlier objects forming are also the densest.

49. Location of first stars (detailed simulation)
150,000 light years
Strongly Clustered
Notice, objects forming at high-z are highly clustered because they form within larger over-dense regions.
Figure from Yoshida et al 2003 Astrophysical Journal (University of Chicago Press)
You might also get permission directly from Naoki Yoshida who is at the National Astronomical Observatory of Japan (Nagoya University)
First stars form in densest dark matter regions

50. More quantitative approach
If we want to ask when a region of size ~r will collapse, we simply ask whether δρ/ρ > 1 when averaged over the region.
If the answer is “yes” then it will soon collapse and virialize (possibly including many smaller previously virialized objects).
It is important to realize that the relative number of regions of size r that are about to collapse depends strongly on r. In general, at any given time, fewer regions of larger r will be collapsing.
We can get some sense of this by asking what is the variance (i.e. σ2) of δρ/ρ for an ensemble of spheres of radius r.
When σ2 ~ 1, then a significant fraction are ready to collapse.
This is illustrated in the next figure.

51. Density spread depends on region size
Collapse
threshold
100 M lyr
small regions
under-
dense
over-
larger regions
some
regions
collapse –1
–0.5
0.5 1
 /  no
regions
collapse
Mottled pattern from:
The last image on the bottom: redshift z18.3 (1024 x 768)
The other images are mine –1
–0.5
0.5 1
 / 
The Universe at 200 Myr

52. More quantitative approach
Placing spheres of radius r at random and evaluating δρ/ρ within the sphere is the same as convolving the density field by a spherical “top hat”. The variance of this smoothed distribution is what we’re after.
Recall: convolving in real space can be done in Fourier space:
Where W(kr) is the Fourier transform of the spherical top hat:
Since this is a relatively peaked function at kr ~ 1, then it turns out:

53. Dimensionless power spectra: Δ2(k) and σ2(r)
Note: Δ2(k) ~ k3 P(k) is a dimensionless version of the power spectrum, and is often used instead of P(k).
When Δ2(k) reaches 1 then regions of size r ~ 1/k will soon collapse.
My diagram.

54. Region size collapse sequence
Critical  for breakaway & collapse
1.0
1.0
Rapid initial growth in size of collapsing regions
Variance in δρ/ρ: σ2(r)
Myr
Time
1000
800
600
400
200
σ2(r) ≈ 2(k) ≅ k3  P(k)
My diagram.
larger
regions
smaller
regions
Region size: r

55. web clusters galaxies stars Roughness: 2(r) Region size: r (Mly)
collapse
threshold
Roughness: 2(r)
time
104
1000
100 10 1
0.1
0.01
1 Gly
10 kly
Region size: r (Mly)

56. Growth of roughness (expansion not shown!)
The credit is to Volker Springel:
However, I can’t find the movie on his website anymore. You may have to contact him directly, and specify that
You’re wanting the time evolution of the galaxy structure, not the flythrough. There are several resolutions available -- get the best, the one that I have here was a lower resolution movie.

57. Formation of filaments
I can’t find an appropriate image -- please provide one -- thanks.
Following symmetric peak collapse, next is asymmetric ridge collapse.

58. Fly-thru millennium simulation
From the millenium simulation.
Take the movie called “Fast Flight” (60 MB .avi format).

59. End of Growth of Structure