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Published byLaura Stevenson Modified over 9 years ago
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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:
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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?
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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.
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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
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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.
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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.
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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.
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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
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Super-Horizon Growth Lower density Higher density C A B Average
Horizon Size
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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!
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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.
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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.
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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
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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)
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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.
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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.
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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%
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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
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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
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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
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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
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advanced illustrations
Slightly more advanced illustrations
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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.
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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
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Non-linear effects: high k side less steep
Spatial Frequency: k
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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
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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
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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
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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.
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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)
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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
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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.
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Rapid growth of baryon P(k)
1 3 5 7
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CDM & baryon evolution to z ~ 30
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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.
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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
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Collapse of denser regions
Average Density Region Time Higher Density Region expansion turnaround collapse Star forms provisional
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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
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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).
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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.
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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
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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.
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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.
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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
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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.
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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
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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:
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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.
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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
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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)
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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.
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Formation of filaments
I can’t find an appropriate image -- please provide one -- thanks. Following symmetric peak collapse, next is asymmetric ridge collapse.
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Fly-thru millennium simulation
From the millenium simulation. Take the movie called “Fast Flight” (60 MB .avi format).
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End of Growth of Structure
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