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 00 winter school, which is a briefer version of his textbook: http://ned.ipac.caltech.edu/level5/sept0/peacock/peacock_contents.html 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 ~ (i.e. ρ ~ <ρ>), a region breaks away from Hubble expansion, slows, turns around, and s this is now object formation (stars, galaxies, clusters). We describe δ(r) = δρ/<ρ>(r) using its Power spectrum, P(k):!(r) =!! k e ik.r and P(k) =! k where k = 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? Growth in roughness: δρ/ρ 0 Position (e.g light years) 00 00 600 800 000 Density Variations λ δρ δρ δρ ρ Sum all sine Density: ρ ρ Roughness: δρ ρ Wavelength: λ Wavenumber: k = π λ Both co-moving. Position Density : ρ ρ Position Even though ρ is dropping due to expansion, δρ/ρ can increase. P(k) 0 Long strong long λ medium medium λ weak short λ 00 00 00 00 500 Short Example -D power spectrum, P(k), and its density field, δρ/ρ. In this, and all other cosmological examples, the phases are random Roughness P(k) Today s measured power spectrum, P(k) Long Peak λ 00 Mly wiggles P(k) /k Short Power P Roughness δρ/ρ Simple -D example of today s P(k) Long Position (light-years) Peak P(k) /k Density variations Short Largest features (super-clusters/voids) have scale corresponding to peak of P(k). Larger scales are progressively smoother.
Super/Sub-Horizon Growth P(k) for the density field contains both short and long λ modes. Example of two component density pattern. Position (light years) 0 00 00 600 800 000 Actual roughness In linear growth (when δρ/ρ << ), 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. Roughness: δρ/ρ Large wavelength component (λ ~ 00 ly) Super-horizon roughness Small wavelength component (λ ~ 0 ly) Horizon size at 0 years Initially, we are only interested in the dark matter component. This ultimately dominates the matter components It is pressureless, so only responds to gravity. Sub-horizon roughness 0 00 00 600 800 000 Position (light years) Super-Horizon Growth Lower density Higher density C A B Average Region Size, R Different expansion rates amplify density variations δr R δr R = % δr R δr R = 5% C: under-dense B: average A: over-dense δr R δv V δρ ρ all increase Horizon Size Big Bang During radiation era: δρ/ρ! t, so from inflation (0-6 s) to 000 yrs δρ/ρ grows by factor ~0 6 (on all scales). Incredible growth! Horizon Size Sub-Horizon Growth Higher density A Lower density B C Average Local forces at work gravity pulls material into denser regions away from less dense regions density contrast grows. Perturbation growth in an expanding medium This process has a long history the Newton-Bentley dialogue: Newton realized infinite universe unstable to local. For static system, proceeds exponentially: δρ/ρ ~ e t. However, expansion reduces the density, slowing the growth rate: Static time: t c ~ (Gρ m ) -/, Cosmic expansion timescale: /H ~ (Gρ c ) -/ Isaac Newton 6-77 Richard Bentley 66-7 During the matter era: ρ m ρ c and δρ/ρ! a! t / During the radiation era: ρ m << ρ c so δρ/ρ ~ frozen In fact, super-horizon growth slowly redshifted away, so δρ/ρ! ln a. This is the Mésáros effect.
Density Contrast Growth Rates. Position (light years) 0 00 00 600 800 000 Actual roughness Roughness Growth Rates Super-horizon Sub-horizon Radiation Era δρ/ρ a t δρ/ρ ln a (frozen) Matter Era δρ/ρ a t / δρ/ρ a t / Dark Energy Era δρ/ρ = const (frozen) δρ/ρ = const (frozen) Roughness: δρ/ρ 0 Large wavelength component (λ ~ 00 ly) Super-horizon roughness Small wavelength component (λ ~ 0 ly) Sub-horizon roughness Horizon size at 0 years 00 00 600 800 000 Position (light years) Roughness: δρ/ρ Same y-scale Short λ: Long λ:.5 Short λ: Long λ: Short λ: Long λ: 6 Position Rescaled to show y-range Because sub-horizon scales have suppressed growth, the overall nature of the density pattern changes over time. Evolution of P(k) Quantum fluctuations during (exponential) inflation provide an Initial Power Spectrum (IPS): P(k) = A k n with n = This is also called the Harrison-Zel dovich spectrum, and/or the scale-invariant spectrum. This latter term is used because: Δ k (Φ) = k P(k) = constant. This means that the rms variation in gravitational potential, Φ, within regions of size r ~ /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. 0.0% δφ φ -0.0% 0.0% δφ φ -0.0% 0.0% δφ φ -0.0% Gravitational roughness is similar on all scales Gly box 0 Gly Mly box 0 Mly kly box 0 kly Power P Roughnenss δρ/ρ The Initial Power Spectrum and its corresponding density (-D) Weak long Position (light-years) Density variations The Initial Roughness spectrum: Strong short
Log Size long 5 medium short Big Bang Expansion of different sized Radiation Era expansion Horizon size c t Matter Era wave enters horizon Equality Log Now 5 long medium short Roughness: P(k) short medium long 5 Big Bang Growth of roughness for different sized Radiation Era wave enters horizon Initial roughness spectrum: Equality Matter Era Now medium short 5 long Final roughness spectrum: peaked Roughness Log P(k): Matter era Radiation era CDM P(k) from CMBFAST models Age/yr 0 0 0 9 0 8 0 7 0 6 0 5 0 000 early universe today Slightly more advanced illustrations 00 Long Spatial Frequency: Log k Short
Change in gradient by : n = to - Roughness from the Big Bang to today It is relatively easy to see why there is a change by in the gradient of log P(k) vs log k due to suppressed growth in the radiation era: Consider two, dex apart in k. Since wavelengths grow ~ t / 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. 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) ). 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 - part, and (ii) non-linear effects make the high-k side less steep anyway. Roughness spectrum: P(k) Initial roughness spectrum from inflation: Long Horizon size near equality Final roughness spectrum Growth of short suppressed during the radiation era Short Non-linear effects: high k side less steep Review using -D examples: The Initial Power Spectrum Position (light-years) Power P Roughnenss δρ/ρ Weak long Density variations The Initial Roughness spectrum: Strong short 5
Power P Roughness δρ/ρ Long Today s Power Spectrum Position (light-years) Peak P(k) /k Final roughness spectrum Density variations Short Density: δρ/ρ Roughness: P(k) long Primordial roughness Roughness spectrum Position: x Slope: n = Spatial frequency: k short time long Today s roughness Galaxy clusters s Roughness spectrum Position: x Primordial Spatial frequency: k short 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. Since λ J c s (π/gρ) / and c s c/, then λ J 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. Roughness: δρ/ρ Horizon entry Growth of roughness in dark & atomic matter Dark Matter Atomic Matter foggy clear If no dark matter 0 0 0 0 5 0 6 0 7 0 8 (years) Variations in density at time of CMB Dark matter Density % variation in dark matter density % Dark matter Atomic matter Atomic matter s pattern quickly matches dark matter 00,000 years 5 million years 0 million years 0.0% variation in atomic matter density Before CMB: atomic matter oscillates in the DM potentials After CMB: atomic matter falls into the DM potentials takes on its -D density pattern. 6
Rapid growth of baryon P(k) 5 7 CDM & baryon evolution to z ~ 0 Turnaround and Collapse Previous discussion was for linear growth regime, so up to δρ/ρ ~. After that, a region deviates significantly from the Hubble flow, and ultimately halts expansion, turns around and s. Here are some figures illustrating the situation. Global expansion & local very smooth Global & Local First stars form inside these regions Expansion smooth very rough Global expansion Local expansion Global expansion rough Average Density Region Higher Density Region Collapse of denser regions expansion turnaround Star forms Local 7
Dark matter s inefficient Collapse of dark matter region: violent relaxation Region Size Big Bang C: lower density B: average A: higher density R turn violent relaxation 00 density difference / R turn initial big crunch first rebound final state Once starts in the early Universe, it is a rapid process early structure formation is fast. The reason is simple: t coll ~ / Gρ, and since ρ is high at early times, times are high (roughly, t age ). Hierarchy of Collapse Adult galaxy The first objects form where the density is highest. & merge 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. & merge Infant galaxies Stars Full density pattern density Sum Sum Short (Stars) Medium (Infant galaxies) Long (Adult galaxy) This is often shown as a merger tree. Merger Tree Collapse = Cosmic Age to turnaround to Overall average expansion Region Size Highest density s first Then next highest Then next highest Big Bang Notice that the earlier objects forming are also the densest. 8
Location of first stars (detailed simulation) More quantitative approach Strongly Clustered Notice, objects forming at high-z are highly clustered because they form within larger over-dense regions. If we want to ask when a region of size ~r will, we simply ask whether δρ/ρ > when averaged over the region. If the answer is yes then it will soon 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 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. σ) of δρ/ρ for an ensemble of spheres of radius r. When σ ~, then a significant fraction are ready to. 50,000 light years First stars form in densest dark matter regions This is illustrated in the next figure. Density spread depends on region size 00 M lyr Collapse threshold small regions underdense some regions overdense 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:! (r) = 0.5 0 0.5 δρ / <ρ> larger regions underdense (" )! P(k) W (kr) " k dk Where W(kr) is the Fourier transform of the spherical top hat: no regions overdense W (kr) = [sin(kr)! kr cos(kr)] (kr) Since this is a relatively peaked function at kr ~, then it turns out: 0.5 0 0.5 δρ / <ρ>! (r)! " (k) # The Universe at 00 Myr Dimensionless power spectra: Δ(k) and σ(r) k P(k) where k! " r Region size sequence When Δ(k) reaches then regions of size r ~ /k will soon. Variance in δρ/ρ: σ(r).0 Note: Δ(k) ~ k P(k) is a dimensionless version of the power spectrum, and is often used instead of P(k). Critical δρ/ρ for breakaway &.0 Rapid initial growth in size of collapsing regions Myr 000 800 600 00 00 larger regions σ(r) Δ(k) k P(k) Region size: r smaller regions 9
clusters Roughness: σ(r) web galaxies stars Growth of roughness (expansion not shown!) threshold time 0 000 Gly 00 0 0. Region size: r (Mly) 0.0 0 kly Formation of filaments Fly-thru millennium simulation Following symmetric peak, next is asymmetric ridge. End of Growth of Structure 0