Large Scale Structure

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1 Large Scale Structure L2: Theoretical growth of structure Taking inspiration from - Ryden Introduction to Cosmology - Carroll & Ostlie Foundations of Astrophysics

2 Where does structure come from?

3 Initial density fluctuations: Adiabatic & Isothermal Fluctuations of all species trace one another Different species distributed differently No energy exchange Sum of 4-density fluctuations is zero Complicated growth history before decoupling Observa<ons strongly suggest fluctua<ons in our universe are adiaba<c Potential for density increase (e.g. pressure) inhibited by interaction with the radiation field Could only grow after decoupling All types of fluctua<ons can be described by a superposi<on of these two h"p:// perturba<ons.html

4 Definition of a density fluctuation δ ~ 200! = "! " " δ ~10 5 δ ~10 30 δ ~10 30 Rough δ ~ 10 30

5 Linear vs non-linear Linear: δ << 1 Non-linear: δ 1 Very non-linear: δ > 1 Perturbation thry applies To solve: To solve:

6 How do density fluctuations grow? Sta<c Solu<on (ρ = const) Grav accel at surface (due to excess mass)!! R R =! 4!G 3 "#! R! Conservation of mass Leaves differential equation See Ryden - Ch12 of Introduction to Cosmology, R!! R! "!!! 3!!! = 4"G#! Which has solution: Growing mode Decaying mode!(t) = A 1 e t/t dyn + A 2 e!t/t dyn t dyn = (4"G#)!1/2 Coefficients determined by ini<al condi<ons

7 Dynamical timescale t dyn = (4!G")!1/2 The density in this room is ~1kg/m 3 On what timescale will it collapse gravitationally? G = 6.67x10-11 kg -1 m 3 s -2 About 9 hours!

8 Pressure to the rescue Hydrostatic equilibrium = when pressure balances gravity Pressure builds up at the speed of sound, so Where c s! dp d! = c w Equa<on of state t pre! R c s For pressure to save the day, need t pre < t dyn R / c s < (4!G")!1/2 i.e. radius less than Jeans length R < c s 4!G" = # J Jeans Length

9 Atmosphere is stable The density in this room is ~1kg/m 3 What is the Jeans length of the air in this room? c s ~330m/s G=6.67e-11 kg -1 m 3 s -2! J = c s 4"G# About 20,000km! (Stratosphere goes to al<tude of about 50km.) Note that when sound speed goes to zero, so does the Jeans length, so all overdensities collapse.

10 Sound speed In general, sound speed given by: c s! dp d! = c w Equa<on of state Relativistic fluid has w=1/3 c s = c 3! 0.58c! J = c s 4"G# Jeans Length is greater than horizon size Including Baryons c c s = 3(1+ 3! b / 4!! ) No Collapse!

11 Evolution of sub-horizon adiabatic fluctuations Once adiabatic fluctuation becomes sub-horizon, but before recombination Example of an ini-al fluctua-on propaga-ng as sound wave Pressure supported Experiences acoustic oscillations Sound waves traverse it at nearly light speed (c/ 3) to leave imprint on CMB Dan Eisenstein h"p://cmb.as.arizona.edu/ ~eisenste/acous<cpeak/

12 Formation of acoustic peak

13 Calculate the BAO scale Time between big bang and last scattering = 380,000yr Speed of sound wave = 0.58c 1 pc = 3.1x10 16 m 1 yr = 3.56x10 7 s c = 3x10 8 ms -1 d = c s t = 0.58 x 3x10 8 ms - 1 x 380,000 yr x conversion factors = 74 Mpc

14 Can only be seen statistically The initial fluctuations happened on all scales, so reality a little more complicated than a single peak

15 Silk Damping Photons leaked out of small regions Small fluctuations did not survive Minimum mass to survive = mass within sphere of radius d = about M = about an elliptical cd galaxy or small galaxy cluster

16 Silk Damping scale Photons leaked out of small regions Small fluctuations did not survive! = 6.65e! 24m 2 " b (z dec ) = 5.4e!19kgm!3 m H =1.7e! 24g n = " b / m H = 3.2e8m!3 Mean free path: l = 1 n! e - #- density e - sca"ering cross- sec<on Time between collisions: t = l / c # of collisions before decoupling: N = t dec t = n! ct dec d Distance travelled: d = l N = ct dec n! =0.014Mpc Minimum mass to survive = mass within sphere of radius d = about M = about an elliptical cd galaxy or small galaxy cluster

17 Characteristic expansion time Hubble time t H = 1 H = 3 8!G" = 3 2 t dyn! 1.22t dyn Comparable to the dynamic <me => both are important! J = c s t dyn = c wt dyn! J = c H 2 3

18 Growth including expansion c/h Expanding Solu<on (due to Grav accel at surface total mass) R!! R = 4!G 3 R"(1!#) Conservation of mass R!! R! a!! a "!!! 3 " 2 3 # % $!a a & (! ' Leaves differential equation!!! + 2H!! = 4"G#! λ J! R! Same as sta<c case, but with an extra term, 2H!! known as the Hubble fric<on term!!! + 2H!!! 3 2 " M H 2! = 0 See Ryden - Ch12 of Introduction to Cosmology,

19 Radiation dominated!!!!! +! 2H + 2H!!! 3 = 4"G#! 2 " H 2 M! = 0 Radiation dominated: Ω M <<1, H=1/2t!!! + 1 t!! = 0 Has solution: Constant Growing mode!(t) = B 1 + B 2 ln(t) Grows logarithmically (i.e. slowly)

20 Lambda dominated!!!!! +! 2H + 2H!!! 3 = 4"G#! 2 " H 2 M! = 0 Lambda dominated: Ω M <<1, H=H Λ!!! + 2H!!! = 0 Has solution: Constant Decaying mode!(t) = C 1 + C 2 e!2h "t Density contrast is constant

21 Matter dominated!!!!! +! 2H + 2H!!! 3 = 4"G#! 2 " H 2 M! = 0 Matter dominated: Ω M =1, H=2/(3t)!!! + 4 3t!!! 2 3t 2! = 0 Has solution: Growing mode Decaying mode!(t) = D 1 t 2/3 + D 2 t!1 Grows as t 2/3, linearly propor<onal to scalefactor Only during matter domination can density fluctuations grow significantly

22 Another way to calculate Jeans condition (3/2)NkT -(3/5)GM 2 /r The virial theorem says: 2K + U = 0 2K > 0 -> a stable, gravitationally-bound system. -> pressure dominates U > 2K -> gravity dominates (collapses) Estimate Jeans mass after Recombination Mass above which a spherical system with constant density collapses T=2970 K, ρ=5.4e-19 kg m -3, µ=0.584 for composition of Hydrogen: X=0.77, and Helium: Y=0.23 (see Carroll and Ostlie Eq ) M J!! " # 5kT Gµm H $ % & 3/2! " # 3 $ 4'( % & 1/2! 1.9 ) 10 6 M " ~ Mass of globular star cluster

23 Two characteristic mass scales Jeans mass after recombination ~10 6 M Fluctuations M > 10 6 M are amplified after recombination Recall you needed M > M to survive Silk damping 10 6 M to10 13 M neatly spans a range of structures (globular cluster to massive elliptical galaxy) Could these higher-density regions have produced the structures we observe today?

24 Timing of Structure Formation (see Carroll and Ostlie Example ) Record for most distant QSO: z=6.43 (out of date!) Estimate density fluctuation needed at recombination to form a quasar this soon Gravitational collapse begins when δ ~ 1 In matter era, δ = δ i (t/t i ) 2/3 t = 2/(3H 0 ) a 3/2 (~lower limit to age) H 0 = 70 km s -1 Mpc -1 = Gyr -1 a = 1/(1+z) = t = 2/(3H 0 ) a 3/2 = 462 Myr let t = t dec = 0.38 Myr (WMAP) => δ = There must have been density fluctuations of several tenths of a per cent at decoupling. But that doesn t match the CMB observations! DARK MATTER to the rescue!!

25 Timing of Structure Formation More evidence that dark ma"er is not baryonic CMB fluctuations Observed fluctuations could not have grown fast enough to make galaxies and quasars! Solution: non-baryonic dark matter Negligible interaction with radiation Could start collapsing long before decoupling, at start of matter era Could grow during matter era Could reach required level by recombination with no trace in CMB signal After decoupling Baryonic matter attracted to the large clumps of dark matter until they share the same density They grow together to form objects we see today Figures: D. Eisenstein (U. of Arizona)