Senior : lecture 9 (ADV)

1 Senior Astrophysics

› Scott Croom: [email protected] rm 561A - Lectures 1 to 9: Cosmology - 1st assignment given: 20th September, due 4th October - Lecture 9 ADV/REG split (6th Oct).

› Helen Johnston: [email protected] rm 563 - Lecture 10 to 19: Stellar evolution and the end states of . - 2nd assignment given: 13th Oct, due 27th October - Lecture 19 ADV/REG split.

2 Lecture 8: Structure formation

Revision from lecture 8: Concordance cosmology Q: How do get get from the primordial fluctuations in the CMB to the galaxies we see today? 1. How do we form galaxies? 2. Growth of linear fluctuations. 3. The Jeans Mass and

Wikipedia: Jeans Instability, Jeans length, Structure formation

Books: Peacock, “Cosmological Physics”, p460

3 Revision: Concordance cosmology

› Measured of stars is not sufficient for critical density, Galaxy cluster Abell 1689

Ωstar~0.005-0.01 › Nucleosynthesis gives 2 0.016<ΩBh <0.024. › Galaxy clusters allow us to measure the baryon to total mass ratio

› This results in ΩB/ΩM ~ 0.1, which means that ΩM ~ 0.3-0.5.

X-ray: NASA/CXC/MIT/E.-H Peng et al; Optical: NASA/STScI.

4 Revision: Concordance cosmology

› Type 1a supernovae are good “standard candles” › Used to determine the

luminosity distance, dL. › Inconsistent with an Einstein-de Sitter . › Evidence for a cosmological constant, or more generally “dark energy”.

Nobel prize for 2011!!!!!

5 Revision: Concordance cosmology

› CMB structure gives us a “standard ruler” to examine the expansion history to z~1100. › The first peak in the power spectrum gives us the horizon scale to wave at decoupling.

Peak~10

6 Revision: Concordance cosmology

7 9.1 How do galaxies form?

Z=1100, ΔT/T~10-5

z=0, Δρ/ρ~1 or more 9.1 How do galaxies form?

Initial Collapse: Hierarchical Merging:

Elliptical Spiral Irregular 9.2 Linear growth of structure

› Key question is: How fast does structure grow in an expanding Universe? › We will consider this using linear theory, i.e. assume that deviations away from the mean density are small. › Can define: ρ(x,t) − ρ (t) Δρ(x,t) δ(x,t) = 0 = ρ0(t) ρ0 (t)

Where ρ0 (t) is the mean density and ρ(x,t) is the density at some position x.

› The we “simply” have to find the evolution of δ with time.

€ 10 9.2 Linear growth of structure

› Start from fluid equations (assuming no ):

⎛ ∂ρ⎞   ⎜ ⎟ + ∇ r ⋅ (ρu ) = 0 Continuity equation (mass conservation) ⎝ ∂t ⎠ r  ⎛ ∂u ⎞     ⎜ ⎟ + (u ⋅ ∇ r )u = −∇ rΦ Euler equation (momentum conservation) ⎝ ∂t ⎠ r 2 4 G Poisson Equation (gravitational potential) ∇ rΦ = π ρ

› This is defined in a fixed (“Eulerian”) coordinate system, but we need to be in a comoving frame: € r a˙ x = and v = u − r Subtracting Hubble a(t) a flow (=Hr) to leave “peculiar velocity”

11 € 9.2 Linear growth of structure

› We also want to consider small perturbations so also define:  ρ = ρ [1+δ(x ,t)] › With fair bit of effort (we won’t do it here), the fluid equations become: ∂δ 1   € + ∇⋅ [(1+δ)v ] = 0 (Continuity) “Hubble friction” ∂t a  term ∂v 1    a˙  1  + (v ⋅ ∇) v + v = − ∇ φ (Euler) ∂t a a a ∇2φ = 4πGρ a2δ (Poisson)

› Now all in comoving coordinates. › Also we have redefined the potential to be just the part due to the € non-uniform density (Φ φ).

12 9.2 Linear growth of structure

› We can cancel 2nd order terms as we are considering small changes from the mean (i.e. v2, δv etc.), so that the equations become: ∂δ 1   ∂δ 1   1: + ∇⋅ [(1+δ)v ] = 0 ⇒ + ∇⋅ v = 0 ∂t a ∂t a   ∂v 1    a˙  1  ∂v a˙  1  2 : + (v ⋅ ∇) v + v = − ∇ φ ⇒ + v = − ∇φ ∂t a a a ∂t a a 3: ∇2φ = 4πGρ a2δ

› We can now eliminate v by: i) taking time derivatives of 1st equation, ii) 1/a x divergence of 2nd equation iii) subtract 1st and 2nd equations iv) substitute continuity (1st) and Poisson (3rd) equation. € › Try this – note: derivation of growth equation is not examinable (solving it is). › For further details see: Cosmological Physics, by Peacock, p460.

13 9.2 Linear growth of structure

› The result of all that is a second order differential equation for the density contrast, δ: ∂ 2δ a˙ ∂δ + 2 = 4πGρ δ ∂t 2 a ∂t Linear growth equation

› How do we solve this? › Consider our favourite EdS Universe. The scale factor is a power- law in t: a~t2/3, so try power-law solutions: € δ ∝ t n

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€ 9.2 Linear growth of structure

› The growing mode solution in the EdS Universe is: 2 1 δ(t) ∝ t 3 ∝ a(t) ∝ (1+ z)

› After all that, the growth of density fluctuations has a remarkably simple relation to scale-factor and redshift (at least for an EdS Universe).

€ › Some more complicated solutions can be derived analytically, but in general for multiple components (e.g. radiation, dark energy +…) this needs to be solved numerically. › Generally decreasing matter density decreases the growth rate.

15 9.2 Linear growth of structure

› Can use this to estimate the growth from the CMB until now (Universe close to EdS for most of that time). › CMB decoupling at z=1100: ΔT δ(z =1100) ~ ~ 10−5 T ΔT δ(z = 0) ~ × (1+ z ) ~ 10−2 ??? T dec › What’s going on? We should only have ~1% density fluctuations at the present if the CMB fluctuations (baryons + photons) grew according to linear theory. €

16 9.3 Gravitational collapse

› Which ripples will collapse ?

ρ

› Gravity pulls matter in. L › Pressure pushes it back out. › When pressure wins -> oscillations (sound waves). › When gravity wins -> collapse. › Cooling lowers pressure, triggers collapse. › Applies to both formation! and galaxy formation. 9.3 Gravitational collapse: Jeans length

N molecules of mass m in box of size L at temp T. G M M M = N m › Gravitational Energy: EG ~ " 3 - (self-gravitating) L ~ L "

› Thermal Energy: ET ~ N k T

2 G " L3 m # & 2 › Ratio: EG G M ( ) L ! ~ ~ = % ( E L N k T L k!T L !T $ J '

1/ 2 # k T & Developed by › Jeans Length: LJ ~ % ( James Jeans in ! $ G " m' early 1900s.

› Gravity wins when L > LJ .

! 9.3 Gravitational collapse: Jeans length

› Gravity tries to pull material in.

› Pressure tries to push it out.

› Gravity wins for L > LJ

› ----> large regions collapse.

› Pressure wins for L < LJ

› ----> small regions oscillate (dissipate). 1/ 2 › Jeans Length: ⎛ k T ⎞ LJ ~ ⎜ ⎟ ⎝ G ρ m⎠

› Large cool dense regions collapse.

€ 9.3 Gravitational collapse: timescales

› Ignore Pressure. Time to collapse = free fall time (tG). 1 › Gravitational acceleration: [s = at 2 ] 2 L GM gt ~ 3 G g ~ 2 M ~ L " tG L

› Time to collapse: L L3 1 tG ~ ~ ~ ! g G M! G "

› Gravitational timescale, or dynamical timescale.

› Denser regions collapse faster. !› Same collapse time for all sizes. 9.3 Gravitational collapse: timescales

› Ignore Gravity. Pressure waves travel at sound speed. 1/ 2 # P & # k T &1 / 2 cS ~ % ( ~ % ( $ " ' $ m ' › Sound crossing time: 1/ 2 L " m % ! tS ~ ~ L $ ' cS # k T & › Small hot regions oscillate more rapidly.

› Note: before decoupling radiation pressure >> gas pressure so: 1 ! c ~ c S 3

! 9.3 Gravitational collapse: timescales

› Collapse time: Sound crossing time: 1 L " k T %1 / 2 t = tS = G cs ~ $ ' G " cS # m & › Ratio of timescales: # &1 / 2 tS L G " G " m L ! ~ ! ~ L %! ( ~ tG cS $ k T ' LJ

› Jeans length (again!): cS LJ ~ G " !

! 9.3 Gravitational collapse: when and where?

L tS = cS timescale ! oscillate 1 tG = G " collapse

! L J size

! 9.3 Gravitational collapse: Jeans mass

Jeans Length : (smallest size that collapses) 1/ 2 # k T & LJ ~ % ( $ G " m'

Jeans Mass: (smallest mass that collapses)

# & 3 / 2 ! 3 k T 3 / 2 *1/ 2 MJ ~ " LJ ~ " % ( )T " $ G " m'

› Need cool dense regions to collapse stars, › But galaxy-mass regions can collapse sooner. ! 9.3 Gravitational collapse at decoupling

› Today:

#28 -3 T0 = 2.7 K "0 =10 kg m

› Expanding Universe (matter dominated): T R#1 R#3 T 3 ! "! $ " " T = 3000 K

› At decoupling: $ 3000' 3 " =10#28 & ) =1.4!* 10#19 kg m-3 ! % 2.7 ( #3 + 2 Msun pc

! 9.3 Gravitational collapse at decoupling

› Prior to decoupling, radiation pressure dominates. › Sound speed is close to the speed of light. 1 c ~ c › Jeans length is close to the horizon scale. S 3

cS c c 14 LJ ~ ~ ~ −15 ~ c × 2 ×10 m G ρ 3 × 6.7 ×10−11 ×1.4 ×10−19 5 ×10 5 7 13 dH ~ 3ct ~ c × 9 ×10 × 3 ×10 ~ c × 3 ×10! m › So baryons cannot collapse  baryon-photon fluid oscillates (baryon acoustic oscillations – BAO). € › After decoupling, the sound speed drops dramatically, allowing baryons to collapse on sub-horizon scales.

26 9.3 Gravitational collapse at decoupling

› Physical properties: T = 3000 K $19 -3 -3 " =1.4 #10 kg m % 2 Msun pc › Jeans Length : 1/ 2 1/ 2 *23 *1 # k T & # 1.4 )10 J K (3000K) & L ~ = % ( ) ( J % ( % *11 3 *1 *2 *19 *3 *27 ( $ G " m' ! $ (6.7 )10 m kg s )(1.4 )10 kg m )(1.7 )10 kg)' 1.6 )1018 m = = 50 pc 3.2 )1016 m/pc

3 -3 3 › Jeans Mass: MJ ~ " LJ ~ (2 Msun pc )(50 pc) = 3#105 M ! sun › More than a star, less than a galaxy. › Close to a globular cluster mass. ! 9.4 Structure formation: summary

Hold the oldest stars. Orbit in the Halo. 9.4 Structure formation: summary

› Dark matter and baryons Growth rate for given Fourier mode: modes grow until they enter the horizon › Dark matter does not couple to photons, so Dark matter continues to grow. › Baryons caught in oscillating mode, can’t grow. (recall BAO from Baryons previous lecture and see below). › After decoupling baryons fall into the potential wells of the dark matter, Dark matter to the rescue!!! catching up!

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