7. Structure Formation Flashcards
How do you find Jean’s length when given the fluctuation equation?
Set the 2 terms on the RHS equal
How does expansion affect structure formation?
Expansion makes it hard for fluctuations to grow
What is delta?
Scaled fluctuation
(Scaled to background density)
Equation for delta?
delta = deltap/p
What are the terms on the RHS of the fluctuation equation? What do they do?
Self-gravity (causes collapse)
Pressure (resists collapse)
Which has a later a_eq, an open (Ω=0.3) or flat (Ω=1) universe?
Open has a later matter-radiation equality (i.e., z lower)
Since there is less matter
Is recombination at the same z for different Ω values?
Yes (around z~1100)
What dominates during recombination?
Matter
Difference in radiation density for Ω0=1 and Ω0=0.3?
Same - since this is set by CMB
Difference in matter density for Ω0=1 and Ω0=0.3?
Ω=0.3 has lower (open)
Does a_eq occur before or after recombination?
Before
For matter fluctuation equation, why is sound speed ≠ 0 if p = 0?
Baryons feel radiation pressure until recombination
Equation for sound speed?
cs^2 = dp/drho
What is the only thing that has no pressure after the matter-dominated era?
CDM
(no sound speed)
What happens if λ > λJ?
Gravity wins
What happens if λ < λJ?
Pressure wins
Why does pressure win for small things (i.e., λ < λJ)?
They can feel each others pressure
Why can we compare fluctuation equations, Jeans’ length to comoving horizons?
Because everything, k, λJ is comoving
What is λJ / λH?
sqrt(3)/2 = 0.85
Why is λJ ~ λH? And when?
Radiation dominated era
Structure only forms on scales above MFP of photons/neutrinos, otherwise particles involved effectively suppress any growth, smearing out the fluctuations in the density.
The radiation is gravitationally dominant at this time, so if fluctuations in the radiation are smeared out, the same is true for any matter present
Why can’t we have pressure effects above the horizon?
As this would be faster than the speed of light
After a_eq, what happens to CDM and why?
It is non-interacting; it has no pressure.
So λJ = 0
CDM fluctuations can grow on all scales
When is λJ=0?
After a_eq for CDM
When do baryon fluctuations grow?
When recombination occurs
Before they are still coupled to radiation by Compton scattering
What is Jeans length for baryons?
Same as it was at a_eq until recombination
What happens to radiation at recombination? What does this mean in terms about what it can tell us?
After recombination, decoupling.
So radiation can do its own thing; nothing else in terms of growing earth.
So CMB is a powerful as to the early universe
Do CDM fluctuations have gravitational potential?
Yes
What happens once baryons are decoupled from radiation?
Baryons fall into potential wells that have been forming since a_eq
What effect does CDM presence have on baryon fluctuations?
CDM boosts their growth via potential wells
So fluctuation in baryons can be significantly larger than fluctuations in CMB
Difference in baryon-only and baryon and CDM universes?
Baryon and CDM: fluctuation in the baryons can be significantly larger than the fluctuations in the CMB.
Baryon-only: fluctuations in the matter and radiation similar
Rough size of structures in baryon-only universe?
More like horizon size at recombination
Do we still see fluctuations in the CMB?
Yes
Significance that the Jean’s length at recombination is a sphere about 10^5 solar masses?
This is the size of a globular cluster
These are the oldest things that we see nearby (13-14 billion years old)
Consistent mass
What was the first thing that was formed in the universe?
Globular clusters - since jeans’ length of baryons, so baryons can’t form anything smaller than this as they’re held up by pressure
Does baryon pressure affect large scale structure?
No (jean’s length only 8kpc)
If Ω is lower, when does a_eq occur?
Later
Would a wavelength above the horizon (think of diagram) have potential for growth?
Always
If λ > λJ, is it necessarily growing?
No - expansion can affect
Briefly, how to solve fluctuation equations?
Slightly different for radiation and matter
Assume Ω0=1, so K=0 and Λ=0
Take growth solution (λ»λJ - self gravity dominates)
Rearrange Friedmann to subs in the 8πG*rho
Differentiate a solution for å/a
Take solution in form At^m, multiply by t^2 to see and solve differential equation
Compare t^m value to H and a
What does the no-growth solution regard?
For either baryons, or during radiation domination
(as you need to have a sound speed)
What is the outcome of the no-growth solution?
Solution is a damped oscillator (amplitude decreases with time)
Is CDM affected by no-growth solution?
No - as it doesn’t have pressure, so doesn’t have sound speed, so is not damped and not affected after a_eq
What type of universe does the no-growth solution affect the most?
Baryon only
What happens to fluctuation growth if Λ≠0?
Expansion is driven faster
This stops the growth - no gravitational collapse more quickly than it pulls apart
Are fluctuations self-gravitating? What effect does this have?
No
So they can be really easily pulled apart
What happens to fluctuation growth if Ω0<1?
Same effect as Λ≠0
Universe expands a bit more quickly than matter
How can we approximate fluctuation growth if Ω0<1?
Empty universe
Get rid of background density on RHS of eqn
Leaves deltas
Solution for empty universe fluctuation equation?
m=0 or m=-1
So fluctuations stop growing
Is structure hard to form in a low or high density universe?
Low
What initial fluctuation ^2 prop to?
k^n
What contains n for the initial fluctuation spectrum?
Observations
What does the D(z) factor show?
How much a fluctuation with infinite wavelength (k = 0) grows between z = ∞ and z
Do we observationally measure transfer function?
No
What does it mean is the transfer function <1?
λ < λJ at some point - either due to damping or been stopped from growing
What is P(k)?
The power spectrum i.e., delta_final(k,z)^2
What shape is the power spectrum?
k^n, and k^n-4
When does T(k)=1?
For wavelengths that never enter the horizon before a_eq so 𝑷(𝒌) ∝ 𝒌^𝒏
When is T(k) prop to x^2?
T(k) =(𝜆𝑒𝑞/𝜆𝐻(𝑘))^2 for wavelengths that enter the
horizon before 𝑎𝑒𝑞
What is 𝜆H^2(k) in the power function?
The horizon size when the individual k mode has λ<λJ
What is λeq?
Horizon wavelength at matter-radiation equality
Why is CMB so important?
After recombination, fluctuations in CMB essentially don’t change
Anything we see in the CMB reflects the state of the baryons in recombination
What do polarisation peaks being out of phase with the intensity spectrum tell us?
Thomson scattering whacking photons in our line of sight is correct
I.e., Thomson scattering polarises light, max P at max electron velocity
What does Planck data tell us the best model is?
lambda-CDM
Why is the best model lambda-CDM?
CDM: deltaT/T is too small for pure baryon fluctuations have grown
(i.e., not enough time for them to have grown)
Lambda: Don’t want to have enough CDM to make universe flat
Recomb peak exactly where you’d expect if Ω_total=1
Why doesn’t Ω_lambda change the position of the recombination peak, if angular position of peak is related to Ω?
Lambda doesn’t affect recombination, but affects distance to horizon of recombination
(move back row further back)
Lose strong dependence in Ω on measuring the angle (need to use features instead)
What happens to peaks when baryon density increased? Why?
Accentuated
Since suppressing stuff within Jeans length after a_eq more
Rarefaction effect changes as well
What happens to peaks when matter density (CDM) increased? Why?
Changes odd/even peaks
Due to effectively mass loading - making spring hold more stuff
Change relationship between compression and rarefaction
What happens to peaks if Ωtotal=1, with Ω_lambda varying? Why?
(or allow Ω with no lambda to vary)
Spatial curvature and dark energy both change the angular diameter distance to recombination and hence shift the angular location of the peaks left and right
What is the transfer function?
Encapsulates the way different wavelengths grow.
Equation for final fluctuation spectrum?
δk(now) = δk(initial)T(k)D(z)
How is T(k) calculated?
Calculated numerically, though some of the basic features seen can be understood from first principles.
What is δk(initial)?
An assumption that can be tested observationally.
What differences do we expect to see in the real galaxy distribution when considering the baryon vs CDM models?
(sort out this cards)
CDM: model with the largest “small scale power” since it has the largest value of T (k) at large k (small λ). This is because CDM fluctuations can grow continuously since matter-radiation equality. You can see the change in the slope of the curve at about k = 0.3. This is the split between the continuous growth for all time (small k) and only after matter radiation-equality (large k).
The model with the least small scale power is the baryon only one. That is due to the fact it only grows continuously on all scales after recombination.
This means the large scale distribution of galaxies will show fewer larger scale features in a CDM model than a baryonic one, and rather more on smaller scales. The actual observed distribution is somewhere in between these two cases.
δ ∝ λ2Horizon for modes that are always growing. Which part of the T(k) plot is this?
Smallest k = largest λ
Continuously growing modes are those at largest λ (outside the horizon at all times before recombination for baryons, or matter-radiation equality for CDM)
So in this case T (k) = 1 (this is basically the mode we use to calculate the D(z) term).
Why is there a break in the slope of T(k)?
Shorter wavelengths don’t grow once they enter the horizon
So there is a λ^2 dependence in terms of the difference in growth between those that enter the horizon before matter-radiation equality and after
The appearance of λeq there is because all CDM fluctuation modes grow equally after aeq
So the suppression is basically from when a mode with given k enters the horizon (λHorizon(k)) to aeq. Using the 10Mpc example from before, growing modes increase as a2 as the universe expands. This mode enters the horizon around a = 10−5, so if we said aeq = 10−4 (just as an example), this mode misses out on (10−4/10−5)2 of growth compared to the unimpeded k = 0 mode. The difference in T(k) between the “suppressed” mode and the constantly growint mode is there just = (λeq/λHorizon(k))2 since that is the same as a2eq/a2k,Horizon where ak,Horizon is the scale factor at which the mode with wavenumber k enters the horizon. (Make sure you understand this argument!)
Does matter cool more quickly after recombination?
Yes
