2. Models for Cosmology Flashcards
When can we use Newtonian theory?
Geometry unimportant (CP)
Already know answer from relativity - fudge residual bits
What is the equation for co-moving coordinates? What does each term mean?
x = a(t) * r
x - real distance (ruler coordinate)
a - scale factor
r - co-moving distance separation (stays fixed with time)
Is the milky way expanding?
No - gravity holds it together
How is a usually defined?
a0 = a(now) = 1
Show density is only dependent on scale factor - not position.
n = N/V = N/x^3 = N/a^3r^3 prop to 1/a^3
N = fixed number of particles
What is dr/dt equal to?
0
Briefly, how is the fluid equation derived?
Adiabatic expansion of universe
W + ∆U = Q = 0 (1st law of thermodynamics)
What does the fluid equation govern?
How the density of a gas changes as the universe expands
What is an equation of state?
Relationship between pressure and density
Pressure value for ordinary matter?
0
Why do we set p=0 for matter?
Interactions happen infrequently enough that they do not affect the state of the gas
(rhoc^2»_space; p)
How to think of ρ_radiation?
Energy density ε = Energy / V
Analogy with E=mc^2
ε_r = ρ_r c^2
p for relativistic matter?
p = ρ c^2 /3
General equation of state?
p = wρc^2
w for matter and radiation?
w=0 matter
w=1/3 radiation
Upper limit on w?
Sound speed (w=1)
Equation for sound speed in an ideal gas?
cs^2 = dp/dρ = wc^2
Upper value of w?
1
Why is the CP important for the derivation of the Friedmann equation?
It only feels the effect of the stuff within the sphere
Stuff around cancels out - ρ constant
Without the CP, this wouldn’t work
What does the Friedmann equation govern?
Change in size of the universe with time
What are the terms in the Friedmann equation?
å/a ^2 - normalised expansion rate
8πGρ/3 - self gravity
Which density do we use for the acceleration equation?
ρ_total = ∑ρi (matter, radiation etc)
Is an expanding, homogeneous, isotropic universe Euclidean?
No
What coordinate system should we use? Why?
Spherical - isotropy of universe
Briefly, how to get the Robertson-Walker metric?
Line element in spherical coordinates dl^2
Add in time with spacetime ds^2
Add in arbitrary curvature K as it is expanding
Properties of K > 0?
Spherical
Sum of degrees in a ∆ >180
Space bounded
Has a max since Kr^2 < 1
ds (path through spacetime) for photons?
ds = 0 - not necessarily straight line
Properties of K < 0?
Hyperbolic
Sum of degrees in a ∆ <180
r is unbounded
Properties of K = 0?
Same as euclidean space
Flat
Sum of degrees in a ∆ =180
r is unbounded
What is Hubble’s constant in comoving coordinates?
H = å/a
Why does H = å/a?
v = dx/dt = år = Hx = Har
How do we get ρ_crit,0?
Subs H = å/a into Friedmann
Rearrange for K
Set K=0
Rearrange for ρ_crit,0
What is ρ_crit,0?
A critical density at the current epoch
Equation for ρ_crit,0?
3H0^2/8πG
Value of K for ρ_crit,0?
0
Difference between H and H0?
H Hubble parameter
H0 Hubble constant
Equation for Ω?
ρ/ρ_crit
What is Ω?
Density parameter
If K=0, what is Ω?
Ω=1 for all time
Present values of Ω radiation and matter?
Ω_r is tiny
Ω_m ~ 0.3
If the deceleration parameter is negative, what happens?
Acceleration
In the RW metric, what do photons obey?
dø = dtheta = 0 and ds = 0
No angular motion
How do we get that 1+z1 = 1/a1?
Use RW metric with ds = dø = dtheta = 0
Photons at r1, frequency f1 emitted within a time
Seen by us at r0, f0
Integrate
Use v prop to 1/∆t and def of z
What are the ‘z revisited’ equations?
f1/f0 = a0/a1
1+z1 = a0/a1 = 1/a1
General solution for fluid equation?
If p = wρc^2
ρ = ρ0 a^(-3(1+w))
How to define ρ0 when solving equations?
ρ = ρ0 when a = a0 = 1
Why is solution for ρ_matter prop to 1/a^-3?
Number of particles must decrease as 1/a^3
Why is solution for ρ_matter prop to 1/a^-4?
Number density of photons varies with 1/a^3
But photons can also redshift - actual phonon radiation ρ prop to 1/a
w value for cosmological constant?
-1
Which w leads to ‘natural’ behaviour?
w>-1
What happens to energy if universe expands and w<=-1?
More energy present if universe expands (making energy for free)
What value of w sets a natural division?
w > -1/3 - real physical materials, self gravity and brake on expansion
w < -1/3 accelerant
What is 8πG equal to?
H0^2 / ρ_crit,0
What is the matter-radiation equality?
SF at which density of matter and radiation are equal
Value for ρ in empty universe?
0
Derive a_eq
See notes
What is the dependence of t on a in a matter-dominated, flat, universe?
K=0, p=0, ρ prop to a^-3
Rearrange Friedmann
Integrate
a ∝ t^2/3
(H ∝ 2/3 t^-1)
What is free expansion?
An open universe, tends towards an empty universe
What is the dependence of t on a in a radiation-dominated, flat, universe?
K=0, p=0, ρ prop to a^-4
Rearrange Friedmann
Integrate
a ∝ t^1/2
(H ∝ 1/2 t^-1)
Generic equation for a-t dependence (K=0)?
t = 2/3(1+w) * (8πGρ0/3)^-1/2 * a^[3(1+w)/2]
Equation for age of universe from Friedmann?
H = 2/3 t^-1
(since H = å/a)
Friedmann: do we solve matter and radiation at the same time?
No
Need to look which term is dominating
Fluid: do we solve matter and radiation at the same time?
Yes
When does K become important?
See slide 139 or 2.7.7.
Friedmann equation solution when curvature dominates (K<0)?
a ∝ t
free expansion
(ignore matter)
Friedmann equation solution when K > 0?
Reaches H=0
Gravitational attraction dominates, universe recollapses
For K<0, what is the t - Ωm relation?
t ∝ 1/sqrt(Ωm)
Implication of t ∝ 1/sqrt(Ωm) for K<0?
Low density universes are older, high density younger
We measure H0 now, current day Hubble constant, a denser universe expands more quickly at the start and then puts brakes on
[More stuff = more self gravity = decelerating more. Must’ve been higher velocity in the past. So denser = younger.]
In the acceleration equation, if p and ρ are +ve, what sign is the bracket?
+ve
In the acceleration equation, what does the sign of the bracket mean? What is the turning point?
() +ve - deceleration
() -ve - acceleration
w = -1/3
Do you get more deceleration from radiation or matter?
Radiation
