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Cosmology > 2. Models for Cosmology > Flashcards

2. Models for Cosmology Flashcards

1
Q

When can we use Newtonian theory?

A

Geometry unimportant (CP)

Already know answer from relativity - fudge residual bits

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2
Q

What is the equation for co-moving coordinates? What does each term mean?

A

x = a(t) * r

x - real distance (ruler coordinate)
a - scale factor
r - co-moving distance separation (stays fixed with time)

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3
Q

Is the milky way expanding?

A

No - gravity holds it together

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4
Q

How is a usually defined?

A

a0 = a(now) = 1

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5
Q

Show density is only dependent on scale factor - not position.

A

n = N/V = N/x^3 = N/a^3r^3 prop to 1/a^3

N = fixed number of particles

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6
Q

What is dr/dt equal to?

A

0

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7
Q

Briefly, how is the fluid equation derived?

A

Adiabatic expansion of universe

W + ∆U = Q = 0 (1st law of thermodynamics)

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8
Q

What does the fluid equation govern?

A

How the density of a gas changes as the universe expands

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9
Q

What is an equation of state?

A

Relationship between pressure and density

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10
Q

Pressure value for ordinary matter?

A

0

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11
Q

Why do we set p=0 for matter?

A

Interactions happen infrequently enough that they do not affect the state of the gas

(rhoc^2&raquo_space; p)

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12
Q

How to think of ρ_radiation?

A

Energy density ε = Energy / V

Analogy with E=mc^2

ε_r = ρ_r c^2

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13
Q

p for relativistic matter?

A

p = ρ c^2 /3

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14
Q

General equation of state?

A

p = wρc^2

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15
Q

w for matter and radiation?

A

w=0 matter

w=1/3 radiation

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16
Q

Upper limit on w?

A

Sound speed (w=1)

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17
Q

Equation for sound speed in an ideal gas?

A

cs^2 = dp/dρ = wc^2

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18
Q

Upper value of w?

A

1

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19
Q

Why is the CP important for the derivation of the Friedmann equation?

A

It only feels the effect of the stuff within the sphere

Stuff around cancels out - ρ constant

Without the CP, this wouldn’t work

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20
Q

What does the Friedmann equation govern?

A

Change in size of the universe with time

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21
Q

What are the terms in the Friedmann equation?

A

å/a ^2 - normalised expansion rate

8πGρ/3 - self gravity

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22
Q

Which density do we use for the acceleration equation?

A

ρ_total = ∑ρi (matter, radiation etc)

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23
Q

Is an expanding, homogeneous, isotropic universe Euclidean?

A

No

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24
Q

What coordinate system should we use? Why?

A

Spherical - isotropy of universe

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25
Q

Briefly, how to get the Robertson-Walker metric?

A

Line element in spherical coordinates dl^2

Add in time with spacetime ds^2

Add in arbitrary curvature K as it is expanding

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26
Q

Properties of K > 0?

A

Spherical

Sum of degrees in a ∆ >180

Space bounded

Has a max since Kr^2 < 1

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27
Q

ds (path through spacetime) for photons?

A

ds = 0 - not necessarily straight line

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28
Q

Properties of K < 0?

A

Hyperbolic

Sum of degrees in a ∆ <180

r is unbounded

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29
Q

Properties of K = 0?

A

Same as euclidean space

Flat

Sum of degrees in a ∆ =180

r is unbounded

30
Q

What is Hubble’s constant in comoving coordinates?

A

H = å/a

31
Q

Why does H = å/a?

A

v = dx/dt = år = Hx = Har

32
Q

How do we get ρ_crit,0?

A

Subs H = å/a into Friedmann

Rearrange for K

Set K=0

Rearrange for ρ_crit,0

33
Q

What is ρ_crit,0?

A

A critical density at the current epoch

34
Q

Equation for ρ_crit,0?

A

3H0^2/8πG

35
Q

Value of K for ρ_crit,0?

A

0

36
Q

Difference between H and H0?

A

H Hubble parameter

H0 Hubble constant

37
Q

Equation for Ω?

A

ρ/ρ_crit

38
Q

What is Ω?

A

Density parameter

39
Q

If K=0, what is Ω?

A

Ω=1 for all time

40
Q

Present values of Ω radiation and matter?

A

Ω_r is tiny

Ω_m ~ 0.3

41
Q

If the deceleration parameter is negative, what happens?

A

Acceleration

42
Q

In the RW metric, what do photons obey?

A

dø = dtheta = 0 and ds = 0

No angular motion

43
Q

How do we get that 1+z1 = 1/a1?

A

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

44
Q

What are the ‘z revisited’ equations?

A

f1/f0 = a0/a1

1+z1 = a0/a1 = 1/a1

45
Q

General solution for fluid equation?

A

If p = wρc^2

ρ = ρ0 a^(-3(1+w))

46
Q

How to define ρ0 when solving equations?

A

ρ = ρ0 when a = a0 = 1

47
Q

Why is solution for ρ_matter prop to 1/a^-3?

A

Number of particles must decrease as 1/a^3

48
Q

Why is solution for ρ_matter prop to 1/a^-4?

A

Number density of photons varies with 1/a^3

But photons can also redshift - actual phonon radiation ρ prop to 1/a

49
Q

w value for cosmological constant?

A

-1

50
Q

Which w leads to ‘natural’ behaviour?

A

w>-1

51
Q

What happens to energy if universe expands and w<=-1?

A

More energy present if universe expands (making energy for free)

52
Q

What value of w sets a natural division?

A

w > -1/3 - real physical materials, self gravity and brake on expansion

w < -1/3 accelerant

53
Q

What is 8πG equal to?

A

H0^2 / ρ_crit,0

54
Q

What is the matter-radiation equality?

A

SF at which density of matter and radiation are equal

55
Q

Value for ρ in empty universe?

A

0

56
Q

Derive a_eq

A

See notes

57
Q

What is the dependence of t on a in a matter-dominated, flat, universe?

A

K=0, p=0, ρ prop to a^-3

Rearrange Friedmann

Integrate

a ∝ t^2/3

(H ∝ 2/3 t^-1)

58
Q

What is free expansion?

A

An open universe, tends towards an empty universe

59
Q

What is the dependence of t on a in a radiation-dominated, flat, universe?

A

K=0, p=0, ρ prop to a^-4

Rearrange Friedmann

Integrate

a ∝ t^1/2

(H ∝ 1/2 t^-1)

60
Q

Generic equation for a-t dependence (K=0)?

A

t = 2/3(1+w) * (8πGρ0/3)^-1/2 * a^[3(1+w)/2]

61
Q

Equation for age of universe from Friedmann?

A

H = 2/3 t^-1

(since H = å/a)

62
Q

Friedmann: do we solve matter and radiation at the same time?

A

No

Need to look which term is dominating

63
Q

Fluid: do we solve matter and radiation at the same time?

A

Yes

64
Q

When does K become important?

A

See slide 139 or 2.7.7.

65
Q

Friedmann equation solution when curvature dominates (K<0)?

A

a ∝ t

free expansion

(ignore matter)

66
Q

Friedmann equation solution when K > 0?

A

Reaches H=0

Gravitational attraction dominates, universe recollapses

67
Q

For K<0, what is the t - Ωm relation?

A

t ∝ 1/sqrt(Ωm)

68
Q

Implication of t ∝ 1/sqrt(Ωm) for K<0?

A

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.]

69
Q

In the acceleration equation, if p and ρ are +ve, what sign is the bracket?

A

+ve

70
Q

In the acceleration equation, what does the sign of the bracket mean? What is the turning point?

A

() +ve - deceleration

() -ve - acceleration

w = -1/3

71
Q

Do you get more deceleration from radiation or matter?

A

Radiation

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