2. Models for Cosmology

Question 1

When can we use Newtonian theory?

Answer 1

Geometry unimportant (CP)

Already know answer from relativity - fudge residual bits

Question 2

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

Answer 2

x = a(t) * r

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

Question 3

Is the milky way expanding?

Answer 3

No - gravity holds it together

Question 4

How is a usually defined?

Answer 4

a0 = a(now) = 1

Question 5

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

Answer 5

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

N = fixed number of particles

Question 6

What is dr/dt equal to?

Answer 6

0

Question 7

Briefly, how is the fluid equation derived?

Answer 7

Adiabatic expansion of universe

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

Question 8

What does the fluid equation govern?

Answer 8

How the density of a gas changes as the universe expands

Question 9

What is an equation of state?

Answer 9

Relationship between pressure and density

Question 10

Pressure value for ordinary matter?

Answer 10

0

Question 11

Why do we set p=0 for matter?

Answer 11

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

(rhoc^2&raquo_space; p)

Question 12

How to think of ρ_radiation?

Answer 12

Energy density ε = Energy / V

Analogy with E=mc^2

ε_r = ρ_r c^2

Question 13

p for relativistic matter?

Answer 13

p = ρ c^2 /3

Question 14

General equation of state?

Answer 14

p = wρc^2

Question 15

w for matter and radiation?

Answer 15

w=0 matter

w=1/3 radiation

Question 16

Upper limit on w?

Answer 16

Sound speed (w=1)

Question 17

Equation for sound speed in an ideal gas?

Answer 17

cs^2 = dp/dρ = wc^2

Question 18

Upper value of w?

Answer 18

1

Question 19

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

Answer 19

It only feels the effect of the stuff within the sphere

Stuff around cancels out - ρ constant

Without the CP, this wouldn’t work

Question 20

What does the Friedmann equation govern?

Answer 20

Change in size of the universe with time

Question 21

What are the terms in the Friedmann equation?

Answer 21

å/a ^2 - normalised expansion rate

8πGρ/3 - self gravity

Question 22

Which density do we use for the acceleration equation?

Answer 22

ρ_total = ∑ρi (matter, radiation etc)

Question 23

Is an expanding, homogeneous, isotropic universe Euclidean?

Answer 23

No

Question 24

What coordinate system should we use? Why?

Answer 24

Spherical - isotropy of universe

Question 25

Briefly, how to get the Robertson-Walker metric?

Answer 25

Line element in spherical coordinates dl^2 Add in time with spacetime ds^2 Add in arbitrary curvature K as it is expanding

Question 26

Properties of K > 0?

Answer 26

Spherical Sum of degrees in a ∆ >180 Space bounded Has a max since Kr^2 < 1

Question 27

ds (path through spacetime) for photons?

Answer 27

ds = 0 - not necessarily straight line

Question 28

Properties of K < 0?

Answer 28

Hyperbolic Sum of degrees in a ∆ <180 r is unbounded

Question 29

Properties of K = 0?

Answer 29

Same as euclidean space Flat Sum of degrees in a ∆ =180 r is unbounded

Question 30

What is Hubble's constant in comoving coordinates?

Answer 30

H = å/a

Question 31

Why does H = å/a?

Answer 31

v = dx/dt = år = Hx = Har

Question 32

How do we get ρ_crit,0?

Answer 32

Subs H = å/a into Friedmann Rearrange for K Set K=0 Rearrange for ρ_crit,0

Question 33

What is ρ_crit,0?

Answer 33

A critical density at the current epoch

Question 34

Equation for ρ_crit,0?

Answer 34

3H0^2/8πG

Question 35

Value of K for ρ_crit,0?

Answer 35

0

Question 36

Difference between H and H0?

Answer 36

H Hubble parameter H0 Hubble constant

Question 37

Equation for Ω?

Answer 37

ρ/ρ_crit

Question 38

What is Ω?

Answer 38

Density parameter

Question 39

If K=0, what is Ω?

Answer 39

Ω=1 for all time

Question 40

Present values of Ω radiation and matter?

Answer 40

Ω_r is tiny Ω_m ~ 0.3

Question 41

If the deceleration parameter is negative, what happens?

Answer 41

Acceleration

Question 42

In the RW metric, what do photons obey?

Answer 42

dø = dtheta = 0 and ds = 0 No angular motion

Question 43

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

Answer 43

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

Question 44

What are the 'z revisited' equations?

Answer 44

f1/f0 = a0/a1 1+z1 = a0/a1 = 1/a1

Question 45

General solution for fluid equation?

Answer 45

If p = wρc^2 ρ = ρ0 a^(-3(1+w))

Question 46

How to define ρ0 when solving equations?

Answer 46

ρ = ρ0 when a = a0 = 1

Question 47

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

Answer 47

Number of particles must decrease as 1/a^3

Question 48

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

Answer 48

Number density of photons varies with 1/a^3 But photons can also redshift - actual phonon radiation ρ prop to 1/a

Question 49

w value for cosmological constant?

Answer 49

-1

Question 50

Which w leads to 'natural' behaviour?

Answer 50

w>-1

Question 51

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

Answer 51

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

Question 52

What value of w sets a natural division?

Answer 52

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

Question 53

What is 8πG equal to?

Answer 53

H0^2 / ρ_crit,0

Question 54

What is the matter-radiation equality?

Answer 54

SF at which density of matter and radiation are equal

Question 55

Value for ρ in empty universe?

Answer 55

0

Question 56

Derive a_eq

Answer 56

See notes

Question 57

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

Answer 57

K=0, p=0, ρ prop to a^-3 Rearrange Friedmann Integrate a ∝ t^2/3 (H ∝ 2/3 t^-1)

Question 58

What is free expansion?

Answer 58

An open universe, tends towards an empty universe

Question 59

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

Answer 59

K=0, p=0, ρ prop to a^-4 Rearrange Friedmann Integrate a ∝ t^1/2 (H ∝ 1/2 t^-1)

Question 60

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

Answer 60

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

Question 61

Equation for age of universe from Friedmann?

Answer 61

H = 2/3 t^-1 (since H = å/a)

Question 62

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

Answer 62

No Need to look which term is dominating

Question 63

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

Answer 63

Yes

Question 64

When does K become important?

Answer 64

See slide 139 or 2.7.7.

Question 65

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

Answer 65

a ∝ t free expansion (ignore matter)

Question 66

Friedmann equation solution when K > 0?

Answer 66

Reaches H=0 Gravitational attraction dominates, universe recollapses

Question 67

For K<0, what is the t - Ωm relation?

Answer 67

t ∝ 1/sqrt(Ωm)

Question 68

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

Answer 68

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.]

Question 69

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

Answer 69

+ve

Question 70

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

Answer 70

() +ve - deceleration () -ve - acceleration w = -1/3

Question 71

Do you get more deceleration from radiation or matter?

Answer 71

Radiation