Lecture 5: Age of Universe & Special Cases

Question 1

the current age of the universe will be t0=

Answer 1

intengral between 0 and t0 dt

dt was derived in previous lecture by changing variable from time to redshift

Question 2

is a single treatment of E(z) valid for all z approaching infinity

Answer 2

if radiation parameter important, this needs to be included

this is only important at such high redshifts that we can approximate without

Question 3

final form of t0 after approximation of ignoring radiation term

Answer 3

t0=1/H0 F(Ωm0 , ΩΛ0, Ωk0)

where H0^-1 has units of time and F is dimensionless

Question 4

can convert H0 from usual units of kms^-1Mpc^-1 to

Answer 4

inverse years

Question 5

writing H0=100h we find that to=

Answer 5

9.78 h^-1 F(Ωm0, ΩΛ0, Ωk0) Gyr

Question 6

The pre-factor 9.78h−1 Gyr is often referred to as

Answer 6

the hubble time and it sets the timescale for the expansion of the universe

Question 7

To get a precise value for the age, we need to specify

Answer 7

the values of h and F

this means adopting a particular Friedmann model universe

Question 8

If we take first take h = 0.7 (as current observations indicate) and if we assume that F ∼ 1, then t0 is

Answer 8

approx 14 billion years which is compatible with the ages of stars and galaxies

Question 9

If we adopt the standard ‘Concordance Model’ values of the density parameters, then we can
evaluate the integral F

Answer 9

numerically and find that it is around 0.97

(This consistency between cosmological and astrophysical age estimates is another aspect of the ‘concordance’ that gives the Concordance model its
name).

Question 10

Recall from Cosmology I that for a matter-dominated critical density universe (known as the ‘Einstein de Sitter’ case) we had

Answer 10

a(t) prop to t^2/3

to=2/3 Ho^-1

requires F=2/3 and therefore to=6.52h^-1Gyr

Question 11

einstein de sitter model with to=6.52h^-1Gyr means that for h>0.8

Answer 11

the ages of the universe is less than 8 billion years which conflicts astrophysical estimates

Question 12

The discovery of the accelerated expansion, and the realisation that we live in a flat universe but with positive Λ, has further reduced the age problem since

Answer 12

(with h = 0.7) the presence of the ΩΛ0 term adds more than
two billion years to the theoretical age of the universe compared with an open model with Ωm0 = 0.3.

Question 13

A lower density, matter-dominated universe with Λ = 0 is decelerating more slowly than

Answer 13

a critical density matter-dominated universe, so it takes the lower density universe a longer time to slow down to the expansion rate that we currently measure.

Question 14

In the universe with Λ = 0 it has always been

Answer 14

decelerating to reach its current expansion
rate.

Question 15

In a universe with the same value of the matter density but with Λ positive, the expansion is currently

Answer 15

accelerating so it must have first decelerated to less than its current expansion rate before undergoing a period of acceleration

This extended period of earlier deceleration, followed by acceleration, increases the age of the universe compared with the Λ = 0 case.

Question 16

einstein de sitter: the perturbation to the constant scale factor will

Answer 16

increase exponentially; no matter how small it might be to begin with, it will eventually dominate the behaviour of the scale factor

Question 17

einstein de sitter: a universe where Λ has a value slightly larger than the Einstein static value.

Answer 17

initially expand until
it reached the scale factor given by the Einstein static solution but would then remain in a quasi-static state for a long time, oscillating in size about the static solution; “loitering universe”

Question 18

einstein de sitter universe

Answer 18

a pressureless, matter-dominated model with critical density and zero cosmological constant

Question 19

de sitter universe

Answer 19

cosmological constant that dominates completely.

p=0,k=0

FE reduces to H^2=Λ/3 so

Question 20

de sitter universe - ΩΛ=

Answer 20

1

Question 21

de sitter: Taking the square root of both sides of this equation (and taking the positive square root, since the universe is currently expanding) gives us a

Answer 21

a separable differential equation that is straightforward to
integrate, because Λ is constant

Question 22

At early times our universe behaved
like a de Sitter model so if dark energy really is described by a constant Λ then it is

Answer 22

tending towards a de sitter model in the future, as the mass density becomes less and less important

Question 23

provided we assume that Λ = 0, we can solve

Answer 23

the Friedmann equation analytically for k = ±1.

Question 24

the k=0, Λ=0 case describes

Answer 24

either a matter dominated universe with einstein de sitter solution

or radiation dominated universe

Question 25

solution to the k=+/- 1 can be found by

Answer 25

parameterising (by which we mean ‘turning the equation into something more useful’) in terms of either θ or χ before integrating

Question 26

visualation of non-zero curvature solutions

Answer 26

k=1 semicircle then bounces back up k=-1 steep k=0 less steep

Question 27

does a k=0 universe expand?

Answer 27

yes, just not as fast as the k<0 solution

Question 28

we can expect that if k = 1, then for any value of ρ there is a value of a such that

Answer 28

H=0 because the matter and curvature terms are of opposite signs in the Friedmann equation

Question 29

the big crunch

Answer 29

expansion will end and gravitational attraction will eventually result in the recollapse of the universe

Question 30

why mass or energy density being constant is unexpected?

Answer 30

volume of universe increasing so extra mass coming from somewhere

Question 31

planck time

Answer 31

time at which quantum and gravitational effects are of equal importance

Question 32

vaccum density compared to planck density

Answer 32

difference of more than 120 orders of magnitude the observed equivalent mass density of the vacuum energy is so incredibly small compared with its natural, theoretically predicted value

Question 33

does re-casting the cosmological constant as a dark fluid solve the cos const problem?

Answer 33

no leaves the puzzle of why we appear to be living at a special time close to the point where universe switches from matter dominated

Question 34

quintessence

Answer 34

theoretical models in which ω is a function of time or, equivalently, redshift

Question 35

why invoking a redshift dependence for the equation of state of our dark fluid could be useful

Answer 35

it might provide us with a plausible mechanism for how ρΛ and ρm ‘track’ each other as the universe expands, thus removing/reducing the question about whether we live at this ‘special time’