Part1 - Cosmological Models

Question 1

simple pressureless model of the universe starting point

Answer 1

homogeneous and isotropic universe filled with dust of uniform density

Question 2

considering a galaxy of mass m, proper distance r from centre of sphere containing many galaxies

galaxy is gravitationally attracted by other galaxies and this force is equivalent to

Answer 2

that form a point mass at the centre, equal to the mass of the sphere

Question 3

we talk about a pressureless universe such that its pressure P=0 because

Answer 3

although it contains mass and is dominated by it, it is the expansion of space-time that causes changes in mass density and not any pressure associated with mass

Question 4

we often refer to the pressureless universe as

Answer 4

‘dust-filled’

Question 5

if we let the universe expand, dust is carried

Answer 5

radially outward from the origin

Question 6

a key implication of CP is that the universe’s expansion proceeds

Answer 6

the same way for all shells

Question 7

the radius of a particular shell at any time can be written

Answer 7

r(t)=a(t)s

r(t) is the coordinate distance and a(t) is a dimensionless scale factor identical for all shells

Question 8

kinetic energy of the galaxy

Answer 8

k=1/2m rdot^2 = 1/2 m adot^2 s^2

Question 9

potential energy of the galaxy

Answer 9

U=-GMm/r = -4/3 pi a^2 s^2 Gpm

Question 10

friedmann’s equation

Answer 10

total energy = constant

describes how gravity acts against the expansion of the Universe

Question 11

constant k in Friedmann’s equation defines

Answer 11

the geometry or curvature of the Universe

Question 12

k>0

Answer 12

the universe is closed with positive curvature

Question 13

k<0

Answer 13

the universe is open with negative curvature

Question 14

k=0

Answer 14

the universe is flat with zero curvature

Question 15

the analytic solution of Friedmann’s equation is straightforward only for

Answer 15

the case of a flat universe (k=0)

Question 16

solution to Friedmann for k=0

If we assume a matter dominated universe and mass conserved, then (da/dt)^2=

Answer 16

A/a where A is constant

Question 17

scale factor a is related to redshift

Answer 17

a=1/1+Z

Question 18

redshift in a flat universe
p(z)=

Answer 18

p0(1+z)^3

valid only for a universe filled with preasureless dust

Question 19

deriving the critical density of the universe

Answer 19

when k=0, friedmann reduces to

(adot / a)^2= 8piGp/3

subbing in H=a dot/a

p=3H^2/8piG

Question 20

critical density is

Answer 20

the density required to just close the universe

Question 21

if p>pc

Answer 21

the universe recollapses

Question 22

if p<pc

Answer 22

the universe expands indefinitely

Question 23

present day values of pc

Answer 23

approx6 hydrogen atoms per cubic metre

Question 24

density paramter

Answer 24

omega(t)=p(t)/pc(t)

Question 25

omega >1

Answer 25

universe closed

Question 26

omega<1

Answer 26

universe open

Question 27

omega=1

Answer 27

universe flat

Question 28

although omega can change with time, it can be shown that

Answer 28

its state of being closed, open or flat cannot change (illustrated in graph)

Question 29

Although General Relativity is really the correct theoretical framework for discussing the expanding universe, we will use

Answer 29

the assumption of homogeneity to derive all the important results using Newtonian gravity. This is because, in a homogeneous universe, any fluid element is representative, so we can choose it so small that GR effects can be ignored

Question 30

the ideal fluid model can be adopted if

Answer 30

we imagine that the mean free path of particles is short.

Question 31

We use simple fluid models incorporating special relativity. In this context, we have a first fluid equation governing

Answer 31

the flow and continuity of mass

Question 32

the mass continuity equation tells us

Answer 32

that in a steady state, the rate at which mass enters a system is equal to the rate at which mass leaves the system

Question 33

equation of state

Answer 33

P(p) = kp^5/3 in the adiabatic case, kp in the isothermal case

Question 34

in the more general case, P(p)=

Answer 34

kp^gamma

Question 35

Birkhoff's theorem

Answer 35

spherical shell produces no gravitational force within it

Question 36

in a matter-dominated universe, p is prop to

Answer 36

a^-3

Question 37

in a radiation-dominated universe, p is prop to

Answer 37

a^-4

Question 38

solution to momentum equation for k<0

Answer 38

a dot approaches c root(-k) open, hyperbolic universe

Question 39

solution to momentum equation for k>0

Answer 39

a dot=0 at some critical radius and a collapse phase starts after a=ac closed universe

Question 40

momentum equation for k=0

Answer 40

a dot approaches 0 at t approaching infinity flat universe

Question 41

form of Friedmann's equation which allows physical interpretation

Answer 41

kc^2=H0^2(omega-1)

Question 42

flat universe k=0

Answer 42

omega=1 p0=pc0 so universe jsut unbounded since the density is such that the PE is approx KE

Question 43

open universe, k<0

Answer 43

omega<1 and p0

Question 44

closed universe, k>0

Answer 44

omega>1 and p0>pc0 so the universe is bounded it expands and then recollapses density high enough for gravity to halt expansion

Question 45

closed universe - at turnaround point

Answer 45

a dot=0 amax=omega0/omega0-1

Question 46

Einstein-De Sitter universe

Answer 46

special case of a homogeneous, isotropic universe of zero curvature (k=0) described by Newtonian gravity

Question 47

result of einstein-de sitter universe

Answer 47

a(t) prop to t^2/3 for matter dominated, t^1/2 for radiation dominated

Question 48

in a near-empty universe, omega appraoches 0 so Friedmann's equation becomes

Answer 48

a dot=H0

Question 49

simple relation for age of the universe

Answer 49

found by integrating a dot =H0 this gives 1=H0t0

Question 50

effect of the pressure term in momentum equation

Answer 50

slows down the expansion

Question 51

the equivalent mass of the particles kinetic energy creates

Answer 51

a gravitational attraction slowing down expansion

Question 52

decleration parameter

Answer 52

q(t) dimensionless

Question 53

q(t) for a pressurless universe

Answer 53

q(t)=1/2 omega(t) thus for a flat, pressureless dust universe, q0=0.5