Lecture 2: Acceleration & Cos Constant

Question 1

how to obtain acceleration equation?

Answer 1

differentiate the Friedmann equation wrt time

(then sub in fluid equation and simplify)

Question 2

the acceleration equation (and hence acceleration) is independent of

Answer 2

curvature

Question 3

from the acceleration equation why is the expansion of the universe decelerating

Answer 3

if we have normal matter with p>0 and pressure >or=0
then acceleration equation is overall negative

Question 4

einstein related mass density and energy density via

Answer 4

ε=pc^2
making a direct connection between mass and energy

Question 5

using natural units with c=1 would cause

Answer 5

dimensions of k to be [T^-2]

length and time interchangeable

Question 6

The notion of curvature of space-time demands that we introduce

Answer 6

a new metric, capturing the structure of the universe and, therefore, to be used
to define time, distance, curvature and so on.

Question 7

In 4-dimensional space-time we generalise the notation of co-ordinates as

Answer 7

dx^u or dx^v with u and v being indices defining the dimension of space-time to be considered

Question 8

dx^0

Answer 8

dt
time

Question 9

dx^1

Answer 9

dx
x-dimension

Question 10

dx^2

Answer 10

dy
y-dimension

Question 11

dx^3

Answer 11

dz
z-dimension

Question 12

space time interval between two points

Answer 12

use the einstein summation convention
sum of guv dx^u dx^v

Question 13

einstein summation convention permits us to

Answer 13

drop the summation symbol and sum over repeated indices u,v=0,1,2,3

Question 14

guv

Answer 14

metric tensor

Question 15

robertson-walker metric

Answer 15

the most general of homogeneous and
isotropic metrics that also allows for curved space-time

Question 16

radial geodesic

Answer 16

The shortest distance between two points in curved space is then given
by the line dθ = 0, dφ = 0:

Question 17

Observations show that the expansion of the universe is

Answer 17

accelerating

Question 18

what does the universe accelerating tell us mathematically

Answer 18

expect an extra term in the Friedmann equation - cosmological constant

Question 19

an accelerating universe requires

Answer 19

Λ>0

Question 20

a positive cosmological constant gives

Answer 20

a positive contribution to acceleration, acting as
a repulsive force (compared to the force of gravity).

Question 21

if Λ is sufficiently large, it can

Answer 21

overcome gravitational attraction, leading to an accelerating universe (with cosmological acceleration) having already been measured.

Question 22

can define an equivalent dimensionless density parameter for Λ which varies with

Answer 22

time even though Λ is fixed due to the existence of H in the denominator

Question 23

how are density, geometry and cosmological constant interconnected

Answer 23

Ωm+ΩΛ +Ωk=1