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Lecture 14
Rest of the course continues with Cosmology Will star t by reminding ourselves about the Friedmann-Rober tson-Walker metric and its derivation Going to assume that from Par t II Astrophysics we know what the spatial metric is for a space of constant curvature, and then show how to join this to a time par t to get a metric for the Universe This par t just repeats what we did in Par t II Astro -- however then we will move on to new concepts, principally the concept of how we measure distance in the universe This turns out to be quite surprising. Will study it mathematically today, but next time will link with direct observations which aim to probe the geometry (closed, open or flat?) of the universe, and its dynamics












Now know a version of the Metric for the universe, and want to work out consequences for things we can observe -- distances, angles etc. -- the redshift. Our information about the universe comes primarily from photons moving radially. Thus would be nice if the radial par t of the metric was simple. To achieve this define a new comoving radial coordinate

via k = +1, k = 0, k = -1.
(3 )

The overall FRW metric then becomes

sin-1 d , i.e. = d = 2 1 - k sinh-1

R2 (t) 2 d2 + [S ()] (d2 + sin2 d2 ) , ds2 = dt2 - 2 c
(4 ) where

S () =

s in

k = +1, k = 0, k = -1. r , then ),
(5 )

sinh

This form of the metric, which we have arrived at after two radial coordinate transformations (first remember it in.
3

is essentially our final form, and is probably the form to


P H OTO N P RO PAG AT I O N -- T H E R E D S H I F T
Suppose a photon is travelling to us from an external galaxy. Its motion is radial, so we can ignore the and coordinates.

The condition for photon propagation is that the interval should vanish. Thus
ѕ ѕ

в
and so this implies

ь



йѕ ґьµ ѕ йґьµ

ѕ

ь
with time.



for an incoming photon. This is how its

coordinate changes

Suppose the photon was emitted at cosmic time ьЅ and is received by us at time ьј . (Conventionally ьј is always taken as the time here now.) Clearly the is given by
ьј

coordinate of the emitter

Ѕ

ь йґьµ

ьЅ

The crucial point is that if the galaxy can be treated as a fundamental observer (i.e. moving with the cosmic fluid without a peculiar velocity of its own), then its fixed. This enables us to work out the redshift. coordinate is

1


Analysing this situation (see Section 4.1 of Par t II Astrophysics Lecture 20 Handout for details) leads to:

ф ьЅ йґьЅ µ
Now frequency
ј Ѕ
Ѕ фь

ф ьј йґьј µ
, and hence

йґьј µ йґьЅ µ. With redshift following our usual

definition of ґ с в



яµ

я , we derive

ч
i.e.
Ѕ·ч

ґј


Ѕ

Ѕµ

йґьј µ йґьЅ µ



Ѕ

йґьј µ йґьЅ µ

scale factor of universe on reception scale factor of universe when emitted

2