Our Goal: take R(t) and physics (gravity) to calculate how R(t) varies with time.

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Our Goal take R(t) and physics (gravity) to
calculate how R(t) varies with time. Then plug
back into (cdt)2 R(t)2dr2/(1-kr2) Get t versus
R(t) and derive age of universe (t0) versus W0
and H0 Simple estimate of t0 1/H0 H0 50-70
km/sec-Mpc gt 1/H0 has units of time 19-14
billion years Mpc megaparsec 3 million
lt-years 3 x 1024 cm
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• Want to show where the following come from
• H0 expansion rate for universe today
• rc critical density 3H0/8pG
• W0 r0/rc ltgt k relation
• q0 de-acceleration parameter
• L cosmological constant ltgt pressure and why
  positive L (and WL causes an accelerating
  universe)

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And, R(to)r for the observed object translates
into a distance to the object today, and our goal
is to figure out how to calculate R and r
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For the related figures, see page 217 (shows
geometry) , 283 (shows R changing in different
ways), and 299 (shows R for k -1, 0, 1)
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For the math we will do, assume that there is no
dark energy (cosmological constant) until further
notice
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Predicting the Future from the past
A primary goal of the cosmologist is to tell us
what will happen to R as function of time, based
on fitting models to the data
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Predicting the Future from the past

• Measure R(t) by looking back in time
• Measure how the geometry of the universe affects
  our measure of distance or apparent size.

R(t0)/R(t) 1 z t the age of the universe
when light left the object t0 age of the
universe today by definition cf. pages 374-376
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Predicting the Future from the past
Also, R(t0)/R(t) lob/lem lambda(observed)/lamb
da(emitted). the universe is expanding R(t0) is
always greater than R(t) (for us today)
lambda(observed) must always be gt lambda
(emitted) longer lambda (now this means
wavelength of light) means redder, we call this a
redshift!
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How to get R(t)
We need to relate R(t) to some force
The Universe affects itself. It has self
gravity
Self-gravity will slow down expansion
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Equate potential energy (GMm/R) with kinetic
energy (1/2) mv2
M is the self-gravitating mass of the universe R
is the scale factor of the universe.
M density(r) x volume(4/3) x p R3)
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How to get R(t), part 1, cont.
density r volume 4pR3
gt M r4pR3
.
v R
Aside A subscript 0 means today (R(t0) R0 )
to keep from writing R(t) or R(t0).
.
gt (1/2)mv2 (1/2)mR2 and GMm/R Gr4pR3m/3R
Gr4pmR2/3
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How to get R(t), part 1, cont.
KE gt PE, we get escape KE lt PE, the universe
will collapse on itself.
(1/2)mv2 GMm/R, KE PE

The little ms cancel out.
Put an energy term on the KE side to allow us to
describe to escape or not to escape
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R2 G8prR2/3 , now adding in the extra term
.
Yes! The k we used for our geometry and c is the
speed of light.

.
R02 kc2 G8pr0R02 /3 today
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The KE, kc2, and PE connection
.
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2
R0 G8pr0R0 /3 - kc2
So, k -1 means the KE is more than the PE, and
we get escape, and vice versa
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Critical density when pull of gravity (PE) just
balances the BB push (KE), i.e. the density when
k 0 !

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How to get R(t), part 1, cont.
So, rc as it is called is when k 0 and we have
rc 3R0/(8pGR0), but R0/R0 H0 !
(another old friend) the expansion rate of the
universe today
.
.
2 2
2
2
Or, rc 3H0/8pG
2
2
Or, 1 kc2/(H0 R0 ) r0/rc ?
W0
2
2
Or kc2/(H0R0) W0 -1
We see the relationship between k and W0 and the
fate of the universe!
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• Aside on H0
• How to use to get distances (good to 1z of
  about 1.2)
• D v/H0 where v velocity of recession
• use km/sec along with H0 50 km/sec-Mpc for
  example
• D v/H0 is the Hubble Relation
• Observation of this told us Universe is
  expanding
• For z ltlt 1, z v/c (approximately) z ltgt v