harrison b. prosper florida state university...
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Harrison B. Prosper Florida State University
YSP
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Island Universes The Expanding Universe The Universal Scale Factor Models of the Universe Summary
Topics
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Henrietta Leavitt 1912
Luminosity-Period Relation of Cepheid Variables
Cepheid Variables
She deserved, but did not get, a Nobel Prize
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Island Universes
1924 – Edwin Hubble Using the work of
Henrietta Leavitt, Hubble measured the distances to several galaxies and found that they are immense star systems very far from Earth
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The Expanding Universe
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The Expanding Universe
1 Mpc (Mega-parsec) = 3.26 x 106 light years (ly)
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The Universal Scale Factor
a < 1
D(t1)
t1 = past
D(t0)
a = 1
t0 = today
a > 1 D(t2)
t2 = future
a(t) is the scale factor of the Universe
t is cosmic time
λe = a(t) λo
D(t) = a(t) D(t0)
z = (λo - λe) / λe a = 1
1+ z
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The Hubble Parameter
a > 1 d(t2)
t2 = future
D(t) = a(t)D(t0 )D(t) = a(t)D(t0 )D(t)
D(t)=a(t)a(t)
≡ H (t)
H(t) is called the Hubble parameter The Hubble constant H0 is simply the Hubble parameter H(t0) at the present epoch
D(t) ≡ dD
dt= v(t)
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Why Can We Assign a Cosmic Time?
We have learned that your now and my now do not coincide as we move relative to each other.
However, since our relative speeds are small relative to c, it is a very good approximation to take our nows to be the same.
The same is true for galaxies. Their motions relative to space are << c. Consequently, we can assign each galaxy approximately the same cosmic time.
9
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Models of the Universe
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Distribution of Galaxies
APM Galaxy Survey, Steve Maddox, Will Sutherland, George Efstathiou & Jon Loveday
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Models of the Universe – II
ds2 = c2dt2 − a2 (t)dl2
D(t) = a(t)dl= a(t)D(t0 ) where t0 is the lifetime of
the Universe
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How Far Is Far ?
D1, t1
D0, t0
L = c (t0 – t1)
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Models of the Universe – III
E = 1
2 mv2 − GmMD
According to Newton’s laws, the total energy E of a galaxy of mass m at a
distance D is given by
v
D where M is the total mass enclosed within the sphere of radius D
M = ( 43πD3)ρ
D(t) = a(t)d0writing
aa
⎛⎝⎜
⎞⎠⎟
2
= 8π3
Gρ + 2Emd0
2
⎛
⎝⎜⎞
⎠⎟1a2gives
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Models of the Universe – IV Alexander Friedmann 1888 - 1925
aa
⎛⎝⎜
⎞⎠⎟
2
= 8π3
Gρ − Kc2
a2
We can write this differential equation as
ρ0 ≡3H0
2
8πG, Ω0 ≡ ρ(1) / ρ0
Ω(a) ≡ [ρ(a) / ρ0 + (1− Ω0 ) / a2 ]
aa= H0 Ω(a)
where
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Models of the Universe – IV
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Models of the Universe – V
Since the Friedmann equation is a 1st order differential equation
we can re-write it as follows
where C is a constant determined by the initial conditions
aa= H0 Ω(a)
where
H0t =
daa Ω(a)∫ + C
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Models of the Universe – VI Georges Lemâitre 1927
Assume Ω0 = 1 a(0) = 0 Ω(a) = Ω0 / a3
and remember that a(t0) = 1
H0t =
daa Ω(a)∫ + C
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Summary
Expansion of Space In 1929, Hubble discovered the expansion of the Universe.
The Friedmann Equation This equation describes how the scale factor a(t) varies with cosmic time.
Different cosmological models give different predictions for a(t)