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II. The Universe Around Us
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Cosmology : II. The Universe Around Us
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Some Units Used in Astronomy
•  1 parsec ≡ distance at
which parallax angle is 1”;
1 pc = 3.086×1016 m
(≈3.26 light years; 1 kpc =
3.086×1019 m, 1 Mpc =
3.086×1022 m)
•  1 M = 1.989×1030 kg, 1L
= 3.839×1026 W
•  1 Å = 10-10 m (0.1 nm)
•  1 eV = 1.602×10-19 J
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The Visible Universe
• 
• 
• 
• 
• 
Stars (Sun: 1 M, 1 L)
Galaxies (MW: 1011 – 1012 M)
Local Group (MW + M31 + ...): scale ~1 Mpc
Clusters/superclusters of galaxies, voids: scale ~100 Mpc
~Smooth on larger scales...
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Other Wavelengths of “Light”
•  Microwaves: the Cosmic
Microwave Background (CMB),
black body spectrum
corresponding to 2.7°K
•  Radio, Infrared: can penetrate (IR
can be emitted by) dust, see
obscured star formation / objects
at high redshift (+...)
•  X-ray: hot gas, important for
measuring the mass of galaxy
clusters (+...)
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The Cosmological Principle
Considering the largest scales in the Universe, we make the following
fundamental assumptions:
1) Homogeneity: On the largest scales, the Universe has the same physical
properties
Every region has the same physical
properties (mass density, expansion
rate, visible vs. dark matter, etc.)
2) Isotropy: On the largest scales, the Universe looks the same in any
direction
We should see the same largescale structure in any direction
3) Universality: The laws of physics are the same everywhere in the Universe
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Homogeneity and Isotropy
•  Cosmological principle: on large scales
the Universe is homogeneous and
isotropic
–  homogeneous (Universe looks the same at
each point) ≠ isotropic (Universe looks the
same in all directions)
–  but isotropic at every point = homogeneous
–  large scales: >≈ 100 Mpc
CMB
•  Perfect cosmological principle: the
Universe is homogeneous and isotropic
in space and time  Steady State
Universe (not true!)
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Olbers’ Paradox
•  Heinrich Olbers (1826): Why is the night sky dark?
–  n = mean number density of stars, L = mean stellar
luminosity
L
–  flux at Earth:
4 πr 2
L
2
–  power (unit area)-1 (steradian)-1: dJ(r) =
×
n
×
r
dr
2
4 πr
nL ∞
–  total intensity of starlight:
J=
dr = ∞
∫
r=
0
4π
f (r) =
•  Yet the night sky is dark...?
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The Expanding Universe
Blueshift
•  Redshift z : z ≡ λobs − λem
λem
z≈
Redshift
(SR:
(1+ z) =
1+ v /c
1− v /c
)
•  1920’s: Hubble (and
Humason) discovered a
proportionality between a
galaxy’s redshift and its
distance (Hubble Law)
z=
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v
c
H0
r
c
Cosmology : II. The Universe Around Us
v = H0r
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The Hubble Constant
•  Hubble found a value for the Hubble Constant
H0 ≈ 500 km s-1 Mpc-1 (bad calibration!)
•  For decades H0 disputed (50 – 100); current
consensus H0 ≈ 70 km s-1 Mpc-1
•  Outside the Local Group, virtually every galaxy
is moving away from us -- why doesn’t this
violate the Cosmological Principle? •  If no acceleration/deceleration, galaxies were
together at time:
Original Hubble Diagram
More Recent Version
r
r
−1
(Hubble Time)
=
= H0
v H0r
c
H 0 ≈ 70 ⇒ H 0−1 ≈ 14Gyr,
≈ 4300Mpc (Hubble Distance,
H0
or horizon distance)
t0 =
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How can the Hubble Law be Isotropic?
•  Three galaxies (1,2,3) in a triangular
configuration, with sides r12, r23, r31
•  Homogeneous, uniform expansion
means shape preserved 
expansion law of form r12(t) = a(t)r12(t0)
•  a(t): scale factor; a = 1 @ t= t0
•  At time t, an observer in galaxy 1 will
see the other galaxies receding with
velocity:
dr
a˙
1
r12 ( t )
dt
a
dr
a˙
v 31 (t) = 31 = a˙ r31 ( t 0 ) = r31 ( t )
dt
a
v12 (t) =
12
2
= a˙ r12 ( t 0 ) =
3
 the velocity distance relation takes
the linear form v = Hr, with H=å/a
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What’s the Universe Made Of?
Energy of a particle
2
E total
= m 2c 4 + p 2c 2
E total
2 ⎞1/ 2
⎛
p
1 p2
2
2
= mc ⎜1+ 2 2 ⎟ ≈ mc +
2 m
⎝ mc ⎠
Baryons
E γ = hf ( hν )
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in the nonrelativistic limit
Energy of a photon
Particle
Symbol
Rest Energy (MeV) Charge
proton
p
938.3
+1
neutron
n
939.6
0
electron
e-
0.511
-1
neutrino
νe,νμ,ντ
?,?,?
0
photon
γ
0
0
dark matter?
?
?
0
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Sneak Peek: What is the Universe Made of?
0.6% in Stars
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Blackbody Radiation
•  In thermodynamic equilibrium, photons have a
energy densityεin the frequency interval df around
frequency f given by the blackbody (BB) function:
8πh
f 3 df
ε( f ) df = 3
c exp(hf /kB T) −1
2 4
π
kB
4
εγ = αT , α =
15h 3c 3

≈7.565×10-16 J m-3 K-4
•  The number density of photons is:
3
2.404
k
B
7
-3 -3
nγ = βT 3 , β =
2
3 3 ≈2.03×10 m K
π hc
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Blackbody Radiation II
•  Peak frequency of BB distribution: fpeak ≈ 2.8kBT/h
•  Peak energy of BB distribution: Epeak = hfpeak
≈2.8kBT
•  Mean photon energy: Emean = hfmean ≈ 2.7kBT
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The Cosmic Microwave Background
•  Discovered in 1965 by
Penzias and Wilson, BB
spectrum with
T=2.725±0.001 K
•  Big Bang vs. Steady
State
•  Once dipole (motion
towards Hydra, 630 km
s-1) and Galactic
emission subtracted,
extremely isotropic /
homogeneous (10-5)
•  Note: ~1% of the static on TV is due to the CMB!
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How is the CMB a Relic of the Big Bang?
•  “Hot” Big Bang: Early
Universe very dense and
very hot (T>>104 K)
baryonic matter completely
ionised, free electrons
made Universe opaque to
photons
•  As Universe expanded, T;
when T~3000°K, neutral
atoms formed, no longer
many free electrons 
photons free to go
•  So why is the CMB a 2.7°K
BB, not a 3000°K BB?
εγ
3
dE
dV
dQ = dE + PdV, dQ = 0,
= −P ( t )
dt
dt
4
α
T
E = εγ V = αT 4V, P = Pγ =
3
⎛ 3 dT
1 4 dV
4 dV ⎞
α⎜ 4T
V +T
⎟ = − αT
⎝
dt
dt ⎠
3
dt
1 dT
1 dV
d
d
3
=−
; V ∝ a( t ) ⇒ (lnT ) = − (ln a)
T dt
3V dt
dt
dt
V ∝ a( t ) , εγ = αT 4 , Pγ =
3
T(t) ∝ a(t)−1
•  The CMB was a 3000°K BB when the Universe was ~1100× smaller!
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Looking at the Last Scattering Surface
•  Today, at z = 0 the universe is fairly
transparent
•  At higher redshift, z (looking backward
in time) the universe was denser (ρ =
ρ0×(1+z)3) and hotter (T=T0×(1+z))
•  At z ≈ 1100, the universe was so
dense that T >≈ 3000°K •  At z > 1100 there is a transition: the
universe becomes completely ionised
and opaque to visible light − the last
scattering surface
•  The universe was ~350,000 years old
at z≈1100 ASTR378
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Should the CMB be Smooth ?
•  There are people,
planets, stars, galaxies,
galaxy clusters and
galaxy superclusters
today, so we expect
some non-uniformities
(wiggles, etc.) in the
CMB
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The Cosmic Background Explorer (COBE)
Objectives:
•  Accurately measure the CMB
temperature
•  Find expected CMB
fluctuations
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Basic results from COBE
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More Results from COBE
•  The Earth is moving with
respect to the CMB 
detectable Doppler shift!
–  Earth’s motion around the Sun
–  Sun’s motion around the Galaxy
–  the Galaxy’s motion w/rt other
galaxies (large scale flows)
•  Microwave emission from the
Galaxy
•  Fluctuations in the CMB
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The CMB Today
COBE (1990’s)
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WMAP (2000’s)
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The Latest: Planck (2009 – )
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Olbers’ Paradox Revisited
•  What are our assumptions?
•  Unobstructed line of sight to every star in the Universe?
–  not true – some could be blocked by foreground stars, intervening
dust – but still night sky should look like the surface of a star!
• 
• 
• 
• 
Number density n and mean luminosity L constant w/rt r ?
Universe is infinitely large? (And filled with stars?)
Universe is infinitely old?
Flux from distant sources follows inverse square law?
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