Lecture 14
The early universe and the cosmic microwave
background
Please read chapters 10–12 in the book by Liddle
Key concepts in this lecture
The cosmic microwave background
Matter-radiation equality
Decoupling
The surface of last scattering
Big Bang nucleosynthesis
Recap
In the last lecture we derived the acceleration equation from the Friedmann and
fluid equations
ä
=
a
4⇡G
(⇢ + 3p) .
3
We could immediately see that the Universe will decelerate if it only contains matter,
⇢m > 0 ) ä < 0, which is of course not surprising. If we want to have anything
else than deceleration, the Friedmann equation needs to be modified. Historically,
Einstein added the cosmological constant, ⇤, to the equation in order to obtain a
static solution. Although, we today know that the Universe is not static, there are
still two reasons why the constant may be kept: (i) there is nothing that forbids ⇤
so we can keep it and let the observations decide its value and (ii) we know from
particle physics that empty space is not empty, since the lowest energy state in
Particle & nuclear physics, astrophysics and cosmology, FK5024 Lecture 14
quantum mechanics is not zero. The fact that it has been shown that the Universe
is accelerating is a very, very good reason to keep it.
With a cosmological constant the equations take the form
✓ ◆2
ȧ
8⇡G
k
⇤
=
⇢
+
a
3
a2
3
ä
4⇡G
⇤
=
(⇢ + 3p) + .
a
3
3
Although we have used observations to motivate the formulation of these equations,
we have not been discussing how to measure the parameters. The way they are
written above is not very useful for such a discussion since a(t) is not easy to
measure.
We can rewrite the Friedmann equation by first expressing the densities in terms of
the critical density, ⇢c , i.e. ⌦ = ⇢/⇢c , and introduce similar parameters for k and
⇤. The equation then becomes
⌦k (a) + ⌦(a) + ⌦ (a) = 1 .
This is a useful form to see how the di↵erent species a↵ect the geometry of the
Universe.
The next step is to rewrite the equation and get rid of the variable a since this is
not something that can be measured directly. We do however know how to measure
the relative scale factor, 1 + z = a0 /a, and we also know how the species evolve
with the scale factor, ⇢m / 1/a3 and ⇢ / 1/a4 . Combining these gives
⇥
⇤
H(z)2 = H02 ⌦M (1 + z)3 + ⌦ (1 + z)4 + ⌦K (1 + z)2 + ⌦⇤ .
If we can come up with a way to either measure the expansion rate H as a function
of z (we know how to measure the local rate, H0 ), or to rewrite this in terms of only
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Particle & nuclear physics, astrophysics and cosmology, FK5024 Lecture 14
measurable variables and cosmological parameters, we can use dynamical studies to
determine the latter. This will be the topic of next week’s seminar.
There are however, astronomical measurements we can do to measure ⌦M . We can
for example measure the stellar mass from the brightness which gives ⌦stars ⇠ 0.005–
0.01. The gravitational potential of galaxies and galaxy clusters can be studied
by observing the motion of objects in these and use the Keplerian laws. The
gravitational potential of galaxy clusters can also be measured by studying the
temperature of the gas that has fallen into them. This gives ⌦M ⇠ 0.3 which is
much larger than the stellar mass above. In other words, most of the matter in the
Universe is dark.
The cosmic microwave background
Is there any non-dynamical way to measure ⌦ ? We could try to use a similar
technique as we did for ⌦stars , that is, just go out and try to measure how much
radiation there is in a given volume of space. The expansion of the Universe is
a strong argument for the Big Bang model by playing the movie backwards, and
extrapolating to a ! 0. If the early Universe had a temperature, the adiabatic
expansion would cool the Universe, but it should still have a temperature today.
What this means in practice is that the heat radiation of the Universe should be
measurable. Further, if the Universe is homogeneous and isotropic, it should have
the same temperature independently of what direction we observe.
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The horizon of the observable
universe is given by the age of
the universe, i.e. how far a
light ray can travel since the
Big Bang
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Where is it?
The observable universe
The CMB was emitted in
all directions. No matter
which direction we are
observing we will detect
the light that was emitted
13.7 billion years ago.
Observer
This is when the universe
became transparent to
light. The universe was
400 000 years old.
The CMB has been
travelling through the
universe ever since it
became transparent
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FK5024 Lecture 10
The Cosmic Microwave Background
Time
Hydrogen
atom
Proton
3000 K
A LONG TIME AGO
The temperature of the universe has
dropped enough to allow electron and
protons to form neutral hydrogen atoms.
RELATIVE SIZE OF
THE UNIVERSE
A LONG, LONG TIME AGO
Electron
Temperature
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FK5024 Lecture 10
CMB is black body radiation!
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FK5024 Lecture 10
If you really like the CMB…
Nobel Prize 2006!
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FK5024 Lecture 10
COBE (1992)
John Mather
George Smoot
The CMB is isotropic to first order - the
best blackbody known in nature
Subtract 2.728 K and amplify the
contrast by a factor1000. The Doppler
shift caused by the motion of the Solar
System relative the CMB
Vintergatan
Subtract the dipole and amplify the
contrast by 5 orders, we can see
fluctuations in the CMB (the centre band
is the Milky Way)
NASA (http://space.gsfc.nasa.gov/astro/cobe)
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1992
FK5024 Lecture 10
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FK5024 Lecture 10
WMAP (2010)
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FK5024 Lecture 10
Planck (2015)
2015
ESA
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The cosmic microwave background radiation (CMB) was discovered by Penzias and Wilson in 1965 hand has now been measured to a phenomenal precision.
The temperature, T0 , today of the Universe is
T0 = 2.725 ± 0.001 K .
and follows a black-body curve, suggesting that the Universe has been in thermal
equilibrium from very early on. The black-body nature of the radiation also means
that we can estimate its energy density using the Stefan-Boltzmann law as
⇢ (a0 ) =
with
0
= 4.7 · 10
3
0
· T04 = 4.17 ⇥ 10
14
Jm
3
= 0.25 eV/cm3 ,
eV/cm3 /K4 . This is of course not all the radiation in the
Universe, since we also have the radiation produced in the galaxies. However, most
of the night sky is dark, and the CMB has a constant density per unit volume so it
will dominate the radiation density completely.
From Laura’s tutorial we know that the mean energy of a photon in a black-body
distribution is Emean ⇠ 3kB T , which for the CMB today gives Emean (a0 ) = 7.0 ⇥
10
4
eV. Using this with the value of ⇢ (a0 ) = 0.25 ev/cm3 we can estimate the
photon number density today as
n (a0 ) =
⇢ (a0 )
0.25
⇡
E (a)
7 ⇥ 10
4
] ⇡ 360 cm
3
.
Back in the day of analog TV, about 1 % of the noise came from the CMB.
It is also very straight-forward to see how temperature decreases as the universe
expands,
⇢ / T4
⇢ /
a0 4
a
9
>
=
;
= (1 + z)4 >
4
) T / (1 + z) /
1
.
a
Particle & nuclear physics, astrophysics and cosmology, FK5024 Lecture 14
The black body nature of the hot initial radiation is preserved, while its e↵ective
temperature falls proportionally to a 1 .
Similarly we have that
⇢c (a0 ) =
3H02
⇠ 104 eV/cm3
8⇡G
)
⌦ ⇡ 10
5
.
In other words, this is significantly smaller than ⌦stars and ⌦M , which confirms that
radiation today is negligible for the dynamic evolution of the Universe. We can now
ask our self at what redshift, zeq , we had ⌦m (z) ⇠ ⌦r (z) (with ⌦r (0) = ⌦ ). Since
we know that
⌦m (z) = ⌦M (1 + z)3
⌦r (z) = ⌦ (1 + z)4 ,
we have
1 + zeq ⇠
⌦M
⇠ 104 .
⌦
If we had done the same calculate but instead with the baryon density, ⌦b , about
an order of magnitude lower ⌦stars ⇠ 0.01+gas, zeq ⇠ 103 . We can then easily
compare the number densities between the photons and baryons, n /nH , assuming
that all the baryonic matter is in non-relativistic protons
n
⇢ /E
mH
⇡
=
.
nH
⇢M /mH
E
since ⇢ ⇠ ⇢b . Here E is the average energy at zeq . We know that
Emean ⇠ 3kB T = 3kB T0 (1 + z
eq) ⇠ 1 eV.
The proton mass is ⇠ 1 GeV which gives
n
mH
1 GeV
⇡
⇠
⇠ 109 .
nH
E
1 eV
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Particle & nuclear physics, astrophysics and cosmology, FK5024 Lecture 14
There are about 109 photons for every baryon! The photon is in fact the most
common particle in the Universe.
Baryon-photon decoupling
The sky is literally glowing of CMB photons. These can propagate through the
Universe without anything stopping them, but this has not always been the case.
In the early Universe the temperature, i.e. the photon energy, was high enough to
ionize any neutral hydrogen, if it existed. This would create free electrons, and the
cross section for photon-electron scattering (Thomson scattering) is high for such
a scenario. Matter (mainly protons and electrons) was kept in thermal equilibrium
with radiation through interactions
+e
H+
!
+e
! p+e
The ionization energy for neutral hydrogen is 13.6 eV so as the Universe is expanding, the temperature will drop, and so will the average photon energy. At some
point, the equilibrium will be broken and the last reaction above can only go in one
direction, i.e.
!)
.
As a result, we will have neutral hydrogen and relic radiation. Photons travel
unscattered through the universe (there are no longer any free charged particles).
We have decoupling between radiation and matter.
Since there are many more photons than baryons, the decoupling does not take
place at the e↵ective temperature corresponding to 13.6 eV (the ionization energy
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Particle & nuclear physics, astrophysics and cosmology, FK5024 Lecture 14
of hydrogen) but rather at 0.25 eV, because of the tail of the black-body distribution.
This energy corresponds to the temperature T = 0.25/kB ⇡ 3000 K. Since T /
1 + z, this means that decoupling happened at zdec ⇠ 1000.
We have seen that after z ⇠ 104 the Universe has been mostly dominated by matter,
and we derived in Lecture 12 a relation for the time-dependence of the scale factor
2
under such assumptions, a / t 3(1+w) = t2/3 for w = 1/3. Since we just saw that
T / 1/a we can combine these to get a relation between the temperature and time
✓
◆2/3
T
4 ⇥ 1017 s
=
,
2.725 K
t
where we have assumed k = 0 and ⇤ = 0. The proportionality factors were set
by the temperature of the Universe today and its current age. For the latter we
use 12 billion years to compensate for ignoring ⇤. Since decoupling happened after
matter-radiation equality at a temperature T ⇡ 3000 K, this formula applies and
we get that
tdec ⇡
✓
2.725
3000
◆2/3
1
s = 350 000 years after the BB.
4 ⇥ 1017
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Particle & nuclear physics, astrophysics and cosmology, FK5024 Lecture 14
The early universe
At the very early epochs we can use the Friedmann equation directly together with
the expression of ⇢ from the Stefan-Boltzmann’s law. In this era we can safely
neglect non-relativistic matter, dark energy and curvature and we therefore have
✓ ◆2
ȧ
8⇡G
8⇡G 4
2
H =
=
⇢ =
T .
a
3
3
p
For radiation domination we have a(t) / t which gives
ȧ 1 1
1
1
H =
= p ·p =
a 2 t
t 2t
✓ ◆2
1
8⇡GN 4
=
T ,
2t
3
which after inserting all constants yields
1
T (MeV) ⇡ p
.
t(sec)
In other words, ⇠ 1 s after the Big Bang, the temperature of the Universe was
kB T ⇡ 1 MeV. We can convert from MeV to K by using the value of kB ,
kB = 1.38 · 10
23
J/K = 8.62 · 10
5
eV/K
)
T = 1 MeV ⇡ 1010 K .
For comparison, the energy at LHC, ECMS , is
ECMS ⇠ 10 TeV ) 10
13
s after the Big Bang!
Measuring the temperature in MeV is very useful to get an understanding of the
processes that are taking place in the Universe for the di↵erent epochs. Seconds
to minutes of after the Big Bang the temperatures were at the MeV scale which
is typically the energies for nuclear reactions, while ⇠ 350 000 years later it had
dropped to the eV scale which is the energy levels of atomic physics (as we have
already seen during decoupling, H +
! p + e ).
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Particle & nuclear physics, astrophysics and cosmology, FK5024 Lecture 14
Big Bang nucleosynthesis
At very early times, < 1 s after the Big Bang, all nuclear processes were in thermal
equilibrium. That is, all reactions go equally fast. At this point, the number density
of the particles, n, follow the Maxwell-Boltzmann distribution (if you have not seen
this before, you simply have to accept it)
n / m3/2 exp
✓
mc2
kB T
◆
,
where m is the particle mass. The relative number densities of e.g. protons and
neutrons can then be written as
✓ ◆3/2
✓
◆
nn
mn
(mn mp )c2
=
exp
.
np
mp
kB T
Given that the mass di↵erence between protons and neutrons is small, 1.3 MeV,
with respect to their weights, ⇠ 1 GeV, the ratio above will be very close to 1 when
kB T
(mn
mp )c2 .
From the particle physics lecture we know that allowed reactions to covert between
protons and neutrons are
n + ⌫e
! p+e
n + e+
! p + ⌫¯e
where charge and lepton numbers are conserved. As the temperature drops, and
approaches the mass di↵erence between the particles, the equilibrium will be broken,
and at a temperature of 0.8 MeV the conversion from protons to neutrons stop (the
reason why this is lower than 1.3 MeV is due to tail e↵ects similar to the ones we
have seen for the CMB), giving
nn
⇡ exp
np
✓
1.3
0.8
9
◆
⇡
1
.
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Particle & nuclear physics, astrophysics and cosmology, FK5024 Lecture 14
At this point, the abundance will be a↵ected by the free neutron decay. When the
temperature has dropped sufficiently the lightest elements can form,
p+n
! D
D+p
!
3
He
D+D
!
4
He
and the abundance of these elements will be determined by the available free neutrons. The last reaction will take place when the temperature has dropped to
0.06 MeV which will happen
t(sec) ⇡
1
0.82
1
⇡ 300 s later.
0.062
The neutron half-life is ⇡ 600 s and using the results from Per-Erik’s lecture we can
calculate the abundance at that point to

nn
1
300 · ln 2
1
⇡ exp
⇡ .
np
5
600
7
If all the neutrons at this point ends up in 4 He atoms, we can calculate the mass
fraction of 4 He, Y4 . We will have n4 He = nn /2 and each 4 He will weigh about four
times the proton mass, giving
Y4 =
4n4 He
2nn
2
=
=
= 2/8 ⇡ 25 % .
nn + np
nn + np
1 + nn /np
A more careful calculation gives 24 %. This matches well with what is observed in
the Universe, and it is not possible to obtain this high mass-fraction from stellar
fusion for the limited life-time of the Universe.
The spectacular match between the prediction of the abundances of the light elements from the Big Bang nucleosynthesis with observations is one of the three
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Particle & nuclear physics, astrophysics and cosmology, FK5024 Lecture 14
corner stones of the Big Bang theory. The other two are the prediction and observation of the CMB, and the discovery of the expanding Universe.
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