Lecture 4:
The Cosmic Microwave
Background Radiation (CMB)
Key developments in our knowledge of the CMB
1948 
Predicted in 1948 by Alpher, Bethe, Gamow but largely ignored
1965
Discovered serendiptously by Penzias and Wilson (in parallel, Princeton
group building antenna to look for it) Nobel Prize
1990 
COBE satellite confirms perfect Planck spectrum and detects fluctuations on
large scales (Mather & Smoot) Nobel Prize
2000
Boomerang Balloon flight measures peak in Cl spectrum at 1 deg,
establishing Universe is flat
2002 
WMAP satellite produces very detailed spectrum and also evidence for
foreground screen, bringing in “precision cosmology”
~2014 Release of data from Planck satellite, including first extensive polarization
data
Temperature as f(R)
We derived R(T) for blackbody radiation field as T ∝R-1 from the RW metric.
What about the temperature of matter?
Can apply the Ideal Gas Law for adiabatic expansion of gas: TVγ-1 = constant;
γ = 5/3 for ideal gas c.f. γ = 4/3 for photons
T ∝V 1−γ
∝ R 3(1−γ )
∝ R −2
for non-relativistic matter
∝ R −1 for radiation and relativistic matter
Can also consider redshifting the wavelengths and de Broglie waves for radiation
and matter respectively to give same result
T ∝ p 2 ∝ λ −2 ∝ R −2
for non-relativistic matter
T ∝ p1 ∝ λ −1 ∝ R −1 for radiation and relativistic matter
density of water
Coupling of matter and radiation
So, matter and radiation will follow different T(R) unless they are closely coupled.
Coupling between matter and radiation can occur through
•  creation-annihilation reactions for kT > mic2
•  Compton scattering of photons off of charged particles in a plasma
N.B. Thomson scattering changes direction but with negligible
exchange of energy. Compton ceases at z < 105. This is when
matter and radiation formally decoupled.
Note that the ratio of the number densities of photons and (non-relativistic)
particles is constant during the expansion, since both vary as R-3, and is a very large
3
number.
"
%
T
−3
nγ = 3.78 ×108 $
' m
# 2.73K &
" ΩB,0 h 2 % −3
ρB
nB =
= 0.22 $
'm
η is so small that the heat capacity of
mp
# 0.02 &
the coupled fluid is dominated by the
nB
−10
η = ~ 5.8 ×10
photons. Therefore, for as long as
nγ
there is thermal coupling, the matter
−1
9
η ~ 1.7 ×10
will follow the T(R) of the photons.
Pair Creation and Annihilation
For any particle species pi of mass mi there is a threshold temperature Ti above which
•  the particles will be relativistic and contribute to the ρr
•  particle anti-particle pairs can be created (at least on energetic grounds). For
charged particles interacting electromagnetically, we would expect equilibrium
in the reaction p + p ↔ γ + γ with roughly equal number densities of
particle, anti-particles and photons (see later).
Below Ti, annihilation reactions will dominate and the number density of particles
will drop catastrophically. Energy that was in the particle/anti-particle pairs will go
into the photons, raising their temperature slightly.
If there was exactly equal amounts of matter and anti-matter, then all** particle/antiparticle pairs of that species would annihilate. Small imbalance leads to non-zero
particle-photon ratio, i.e.
** see next slide
np
np
~ 1+ η
T >Ti
An aside that will be important later
All reactions, including annihilations, have a characteristic rate (or equivalently a
timescale t), that usually depends on the density of reagents n, their speed v and the
reaction cross-section σ, the latter defined so that
−1
rreac = τ reac
= nσ v
where the reaction rate rreac is the incremental chance that a given particle undergoes
a reaction in unit increment of time: dp = rreac dt
v
σ
vdt
Key concept: If the reaction timescale τreac is longer than the age of the Universe, i.e.
τreac >> H-1 ~ τH, then the reaction will not be happening to a significant degree, even
if it is energetically favourable.
An example (more to follow)
Neutrino annihilation
At epochs after τ > 10-4 sec, neutrino annihilations take place via the neutral current
weak interactions with e+e-
ν i + ν i ↔ e− + e+
with i = e, µ, τ
Since e+e- are closed coupled to the photons this keeps Tν = Tγ
31 3
−3
But both n and σ are rapidly dropping with T. It turns out that n ~ 2 ×10 T10 cm
σ ~ 10 −44 T102 cm 2
τ reac ~ 160T10−5 s
Since τH ~ T10-2 at this epoch, these annihilation reactions will effectively cease when the
temperature falls to 5×1010K, i.e. when τH ~ 0.04s. All three neutrino species are
definitely still highly relativistic at this point, as are the electrons and positrons
Two consequences (testable, but not yet):
•  There must be still a neutrino/anti-neutrino background today with number density
comparable to the photons
•  The temperature of the ν background will be lower than that of the photons because the
latter were “re-heated” when the e+e- annihilated (at 5×109K). Predict Tν ~ Tγ/1.4 ~ 1.9K
Loss of equilibrium between matter and radiation
The annihilation of e+e− at 5×109 K (about 4 seconds after the Big Bang) marks the end
of the period when the matter and radiation in the Universe were in thermal equilibrium
and the last significant energy injection into the photons.
For as long as the matter is ionized, Compton scattering maintains a thermal coupling
between particles and photons, maintaining Tmatter ~ Tγ because of the much higher heat
capacity of the photons (reflecting the very high photon/particle ratio).
All coupling between particles and photons is lost as neutral atoms are able to
form, as the temperature drops to about 4000 K, which happens after of order 105
years, at z ~ 1500. This is called (oddly) recombination.
Why does recombination not happen at T ~ 150,000 K, when kT ~ 13.6eV the ionization
potential of Hydrogen?
•  nγ >> np so you only need a few high energy photons in the tail of the Planck
spectrum to keep Hydrogen fully ionized
•  The physics of recombination is quite subtle, as we’ll see
Small complication we’ll ignore: 25% of baryonic mass is in 4He (see later).
recombination occurs earlier because of higher ionization potential.
He
The physics of Hydrogen Recombination
The difficulty in forming neutral atoms:
•  Neutral atoms are mostly formed by the
cascade of electrons down through the
energy levels.
•  The last step from n = 2 to n = 1 mostly
occurs from 2P to 1S through emission of
a Lyα photon (121.6 nm).
•  The problem is that this will generally be
immediately absorbed by another neutral
atom, exciting it to n = 2, from which it
can be easily re-ionized.
2P
2S
Two-photon
emission
Lyman α photon
1S
•  The primary channel for production of atoms in ground-state is in fact the “leakage”
from 2S to 1S due to rare 2-photon emission, which produces photons that cannot reexcite a ground-state atom.
The physics of Hydrogen Recombination
Introduce x as the ionized fraction.
Assuming 2-photon emission is the
dominant channel, we can write
Λ 2γ
d (nx )
= − R(T ) (nx ) 2
dt
Λ 2γ + Λν (T )
Rate of influx into 2S
state from ionized
population – NB square
of nx because two body
process p+ + e−
2P
2S
Two-photon
emission
Fraction of electrons
that de-excite by twophoton instead of reexcitation out of 2S
through collisions
and/or photoexcitation
Lyman α photon
1S
Rate coefficient: R(T) ~ 3 × 10-17 T-0.5 m3s-1, i.e. proportional to z-1/2
Our primary interest is calculation of how this will depend on cosmological parameters
(e.g. ΩB,0, Ωm,0 etc).
The physics of Hydrogen Recombination
Λ 2γ
d (nx )
= − R(T ) (nx ) 2
dt
Λ 2γ + Λν (T )
We are interested mostly in dependences on parameters. Also, take (1+z) ~ z
•  R(T) and Λ(T) depend only on T, i.e. only on redshift z
•  n scales as the baryon density, ρB, i.e. as ΩB,0H02 z3
•  To convert the above equation into a differential equation in z, we need to know
that
dz
R
=− 2
dt
R
= H (1+ z)
5/2
= H 0Ω1/2
m,0 z
Assuming (reasonably, it turns out) that the
last (Λ-ratio) term is approximately unity,
and using H0 = 100h kms-1Mpc-1, simple
manipulation and evaluation gives
at high z, using H (z) from Lecture 3
" Ω h2 %
d ln x z dx
=
= 60xz $$ B,0
''
1/2
d ln z x dz
# Ωm,0 h &
Key point: This (simplified) analysis suggests (a) x(z) will be very steep and
(b) that it must have the dependence of ΩB,0, Ωm,0, and h as given
It turns out that a good analytic approximation
for the x(z) that you compute around z ~ 1000
with a full treatment of all effects is given by
12.75
2 1/ 2
⎛ z ⎞
− 3 (Ωh )
x( z ) = 2.4 × 10
⎟
2 ⎜
Ω B h ⎝ 1000 ⎠
Now we know x(z) we can compute the probability that a photon is Thomson scattered
Key concept of optical depth τ
Consider a source of light surrounded by a scattering medium.
Define dτ is the chance that photon is scattered within a given interval of path length ds
Let P(s) be the probability that a photon survives unscattered to a distance s from a given
source of light
Source
s
dP(s) = −P
dτ
ds
ds
P(s) = e−τ (s)
Us
Knowing free electron density ne along the line of sight it
is easy to calculate the optical depth at different distances
dτ = neσ T ds
s
τ (s) =
∫ n (s') σ
e
0
T
ds'
Key concept of optical depth τ
Exactly the same concept applies to calculating when photons from an isotropic
background were last scattered onto a given line of sight to us, since we again need the
probability that a photon was then unscattered from some point to us.
Us
z
Looking at all photons that come to us along a given line of sight, P(z) now gives the
probability distribution of where they were last scattered, and is easily derived from τ(z)
14.25
! z $
τ (z) = 0.37 #
&
" 1000 %
P(< z ) = e−τ ( z )
Optical depth to scattering of the CMB
We can of course compute τ(z) for electron scattering of CMB photons either in
physical space, or in comoving space, to get the same answer
dτ = neσ T ds
z
In physical space
τ (z) =
∫
0
z
! 1 dω #
3#
!
x(z') "n0 (1+ z) $ σ T %
&
" (1+ z) dz $
dω
2
In comoving space τ (z) = ∫ x(z') n0 !"σ T (1+ z) #$
dz
0
Now, we know from
Lecture 3 the
expression for dω/dz
and we know n0 from
ΩB,0 h2
dω
c $
2
3
4 &−0.5
=
(1−
Ω
)(1+
z)
+
Ω
+
Ω
(1+
z)
+
Ω
(1+
z)
tot,0
Λ,0
m,0
r,0
%
'
dz H 0
c −1/2
Ωm,0 (1+ z)−3/2 at high z
H0
( 3ΩB,0 H 02 + 1 ( n p +
-n0 ~ *
- **
) 8π G , m p ) nn + n p ,
=
We had before
12.75
2 1/ 2
⎛ z ⎞
− 3 (Ωh )
x( z ) = 2.4 × 10
⎜
⎟
Ω B h2 ⎝ 1000 ⎠
Important consequence: There will be no
dependence of τ(z) on ΩB,0, Ωm,0 or H0. So, the
location of the Last Scattering Surface in z is fixed
Evaluating the integral
14.25
! z $
τ (z) ~ 0.37 #
&
" 1000 %
The probability distribution of where the
photons were last scattered is dP(z)/dz is
roughly Gaussian with a mean z of 1065
and σ ~ 80.
50% of
photons
The Last Scattering Surface (LSS) of the
CMB is well-defined with a relatively
narrow width in redshift space (of order
10%).
Δz=120
Note that τ ~ 1 is reached at z ~ 1080, at
which x is already only 0.06 << unity.
Emphasizes that LSS is an optical depth
effect which is not (exactly) the same as
when the Universe recombined.
Small anisotropies in the CMB distribution reflect
inhomogeneities in the Universe on the (fuzzy)
Last Scattering Surface
The Last Scattering Surface is the transition
between opaque and transparent Universe
Transparent Universe
Inhomogeneities at later times not seen
because Universe is transparent (except for
e.g. small gravitational lensing effects)
Opaque
Universe
Inhomogeneities at earlier times
not seen because all positional
information is washed out by
multiple scattering
Small anisotropies in the CMB distribution reflect
inhomogeneities in the Universe on the (fuzzy)
Last Scattering Surface
Each Fourier mode in the density distribution of
the Universe at LSS will correspond to a Fourier
mode in the brightness distribution of the CMB
How do inhomogeneities in the Universe at time of LSS get
imprinted as brightness variations in the CMB?
Note that the three main effects all simply change the observed temperature in that
direction. We will see how to calculate these later, but for the moment:
(1) Adiabatic compression of the photon fluid
δT 1 δρ
~
T 3 ρ
(2) Doppler effect from relative velocities
δT v
~
T c
(3) Gravitational redshifts (Sachs-Wolfe effect)
δT 1 δφ
~
T 3 c2
Rather than Fourier modes, it is
convenient to characterise CMB
brightness variations in terms of
spherical harmonics
Cl = al2,m
ΔT
(θ, ϕ ) = ∑ al,m Yl,m (θ, ϕ )
T
l,m
The Last Scattering Surface has finite thickness
Thickness of LSS (of order 40 comoving
Mpc) is however quite large compared to
structure seen in the Universe today
(equivalent to Δz = 0.01)
dω =
c −1/2
Ωm,0 (1+ z)−3/2
H0
at high z
~ 0.257dz Mpc
Effect of Fourier modes on smaller scales will be washed out on LSS due to superposition
of multiple modes. i.e. variations in CMB brightness will be suppressed on angular scales
below smaller than the angular size of the thickness of the LSS.
D(z) = 3(c/H0) ~ 12,000 Mpc at high redshifts (see Lecture 3). So this angular size (in a
flat Universe) is just
θ~
40
~ 0.2 deg
12, 000
The thickness of LSS (of order 40 comoving Mpc) is also quite large compared to
structure seen in the Universe today (equivalent to Δz = 0.01).
So the structure we “see” in the CMB are on very large spatial scales, where the
Universe today is still relatively homogeneous today with δρ/ρ << 1
What if the Universe is reionized at some later time?
Massive stars and black hole accretion disks (in active galactic nuclei) both emit
radiation at hν > 13.6 eV which can therefore ionize H
recombination
reionization
Simply need to redo calculation of optical depth τ(z) with estimate of x(z)
•  What density of gas in intergalactic medium?
•  What fraction of ionizing photons “escapes” from galaxies?
•  How quickly do ionized atoms recombine?
Because of the form of the integral for τ(z), most scattering will in fact occur at the
highest redshifts at which the universe was reionized, priducing a broad but finite
foreground “screen”
What if the Universe is reionized at some later time?
An electron in the foreground screen
“sees” CMB photons from their own
last scattering surface.
Some fraction of the photons
coming to us are scattered out of
the path, and are replaced by others
scattered in, averaging over the
whole LSS of the scattering
electron.
This reduces the amplitude of fluctuations because
of the averaging. There will be also a small
polarization effect if the CMB “seen” by the
electron is anisotropic
Effects of non-zero foreground τ are
•  To more or less uniformly reduce the amplitude of temperature fluctuations on all
scales by of order e-τ ~ (1-τ)
•  To produce a polarization signal that is correlated on quite large angular scales
Both effects are seen in WMAP data, leading to estimate that τforeground ~ 0.09
This implies reionization occurred around about z ~ 10.
A schematic
model of x(z) and
τ(z) from WMAP
data
Important points
•  The CMB is the relic radiation from a period when matter and radiation where in thermodynamic
equilibrium via annihilation-creation reactions.
•  Above kT ~ mic2 a given species will be relativistic, with a number density comparable to that of
photons, and will therefore contribute comparably to the radiation density (see later).
•  When they become non-relativistic, the particles will (generally) annihilate, re-injecting energy
into the remaining relativistic species. The finite photon-particle ratio today is a measure of the
small excess of matter over anti-matter in the relativistic phase.
•  The CMB photons last scatter at a redshift of about z ~ 1050, soon after most of the electrons are
taken up into neutral atoms. This marks the transition from an opaque to a transparent Universe.
•  The Last Scattering Surface(LSS) is quite sharply defined in distance, but nevertheless has a
width quite large compared with length scales in today’s Universe.
•  Spatial inhomogeneities in the Universe at the time of the LSS appear as temperature anisotropies
in the CMB.
•  Subsequent reionization of the Universe can (and does) produce a foreground screen with τ ~
0.09 at z ~10 which modifies the appearance of the LSS.