The Cosmic Microwave Background
Historical Notes
1946-48: George Gamov and Ralph Alpher wonder if
nucleosynthesis during a hypothetical hot early Universe might
explain cosmic abundances. (Alpher, Bethe, Gamov 1948)
PROBLEM: lack of stable nuclei between A=5,8 means no way
to make elements heavier than Helium.
1948: Alpher and Ralph Herman predict a universal background
radiation field with T  5 K. Detection seems improbable.
1957: Burbidge2, Fowler, and Hoyle show how reactions inside
stars can make heavy elements, but can’t explain He abundance.
1964: Robert Dicke and P.J.E. Peebles independently predict
existence of a universal CMB, and decide to look for it.
1964-65: Arno Penzias and Robert Wilson unknowingly detect
the CMB with a satellite communications antenna. Discover
predictions of Dicke & Peebles and make connection.
Investigating the CMB
How rapidly does the Universe cool?
Consider black body radiation of energy density u. Two effects
reduce u as the Universe expands:
1. All volumes become larger (remember that r(t) = R(t)r0).
2. Cosmological redshift: Time intervals measured by photon
frequencies become larger as the Universe expands. Photon
wavelengths therefore become larger.
Combine these effects to see how the energy density of photons
in a small wavelength interval evolves:
u0 d0  R 3  R   u(t )d (t )
Assume that, like photon wavelengths, the temperature depends
on time. At time t the energy density is
8hc (t ) 5
u (t ) 
exp hc /  (t )kT (t )   1 .
Since the cosmological redshift means that
 (t )  0 R ,
the energy density becomes
8hcR0 
u (t ) 
exp hc / R0 kT (t )   1 .
5
Therefore, the present energy density in interval d0 is given by
8hcR0 
u0 d0  R 
 Rd 0
.
exp hc / 0 k RT (t )  1
5
4
Clearly, however, the present energy density must be given by
8hc05
u0 
exp hc / 0 kT0   1 ,
so evidently we must conclude that
T0  R  T (t ) ,
or, in other words, that
RT(t) = constant.
Estimating the Present Temperature
Use likely values for conditions during the synthesis of He:
T  109 K, and   10-5 gram cm-3.
These values are chosen because:
 Nucleosynthesis can’t proceed at much lower temperature.
 Deuterium, the fuel for Helium production, is photodissociated at much higher temperature.
 Present isotopic abundance ratios of He and H can’t be
produced at much different densities.
Recall that equation 8) states that
 0  R
3
,
and the present baryon density is believed to be
 B,0  0.6  1030
grams cm-3.
So the value of R during Helium production was about
 
R   0 
  
1/ 3
 0.6  10  30 

 
5
10


1/ 3
 0.6  10  25 
1/ 3
 3.9  10  9
Therefore, since we’ve just found that
T0  R  T (t ) ,
we predict that
T0  3.9  10 9  10 9  3.9 K.
When expressed in terms of intensity per unit wavelength, B,
black body radiation of this temperature peaks at
MAX 
2.9
 0.7 mm.
T0
The Cosmic Background Explorer
Nov 1989
launch
COBE nails the temperature of the CMB
The CMB (top) and the Dipole Anisotropy
Interpreting the Dipole Anisotropy of the CMB
Suppose that the Sun is moving with peculiar velocity v with
respect to the Hubble flow. CMB radiation received on earth
from direction  with respect to the vector v will be Doppler
shifted, and the spectrum of light from this direction will be that
of a black body of temperature
v2
1 2
c
T ( )  TREST
v
1  cos  .
c
Assuming that v << c, we can represent the numerator and
denominator on the RHS by binomial series, retaining lowest
order terms in v/c. Thus

v2 
 1 2  
2
  v
1
v
v

c

  1 


1

cos


1

cos


2 

v
c
1  cos   2 c   c

,
 c



so that
v


T ( )  TREST  1  cos   .
c


The amplitude of the observed anisotropy is about 1 part in 500,
i.e.
T (0 deg .)
TMAX
1

1
T (180 deg .) TMIN
500 .
Evaluate T() at its maximum and minimum directions:

TMAX  TCMB 1 

v
v

T

T
1


.
MIN
CMB 
c  , and
c


Form the ratio TMAX / TMIN and solve for the ratio v / c. The
result is
TMAX
1 1
v TMIN

 500  0.001
c TMAX  1
2
.
TMIN
Therefore, the Sun’s peculiar speed must be roughly
v = 0.001c = 300 km s-1.
After correcting for the (not well known) velocity of the Sun in
the Local Group rest frame, one finds the peculiar velocity of the
Local Group to be
vLG  600 km s-1 towards (RA  10.5h , DEC  -26).
This represents the motion of the Local Group with respect to a
reference frame at rest in the Hubble flow.
The Radiation and Matter Eras
We have found that the radiation density decreases according to
u (t )  R 4 ,
while we already knew that the density of ordinary matter must
decrease as
  R 3 .
At very early times (i.e. small R), therefore, it seems that
radiation might have accounted for most of the total massenergy density of the Universe. We want to find out when the
two densities were equal and what conditions were like at that
time.
The total energy density of black body radiation at temperature
T is
u
4 4
T ,
c
where  is the Stefan-Boltzmann constant. The equivalent mass
density can be obtained using Einstein’s famous relation E=mc2.
Thus
 RAD
u 4 4
 2  3 T .
c
c
Combining the above with the known dependence of radiation
density on R shows that
 RAD (t ) 
 RAD,0
R4

4 4
T
4 3 0 ,
Rc
while for ordinary matter
 (t ) 
0
R3 .
When were the two densities equal?
 RAD (t )   (t )
4 4  0
T  3
4 3 0
R c
R
So the two densities equal when
4T04
R
0c3 .
But R = (1+ z)-1, so the this was the time when
0c3
1 z 
4T04 .
Replace 0 with more convenient quantities by noting that
 0   0  C ,0
3H 02
 0 
8G ,
so that
 3c 3

2.231  106
2
2

1  z  


H



h
0
0
0
4 
4
.
32

G

T
T
0 
0

Stick in T0 =2.726 K, h = 0.71, and 0 = 1 to find that
z  20,000 .
The temperature at this time was
T0
 T0  1  z   2.726  2  10 4   5.5  10 4 K,
R(t )
and the age of the Universe, for h = 0.71, was
T (t ) 
t
3 / 2
2
2
3 / 2
t H  1  z 
  9.78  10 9 h 1  2  10 4 
 3200 yr.
3
3
The Recombination Epoch
Because of photon scattering by free electrons, a brilliant plasma
“fog” filled the early Universe. As the temperature declined,
however, the fog eventually cleared.
Clearing happened when free electrons could recombine with
protons without immediately being re-ionized. Let’s assume
this happened when half the Hydrogen was neutral.
When did this happen, and what was the Universe like then?
Estimate the value of R as follows:
Assume that Hydrogen ionization could be approximately
described by the Saha Equation:
N II
2Z II  2me kT (t ) 



NI
ne ( t ) Z I 
h2

3/ 2

exp    I
kT (t )  .

where
I is the ionization energy of state I (neutral Hydrogen),
ZI, ZII are the partition functions for states I and II, respectively
(number of different electron configurations having the same
energy),
ne, me are the number density of electrons and their mass.
NOTE: assume that both ne and T were changing with time.
Recall that
T (t ) 
T0
R(t ) .
Also, for a pure, half-ionized Hydrogen gas (neglecting other
elements) the electron number density and total density are
related through
  2  ne m H .
and the total density declines with R as
   0 R 3 .
Substitute into the Saha equation to express ne and T in terms of
R, assuming that ZI and ZII are 2 and 1, respectively, and find
N II 2mH R 3  2me kT0 



2
NI
 B ,0  h R 
3/ 2

exp    I R
kT0  .

If there are equal numbers of neutrals and ions, then NII/NI = 1.
As a function of R the Saha equation now has the form
R 3 / 2  M  exp NR  ,
where M and N are positive constants. The book claims that the
solution is
RR  7.2  10 4 ,
but C&O appear to neglect the factor of 2 relating ne and .
We’ll go with their result anyway.
This scale factor corresponded with a redshift of
1
zR 
 1  1400
,
RR
and the temperature and age of the Universe were, respectively
TR 
tR 
T0
RR

2.726
 3800 K, and
7.2  10 4
2
3 / 2
 9.78  10 9 h 1  1  1400
 180,000 years,
3
for h = 0.71 and k = 0.
FOOTNOTE: With Recombination, the Universe entered the
so-called “Dark Ages”, which lasted until the first generations of
stars reionized intergalactic space.
Higher-order Fluctuations in the CMB
After subtracting the Dipole fluctuation, COBE maps of the
CMB show fainter ripples having
angular size  10 (= COBE resolution), and
T/T  a few  10-5.
A 10 feature seen at z  1000 has linear size 1000/(0h) Mpc,
so these fluctuations are much larger than galaxies or galaxy
clusters.
Structure in the COBE map
Mean-subtracted and
Zodiacal light removed
Dipole-subtracted
Milky Way-subtracted
Describing the Fluctuations
Use spherical harmonics (orthogonal functions on a spherical
surface):
 l
T  ,    T0
T
 ,   
   almYlm  ,   .
T
T0
l 0 m   l
The Ylm are functions of Sin, Cos and exp(im), and the alm
specify the amplitudes of the various terms.
Think of Ylm as dividing the sphere into cells of dimension /l
radians. The index l is called the multipole.
COBE resolution: 10  l  20. Smaller angular features
(larger l) can’t be resolved by COBE.
The Power Spectrum Cl
1
2

Cl 
a

a


a

lm
lm
lm  ,
2l  1 m
which means that Cl measures the significance of fluctuations
with angular sizes of /l radians.
Synthetic “CMB Fluctuations” for Multipoles 2 and 16
Interpreting the Fluctuations – the Sachs-Wolf
Effect
In 1967, only 2 years after the discovery of the CMB, Sachs and
Wolf predict that it should contain large-scale fluctuations.
Photons become slightly redshifted while climbing out of the
large but shallow potential wells of random density fluctuations.
Let k  fluctuation wavenumber = 1/(size of fluctuation), and
assume that the spectral power P has the form
P( k )  k n .
Larger P indicates more numerous fluctuations. The angular
size of the redshift-induced temperature fluctuations can be
shown to look like
1 n 
T
2

,
T
and since there are reasonable grounds for expecting n  1, one
anticipates that
Cl  const .
until fluctuations begin to wash out at angular scales of
  0.2 deg . ,
that is, for l  1000. Smaller fluctuations smear out over the
time needed for recombination to occur.
CONCLUSION: The Sachs-Wolf effect provides information
about the power spectrum of initial fluctuations on large scales –
the origin of the filamentary network recorded by large scale
surveys (e.g. SDSS, 2MASS)
All by itself, then, the signature of LSS should be a nearly
constant value of Cl for l < 1000, which should rapidly decline
for larger multipoles.
The Acoustic Peaks
Fluctuations with angular scales of 0.2 – 4 degrees (900  l 
50) are measured by the recent Boomerang and MAXIMA
balloon experiments, by the DASI radiometer at the South Pole,
and most recently by WMAP.
Important fact: Before recombination, the Jeans length was so
large that perturbations of any size were stable. Rather than
collapsing, all perturbations would propagate as sound waves.
Maximum wavelength of sound waves in a universe of age t is
S  sound _ speed   t 
c
t,
3
and such waves should have angular sizes of

S
 0.310/ 2 degrees.
d
For 0 = 1, those waves should produce a peak in the angular
power spectrum near multipole
l

0.3
57.3
 600
.
More careful calculations predict a peak near l = 200 (0 = 1),
or 430 (0 = 0.2).
Higher multipole peaks and valleys occur on scales
corresponding to max or min “sound” amplitudes at the time of
recombination. Spacings and amplitudes of peaks depend on
various combinations of H0, 0, and .
BOTTOM LINE: The power spectrum of CMB fluctuations
contains powerful constraints on possible world models.
Four Recent Measurements of the CMB
Angular Power Spectrum
Boomerang
MAXIMA at Palestine, Texas
DASI looks up
from the South Pole
Measuring the
Power Spectrum
of the CMB
The Wilkinson Microwave
Anisotropy Probe (WMAP)
Launch 2001 to the L2 point.
13’ resolution
3.2 – 13 mm wavelength coverage
Intensity and polarization mapping
Comparing dipole-subtracted maps of the CMB
WMAP nails the power spectrum
to l ~ 1000
Polarization data for l < 10
demands an epoch of
reionization, and implies
an electron scattering optical
depth of τ = 0.17 ± 0.04
Cosmological constraints from CMB
fluctuations
Total density parameter
Other density: the “Cosmological Constant”
Electron scattering optical
depth
Fluctuation power spectrum
Curves refer to different “priors” –
Baryonic and cold dark matter densities
Sievers et al. 2003 ApJ, 591, 599