### cmb - University of Arizona

```Cosmic Microwave Background
References:
COBE web site
WMAP web site
Web sites of Wayne Hu, Max Tegmark, Martin White,
Ned Wright and Yuki Takahashi
R&L Problem 4.13
Show that an observer moving with respect to a blackbody
field of temperature T will see blackbody radiation
with a temperature that depends on angle according to
1  

T 
2 1/ 2

We showed that
I  2h3

c2
1  cos
I
 3
3
 
I
h/ kT '
e
1

1
T
where
I
B
T
)

(
Planck function
1
2h3 h / kT

e
1

2
c








1


cos

 
2
h




 

I
e

1

The Doppler formula
implies

So, if we define
Then
3

1
(

h
1

cos
)
/
kT
2
c
T

T


1


cos



I
B
T
)

(


for each direction
For β << 1





T

T
1

cos
T
T(1-β) > T
T(1+β) > T
v
T
So because the Earth is moving with respect to the Cosmic
Microwave Background, it appears cooler in the direction towards
which we are going, and warmer in the opposite direction
Map of T across sky, as measured by WMAP
T = 2.725, very accurately Planckian.
Uniform to 1 part in 10^5
Subtracting T=2.75K, what you see is a DIPOLE
The CMB temperature is 3.353 K hotter in one direction,
and 3.353 mK cooler in the opposite direction.
Milky Way
The CMB defines a “cosmic reference frame”.
The Earth is moving with a “peculiar velocity” with respect to
the “Hubble flow” of galaxies.
The observed dipole indicates that the Solar System is
moving at 368+/-2 km/sec relative to the observable
Universe in the direction galactic longitude l=263.85o and
latitude b=48.25o with an uncertainty slightly smaller than
0.1o
Earth’s Motions
(1) Earth spins on its axis
V≈ 0.5 km/sec at equator
(2) Earth orbits Sun
“heliocentric velocity”
V ≈ 30 km/sec
(3) Sun orbits Galactic Center,
in Milky Way Galaxy
V ≈ 225 km/sec
(4) Milky Way falling towards M31
(Andromeda) in the Local Group
V ≈ 100 km/sec
M31, Andromeda
(5) Local Group is falling into the
Virgo supercluster of galaxies
V ≈ 220 km/sec
“Virgocentric infall”
center of the Virgo Cluster of Galaxies
(6) Virgo Supercluster is falling
V ≈ 600 km/sec
towards the Hydro and Centaurus
clusters of galaxies
“The Great Attractor”
~10^16 solar masses at 50 Mpc
(7) Claims of larger scale flows??? Probably not
Lauer-Postman Brightest cluster E’s
SN Ia
COBE data of CMBR
c. 1992
First detection of
temperature
fluctuations, i.e.
CMBR anisotropies
Milky Way
Subtract dipole 
temperature fluctuations
indicates density fluctuations
which later collapse to form
galaxies, clusters, etc.
Since WMAP:
-Balloon experiments at S. Pole
- Boomerang
- Radio telescopes at S. Pole:
DASI
Planck Satellite: Launched 2009.
All sky picture in mm
The History of the Universe from t=0 to t=380,000 years
1. t=0 to the Planck Time.
The Planck Time = 10-43 seconds after the Big Bang.
This is when the cosmological horizon was smaller than an electron.
Before this, we're pretty sure that normal Quantum Mechanics
doesn't apply.
Horizon: Because the universe is t seconds old, parts
of the Universe farther apart than d=ct are not
causally connected.
Remember, the Universe is infinite; “expansion” is not
like a balloon blowing up, or a raissen bread rising, which
has a center of expansion, and where the expansion is
in a directions in x,y,z space.
2. The GUT era; Inflation.
GUT = grand unified theory
When the GUT divided into the strong/electroweak/gravity forces,
the Universe INFLATED very very rapidly.
In 10-36sec, a piece of the Universe the size of an atom expanded
to the size of the Solar System.
INFLATION is a natural explanation of the HORIZON PROBLEM.
Why is the
CMBR temperature
the same in
all directions?
The flatness problem
Why is space so nearly flat?
If Ω were even a tiny bit different than 1, it would by now
be WAY different.
“Inflation” of the Universe happened when the Universe had
cooled far enough that “symmetry breaking” of the GUT force
could occur -- sometimes described as a “phase transition.”
Only a small part of the Universe which was within our Horizon just
prior to inflation remained in our Horizon after inflation.
The huge factor by which the Universe expanded also explains
why we’re nearly flat.
Here’s a 2D
representation of
a 3D surface
that is nearly flat
when expanded.
Note that during inflation the radius of curvature of the
geometry of the Universe increased effectively faster than
the speed of light.
But since the expansion was on the geometry of the Universe itself,
and not the matter, then there is no violation of special relativity.
Our visible Universe, the part of the Big Bang within our horizon, is
effectively a `bubble' on the larger Universe. However, those other
bubbles are not physically real since they are outside our horizon.
We can only relate to them in an imaginary, theoretical sense.
They are outside our horizon and we will never be able to communicate
with those other bubble universes.
Since the Universe was opaque prior to recombination, you might
think that it’s difficult to observe directly anything that happened
before recombination.
However, during inflation, the expansion may have generate
gravitational waves which would cause ripples in the density of
the Universe, which would be reflected in the CMB.
The amplitude of the gravity wave generated is proportional to the
expansion rate H during inflation, which in turn is proportional to the
inflation energy scale squared:
GW amplitude ∝ H ∝ Einf2, where Einf~<1016GeV
So detecting the signature of these gravitational waves in the CMBR
could test the inflation model and measure Einf.
Back to the history of the Early Universe:
3. t=10-38sec to 10-10 second after the big bang:
THE ELECTROWEAK era
4. 10 -10 second to 0.001 second: The Particle Era
The Universe was filled with electrons, neutrinos and quarks.
3 quarks = proton, anti-proton, neutron, antineutron
but the universe was too hot for the quarks to be bound
together as protons or neutrons.
When the Universe finally cooled for the quarks to form
protons, neutrons, etc. the protons and anti-protons rapidly
annihilated, forming photons.
The protons slightly outnumbered anti-protons, so we were left with
protons and photons.
Photons outnumber protons by a billion to one.
There are approximately 400 photons / cm3 currently in the CMBR.
5. 0.001 second to 3 minutes: The era of Nucleosynthesis
The Universe cooled to temperature = 109 K, similar to the core
of the Sun.
As in the core of the Sun, fusion of hydrogen into helium created
a Universe with about 25% helium (by mass), plus very small
amounts of deuterium (proton + neutron) and lithium.
The exact ratios of He/hydrogen, deuterium/hydrogen etc depend
on the density of baryons (protons & neutrons) and the temperature
of the Universe at during this era. These in turn depend on cosmological
parameters such as the expansion rate of the Universe.
Helium-4 is measured
in by looking at
emission lines in
spectra of H II
regions in dwarf
galaxies.
Li/H is measured
by spectroscopy of
old star stellar
atmospheres.
D/H is measured
by looking at
quasar absorption
lines from distant
galaxies.
(6) t=300,000 years:
Recombination.
Electrons and protons recombined, the opacity of the universe
to Thomson scattering went to zero, and the photons were then
free to stream to us.
CMBR Anisotropies
Shortly after the CMB was discovered by Penzias & Wilson,
Sachs & Wolfe (1967 ApJ, 147, 73) and others realized
that there should be angular variations in temperature,
as a result of density inhomogeneities in the Universe at
the time of recombination.
The denser regions cause the CMB photons to be gravitationally
redshifted compared to photons arising in less dense regions.
Another effect is that regions that were overdense recombined
first because the recombination rate depends on n2.
The amplitude of the T fluctuations is roughly 1/3 of the density
fluctuations.
These slightly overdense regions at the time of CMB production
later became gravitationally unstable and collapsed to form galaxies,
clusters of galaxies and all other structures we see in
the Universe today.
From the observed CMBR angular anisotropies in temperature, it
is straight-forward to derive what density fluctuations created them.
WMAP temperature fluctuations
The fluctuations in temperature are at a level of 10-5 T, and
so were difficult to measure – first detection was in 1992.
The angular distribution of the temperature fluctuations
are modeled in terms of spherical harmonics, which are the
convenient functions for describing a scalar field projected
on a sphere:





T
(
,)

a
Y
(
,)


lm
lm
T l

0
m


l
l
This is like describing a light curve in terms of the power series,
i.e. the coefficients of the sine’s and cosines’ of different
frequencies which, when summed, reproduce the observed
light curve.





T
(
,)  l

a
Y
(
,)


lm
lm
T l

0
m


l
If the underlying density fluctuations are described by a gaussian
random process, as inflation predicts, then all the information
is contained in the angular power spectrum,
C
l a
2
lm
The angled brackets indicate the average over all observers
in the Universe; the absence of a preferred direction in the
Universe implies that the coefficients
alm
are independent of m.
2
Usually people don’t plot Cl but
l(l 1)
Cl
2
since you can write
 
l
(
l

1
)
C
2
l

1
2
l



T

T
C

T
d
ln
l

l

4
2
l
2 2
So
l(l 1)
Cl
2
is a measure of the power in the temperature
fluctuations per logarithmic interval in ℓ space
The exact mixture
of material in the
early Universe
(baryonic,
neutrinos, cold dark
matter),
cosmological
parameters (H0,
vacuum
energy) and initial
perturbation
spectrum control
the position and
amplitude of these
peaks and troughs.
The dependence of the peaks on parameters are understandable
in terms of physical processes, but are beyond the scope of
this discussion.
The position of the first peak is sensitive to the total energy
density and can be used to determine the geometry of the
Universe: . It moves to smaller angles as l decreases because
the distance to the last-scattering surface increases
(the expansion slows less in a low-density universe) and
geodesics (i.e. the path of light rays) diverge in negatively
curve spaced (fixed distance on the last-scattering surface
subtends a smaller angle).
Individual
l’s
Cumulative Cl
Temperature differences between points on the sky separated
by angle θ are related to the multipoles with spherical-harmonic
indices around
l
100
/

o
For example, the density fluctuations of wavelength around 2Mpc,
which seed galaxies, subtend an angle θ of around an arcminute;
those of 20Mpc that seed clusters of galaxies subtend about
10 arcminutes; and those of around 200Mpc that seed the
largest structures seen today subtend about 1 degree.
(All of these distances were a thousand times smaller at the
time of last scattering, when the linear size of the universe
was a thousand times smaller... But it is conventional to
quote ``comoving separations'' as they would be now.)
What caused the density fluctuations?
There are basically two models:
(1) Before inflation, there were quantum fluctuations on subatomic
scales which were stretched to astrophysical size during inflation.
These fluctuations became density perturbations when the vacuum
energy that drove inflation decayed into radiation and matter.
(2) The competing theory says that the density perturbations were
seeded by topological defects. Depending upon how the
symmetry is broken during inflation these defects might
be point-like (global monopoles), one-dimensional (cosmic strings), or
three-dimensional (spacetime textures).
It turns out that topological defects create density perturbations
which develop significantly later than the quantum fluctuation
model, and so there is a signature in the CMB.
The current anisotropy data appear to be consistent with inflation and
inconsistent with the topological defect scenario
Hot /cold
Spots in the CMB
Observed
Predicted,
For different
Geometries
Of the Universe
Flat
theory
CMB temperature fluctuations
reality
One thing to keep in mind:
All these maps require that you “clean” out the foreground
emission, i.e. the emission from the solar system, Milky Way
and foreground galaxies. How this is done is a long story,
but essentially they make maps at several frequencies,
and subtract out models for the foreground emission sources.
Zodiacal Light:
thermal emission from
dust in the solar system
Milky Way:
thermal emission from
dust in the Milky Way
COBE
DIRBE
map
The foreground is particularly bad for measurements of
CMBR polarization.
WMAP measurement of polarization: white lines indicate direction
of linear polarization; most of this is caused by B-fields in the Milky Way.
Polarization of the CMBR itself is very interesting.
If the Universe were perfectly homogeneous and isotropic,
Thompson scattering would of course produce linear polarization
as we discussed in class, but the directions of polarization
from all the photons would cancel out – so no net polarization
would be seen.
However, if there are quadrupole anisotropies in the density
of the Universe, then the Thompson scattering will have
preferred direction for each size scale, resulting in a net
polarization.
The % polarization is really small – so the fluctuation map
of fluctuations in polarization will have amplitudes several
orders of magnitude smaller than the temperature fluctuations.
But WMAP, and DASI have recently claimed detection of
the spectrum of polarization fluctuations for the CMB.
Thomson Scattering
Thomson scattering in Isotropic Radiation Field
When an electromagentic wave is incident on a free electron, the
scattered wave is polarized perpendicular to the incidence direction.
If the incident radiation were isotropic or had only a dipole variation,
the scattered radiation would have no net polarization. However, if the
incident radiation from perpendicular directions (separated by 90°)
different intensities, a net linear polarization would result. Such
anisotropy is called "quadrupole" because the poles of anisotropy
hotter
are 360°/4 = 90° apart
colder 
Wayne Hu
What would cause quadrupole anisotropies? 3 possibilities:
Scalar perturbation: Energy density fluctuations
Vector perturbation: Vorticity in the plasma cause Doppler shifts
resulting in the quadrupole lobes in the figure. However, vorticity
would be damped by inflation and is expected to be negligible.
Tensor perturbation: Gravity waves stretch and squeeze space in
orthogonal directions (as shown by the test 'circles' in the figure).
This also stretches the wavelength of radiation, therefore creating
from inflation would produce tensor perturbation.
The NASA press release had this representation of
the WMAP data on polarization:
The polarization pattern in the sky can be decomposed into 2
components:
Curl-free component, called "E-mode" (electric-field like)
Grad-free component, called "B-mode" (magnetic-field like)
The E-mode may be due to both the scalar and tensor perturbations,
but the B-mode is due to only vector or tensor perturbations.
E-mode: <= Scalar / Tensor perturbations, corr. w/ T perturbations
B-mode: <= Vector / Tensor perturbation, not corr. w/ T perturbations
E-mode
B-mode
E-mode polarization fluctuations have been detected by a
number of experiments, most notably DASI.
B-mode polarization fluctuations
Based on the WMAP data, we have a limit on B-mode
polarization -- all we know is that the energy scale of
inflation must have been < 3x1016 GeV.
When we detect the gravity wave signal in the CMBR
polarization maps, we will be able to measure the energy
scale of inflation: so it will be an observation
of the Universe 10-35 seconds after the Big Bang.
```