1. CMB acoustic peaks

2. Potential fluctuations broken up by mode
hill
well
r, or time
Potential fluctuations broken up by mode

3. Fluid oscillations in a potential well
Maximum fluid
rarification
Maximum velocity –
maximum contribution
to the Doppler term
Maximum fluid
compression
Graphic – Wayne Hu

4. Potential fluctuations broken up by mode
hill
well
dT/T
r, or time
Potential fluctuations broken up by mode
baryon-photon fluid
propagated this far
since Big Bang

5. Potential fluctuations broken up by mode
3rd acoustic peak
fluid compression
in potential wells
hill
well
dT/T
time
2nd acoustic peak
fluid compression
in potential hills
Potential fluctuations broken up by mode
1st acoustic peak
fluid compression
in potential wells

6. Quantitative treatment of oscillations
In general, there are three contributions to the observed temperature fluctuations
at any given spatial scale smaller than sound horizon, at the time of recombination
denser fluid is hotter
gravitational
redshifting
90 deg out of of phase
with the other two

7. Quantitative treatment of oscillations
Gravitational instability:
Jeans analysis of small
density perturbations
in the linear regime
(use physical/proper coords)
in k-space, differentiating twice is same
as multiplying by –k*k
assuming Hubble
term can be ignored
fluid fluid DM
general
solution
why not sin(…) solution? recall continuity eqn. from Jeans analysis:
~ sin(…) will give ~ cos(…) which implies non-0 velocities at t=0
get B by using soln
in the oscillator eqn

8. Quantitative treatment of oscillations
Constant A can be obtained by applying
the boundary conditions of the Sachs-Wolfe effect, when t is small, or k is small 
The Doppler velocity term:
for a single
k-mode

9. Quantitative treatment of oscillations
Putting in values for A and B:
The two contributions to temperature (i.e. exlcuding the Doppler term):
The velocity Doppler contribution to temperature (multiplied by i, 90deg out of phase)
line of sight velocity
is a third of full v2

10. Quantitative treatment of oscillations
First, third, etc (odd) acoustic peaks
fluid compression in potential wells  HOT spots in CMB
fluid rarification in potential peaks  COLD spots in CMB
Second, fourth, etc. (even) acoustic peaks
fluid rarification in potential wells  COLD spots in CMB
fluid compression in potential peaks  HOT spots in CMB
enhanced by (1+6R)
because baryons’
inertia makes them
compresses in wells and
move away from peaks
not enhanced: baryons’
inertia resists
rarification in the wells &
compression in the peaks
Doppler velocity term: amplitude is given by +/- of this
amplitude does not
change much;
baryon-loaded fluid
moves slowly

11. Potential and Doppler terms; no baryons
Potential hill
p p p p
k (fixed t)
compressions hot spots
rarifactions cold spots
potential
doppler
Potential well
p p p p
k (fixed t)
Sachs-Wolfe effect
(small k, large scales)

12. Add baryons Baryon drag
decreases the height of even-numbered peaks (2nd, 4th, etc.)
compared to the odd numbered peaks (1st, 3rd, etc.)
Potential well
3rd acoustic peak
1st acoustic peak
Doppler term is also
enhanced, but not as much,
because fluid with baryons
is heavier, moves slower
k (fixed t)
p p p p
2nd peak

13. WMAP 3 year data Convert +ve and -ve temp. fluctuations
to variance in fluctuations,
then add grav & thermal + Doppler terms
in quadrature
3D -> 2D projection effects
and smearing of fluctuations
on small scales due to photon
diffusion out of structures
baryon
drag
damping envelope

14. Why CMB implies dark matter
What we see is the result of baryon-photon fluid oscillations in the potential wells
and peaks of dark matter. DM is not directly coupled to baryons & photons.
DM density fluctuations have been growing independently of baryons & photons.
Need to consider their growth rate first.

15. Growth of small DM density perturbations: Sub-horizon, Matter dominated
Jeans linear perturbation analysis applies (physical/proper coords):
zero
Dark matter has no pressure of its own;
it is not coupled to photons,
so there is almost no restoring pressure force.
Can assume that total density is the same as critical density at that epoch:
growing decaying
mode mode
Two linearly indep.
solutions: growing
mode always comes
to dominate; ignore
decaying mode soln.
Matter dominated epoch:

16. Why CMB implies dark matter
Fractional temperature fluctuations in the CMB are ~1/105
The growth rate of density perturbations in the linear regime
of non-relativistic component is at most as fast as a=1/(1+z)
Recombination took place at z=1000
With no DM, today amplitude of typical fractional overdensities should be 0.01
But, in galaxies and clusters
fractional overdensities ~100 and up
 fluctuations that we see in the CMB
are not enough to give us structure today
Potential fluctuations at recombination
must have been larger than temp. fluct.

17. Why CMB implies dark matter
Evolution of amplitude of a single k-mode
Dark matter is not coupled to
photons and baryons, so its
fluctuations can grow independently.
DM fractional overdensities
are larger at recombination
(but we do not see them directly)
Baryon-photon fluid oscillates in the
potential wells of DM, but fluctuation
amplitude is small – this is what we
see as dT/T ~ 1 part in 105.
After recombination baryons are
let go from photons, and fall into
the potential wells of DM.
Radiation is free-streaming after
recombination.
MRE
additional x
log (fractional overdensity)
log (scale factor)
super-horizon: sub-horizon:

18. Polarization of the CMB
General – applies to primary anisotropy and reionization induced polarization
incoming radiation is from one direction;
radiation scattered by electron is
polarized
incoming radiation is isotropic;
radiation scattered by electron is
not polarized

19. Polarization of the CMB
General – applies to primary anisotropy and reionization induced polarization
Needed for polarization:
component of radiation where
different amplitude of radiation
are coming at the electron
at 90 degrees between them –
this is the property of
quarupole distribution

20. Polarization of the CMB
Polarization in the primary anisotropies:
single mode
potential fluctuation:
Where does the quarupole come from
during recombination?
Convergent velocity field in potential
wells/peaks as the fluid oscillates.
Then, have to get the projection of
these on the plane of the observer’s sky.
Most polarization will be seen on scales that correspond to max. fluid velocity – at 90deg from max. fluid compression.
Polarization peaks will be between
the temperature fluctuation peaks.

21. Polarization fluctuations
Peaks in temp. fluct. correspond to
troughs in polarization fluctuations
Primary polarization, z~1000
Polarization from
the reionization epoch at z~10
Quadrupole radiation field was
provided by the CMB primary anisotropies

22. Observed polarization correlation

23. Measurements of CMB temperature and polarization
temp-temp
auto corr.
temp- E pol
cross corr.
E pol-E pol
auto corr
B pol-B pol