CMB acoustic peaks
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Transcript CMB acoustic peaks
CMB acoustic peaks
Potential fluctuations broken up by mode
hill
well
2
3
r, or time
Potential fluctuations broken up by mode
hill
well
baryon-photon fluid
propagated this far
since Big Bang
2
3
dT/T
r, or time
Potential fluctuations broken up by mode
3rd acoustic peak
fluid compression
in potential wells
hill
well
2nd acoustic peak
fluid compression
in potential hills
1st acoustic peak
fluid compression
in potential wells
2
3
dT/T
time
Fluid oscillations in a potential well
Maximum fluid
rarification
Maximum velocity –
maximum contribution
to the Doppler term
Maximum fluid
compression
Graphic – Wayne Hu
Quantitative treatment of oscillations
In general there are three contributions to the observed temperature:
gravitational
redshifting
denser fluid is hotter
90 deg out of of phase
with the other two
Note: for the Sachs-Wolfe plateau, the Doppler term is insignificant
Quantitative treatment of oscillations
Jeans analysis
of small perturbations
(linear regime)
after applying
assumptions
general
solution
get B by using soln
in the oscillator eqn
Constant A and the absence of the sin(…) term in the solutions
come from applying the boundary conditions of the Sachs-Wolfe effect
velocity negligible
at very early times
and large k (SW effect)
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
Quantitative treatment of oscillations
First, third, etc (odd) acoustic peaks:
enhanced by (1+6R)
because fluid with
baryons compresses well!
Second, fourth, etc. (even) acoustic peaks:
Amplitude of the velocity term is given by +/- of this
fluid rarifaction in
potential wells
(smaller amplitude
than compression)
amplitude does not
change much;
baryon-loaded fluid
moves slowly
Potential and Doppler terms; no baryons
T
T
Potential hill
compressions
3
hot spots
2
3
4
k (fixed t)
rarifactions cold spots 3
Potential well
potential
doppler
3
k (fixed t)
3
Sachs-Wolfe effect
(small k, large scales)
2
3
4
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
T
T
13 (1 6 R)
doppler term is also
enhanced, but not as much,
because fluid with baryons
is heavier, moves slower
1 (1 3R)
3 (1 R)
k (fixed t)
1
3
2
3
4
WMAP 3 year data
convert positive and negative temperature
fluctuations to rms fluctuations
(take squares – all positive)
3D -> 2D projection effects
and smearing of fluctuations
on small scales due to photon
diffusion out of structures
baryon
drag
Growth of small density perturbations:
Sub-horizon, Matter dominated
Jeans linear perturbation analysis applies (proper):
k 2 Hk [cs 2 k 2 4G0 ] k 0
zero
Dark matter has no pressure of its own;
it is not coupled to photons,
so there virtually no restoring pressure force.
Can assume that total density is the critical density at that epoch:
0
4G0
3H 2
8G
2
3t 2
Matter dominated epoch:
a t2/3
a 23 t 1 / 3
a
2
H
a
3t
Ht 23
k 34t k
2
3t 2
k 0
k At 2 / 3 Bt 1
growing
mode
decaying
mode
Two linearly indep.
solutions: growing
mode always comes
to dominate; ignore
decaying mode soln.
DM growing mode solution k a
Why we need dark matter - I
Fractional temperature fluctuations in the CMB are ~1/105
The growth rate of density perturbations in a non-relativistic component
is at most as fast as a=1/(1+z)
Recombination took place at z=1000
Then predict that today amplitude of typical fractional overdensities should be 0.01
But, in galaxies and clusters
observed fractional overdensities ~100
fluctuations that we see in the CMB
are not enough to give us structure today
Why we need dark matter - II
log (fractional overdensity)
Evolution of amplitude of a single k-mode
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.
additional
x 10-100
log (scale factor)
super-horizon:
all grow at
same rate
Because dark matter is not
coupled to photons and baryons
its fluctuations can grow
independently.
sub-horizon:
causal “microphysics”
is now important
DM fractional overdensities
are larger at recombination
(but we do not see them directly)
After recombination baryons are
let go from photons, and fall into
the potential wells of DM.