CMB acoustic peaksCMB acoustic peaks

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2 3

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r, or time

Fluid oscillations in a potential well

Graphic – Wayne Hu

Maximum velocity – maximum contribution to the Doppler term

Maximum fluid compression

Maximum fluid rarification

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dT/T

baryon-photon fluidpropagated this farsince Big Bang

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1st acoustic peakfluid compressionin potential wells

2nd acoustic peakfluid compressionin potential hills

3rd acoustic peakfluid compressionin potential wells

wellhill

time

dT/T

Quantitative treatment of oscillations

90 deg out of of phasewith the other two

In general, there are three contributions to the observed temperature fluctuationsat any given spatial scale smaller than sound horizon, at the time of recombination

gravitationalredshifting

denser fluid is hotter

Quantitative treatment of oscillations

generalsolution

assuming Hubbleterm can be ignored

Gravitational instability:Jeans analysis of small density perturbationsin the linear regime(use physical/proper coords)

get B by using soln in the oscillator eqn

in k-space, differentiating twice is same as multiplying by –k*k

why not sin(…) solution? recall continuity eqn. from Jeans analysis:

~ sin(…) will give ~ cos(…) which implies non-0 velocities at t=0

fluid fluid DM

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

kk vki for a single

k-mode

The Doppler velocity term:

Quantitative treatment of oscillationsPutting 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 velocityis a third of full v2

First, third, etc (odd) acoustic peaks fluid compression in potential wells HOT spots in CMBfluid rarification in potential peaks COLD spots in CMB

Second, fourth, etc. (even) acoustic peaks fluid rarification in potential wells COLD spots in CMBfluid compression in potential peaks HOT spots in CMB

enhanced by (1+6R)because baryons’ inertia makes them compresses in wells andmove away from peaks

Doppler velocity term: amplitude is given by +/- of this amplitude does notchange much; baryon-loaded fluidmoves slowly

Quantitative treatment of oscillations

not enhanced: baryons’inertia resists rarification in the wells &compression in the peaks

Potential and Doppler terms; no baryons

k (fixed t)

3

3

k (fixed t)

3

3

TT

Potential well

Potential hill

compressions hot spots

rarifactions cold spots

Sachs-Wolfe effect(small k, large scales)

potentialdoppler

Add baryons

k (fixed t)

TT

1st a

cousti

c peak

3rd a

cousti

c peak

2nd p

eak

)61(31 R

31

Doppler term is alsoenhanced, but not as much,because fluid with baryonsis heavier, moves slower

Baryon drag decreases the height of even-numbered peaks (2nd, 4th, etc.)compared to the odd numbered peaks (1st, 3rd, etc.)

Potential well

)1(

)31(

3

1

R

R

WMAP 3 year data

baryon drag

damping envelope

3D -> 2D projection effectsand smearing of fluctuationson small scales due to photondiffusion out of structures

Convert +ve and -ve temp. fluctuations to variance in fluctuations, then add grav & thermal + Doppler terms in quadrature

Why CMB implies dark matter

What we see is the result of baryon-photon fluid oscillations in the potential wellsand 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.

0 ]04[2 22 kskk GkcH

Growth of small DM density perturbations:Sub-horizon, Matter dominated

13/2 BtAtkgrowing decaying mode mode

ak solution mode growing DM

2

2

32

0

83

0

4t

GH

G

Can assume that total density is the same as critical density at that epoch:

zero

Dark matter has no pressure of its own; it is not coupled to photons, so there is almost no restoring pressure force.

Jeans linear perturbation analysis applies (physical/proper coords):

Two linearly indep.solutions: growing mode always comesto dominate; ignore decaying mode soln.

32

3/132

3/2

32

Htt

Haa

ta

ta

Matter dominated epoch:

0232

34 ktktk

Why CMB implies dark matter Fractional temperature fluctuations in the CMB are ~1/105

The growth rate of density perturbations in the linear regimeof 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.

Why CMB implies dark matter

log (scale factor)

log

(fra

ctio

nal o

verd

ensi

ty)

Dark matter is not coupled to photons and baryons, so itsfluctuations can grow independently.DM fractional overdensitiesare larger at recombination(but we do not see them directly)

Baryon-photon fluid oscillates in the potential wells of DM, but fluctuationamplitude is small – this is what we see as dT/T ~ 1 part in 105.

After recombination baryons arelet go from photons, and fall intothe potential wells of DM.

Radiation is free-streaming afterrecombination.

super-horizon: sub-horizon:

Evolution of amplitude of a single k-modeEvolution of amplitude of a single k-mode

additional x 10-100

MREMRE

Polarization of the CMB

incoming radiation is from one direction;radiation scattered by electron is polarized

incoming radiation is isotropic;radiation scattered by electron isnot polarized

General – applies to primary anisotropy and reionization induced polarizationGeneral – applies to primary anisotropy and reionization induced polarization

Needed for polarization:component of radiation where different amplitude of radiationare coming at the electron at 90 degrees between them –this is the property ofquarupole distribution

Polarization of the CMBGeneral – applies to primary anisotropy and reionization induced polarizationGeneral – applies to primary anisotropy and reionization induced polarization

Where does the quarupole come fromduring recombination?

Convergent velocity field in potentialwells/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 betweenthe temperature fluctuation peaks.

Polarization of the CMBPolarization in the primary anisotropies:Polarization in the primary anisotropies:

single modepotential fluctuation:

Polarization fluctuations

Peaks in temp. fluct. correspond totroughs in polarization fluctuations

Primary polarization, z~1000Primary polarization, z~1000

Quadrupole radiation field wasprovided by the CMB primary anisotropies

Polarization from Polarization from the reionization epoch at z~10the reionization epoch at z~10

Observed polarization correlation

Measurements

of CMBtemperature

and polarization

temp-tempauto corr.

temp- E polcross corr.

E pol-E polauto corr

B pol-B polauto corr.