1. Cosmic Microwave Background
Primary Temperature Anisotropies
Polarization
Secondary Anisotropies
Scott Dodelson PASI 2006

2. Coherent Picture Of Formation Of Structure In The Universe
Photons freestream: Inhomogeneities turn into anisotropies
t ~100,000 years
Quantum Mechanical Fluctuations during Inflation
Perturbation Growth: Pressure vs. Gravity
Matter perturbations grow into non-linear structures observed today
m, r , b , f
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3. Goal: Explain the Physics and Ramifications of this Plot
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4. Notation Scale Factor a(t) Conformal time/comoving horizon
Gravitational Potential 
Photon distribution 
Fourier transforms with k comoving wavenumber
Wavelength k-1
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5. Photon Distribution
Distribution depends on position x (or wavenumber k), direction n and time t.
Moments Monopole:  Dipole:  Quadrupole: 
You might think we care only about at our position because we can’t measure it anywhere else, but …
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6. We see photons today from last scattering surface at z=1100
 accounts for redshifting out of potential well
D* is distance to last scattering surface
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7. Can rewrite  as integral over Hubble radius (aH)-1
Perturbations outside the horizon
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8. Inflation produces perturbations
Quantum mechanical fluctuations <(k) (k’)> = 33(k-k’) P(k)
Inflation stretches wavelength beyond horizon: (k,t) becomes constant
Infinite number of independent perturbations w/ independent amplitudes
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9. Perturbations in metric induce photon, dark matter perturbations
To see how perturbations evolve, need to solve an infinite hierarchy of coupled differential equations
Perturbations in metric induce photon, dark matter perturbations
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10. Evolution upon re-entry
Pressure of radiation acts against clumping
If a region gets overdense, pressure acts to reduce the density
Similar to height of an instrument string (pressure replaced by tension)
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11. Before recombination, electrons and photons are tightly coupled: equations reduce to
Temperature perturbation
Very similar to …
Displacement of a string
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12. What spectrum is produced by a stringed instrument?
C string on a ukulele
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13. CMB is different because …
Fourier Transform of spatial, not temporal, signal
Time scale much longer (400,000 yrs vs. 1/260 sec)
No finite length: all k allowed!
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14. Why peaks and troughs?
Vibrating String: Characteristic frequencies because ends are tied down
Temperature in the Universe: Small scale modes enter the horizon earlier than large scale modes
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15. Interference could destroy peak structure
There are many, many modes with similar values of k. All have different initial amplitude. Why all are in phase?
First Peak Modes
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16. An infinite number of violins are synchronized
Similarly, all modes corresponding to first trough are in phase: they all have zero amplitude at recombination. Why?
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17. Without synchronization:
First “Trough”
First “Peak”
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18. Inflation synchronizes all modes
All modes remain constant until they re-enter horizon.
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19. How do inhomogeneities at last scattering show up as anisotropies today?
Perturbation w/ wavelength k-1 shows up as anisotropy on angular scale ~k-1/D* ~l-1
Cl simply related to [0+]RMS(k=l/D*)
Since last scattering surface is so far away, D* ≈ η0
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20. The spectrum at last scattering is:
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21. Anisotropy spectrum today
Fourier transform of temperature at Last Scattering Surface
Anisotropy spectrum today
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22. One more effect: Damping on small scales
But So
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23. On scales smaller than D (or k>kD) perturbations are damped
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24. When we see this, we conclude that modes were set in phase during inflation!
Bennett et al. 2003
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25. Polarization Polarization field decomposes into 2-modes E-mode B-mode
B-mode smoking gun signature of tensor perturbations,
dramatic proof of inflation... We will focus on E.
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26. Three Step argument for <TE>
Polarization proportional to quadrupole
Quadrupole proportional to dipole
Dipole out of phase with monopole
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27. Isotropic radiation field produces no polarization after Compton scattering
Modern Cosmology
Adapted from Hu & White 1997
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28. Radiation with a dipole produces no polarization
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29. A quadrupole is needed
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30. Quadrupole proportional to dipole
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31. Dipole is out of phase with monopole
Roughly,
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32. The product of monopole and dipole is initially positive (but small, since dipole vanishes as k goes to zero); and then switches signs several times.
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33. DASI initially detected TE signal
Kovac et al. 2002
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34. WMAP provided indisputable evidence that monopole and dipole are out of phase
Kogut et al.
2003
This is most remarkable for scales around l~100, which
were not in causal contact at recombination.
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35. Parameter I: Curvature
Same wavelength subtends smaller angle in an open universe
Peaks appear on smaller scales in open universe
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36. Parameters I: Curvature
As early as 1998, observations favored flat universe
DASI, Boomerang, Maxima (2001)
WMAP (2006)
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37. Parameters II Reionization lowers the signal on small scales
A tilted primordial spectrum (n<1) increasingly reduces signal on small scales
Tensors reduce the scalar normalization, and thus the small scale signal
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38. Parameters III Baryons accentuate odd/even peak disparity
Less matter implies changing potentials, greater driving force, higher peak amplitudes
Cosmological constant changes the distance to LSS
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39. E.g.: Baryon density Here, F is forcing term due to gravity.
As baryon density goes up, frequency goes down. Greater odd/even peak disparity.
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40. Bottom line Baryon Density agrees with BBN
There is ~5-6 times more dark matter than baryons
There is dark energy [since the universe is flat]
Primordial slope is less than one
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41. What have we learned from WMAP III
Polarization map
Later Epoch of Reionization
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42. Secondary Anisotropies in the Cosmic Microwave Background
Limber
Sunyaev-Zel’dovich
Gravitational Lensing
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43. Secondary Anisotropies Get Contributions From The Entire Line Of Sight
Temperature anisotropy angular distance  from z-axis
Weighting Function
Comoving distance 
Position dependent Source Function; e.g., Pressure or Gravitational Potential
Scott Dodelson PASI 2006

44. What is the power spectrum of a secondary anisotropy?
This is different from primary anisotropies; there Cl depended on  at last scattering
Many modes do not contribute to the power because of cancellations along the line of sight
A wonderful approximation is the Limber formula (1954)
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45. Derivation 2D Fourier Transform of temperature field
In the small angle limit variance of the Fourier transform is Cl
Integrate both sides over l’ and plug in:
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46. Do the l’ integral; use the delta function to do the ’ integral
Fourier transform S and use
to get
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47. Do the  integral to get a delta function and then d2k
Invoke physics
Mode with large kz
Mode with small kz
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48. … Leading To The Correct Answer
Of all modes with magnitude k, only those with kz small contribute
A ring of volume 2kdk/ contributes
This is a small fraction of all modes (4k2dk):
Secondary Anisotropies are suppressed by a factor of order l
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49. Courtesy Frank Bertoldi
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50. This is of the standard form with
Pressure
So we can immediately write the power spectrum:
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51. Computing the pressure power spectrum requires large N-Body/hydro simulations
Need good resolution to pick up low-mass objects and large volume to get massive clusters
Simulations have not yet converged
However, groups agree at l~2000 and agree that Cl very sensitive to 8 ( 87)
GADGET: Borgani et al.
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52. There is a tantalizing hint from CBI of a detection!
Geisbuesch et al. 2004
Bond et al. 2002
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53. If this high signal is due to SZ, amplitude of perturbations is high and matter density is low
Lensing
CBI (1- and 2- sigma)
Contaldi, Hoekstra, Lewis 2003
Komatsu & Seljak 2002
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54. WMAPIII has made thing worse
Amplitude σ8 is currently the most contentious cosmological parameter
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55. There’s more … Notice the difference between these 2 maps
Bennett et al. 2003
Da Silva et al. SZ simulation
Secondary anisotropies are non-Gaussian!
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56. One way to probe non-Gaussianity: Peaks (aka Cluster Counts)
m
8
Battye & Weller 2003
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57. This will provide tight constraints on Dark Energy parameters
South Pole
Telescope
SNAP
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58. Gravitational Lensing
We are used to discrete objects (galaxies, QSOs) being lensed.
How do we study the lensing of the temperature (a Gaussian field) at last scattering?
Einstein 1912!
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59. Gravitational Lensing of the Primordial CMB
Primordial unlensed temperature u is re-mapped to
where the deflection angle is a weighted integral of the gravitational potential along the line of sight
Scott Dodelson PASI 2006

60. Taylor expand … leading to a new term
This has a very familiar form (don’t worry about the prefactor: we know its RMS extremely well!)
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61. The obvious effect of redirection is to smooth out the peaks
Seljak 1996
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62. Leading to percent level changes of the acoustic peaks
Lewis 2005
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63. Again non-Gaussianities lead to new possibilities:
Consider the 2D Fourier transform of the temperature
Recall that ‘
Now though different Fourier modes are coupled! The quadratic combination
would vanish w/o lensing. Because of lensing, it serves as an estimator for the projected potential
Scott Dodelson PASI 2006

64. There have been improvements in reconstruction techniques
Hirata & Seljak 2003
Projected potential
Optimal Quadratic
Likelihood
Requires 1K/pixel noise w/beam smaller than 4’
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65. What can we do with this? Hirata & Seljak 2003
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66. Knox & Song; Kesden, Cooray, & Kamionkowski 2002
Can clean up B-mode contamination and measure even small tensor component
Probe inflation even if energy scale is low
Knox & Song; Kesden, Cooray, & Kamionkowski 2002
Scott Dodelson PASI 2006

67. We have only scratched the surface
Patchy Reionization
Cross-Correlating w/ Other probes
Data Analysis: Generalizing to non-Gaussianities
Observations on the horizon
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68. Conclusions
Primary temperature anisotropies well measured; determine cosmological parameters to unprecedented accuracy
E-mode of Polarization measured; improvements upcoming; B-mode is holy grail
Secondary anisotropies will dominate the upcoming decade
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69. Upcoming Experiments ACT 100 square degrees; 2muK/pixel; 2’ pixels
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70. Available at www.amazon.com
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