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Cosmic Microwave Background

Cosmic Microwave Background. Primary Temperature Anisotropies Polarization Secondary Anisotropies. Coherent Picture Of Formation Of Structure In The Universe. Photons freestream: Inhomogeneities turn into anisotropies. t ~100,000 years.

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Cosmic Microwave Background

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  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 Scott Dodelson PASI 2006

  3. Goal: Explain the Physics and Ramifications of this Plot Scott Dodelson PASI 2006

  4. Notation • Scale Factor a(t) • Conformal time/comoving horizon • Gravitational Potential  • Photon distribution  • Fourier transforms with kcomoving wavenumber • Wavelength k-1 Scott Dodelson PASI 2006

  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 … Scott Dodelson PASI 2006

  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 Scott Dodelson PASI 2006

  7. Can rewrite  as integral over Hubble radius (aH)-1 Perturbations outside the horizon Scott Dodelson PASI 2006

  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 Scott Dodelson PASI 2006

  9. To see how perturbations evolve, need to solve an infinite hierarchy of coupled differential equations Perturbations in metric induce photon, dark matter perturbations Scott Dodelson PASI 2006

  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) Scott Dodelson PASI 2006

  11. Before recombination, electrons and photons are tightly coupled: equations reduce to Temperature perturbation Very similar to … Displacement of a string Scott Dodelson PASI 2006

  12. What spectrum is produced by a stringed instrument? C string on a ukulele Scott Dodelson PASI 2006

  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! Scott Dodelson PASI 2006

  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 Scott Dodelson PASI 2006

  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 Scott Dodelson PASI 2006

  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? Scott Dodelson PASI 2006

  17. Without synchronization: First “Trough” First “Peak” Scott Dodelson PASI 2006

  18. Inflation synchronizes all modes  All modes remain constant until they re-enter horizon. Scott Dodelson PASI 2006

  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 Scott Dodelson PASI 2006

  20. The spectrum at last scattering is: Scott Dodelson PASI 2006

  21. Fourier transformof temperature atLast Scattering Surface Anisotropy spectrum today Scott Dodelson PASI 2006

  22. One more effect: Damping on small scales But So Scott Dodelson PASI 2006

  23. On scales smaller than D (or k>kD) perturbations are damped Scott Dodelson PASI 2006

  24. When we see this, we conclude that modes were set in phase during inflation! Bennett et al. 2003 Scott Dodelson PASI 2006

  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. Scott Dodelson PASI 2006

  26. Three Step argument for <TE> • Polarization proportional to quadrupole • Quadrupole proportional to dipole • Dipole out of phase with monopole Scott Dodelson PASI 2006

  27. Isotropic radiation field produces no polarization after Compton scattering Modern Cosmology Adapted from Hu & White 1997 Scott Dodelson PASI 2006

  28. Radiation with a dipole produces no polarization Scott Dodelson PASI 2006

  29. A quadrupole is needed Scott Dodelson PASI 2006

  30. Quadrupole proportional to dipole Scott Dodelson PASI 2006

  31. Dipole is out of phase with monopole Roughly, Scott Dodelson PASI 2006

  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. Scott Dodelson PASI 2006

  33. DASI initially detected TE signal Kovac et al. 2002 Scott Dodelson PASI 2006

  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. Scott Dodelson PASI 2006

  35. Parameter I: Curvature • Same wavelength subtends smaller angle in an open universe • Peaks appear on smaller scales in open universe Scott Dodelson PASI 2006

  36. Parameters I: Curvature As early as 1998, observations favored flat universe DASI, Boomerang, Maxima (2001) WMAP (2006) Scott Dodelson PASI 2006

  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 Scott Dodelson PASI 2006

  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 Scott Dodelson PASI 2006

  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. Scott Dodelson PASI 2006

  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 Scott Dodelson PASI 2006

  41. What have we learned from WMAP III • n<1 • Polarization map • Later Epoch of Reionization Scott Dodelson PASI 2006

  42. Secondary Anisotropiesin the Cosmic Microwave Background Limber Sunyaev-Zel’dovich Gravitational Lensing Scott Dodelson PASI 2006

  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) Scott Dodelson PASI 2006

  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: Scott Dodelson PASI 2006

  46. Do the l’ integral; use the delta function to do the ’ integral Fourier transform S and use to get Scott Dodelson PASI 2006

  47. Do the  integral to get a delta function and then d2k Invoke physics Mode with large kz Mode with small kz Scott Dodelson PASI 2006

  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 Scott Dodelson PASI 2006

  49. Courtesy Frank Bertoldi Scott Dodelson PASI 2006

  50. This is of the standard form with Pressure So we can immediately write the power spectrum: Scott Dodelson PASI 2006

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