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Lecture 3 - Cosmological parameter dependence of the temperature - PowerPoint PPT Presentation

Lecture 3 - Cosmological parameter dependence of the temperature power spectrum - Polarisation of the CMB - Gravitational waves and their imprints on the CMB Planck Collaboration (2016) Silk+Fuziness Damping Sachs-Wolfe Sound Wave Planck


  1. Effects of Relativistic Neutrinos • To see the e ff ects of relativistic neutrinos, we artificially increase the number of neutrino species from 3 to 7 • Great energy density in neutrinos, i.e., greater energy density in radiation (1) • Longer radiation domination -> More ISW and boosts due to potential decay

  2. After correcting for more ISW and boosts due to potential decay

  3. (2): Viscosity Effect on the Amplitude of Sound Waves The solution is where Hu & Sugiyama (1996) Phase shift! Bashinsky & Seljak (2004)

  4. After correcting for the viscosity effect on the amplitude

  5. Bashinsky & Seljak (2004) (3): Change in the Silk Damping • Greater neutrino energy density implies greater Hubble expansion rate, Η 2 =8 π G ∑ρ α /3 • This reduces the sound horizon in proportion to H –1 , as r s ~ c s H –1 • This also reduces the di ff usion length, but in proportional to H –1/2 , as a/q silk ~ ( σ T n e H) –1/2 Consequence of the random walk! • As a result, l silk decreases relative to the first peak position , enhancing the Silk damping

  6. After correcting for the diffusion length

  7. Zoom in!

  8. (4): Viscosity Effect on the Phase of Sound Waves The solution is where Hu & Sugiyama (1996) Phase shift! Bashinsky & Seljak (2004)

  9. After correcting for the phase shift Now we understand everything quantitatively!!

  10. Two Other Effects • Spatial curvature • We have been assuming spatially-flat Universe with zero curvature (i.e., Euclidean space). What if it is curved? • Optical depth to Thomson scattering in a low-redshift Universe • We have been assuming that the Universe is transparent to photons since the last scattering at z=1090. What if there is an extra scattering in a low-redshift Universe?

  11. Spatial Curvature • It changes the angular diameter distance, d A , to the last scattering surface; namely, • r L -> d A = R sin (r L /R) = r L (1 – r L2 /6R 2 ) + … for positively- curved space • r L -> d A = R sinh (r L /R) = r L (1 + r L2 /6R 2 ) + … for negatively- curved space Smaller angles (larger multipoles) for a negatively curved Universe

  12. late-time ISW

  13. Optical Depth • Extra scattering by electrons in a low-redshift Universe damps temperature anisotropy • C l -> C l exp(–2 τ ) at l >~ 10 • where τ is the optical depth re-ionisation

  14. Important consequence of the optical depth • Since the power spectrum is uniformly suppressed by exp(–2 τ ) at l>~10, we cannot determine the amplitude of the power spectrum of the gravitational potential, P φ (q), independently of τ . • Namely, what we constrain is the combination: exp(–2 τ )P φ (q) ∝ exp( − 2 τ ) A s • Breaking this degeneracy requires an independent determination of the optical depth. This requires POLARISATION of the CMB.

  15. Cosmological Parameters Derived from the Power Spectrum Planck +CMB Lensing [100 Myr]

  16. CMB Polarisation • CMB is weakly polarised!

  17. Polarisation No polarisation Polarised in x-direction

  18. Photo Credit: TALEX

  19. Photo Credit: TALEX horizontally polarised

  20. Photo Credit: TALEX

  21. Necessary and sufficient conditions for generating polarisation • You need to have two things to produce linear polarisation 1. Scattering 2. Anisotropic incident light • However, the Universe does not have a preferred direction. How do we generate anisotropic incident light?

  22. Need for a local quadrupole temperature anisotropy Wayne Hu • How do we create a local temperature quadrupole?

  23. (l,m)=(2,0) (l,m)=(2,1) (l,m)=(2,2) Quadrupole temperature anisotropy seen from an electron

  24. Quadrupole Generation: A Punch Line • When Thomson scattering is e ffi cient (i.e., tight coupling between photons and baryons via electrons), the distribution of photons from the rest frame of baryons is isotropic • Only when tight coupling relaxes , a local quadrupole temperature anisotropy in the rest frame of a photon-baryon fluid can be generated • In fact, “a local temperature anisotropy in the rest frame of a photon-baryon fluid ” is equal to viscosity

  25. Stokes Parameters [Flat Sky, Cartesian coordinates] b a

  26. Stokes Parameters change under coordinate rotation y’ x’ Under (x,y) -> (x’,y’):

  27. Compact Expression • Using an imaginary number, write Then, under coordinate rotation we have

  28. Alternative Expression • With the polarisation amplitude, P , and angle, , defined by We write Then, under coordinate rotation we have and P is invariant under rotation

  29. E and B decomposition • That Q and U depend on coordinates is not very convenient… • Someone said, “I measured Q!” but then someone else may say, “No, it’s U!”. They flight to death, only to realise that their coordinates are 45 degrees rotated from one another… • The best way to avoid this unfortunate fight is to define a coordinate-independent quantity for the distribution of polarisation patterns in the sky To achieve this, we need to go to Fourier space

  30. n = (sin θ cos φ , sin θ sin φ , cos θ ) ˆ “Flat sky”, if θ is small

  31. Fourier-transforming Stokes Parameters? where • As Q+iU changes under rotation, the Fourier coe ffi cients change as well • So…

  32. (*) Nevermind the overall minus sign. This is just for convention Tweaking Fourier Transform where we write the coe ffi cients as(*) • Under rotation, the azimuthal angle of a Fourier wavevector, φ l , changes as • This cancels the factor in the left hand side:

  33. Seljak (1997); Zaldarriaga & Seljak (1997); Kamionkowski, Kosowky, Stebbins (1997) Tweaking Fourier Transform • We thus write • And, defining By construction E l and B l do not pick up a factor of exp(2i φ ) under coordinate rotation. That’s great! What kind of polarisation patterns do these quantities represent?

  34. Pure E, B Modes • Q and U produced by E and B modes are given by • Let’s consider Q and U that are produced by a single Fourier mode • Taking the x-axis to be the direction of a wavevector, we obtain

  35. Pure E, B Modes • Q and U produced by E and B modes are given by • Let’s consider Q and U that are produced by a single Fourier mode • Taking the x-axis to be the direction of a wavevector, we obtain

  36. Geometric Meaning (1) • E mode : Polarisation directions parallel or perpendicular to the wavevector • B mode : Polarisation directions 45 degree tilted with respect to the wavevector

  37. Geometric Meaning (2) • E mode : Stokes Q , defined with respect to as the x-axis • B mode : Stokes U , defined with respect to as the y-axis IMPORTANT : These are all coordinate-independent statements

  38. Parity • E mode : Parity even • B mode : Parity odd

  39. Parity • E mode : Parity even • B mode : Parity odd

  40. Power Spectra • However, <EB> and <TB> vanish for parity- preserving fluctuations because <EB> and <TB> change sign under parity flip

  41. Temperature from sound waves We understand this E-mode from sound waves B-mode from lensing B-mode from GW

  42. Temperature from sound waves We understand this E-mode from sound waves Today’s Lecture B-mode from lensing B-mode from GW

  43. The Single Most Important Thing You Need to Remember • Polarisation is generated by the local quadrupole temperature anisotropy , which is proportional to viscosity

  44. (l,m)=(2,0) (l,m)=(2,1) (l,m)=(2,2) Local quadrupole temperature anisotropy seen from an electron

  45. (l,m)=(2,0) (l,m)=(2,1) L e t ’ s ( l , s m y (l,m)=(2,2) m ) = b ( 2 o , 0 l Hot i ) s e a s Cold Cold Hot

  46. (l,m)=(2,0) (l,m)=(2,1) L e t ’ s ( l , s m y (l,m)=(2,2) m ) = b ( 2 o , 0 l i ) s e a s Polarisation pattern you will see

  47. r L Polarisation pattern in the sky generated by a single Fourier mode

  48. E-mode! r L Polarisation pattern in the sky generated by a single Fourier mode

  49. <latexit sha1_base64="x5ZMo5o+0rukqRjQKVd9YinM=">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</latexit> <latexit sha1_base64="x5ZMo5o+0rukqRjQKVd9YinM=">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</latexit> <latexit sha1_base64="x5ZMo5o+0rukqRjQKVd9YinM=">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</latexit> <latexit sha1_base64="x5ZMo5o+0rukqRjQKVd9YinM=">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</latexit> E-mode Power Spectrum • Viscosity at the last-scattering surface is given by the spatial gradient of the velocity : γ ρ γ = − 32 ¯ ∂ i ∂ j δ u γ σ T ¯ 45 n e • Velocity potential is Sin(qr L ) , whereas the temperature power spectrum is predominantly Cos(qr L )

  50. Bennett et al. (2013) WMAP 9-year Power Spectrum

  51. Planck Collaboration (2016) Planck 29-mo Power Spectrum

  52. South Pole Telescope Collaboration (2018) SPTPol Power Spectrum

  53. [1] Trough in T -> Peak in E because C lTT ~ cos 2 (qr s ) whereas C lEE ~ sin 2 (qr s ) [2] T damps -> E rises because T damps by viscosity, whereas E is created by viscosity [3] E Peaks are sharper because C lTT is the sum of cos 2 (qr L ) and Doppler shift’s sin 2 (qr L ), whereas C lEE is just sin 2 (qr L )

  54. [1] Trough in T -> Peak in E because C lTT ~ cos 2 (qr s ) whereas C lEE ~ sin 2 (qr s ) [2] T damps -> E rises because T damps by viscosity, whereas E is created by viscosity [3] E Peaks are sharper because C lTT is the sum of cos 2 (qr L ) and Doppler shift’s sin 2 (qr L ), whereas C lEE is just sin 2 (qr L )

  55. Polarisation from Re-ionisation

  56. Polarisation from Re-ionisation C lEE ~

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