Lecture 2. - Power spectrum of gravitational anisotropy - Temperature anisotropy from sound waves

Transcription

1 Lecture 2 - Power spectrum of gravitational anisotropy - Temperature anisotropy from sound waves

2 Bennett et al. (1996) COBE 4-year Power Spectrum The SW formula allows us to determine the 3d power spectrum of φ at the last scattering surface from Cl. But how?

3 Sachs & Wolfe (1967) T (ˆn) T 0 = 1 3 (t L, ˆr L ) SW Power Spectrum gives But this is not exactly what we want. We want the statistical average of this quantity.

4 Power Spectrum of φ Statistical average of the right hand side contains two-point correlation function If does not depend on locations (x) but only on separations between two points (r), then consequence of statistical homogeneity where we defined φ and used

5 Power Spectrum of φ In addition, if depends only on the magnitude of the separation r and not on the directions, then Power spectrum! Generic definition of the power spectrum for statistically homogeneous and isotropic fluctuations

6 SW Power Spectrum Thus, the power spectrum of the CMB in the SW limit is In the flat-sky approximation, Perpendicular wavenumber, (qperp) 2

7 SW Power Spectrum Thus, the power spectrum of the CMB in the SW limit is In the flat-sky approximation, For a power-law form,, we get

8 SW Power Spectrum Thus, the power spectrum of the CMB in the SW limit is In the flat-sky approximation, For a power-law form,, we get full-sky correction n=1

9 Bennett et al. (1996) n=1.2 ± 0.3 (68%CL) n=1

10 Bennett et al. (1996) COBE 4-year Power Spectrum

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

12 Planck Collaboration (2016) Planck 29-mo Power Spectrum

13 Planck Collaboration (2016) Planck 29-mo Power Spectrum Clearly, the SW prediction does not fit! Missing physics: Hydrodynamics (sound waves)

14

15 La Soupe Miso Cosmique When matter and radiation were hotter than 3000 K, matter was completely ionised. The Universe was filled with plasma, which behaves just like a soup Think about a Miso soup (if you know what it is). Imagine throwing Tofus into a Miso soup, while changing the density of Miso And imagine watching how ripples are created and propagate throughout the soup

16

17 viscous This is a fluid, in which the amplitude of damps sound waves at shorter wavelength

18

19 When do sound waves become important? In other words, when would the Sachs-Wolfe approximation (purely gravitational effects) become invalid? The key to the answer: Sound-crossing Time Sound waves cannot alter temperature anisotropy at a given angular scale if there was not enough time for sound waves to propagate to the corresponding distance at the last-scattering surface The distance traveled by sound waves within a given time = The Sound Horizon

20 Comoving Photon Horizon First, the comoving distance traveled by photons is given by setting the space-time distance to be null: ds 2 = c 2 dt 2 + a 2 (t)dr 2 =0 r photon = c Z t 0 dt 0 a(t 0 )

21 Comoving Sound Horizon Then, we replace the speed of light with a timedependent speed of sound: r s = Z t 0 dt 0 a(t 0 ) c s(t 0 ) We cannot ignore the effects of sound waves if qrs > 1

22 Sound Speed Sound speed of an adiabatic fluid is given by For a baryon-photon system: - δp: pressure perturbation - δρ: density perturbation We can ignore the baryon pressure because it is much smaller than the photon pressure

23 Sound Speed Using the adiabatic relationship between photons and baryons: [i.e., the ratio of the number densities of baryons and photons is equal everywhere] and pressure-density relation of a relativistic fluid, δpγ=δργ/3, We obtain Or equivalently sound speed is reduced! where

24 Value of R? The baryon mass density goes like a 3, whereas the photon energy density goes like a 4. Thus, the ratio of the two, R, goes like a. The proportionality constant is: where we used for

25 For the last-scattering redshift of zl=1090 Value of R? (or last-scattering temperature of TL=2974 K), The baryon mass density goes like a 3, whereas the photon energy density goes like a 4. Thus, the ratio of the two, R, goes like a. rs = Mpc We cannot ignore the effects of sound waves The proportionality constant is: if qrs>1. Since l~qrl, this means l > rl/rs = 96 where we used where we used rl=13.95 forgpc

26 Creation of Sound Waves: Basic Equations 1. Conservation equations (energy and momentum) 2. Equation of state, relating pressure to energy density P = P ( ) 3. General relativistic version of the Poisson equation, relating gravitational potential to energy density 4. Evolution of the anisotropic stress (viscosity)

27 Energy Conservation Total energy conservation: anisotropic stress: [or, viscosity] velocity potential v = 1 a r u ) C.f., Total energy conservation [unperturbed]

28 Energy Conservation Total energy conservation: Again, this is the effect of locally-defined inhomogeneous scale factor, i.e., The spatial metric is given by ds 2 = a 2 (t)exp( 2 )dx 2 Thus, locally we can define a new scale factor: ã(t, x) =a(t)exp( )

29 Energy Conservation Total energy conservation: Momentum flux going outward (inward) -> reduction (increase) in the energy density ( ) C.f., for a non-expanding medium: + r ( v) =0

30 Momentum Conservation Total momentum conservation v = 1 a r u Cosmological redshift of the momentum Gravitational force given by potential gradient Force given by pressure gradient Force given by gradient of anisotropic stress

31 Equation of State Pressure of non-relativistic species (i.e., baryons and cold dark matter) can be ignored relative to the energy density. Thus, we set them to zero: PB=0=PD and δpb=0=δpd Unperturbed pressure of relativistic species (i.e., photons and relativistic neutrinos) is given by the third of the energy density, i.e., Pγ=ργ/3 and Pν=ρν/3 Perturbed pressure involves contributions from the bulk viscosity: P = P =

32 The reason for this is that Equation of State trace of the stress-energy of relativistic species Pressure of non-relativistic species (i.e., baryons and cold dark vanishes: matter) can be ignored relative to the energy density. μ=0,1,2,3 Τμ μ = 0 Thus, we set them to zero: PB=0=PD and δpb=0=δpd 3X Unperturbed T pressure of relativistic species (i.e., photons and relativistic T neutrinos) i i = +3P + r 2 =0 is given by the third of the energy density, i=1i.e., Pγ=ργ/3 and Pν=ρν/3 Perturbed pressure involves contributions from the bulk viscosity: P = P =

33 Two Remarks In the standard scenario: Energy densities are conserved separately; thus we do not need to sum over all species Momentum densities of photons and baryons are NOT conserved separately but they are coupled via Thomson scattering. This must be taken into account when writing down separate conservation equations

34 Conservation Equations for Photons and Baryons Fourier transformation replaces r 2! q 2 momentum transfer via scattering

35 Conservation Equations for Photons and Baryons Fourier transformation replaces r 2! q 2 what about photon s viscosity?

36 Peebles & Yu (1970); Sunyaev & Zeldovich (1970) Formation of a Photon-baryon Fluid Photons are not a fluid. Photons free-stream at the speed of light The conservation equations are not enough because we need to specify the evolution of viscosity Solving for viscosity requires information of the phase-space distribution function of photons: Boltzmann equation However, frequent scattering of photons with baryons* can make photons behave as a fluid: Photon-baryon fluid *Photons scatter with electrons via Thomson scattering. Protons scatter with electrons via Coulomb scattering. Thus we can say, effectively, photons scatter with baryons

37 Let s solve them! Fourier transformation replaces r 2! q 2

38 Tight-coupling Approximation When Thomson scattering is efficient, the relative velocity between photons and baryons is small. We write [d is an arbitrary dimensionless variable] And take *. We obtain *In this limit, viscosity πγ is exponentially suppressed. This result comes from the Boltzmann equation but we do not derive it here. It makes sense physically.

39 Tight-coupling Approximation Eliminating d and using the fact that R is proportional to the scale factor, we obtain Using the energy conservation to replace δuγ with δργ/ργ, we obtain Wave Equation, with the speed of sound of cs 2 = 1/3(1+R)!

40 Peebles & Yu (1970); Sunyaev & Zeldovich (1970) Sound Wave! To simplify the equation, let s first look at the highfrequency solution Specifically, we take q >> ah (the wavelength of fluctuations is much shorter than the Hubble length). Then we can ignore time derivatives of R and Ψ because they evolve in the Hubble time scale: Solution: SOUND WAVE!

41 Recap Photons are not a fluid; but Thomson scattering couples photons to baryons, forming a photon-baryon fluid The reduced sound speed, cs 2 =1/3(1+R), emerges automatically δργ/4ργ is the temperature anisotropy at the bottom of the potential well. Adding gravitational redshift, the observed temperature anisotropy is δργ/4ργ + Φ, which is given by

42

43

44 Stone: Fluctuations entering the horizon This is a tricky concept, but it is important Suppose that there are fluctuations at all wavelengths, including the ones that exceed the Hubble length (which we loosely call our horizon ) Let s not ask the origin of these super-horizon fluctuations, but just assume their existence As the Universe expands, our horizon grows and we can see longer and longer wavelengths Fluctuations entering the horizon

45 Last scattering Radiation Era Matter Era 10 Gpc/h today 1 Gpc/h today 100 Mpc/h today 10 Mpc/h today 1 Mpc/h today enter the horizon

46 Three Regimes Super-horizon scales [q < ah] Only gravity is important Evolution differs from Newtonian Sub-horizon but super-sound-horizon [ah < q < ah/cs] Only gravity is important Evolution similar to Newtonian Sub-sound-horizon scales [q > ah/cs] Hydrodynamics important -> Sound waves

47 Bardeen, Steinhardt & Turner (1983); Weinberg (2003); Lyth, Malik & Sasaki (2005) ζ: Conserved on large scales For the adiabatic initial condition, there exists a useful quantity, ζ, which remains constant on large scales (super-horizon scales, q << ah) regardless of the contents of the Universe ζ is conserved regardless of whether the Universe is radiation-dominated, matter-dominated, or whatever Energy conservation for q << ah:

48 Bardeen, Steinhardt & Turner (1983); Weinberg (2003); Lyth, Malik & Sasaki (2005) ζ: Conserved on large scales If pressure is a function of the energy density only, i.e.,, then integration constant Integrate

49 Bardeen, Steinhardt & Turner (1983); Weinberg (2003); Lyth, Malik & Sasaki (2005) ζ: Conserved on large scales If pressure is a function of the energy density only, i.e.,, then integration constant For the adiabatic initial condition, all species share the same value of ζα, i.e., ζα=ζ

50 qeq Which fluctuation entered the horizon before the matterradiation equality? qeq = aeqheq ~ 0.01 (ΩMh 2 /0.14) Mpc 1 At the last scattering surface, this subtends the multipole of leq = qeqrl ~ 140

51 Entered the horizon during the radiation era

52 What determines the locations and heights of the peaks? Does the sound-wave solution explain it?

53 Peak Locations? High-frequency solution, for q >> ah VERY roughly speaking, the angular power spectrum Cl is given by [ ] 2 with q -> l/rl Question: What are the integration constants, A and B? Answer: They depend on the initial conditions; namely, adiabatic or not? For adiabatic initial condition, A >> B when q is large [We will show it later.]

54 Peak Locations? High-frequency solution, for q >> ah VERY roughly speaking, the angular power spectrum Cl is given by [ ] 2 with q -> l/rl If A>>B, the locations of peaks are

55

56 The simple estimates do not match! This is simply because these angular scales do not satisfy q >> ah, i.e, the oscillations are not pure cosine even for the adiabatic initial condition. We need a better solution!

57 Better Solution in Radiation-dominated Era Going back to the original tight-coupling equation.. In the radiation-dominated era, R << 1 Change the independent variable from the time (t) to

58 Better Solution in Radiation-dominated Era Then the equation simplifies to where In the radiation-dominated era, R << 1 Change the independent variable from the time (t) to

59 <latexit sha1_base64="durnugb/bp0jcyqiyolgayvouhg=">aaacn3icbvdlsgmxfm34rpu16tlnyjf2fi0ziuhgklpxjrxsazpdyarpg5pjhirtkep/yo2/4u43lhrx6x+ytoomrrdcts45l5t7gogsqrznxvhzxvvf2mxt5bd3dvf2zypdhusxqliooowifucjkwg4roiiubujdmoa4mywvjnqzrewknd2omyr9kpyz6rhefsa6ph3niss5i2giaaknn5f+ypl2x7isvsrlbnpo2ss2h2z4fscwvnlwe1baarv65jpxpejomrmiqqlbltoppweckuqxzo8f0scqtsefdzwkmeqsz+z7t2xtjxttxpc6mounwozhqkmpryhgxagua3kojyl/9pasepd+glhuawwq/nbvzhailvtek0uergpotyaikh0xy00gaiipapo6xdcxzwxqeos4jov9/68ul1o48iby3acssaff6akbken1aecj+avvimp48l4mz6nr7l1xuh7jscfmr5/apurq/a=</latexit> <latexit sha1_base64="durnugb/bp0jcyqiyolgayvouhg=">aaacn3icbvdlsgmxfm34rpu16tlnyjf2fi0ziuhgklpxjrxsazpdyarpg5pjhirtkep/yo2/4u43lhrx6x+ytoomrrdcts45l5t7gogsqrznxvhzxvvf2mxt5bd3dvf2zypdhusxqliooowifucjkwg4roiiubujdmoa4mywvjnqzrewknd2omyr9kpyz6rhefsa6ph3niss5i2giaaknn5f+ypl2x7isvsrlbnpo2ss2h2z4fscwvnlwe1baarv65jpxpejomrmiqqlbltoppweckuqxzo8f0scqtsefdzwkmeqsz+z7t2xtjxttxpc6mounwozhqkmpryhgxagua3kojyl/9pasepd+glhuawwq/nbvzhailvtek0uergpotyaikh0xy00gaiipapo6xdcxzwxqeos4jov9/68ul1o48iby3acssaff6akbken1aecj+avvimp48l4mz6nr7l1xuh7jscfmr5/apurq/a=</latexit> <latexit sha1_base64="durnugb/bp0jcyqiyolgayvouhg=">aaacn3icbvdlsgmxfm34rpu16tlnyjf2fi0ziuhgklpxjrxsazpdyarpg5pjhirtkep/yo2/4u43lhrx6x+ytoomrrdcts45l5t7gogsqrznxvhzxvvf2mxt5bd3dvf2zypdhusxqliooowifucjkwg4roiiubujdmoa4mywvjnqzrewknd2omyr9kpyz6rhefsa6ph3niss5i2giaaknn5f+ypl2x7isvsrlbnpo2ss2h2z4fscwvnlwe1baarv65jpxpejomrmiqqlbltoppweckuqxzo8f0scqtsefdzwkmeqsz+z7t2xtjxttxpc6mounwozhqkmpryhgxagua3kojyl/9pasepd+glhuawwq/nbvzhailvtek0uergpotyaikh0xy00gaiipapo6xdcxzwxqeos4jov9/68ul1o48iby3acssaff6akbken1aecj+avvimp48l4mz6nr7l1xuh7jscfmr5/apurq/a=</latexit> <latexit sha1_base64="durnugb/bp0jcyqiyolgayvouhg=">aaacn3icbvdlsgmxfm34rpu16tlnyjf2fi0ziuhgklpxjrxsazpdyarpg5pjhirtkep/yo2/4u43lhrx6x+ytoomrrdcts45l5t7gogsqrznxvhzxvvf2mxt5bd3dvf2zypdhusxqliooowifucjkwg4roiiubujdmoa4mywvjnqzrewknd2omyr9kpyz6rhefsa6ph3niss5i2giaaknn5f+ypl2x7isvsrlbnpo2ss2h2z4fscwvnlwe1baarv65jpxpejomrmiqqlbltoppweckuqxzo8f0scqtsefdzwkmeqsz+z7t2xtjxttxpc6mounwozhqkmpryhgxagua3kojyl/9pasepd+glhuawwq/nbvzhailvtek0uergpotyaikh0xy00gaiipapo6xdcxzwxqeos4jov9/68ul1o48iby3acssaff6akbken1aecj+avvimp48l4mz6nr7l1xuh7jscfmr5/apurq/a=</latexit> Better Solution in Radiation-dominated Era Then the equation simplifies to where The solution is We rewrite this using the formula for trigonometry: sin(' ' 0 )=sin(') cos(' 0 ) cos(')sin(' 0 )

60 Better Solution in Radiation-dominated Era Then the equation simplifies to where The solution is where

61 Einstein s Equations Now we need to know Newton s gravitational potential, φ, and the scalar curvature perturbation, ψ. Einstein s equations - let s look up any text books:

62 Einstein s Equations Now we need to know Newton s gravitational potential, φ, and the scalar curvature perturbation, ψ. Einstein s equations - let s look up any text books:

63 Einstein s Equations Now we need to know Newton s gravitational potential, φ, and the scalar curvature perturbation, ψ. Einstein s equations - let s look up any text books: Will come back to this later. For now, let s ignore any viscosity.

64 Einstein s Equations Now we need to know Newton s gravitational potential, φ, and the scalar curvature perturbation, ψ. Einstein s equations - let s look up any text books: Will come back to this later. For now, let s ignore any viscosity.

65 Einstein s Equations in Radiation-dominated Era Now we need to know Newton s gravitational potential, φ, and the scalar curvature perturbation, ψ. Einstein s equations - let s look up any text books: non-adiabatic pressure

66 Einstein s Equations in Radiation-dominated Era Now we need to know Newton s gravitational potential, φ, and the scalar curvature perturbation, ψ. Einstein s equations - let s look up any text books: We shall ignore this non-adiabatic pressure

67 Kodama & Sasaki (1986, 1987) Solution (Adiabatic) in Radiation-dominated Era ADI where Low-frequency limit (super-sound-horizon scales, qrs << 1) ΦADI -> 2ζ/3 = constant High-frequency limit (sub-sound-horizon scales, qrs >> 1) ΦADI -> 2ζ damp

68 Solution (Adiabatic) in Radiation-dominated Era ADI where Poisson Equation Low-frequency limit (super-sound-horizon scales, qrs << 1) ΦADI -> 2ζ/3 = constant & oscillation solution for radiation High-frequency limit (sub-sound-horizon scales, qrs >> 1) ΦADI -> 2ζ damp

69 Solution (Adiabatic) in Radiation-dominated Era ADI where Low-frequency limit (super-sound-horizon scales, qrs << 1) ΦADI -> 2ζ/3 = constant High-frequency limit (sub-sound-horizon scales, qrs >> 1) ΦADI -> 2ζ damp

70 Kodama & Sasaki (1986, 1987); Baumann, Green, Meyers & Wallisch (2016) Sound Wave Solution in the Radiation-dominated Era The solution is where

71 Kodama & Sasaki (1986, 1987); Baumann, Green, Meyers & Wallisch (2016) Sound Wave Solution in the Radiation-dominated Era The solution is where

72 Kodama & Sasaki (1986, 1987); Baumann, Green, Meyers & Wallisch (2016) Sound Wave Solution in the Radiation-dominated Era The solution is where i.e., ADI ADI

73 Kodama & Sasaki (1986, 1987); Baumann, Green, Meyers & Wallisch (2016) Sound Wave Solution in the Radiation-dominated Era The adiabatic solution is with Therefore, the solution is a pure cosine only in the high-frequency limit!

74

75 Roles of viscosity Neutrino viscosity Modify potentials: Photon viscosity Viscous photon-baryon fluid: damping of sound waves Silk (1968) Silk damping

76 High-frequency solution without neutrino viscosity The solution is where ' 1 <latexit sha1_base64="bnhabzp0k7zxjmggnjoef5tgyl8=">aaab83icbvbns8naej3ur1q/qh69lbbbu0le0gpri8ck9goaudbbtbp0s1l2n4us+je8efdeq3/gm//gtzudtj4yelw3w8y8uhkmjet+o5wnza3tnepubw//4pcofnzs1wmmco2qlkeqh2jnoro0y5jhtc8vxunias+c3bd+b0qvzql4mjnjgwthgkwmygml359ijcfmj2pkdesnt+kugnajv5iglggp61/+kcvzqouhhgs98fxpghwrwwin85qfasoxmecydiwvoke6ybc3z9gfvuyosputydbc/t2r40trwrlazgsbsv71cve/b5cz6dbimzczoyisf0uzryzfrqboxbqlhs8swuqxeysiy6wwmtammg3bw315nxsvmp7b9b6vg627mo4qnme5xiihn9ccb2hdbwhieizxehmy58v5dz6wrrwnndmfp3a+fwceujft</latexit> <latexit sha1_base64="bnhabzp0k7zxjmggnjoef5tgyl8=">aaab83icbvbns8naej3ur1q/qh69lbbbu0le0gpri8ck9goaudbbtbp0s1l2n4us+je8efdeq3/gm//gtzudtj4yelw3w8y8uhkmjet+o5wnza3tnepubw//4pcofnzs1wmmco2qlkeqh2jnoro0y5jhtc8vxunias+c3bd+b0qvzql4mjnjgwthgkwmygml359ijcfmj2pkdesnt+kugnajv5iglggp61/+kcvzqouhhgs98fxpghwrwwin85qfasoxmecydiwvoke6ybc3z9gfvuyosputydbc/t2r40trwrlazgsbsv71cve/b5cz6dbimzczoyisf0uzryzfrqboxbqlhs8swuqxeysiy6wwmtammg3bw315nxsvmp7b9b6vg627mo4qnme5xiihn9ccb2hdbwhieizxehmy58v5dz6wrrwnndmfp3a+fwceujft</latexit> <latexit sha1_base64="bnhabzp0k7zxjmggnjoef5tgyl8=">aaab83icbvbns8naej3ur1q/qh69lbbbu0le0gpri8ck9goaudbbtbp0s1l2n4us+je8efdeq3/gm//gtzudtj4yelw3w8y8uhkmjet+o5wnza3tnepubw//4pcofnzs1wmmco2qlkeqh2jnoro0y5jhtc8vxunias+c3bd+b0qvzql4mjnjgwthgkwmygml359ijcfmj2pkdesnt+kugnajv5iglggp61/+kcvzqouhhgs98fxpghwrwwin85qfasoxmecydiwvoke6ybc3z9gfvuyosputydbc/t2r40trwrlazgsbsv71cve/b5cz6dbimzczoyisf0uzryzfrqboxbqlhs8swuqxeysiy6wwmtammg3bw315nxsvmp7b9b6vg627mo4qnme5xiihn9ccb2hdbwhieizxehmy58v5dz6wrrwnndmfp3a+fwceujft</latexit> <latexit sha1_base64="bnhabzp0k7zxjmggnjoef5tgyl8=">aaab83icbvbns8naej3ur1q/qh69lbbbu0le0gpri8ck9goaudbbtbp0s1l2n4us+je8efdeq3/gm//gtzudtj4yelw3w8y8uhkmjet+o5wnza3tnepubw//4pcofnzs1wmmco2qlkeqh2jnoro0y5jhtc8vxunias+c3bd+b0qvzql4mjnjgwthgkwmygml359ijcfmj2pkdesnt+kugnajv5iglggp61/+kcvzqouhhgs98fxpghwrwwin85qfasoxmecydiwvoke6ybc3z9gfvuyosputydbc/t2r40trwrlazgsbsv71cve/b5cz6dbimzczoyisf0uzryzfrqboxbqlhs8swuqxeysiy6wwmtammg3bw315nxsvmp7b9b6vg627mo4qnme5xiihn9ccb2hdbwhieizxehmy58v5dz6wrrwnndmfp3a+fwceujft</latexit> ' 1 <latexit sha1_base64="bnhabzp0k7zxjmggnjoef5tgyl8=">aaab83icbvbns8naej3ur1q/qh69lbbbu0le0gpri8ck9goaudbbtbp0s1l2n4us+je8efdeq3/gm//gtzudtj4yelw3w8y8uhkmjet+o5wnza3tnepubw//4pcofnzs1wmmco2qlkeqh2jnoro0y5jhtc8vxunias+c3bd+b0qvzql4mjnjgwthgkwmygml359ijcfmj2pkdesnt+kugnajv5iglggp61/+kcvzqouhhgs98fxpghwrwwin85qfasoxmecydiwvoke6ybc3z9gfvuyosputydbc/t2r40trwrlazgsbsv71cve/b5cz6dbimzczoyisf0uzryzfrqboxbqlhs8swuqxeysiy6wwmtammg3bw315nxsvmp7b9b6vg627mo4qnme5xiihn9ccb2hdbwhieizxehmy58v5dz6wrrwnndmfp3a+fwceujft</latexit> <latexit sha1_base64="bnhabzp0k7zxjmggnjoef5tgyl8=">aaab83icbvbns8naej3ur1q/qh69lbbbu0le0gpri8ck9goaudbbtbp0s1l2n4us+je8efdeq3/gm//gtzudtj4yelw3w8y8uhkmjet+o5wnza3tnepubw//4pcofnzs1wmmco2qlkeqh2jnoro0y5jhtc8vxunias+c3bd+b0qvzql4mjnjgwthgkwmygml359ijcfmj2pkdesnt+kugnajv5iglggp61/+kcvzqouhhgs98fxpghwrwwin85qfasoxmecydiwvoke6ybc3z9gfvuyosputydbc/t2r40trwrlazgsbsv71cve/b5cz6dbimzczoyisf0uzryzfrqboxbqlhs8swuqxeysiy6wwmtammg3bw315nxsvmp7b9b6vg627mo4qnme5xiihn9ccb2hdbwhieizxehmy58v5dz6wrrwnndmfp3a+fwceujft</latexit> <latexit sha1_base64="bnhabzp0k7zxjmggnjoef5tgyl8=">aaab83icbvbns8naej3ur1q/qh69lbbbu0le0gpri8ck9goaudbbtbp0s1l2n4us+je8efdeq3/gm//gtzudtj4yelw3w8y8uhkmjet+o5wnza3tnepubw//4pcofnzs1wmmco2qlkeqh2jnoro0y5jhtc8vxunias+c3bd+b0qvzql4mjnjgwthgkwmygml359ijcfmj2pkdesnt+kugnajv5iglggp61/+kcvzqouhhgs98fxpghwrwwin85qfasoxmecydiwvoke6ybc3z9gfvuyosputydbc/t2r40trwrlazgsbsv71cve/b5cz6dbimzczoyisf0uzryzfrqboxbqlhs8swuqxeysiy6wwmtammg3bw315nxsvmp7b9b6vg627mo4qnme5xiihn9ccb2hdbwhieizxehmy58v5dz6wrrwnndmfp3a+fwceujft</latexit> <latexit sha1_base64="bnhabzp0k7zxjmggnjoef5tgyl8=">aaab83icbvbns8naej3ur1q/qh69lbbbu0le0gpri8ck9goaudbbtbp0s1l2n4us+je8efdeq3/gm//gtzudtj4yelw3w8y8uhkmjet+o5wnza3tnepubw//4pcofnzs1wmmco2qlkeqh2jnoro0y5jhtc8vxunias+c3bd+b0qvzql4mjnjgwthgkwmygml359ijcfmj2pkdesnt+kugnajv5iglggp61/+kcvzqouhhgs98fxpghwrwwin85qfasoxmecydiwvoke6ybc3z9gfvuyosputydbc/t2r40trwrlazgsbsv71cve/b5cz6dbimzczoyisf0uzryzfrqboxbqlhs8swuqxeysiy6wwmtammg3bw315nxsvmp7b9b6vg627mo4qnme5xiihn9ccb2hdbwhieizxehmy58v5dz6wrrwnndmfp3a+fwceujft</latexit>

77 Chluba & Grin (2013) High-frequency solution with neutrino viscosity The solution is where ' 1 <latexit sha1_base64="bnhabzp0k7zxjmggnjoef5tgyl8=">aaab83icbvbns8naej3ur1q/qh69lbbbu0le0gpri8ck9goaudbbtbp0s1l2n4us+je8efdeq3/gm//gtzudtj4yelw3w8y8uhkmjet+o5wnza3tnepubw//4pcofnzs1wmmco2qlkeqh2jnoro0y5jhtc8vxunias+c3bd+b0qvzql4mjnjgwthgkwmygml359ijcfmj2pkdesnt+kugnajv5iglggp61/+kcvzqouhhgs98fxpghwrwwin85qfasoxmecydiwvoke6ybc3z9gfvuyosputydbc/t2r40trwrlazgsbsv71cve/b5cz6dbimzczoyisf0uzryzfrqboxbqlhs8swuqxeysiy6wwmtammg3bw315nxsvmp7b9b6vg627mo4qnme5xiihn9ccb2hdbwhieizxehmy58v5dz6wrrwnndmfp3a+fwceujft</latexit> <latexit sha1_base64="bnhabzp0k7zxjmggnjoef5tgyl8=">aaab83icbvbns8naej3ur1q/qh69lbbbu0le0gpri8ck9goaudbbtbp0s1l2n4us+je8efdeq3/gm//gtzudtj4yelw3w8y8uhkmjet+o5wnza3tnepubw//4pcofnzs1wmmco2qlkeqh2jnoro0y5jhtc8vxunias+c3bd+b0qvzql4mjnjgwthgkwmygml359ijcfmj2pkdesnt+kugnajv5iglggp61/+kcvzqouhhgs98fxpghwrwwin85qfasoxmecydiwvoke6ybc3z9gfvuyosputydbc/t2r40trwrlazgsbsv71cve/b5cz6dbimzczoyisf0uzryzfrqboxbqlhs8swuqxeysiy6wwmtammg3bw315nxsvmp7b9b6vg627mo4qnme5xiihn9ccb2hdbwhieizxehmy58v5dz6wrrwnndmfp3a+fwceujft</latexit> <latexit sha1_base64="bnhabzp0k7zxjmggnjoef5tgyl8=">aaab83icbvbns8naej3ur1q/qh69lbbbu0le0gpri8ck9goaudbbtbp0s1l2n4us+je8efdeq3/gm//gtzudtj4yelw3w8y8uhkmjet+o5wnza3tnepubw//4pcofnzs1wmmco2qlkeqh2jnoro0y5jhtc8vxunias+c3bd+b0qvzql4mjnjgwthgkwmygml359ijcfmj2pkdesnt+kugnajv5iglggp61/+kcvzqouhhgs98fxpghwrwwin85qfasoxmecydiwvoke6ybc3z9gfvuyosputydbc/t2r40trwrlazgsbsv71cve/b5cz6dbimzczoyisf0uzryzfrqboxbqlhs8swuqxeysiy6wwmtammg3bw315nxsvmp7b9b6vg627mo4qnme5xiihn9ccb2hdbwhieizxehmy58v5dz6wrrwnndmfp3a+fwceujft</latexit> <latexit sha1_base64="bnhabzp0k7zxjmggnjoef5tgyl8=">aaab83icbvbns8naej3ur1q/qh69lbbbu0le0gpri8ck9goaudbbtbp0s1l2n4us+je8efdeq3/gm//gtzudtj4yelw3w8y8uhkmjet+o5wnza3tnepubw//4pcofnzs1wmmco2qlkeqh2jnoro0y5jhtc8vxunias+c3bd+b0qvzql4mjnjgwthgkwmygml359ijcfmj2pkdesnt+kugnajv5iglggp61/+kcvzqouhhgs98fxpghwrwwin85qfasoxmecydiwvoke6ybc3z9gfvuyosputydbc/t2r40trwrlazgsbsv71cve/b5cz6dbimzczoyisf0uzryzfrqboxbqlhs8swuqxeysiy6wwmtammg3bw315nxsvmp7b9b6vg627mo4qnme5xiihn9ccb2hdbwhieizxehmy58v5dz6wrrwnndmfp3a+fwceujft</latexit> ' 1 <latexit sha1_base64="bnhabzp0k7zxjmggnjoef5tgyl8=">aaab83icbvbns8naej3ur1q/qh69lbbbu0le0gpri8ck9goaudbbtbp0s1l2n4us+je8efdeq3/gm//gtzudtj4yelw3w8y8uhkmjet+o5wnza3tnepubw//4pcofnzs1wmmco2qlkeqh2jnoro0y5jhtc8vxunias+c3bd+b0qvzql4mjnjgwthgkwmygml359ijcfmj2pkdesnt+kugnajv5iglggp61/+kcvzqouhhgs98fxpghwrwwin85qfasoxmecydiwvoke6ybc3z9gfvuyosputydbc/t2r40trwrlazgsbsv71cve/b5cz6dbimzczoyisf0uzryzfrqboxbqlhs8swuqxeysiy6wwmtammg3bw315nxsvmp7b9b6vg627mo4qnme5xiihn9ccb2hdbwhieizxehmy58v5dz6wrrwnndmfp3a+fwceujft</latexit> <latexit sha1_base64="bnhabzp0k7zxjmggnjoef5tgyl8=">aaab83icbvbns8naej3ur1q/qh69lbbbu0le0gpri8ck9goaudbbtbp0s1l2n4us+je8efdeq3/gm//gtzudtj4yelw3w8y8uhkmjet+o5wnza3tnepubw//4pcofnzs1wmmco2qlkeqh2jnoro0y5jhtc8vxunias+c3bd+b0qvzql4mjnjgwthgkwmygml359ijcfmj2pkdesnt+kugnajv5iglggp61/+kcvzqouhhgs98fxpghwrwwin85qfasoxmecydiwvoke6ybc3z9gfvuyosputydbc/t2r40trwrlazgsbsv71cve/b5cz6dbimzczoyisf0uzryzfrqboxbqlhs8swuqxeysiy6wwmtammg3bw315nxsvmp7b9b6vg627mo4qnme5xiihn9ccb2hdbwhieizxehmy58v5dz6wrrwnndmfp3a+fwceujft</latexit> <latexit sha1_base64="bnhabzp0k7zxjmggnjoef5tgyl8=">aaab83icbvbns8naej3ur1q/qh69lbbbu0le0gpri8ck9goaudbbtbp0s1l2n4us+je8efdeq3/gm//gtzudtj4yelw3w8y8uhkmjet+o5wnza3tnepubw//4pcofnzs1wmmco2qlkeqh2jnoro0y5jhtc8vxunias+c3bd+b0qvzql4mjnjgwthgkwmygml359ijcfmj2pkdesnt+kugnajv5iglggp61/+kcvzqouhhgs98fxpghwrwwin85qfasoxmecydiwvoke6ybc3z9gfvuyosputydbc/t2r40trwrlazgsbsv71cve/b5cz6dbimzczoyisf0uzryzfrqboxbqlhs8swuqxeysiy6wwmtammg3bw315nxsvmp7b9b6vg627mo4qnme5xiihn9ccb2hdbwhieizxehmy58v5dz6wrrwnndmfp3a+fwceujft</latexit> <latexit sha1_base64="bnhabzp0k7zxjmggnjoef5tgyl8=">aaab83icbvbns8naej3ur1q/qh69lbbbu0le0gpri8ck9goaudbbtbp0s1l2n4us+je8efdeq3/gm//gtzudtj4yelw3w8y8uhkmjet+o5wnza3tnepubw//4pcofnzs1wmmco2qlkeqh2jnoro0y5jhtc8vxunias+c3bd+b0qvzql4mjnjgwthgkwmygml359ijcfmj2pkdesnt+kugnajv5iglggp61/+kcvzqouhhgs98fxpghwrwwin85qfasoxmecydiwvoke6ybc3z9gfvuyosputydbc/t2r40trwrlazgsbsv71cve/b5cz6dbimzczoyisf0uzryzfrqboxbqlhs8swuqxeysiy6wwmtammg3bw315nxsvmp7b9b6vg627mo4qnme5xiihn9ccb2hdbwhieizxehmy58v5dz6wrrwnndmfp3a+fwceujft</latexit> non-zero value!

78 High-frequency solution with neutrino viscosity Using the formula for trigonometry, we write where Hu & Sugiyama (1996) Phase shift! Bashinsky & Seljak (2004)

79 High-frequency solution with neutrino viscosity The solution is Thus, the neutrino viscosity will: where (1) Reduce the amplitude of sound waves at large multipoles (2) Shift the peak positions of the temperature power Hu & Sugiyama spectrum (1996) Phase shift! Bashinsky & Seljak (2004)

80 Photon Viscosity In the tight-coupling approximation, the photon viscosity damps exponentially To take into account a non-zero photon viscosity, we go to a higher order in the tight-coupling approximation

81 Yesterday: Tight-coupling Approximation (1st-order) When Thomson scattering is efficient, the relative velocity between photons and baryons is small. We write [d is an arbitrary dimensionless variable] And take *. We obtain *In this limit, viscosity πγ is exponentially suppressed. This result comes from the Boltzmann equation but we do not derive it here. It makes sense physically.

82 Today: Tight-coupling Approximation (2nd-order) When Thomson scattering is efficient, the relative velocity between photons and baryons is small. We write where [d2 is an arbitrary dimensionless variables] And take.. We obtain

83 <latexit sha1_base64="x5zmo5o+00rukqrjqkkkvd9yinm=">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</latexit> <latexit sha1_base64="x5zmo5o+00rukqrjqkkkvd9yinm=">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</latexit> <latexit sha1_base64="x5zmo5o+00rukqrjqkkkvd9yinm=">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</latexit> <latexit sha1_base64="x5zmo5o+00rukqrjqkkkvd9yinm=">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</latexit> Tight-coupling Approximation (2nd-order) Eliminating d2 and using the fact that R is proportional to the scale factor, we obtain Getting πγ requires an approximate solution of the Boltzmann equation in the 2nd-order tight coupling. We do not derive it here. The answer is γ = T n j u γ Kaiser (1983)

84 <latexit sha1_base64="x5zmo5o+00rukqrjqkkkvd9yinm=">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</latexit> <latexit sha1_base64="x5zmo5o+00rukqrjqkkkvd9yinm=">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</latexit> <latexit sha1_base64="x5zmo5o+00rukqrjqkkkvd9yinm=">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</latexit> <latexit sha1_base64="x5zmo5o+00rukqrjqkkkvd9yinm=">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</latexit> Tight-coupling Approximation (2nd-order) Eliminating d2 and using the fact that R is proportional to the scale factor, we obtain Getting πγ requires an approximate solution of the Boltzmann equation in the 2nd-order tight coupling. We do not derive it here. The answer is Given by the spatial gradient of the velocity field - a well-known result in fluid dynamics γ = T n j u γ Kaiser (1983)

85 Damped Oscillator Using the energy conservation to replace δuγ with δργ/ργ, we obtain, for q >> ah, New term, giving damping! where

86 Damped Oscillator Using the energy conservation to replace δuγ with δργ/ργ, we obtain, for q >> ah, New term, giving damping! where Important for high frequencies (large multipoles)

87 <latexit sha1_base64="ruibapbz1adjb7lbqmlyrbmp6vk=">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</latexit> <latexit sha1_base64="ruibapbz1adjb7lbqmlyrbmp6vk=">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</latexit> <latexit sha1_base64="ruibapbz1adjb7lbqmlyrbmp6vk=">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</latexit> <latexit sha1_base64="ruibapbz1adjb7lbqmlyrbmp6vk=">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</latexit> Damped Oscillator Using the energy conservation to replace δuγ with δργ/ργ, we obtain, for q >> ah, New term, giving damping! SOLUTION: Exponential dampling! exp q 2 / T n e H

88 Damped Oscillator Using the energy conservation to replace δuγ with δργ/ργ, we obtain, for q >> ah, New term, giving damping! SOLUTION: Silk Exponential dampling! Silk diffusion length = length traveled by photon s random walks

89 Diffusion Length The mean free path of the photon between scatterings is (σtne) 1 Below this scale, you do not have a photon-baryon fluid The number of scatterings per Hubble time is Nscattering=σTne/H Then, the length traveled by photons by random walks within the Hubble time is (σtne) 1 times Nscatterings The diffusion length is thus (σtne) 1 times Nscatterings = (σtneh) 1/2

90 Diffusion Damping by Wayne Hu Diffusion mixes hot and cold photons -> Damping of anisotropies

91 Planck Collaboration (2016) Silk Damping? Sachs-Wolfe Sound Wave

92 Additional Damping The power spectrum is [ is thus exp( 2q 2 /qsilk 2 ) ] 2 with q -> l/rl. The damping factor qsilk(tl) = Mpc 1. This corresponds to a multipole of lsilk ~ qsilk rl/ 2 = Seems too large, compared to the exact calculation There is an additional damping due to a finite width of the last scattering surface, σ~250 K Fuzziness damping Bond (1996) Landau damping - Weinberg (2001) fuzziness ( )

93 Planck Collaboration (2016) Total damping: qd 2 = qsilk 2 + qfuzziness 2 qd ~ 0.11 Mpc 1, giving ld ~ qdrl/ 2 ~ 1125 Silk+Fuzziness Damping Sachs-Wolfe Sound Wave

94 Recap The basic structure of the temperature power spectrum is The Sachs-Wolfe plateau at low multipoles Sound waves at intermediate multipoles 1st-order tight-coupling Silk damping and Fuzziness damping at high multipoles 2nd-order tight-coupling