Lecture 7: Details of the Acoustic Oscillation
1
The lecture slides are available at
https://wwwmpa.mpa-garching.mpg.de/~komatsu/
lectures--reviews.html
2
What did the stones represent?
•
A stone is dropped when a fluctuation“enters the horizon”.
•
In a decelerating Universe, we can see more of the Universe as time goes by.•
New, longer wavelengthfluctuations keep entering the
horizon, perturbing the photon-baryon fluid.
10 Gpc today 1 Gpc today 100 Mpc today
10 Mpc today 1 Mpc today
“enter the horizon”
Radiation Era Matter Era
today’s scale factor
[c/H(a)]
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a(t) / t
1/2d
H(t) = 2ct / a
2a(t) / t
2/3d
H(t) = 3ct / a
3/23
4
Fluctuations
Entering the Horizon
• The initial impact for a
given wavelength.
Three Regimes
•
Super-horizon scales [q < aH]•
Only gravity is important•
Evolution differs from Newtonian: We need GR•
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 waves5
Super- horizon
Sub- horizon
Part I: Super-horizon Scale:
Conserved Curvature Perturbation
6
The Stone, “ ζ ”
Conserved quantity on the super-horizon scale, q << aH
• 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.
• Derivation: Energy conservation for q << aH:
Bardeen, Steinhardt & Turner (1983);
Weinberg (2003); Lyth, Malik & Sasaki (2005)
7
The “ ζ ”
Conserved quantity on the super-horizon scale, q << aH
• If pressure is a function of the energy density only, i.e.,
Bardeen, Steinhardt & Turner (1983);
Weinberg (2003); Lyth, Malik & Sasaki (2005)
Integrate
integration constant
8
The “ ζ ”
Conserved quantity on the super-horizon scale, q << aH
• If pressure is a function of the energy density only, i.e.,
Bardeen, Steinhardt & Turner (1983);
Weinberg (2003); Lyth, Malik & Sasaki (2005)
integration constant
For the adiabatic initial
condition, all species share the same value of ζ α , i.e., ζ α =ζ
9
q EQ
The wavenumber of the fluctuation that entered the horizon during the equality time
• Which fluctuation entered the horizon before the matter-radiation equality?
• q
EQ= a
EQH
EQ~ 0.01 (Ω
Mh
2/0.14) Mpc
–1• At the last scattering surface, this subtends the multipole of
l EQ = q EQ r L ~ 140
10
100 Mpc today
Entered the horizon during the radiation era
11
What determines the
locations and the heights of the acoustic peaks?
Does the sound-wave solution explain them?
12
Part II: Locations of the Acoustic Peaks
13
Peak Locations?
•
VERY roughly speaking, the angular power spectrum Cl is given by[
] 2
with q -> l/rL.
•
Question: What determines the integration constants, A and B?•
Answer: They are determined by the initial conditions; namely, adiabatic or not.•
For the adiabatic initial condition, A >> B when q is large.High-frequency solution, for q >> aH
[We will show this later.]
14
Peak Locations?
•
VERY roughly speaking, the angular power spectrum Cl is given by[
] 2
with q -> l/rL.
•
If A>>B, the locations of peaks are determined by qrs(tL) = nπ (n=1,2,…):High-frequency solution, for q >> aH
15
16
The simple estimates do not match!
This is 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.
16
A Better Solution in the Radiation-dominated Era
•
In the radiation-dominated era, R << 1 as•
Convenient to change the independent variable from the time (t) toGoing back to the original tight-coupling equation:
17
A Better Solution in the Radiation-dominated Era
18
Then the equation simplifies to
where
•
In the radiation-dominated era, R << 1.•
Convenient to change the independent variable from the time (t) toA Better Solution in the Radiation-dominated Era
19
The solution is
We rewrite this using the formula for trigonometry:
sin(' '
0) = sin(') cos('
0) cos(') sin('
0)
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where
A Better Solution in the Radiation-dominated Era
The solution is
where
20
where
{
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:21
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r
2= 4⇡ G ⇢
C.f., Newtonian (Poisson equation)
( )
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:22
?
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.
23
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.
24
Einstein’s Equations
in the Radiation-dominated Era
•
Now we need to know Newton’s gravitational potential, φ, and the scalar curvature perturbation, ψ.•
Combine Einstein’s equations:“non-adiabatic” pressure
25
Decompose the total
pressure perturbation into the total energy density perturbation and the rest.
Einstein’s Equations
in the Radiation-dominated Era
•
Now we need to know Newton’s gravitational potential, φ, and the scalar curvature perturbation, ψ.•
Choose the adiabatic solution!“non-adiabatic” pressure
We shall ignore this
26
Decompose the total
pressure perturbation into the total energy density perturbation and the rest.
Adiabatic Solution in the Radiation-dominated Era
•
Low-frequency limit (super-sound-horizon scales, qrs << 1)•
ΦADI -> –2ζ/3 = constant•
High-frequency limit (sub-sound-horizon scales, qrs >> 1)•
ΦADI -> 2ζADI
where
27
Kodama & Sasaki (1986, 1987)
The potential decays -> The integrated Sachs-Wolfe Effect
Adiabatic Solution in the Radiation-dominated Era
•
Low-frequency limit (super-sound-horizon scales, qrs << 1)•
ΦADI -> –2ζ/3 = constant•
High-frequency limit (sub-sound-horizon scales, qrs >> 1)•
ΦADI -> 2ζADI
where
Poisson Equation
& oscillation solution for radiation
28
Sound Wave Solution in the Radiation-dominated Era
The solution is
where
Kodama & Sasaki (1986, 1987); Baumann, Green, Meyers & Wallisch (2016)
29
Sound Wave Solution in the Radiation-dominated Era
The solution is
where
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
Kodama & Sasaki (1986, 1987); Baumann, Green, Meyers & Wallisch (2016)
Sound Wave Solution in the Radiation-dominated Era
The complete adiabatic solution is
with
Therefore, the solution is a pure cosine
only in the high-frequency limit!
Kodama & Sasaki (1986, 1987); Baumann, Green, Meyers & Wallisch (2016)
33
Roles of viscosity
•
Neutrino viscosity: Gravitational Impact•
Modify potentials:•
Photon viscosity: Hydrodynamical Impact•
Viscous photon-baryon fluid: damping of sound wavesSilk (1968) “Silk damping”
34
Part III: Damping of the Sound Waves
35
Photon Viscosity
Origin of the Silk damping
• In the tight-coupling approximation, the photon viscosity damps exponentially.
• To take into account a non-zero photon viscosity, we need go higher order in the tight-coupling approximation.
36
The previous lecture: The 1st-order Tight-coupling Approximation
•
When the Thomson scattering is efficient, photons and baryons“move together”; thus, their relative velocity 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.
Peebles & Yu (1970)
Today: The 2nd-order
Tight-coupling Approximation
[d2 is an arbitrary dimensionless variable]
38
•
When the Thomson scattering is efficient, photons and baryons“move together”; thus, their relative velocity is small. We write
[the 1st-order solution]
•
And take . We obtainThe 2nd-order
Tight-coupling Approximation
•
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= 32 45
¯
⇢
T n ¯ e @ i @ j u
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γ γ
39
Kaiser (1983)
The 2nd-order
Tight-coupling Approximation
•
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= 32 45
¯
⇢
T n ¯ e @ i @ j u
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Given by the
spatial gradient of the velocity field
- a well-known result in fluid dynamics
γ γ
40
Kaiser (1983)
Damped Oscillator
•
Using the energy conservation to replace δuγ with δργ/ργ, we obtain, for q >> aH,New term, giving damping!
41
where
Damped Oscillator
•
Using the energy conservation to replace δuγ with δργ/ργ, we obtain, for q >> aH,Important for high frequencies (large multipoles)
42
New term, giving damping!
where
Damped Oscillator
•
Using the energy conservation to replace δuγ with δργ/ργ, we obtain, for q >> aH,43
New term, giving damping!
Exponential dampling!
The new solution is
⇡ exp q 2 / T n ¯ e H
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Damped Oscillator
•
Using the energy conservation to replace δuγ with δργ/ργ, we obtain, for q >> aH,New term, giving damping!
Exponential Silk dampling!
The new solution is
Silk
Silk
“diffusion length”
= length traveled by photon’s random walks
The Diffusion Length
Random walk
• The mean free path of the photon between scatterings is (σ
Tn
e)
–1.
• Below this scale, you do not have a photon-baryon fluid: they are individual particles.
• The number of scatterings per Hubble time is N
scattering=σ
Tn
e/H.
• Then, the length traveled by photons by random walks within the Hubble time is (σ
Tn
e)
–1times √N
scatterings• The diffusion length is thus (σ
Tn
e)
–1times √N
scatterings= (σ
Tn
eH)
–1/2.
45
Silk
“diffusion length”
= length traveled by photon’s random walks
The Diffusion Damping
•
Diffusion mixes hot and cold photons -> Damping of anisotropiesby Wayne Hu
46
Planck Collaboration (2016)
Sachs-Wolfe Sound Wave
Silk Damping?
Additional Damping
fuzziness
( )
•
The power spectrum is[
] 2
with q -> l/rL. The damping factor is thus exp(
–2
q2/qsilk2).
•
qsilk(tL) = 0.139 Mpc–1. This corresponds to a multipole of lsilk ~ qsilk rL/√2 = 1370. Seems too large, compared to the exactcalculation.
•
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)Sachs-Wolfe Sound Wave
Silk+Fuzziness Damping
Total damping:
q
D–2= q
silk–2+ q
fuzziness–2q
D~ 0.11 Mpc
–1, giving
l
D~ q
Dr
L/√2 ~ 1125
Planck Collaboration (2016)
Recap
• The basic structure of the temperature power spectrum is
• The Sachs-Wolfe “plateau” at low multipoles, l(l+1)C
l~ l
n–1• Sound waves at intermediate multipoles
• The 1st-order tight-coupling approximation
• Silk damping and Fuzziness damping at high multipoles
• The 2nd-order tight-coupling approximation
50
Part IV: The Acoustic Oscillation at the Last-scattering Surface
51
Sachs-Wolfe Sound Wave
Silk+Fuziness Damping
Light propagation in a clumpy
Universe
Energy and momentum conservation
Photon viscosity and fuzziness of Last Scat. Surface
Planck Collaboration (2016)
Matching Solutions: Radiation- and Matter-dominated Eras
•
We have a very good analytical solution valid at low and high frequencies during the radiation era:•
Now, match this to a high-frequency solution valid at the last-scattering surface (when R is no longer small)53
Matching Solutions: Radiation- and Matter-dominated Eras
•
We have a very good analytical solution valid at low and high frequencies during the radiation era:•
Now, match this to a high-frequency solution valid at the last-scattering surface (when R is no longer small)Slightly improved solution, with a weak time dependence of R using the WKB method [Peebles & Yu (1970)]
54
Solution at the Last Scattering Surface
•
(*) To a good approximation, the low-frequency solution is given by setting R=0 because sound waves are notimportant at large scales
Weinberg “Cosmology”, Eq. (6.5.7)
where T(q), S(q), θ(q) are “transfer functions” that smoothly interpolate two limits as
q
q << q EQ : q >> q EQ :
with q
EQ= a
EQH
EQ~ 0.01 Mpc
–1, giving l
EQ=q
EQr
L~ 140
“EQ” for “matter-radiation Equality epoch”
q q
55
q q
•
(*) To a good approximation, the low-frequency solution is given by setting R=0 because sound waves are notimportant at large scales
where T(q), S(q), θ(q) are “transfer functions” that smoothly interpolate two limits as
q << q EQ : q >> q EQ :
with q
EQ= a
EQH
EQ~ 0.01 Mpc
–1, giving l
EQ=q
EQr
L~ 140
“EQ” for “matter-radiation Equality epoch”
Due to the decay of
gravitational potential during the radiation dominated era
q q
Weinberg “Cosmology”, Eq. (6.5.7)
56
Solution at the Last Scattering Surface
q q q
•
(*) To a good approximation, the low-frequency solution is given by setting R=0 because sound waves are notimportant at large scales
where T(q), S(q), θ(q) are “transfer functions” that smoothly interpolate two limits as
q << q EQ : q >> q EQ :
with q
EQ= a
EQH
EQ~ 0.01 Mpc
–1, giving l
EQ=q
EQr
L~ 140
“EQ” for “matter-radiation Equality epoch”
Due to the neutrino anisotropic stress
(see “Appendix” of the last lecture slides)
q q
Weinberg “Cosmology”, Eq. (6.5.7)
57
Solution at the Last Scattering Surface
q q q
•
(*) To a good approximation, the low-frequency solution is given by setting R=0 because sound waves are notimportant at large scales
Low-frequency Limit
q -> 0(*)
⇣
5
This should agree with the Sachs-Wolfe result: Φ/3; thus,
= 3⇣ /5
in the matter-dominated eraWeinberg “Cosmology”, Eq. (6.5.7)
58
q q q
High-frequency Limit
q/qEQ >> 1
(1 + R) 1/4 ⇣ cos[qr s + ✓ (q )]
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•
The amplitude of the oscillation on small scales is a factor of5(1+R) –1/4
times the Sachs-Wolfe plateau!Weinberg “Cosmology”, Eq. (6.5.7)
59
q q q
q q q
Effect of Baryons
Shift the zero-point of oscillations
Reduce the amplitude of oscillations
Weinberg “Cosmology”, Eq. (6.5.7)
for
The CMB power spectrum is sensitive to this combination of the parameters.
60
` ⇡ 302 ⇥ qr
s/⇡
No Baryon [R=0]
61` ⇡ 302 ⇥ qr
s/⇡
No Baryon [R=0]
Boost due to decaying potential during the radiation era
62
` ⇡ 302 ⇥ qr
s/⇡
No Baryon [R=0]
Silk damping
63
` ⇡ 302 ⇥ qr
s/⇡
Effect of baryons
64
` ⇡ 302 ⇥ qr
s/⇡
Zero-point shift of the oscillations
Effect of baryons
65
` ⇡ 302 ⇥ qr
s/⇡
WKB factor (1+R)-1/4 and Silk damping compensate the zero-
point shift
Effect of baryons
66
Effect of Total Matter
where T(q), S(q), θ(q) are “transfer functions” that smoothly interpolate two limits as
q << q EQ : q >> q EQ :
with q EQ = a EQ H EQ ~ 0.01 (Ω M h 2 /0.14) Mpc –1
“EQ” for “matter-radiation Equality epoch”
q q
The CMB power spectrum is sensitive to this combination of the parameters.
Weinberg “Cosmology”, Eq. (6.5.7)
67
q q q
` ⇡ 302 ⇥ qr
s/⇡
[Ω
Mh
2=0.07]
Smaller matter density -> More potential decay -> Larger boost
68
Recap
• Decay of the gravitational potential boosts the temperature anisotropy dT/T at high multipoles by 5(1+R)
–1/4compared to the Sachs-Wolfe plateau.
• Where this boost starts depends on the total matter density => We can measure Ω
Mh
2using this.
• Baryon density shifts the zero-point of the oscillation, boosting the heights of the odd peaks relative to those of the even peaks => We can measure Ω
Bh
2using this.
• However, the WKB factor (1+R)
–1/4and damping make the boosting of the 3rd and 5th peaks not so obvious.
69
Not quite there yet…
• The first peak is too low
•
We need to include the “integrated Sachs-Wolfe effect”• How to fill zeros between the peaks?
•
We need to include the Doppler shift of light70
The Doppler Shift of Light
•
Using the velocity potential,we write Line-of-sight direction
Coming distance (r)
v
B is the bulk velocity of a baryon fluidˆ
n · r u B /a
•
In tight coupling,•
Using the energy conservation,The Doppler Shift of Light
•
Using the velocity potential, we writev
B is the bulk velocity of a baryon fluidˆ
n · r u B /a
•
In tight coupling,•
Using the energy conservation,Velocity potential is a
time-derivative
of the energy density:
cos(qr
s) becomes
sin(qr
s)!
Temperature Anisotropy from the Doppler Shift
•
To this, we should multiply the damping factorDamp
73
+Doppler
Doppler shift
reduces the contrast between the peaks andtroughs because it
adds sin
2(qr
s) to cos
2(qr
s)74
(Early) ISW
Hu & Sugiyama (1996)
“integrated Sachs-Wolfe” (ISW) effect
Gravitational potentials decay after the last-scattering because the Universe was not yet completely matter-dominated.
+Doppler +ISW
The Early ISW
affectsonly the first peak because it occurs
after
the last-scattering epoch, subtending a larger angle.Not only it boosts the first peak, but also it makes it “fatter”
76
We are ready!
• We are ready to understand the effects of all the cosmological parameters.
77
Next Lecture!
Appendix: Neutrino Viscosity
78
High-frequency solution without neutrino viscosity
The solution is
where
'<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> 1
79
High-frequency solution with neutrino viscosity
The solution is
where
Chluba & Grin (2013)
non-zero value!
'<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> 1
80
High-frequency solution with neutrino viscosity
Using the formula for trigonometry, we write
where
(Hu & Sugiyama 1996)
(Bashinsky & Seljak 2004)
Phase shift!
81
High-frequency solution with neutrino viscosity
The solution is
where
Hu & Sugiyama (1996)
Phase shift!
Thus, the neutrino viscosity will:
(1) Reduce the amplitude of
sound waves at large multipoles
(2) Shift the peak positions
of the temperature power spectrum
82