Lecture 7: Details of the Acoustic Oscillation

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Lecture 7: Details of the Acoustic Oscillation

1

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2

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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 wavelength

fluctuations 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

^1/2

d

_H

(t) = 2ct / a

²

a(t) / t

^2/3

d

_H

(t) = 3ct / a

^3/2

3

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4

Fluctuations

Entering the Horizon

• The initial impact for a

given wavelength.

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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 waves

5

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Part I: Super-horizon Scale:

Conserved Curvature Perturbation

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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)

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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

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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

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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}

^EQ

^H

^EQ

^{~ 0.01 (Ω}

^M

^h

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

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Entered the horizon during the radiation era

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What determines the

locations and the heights of the acoustic peaks?

Does the sound-wave solution explain them?

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Part II: Locations of the Acoustic Peaks

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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

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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

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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.

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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) to

Going back to the original tight-coupling equation:

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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) to

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A 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

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A Better Solution in the Radiation-dominated Era

The solution is

where

20

where

{

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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)

( )

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Einstein’s Equations

•

22

?

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Einstein’s Equations

•

Will come back to this later.

For now, let’s ignore any viscosity.

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Einstein’s Equations

•

Will come back to this later.

For now, let’s ignore any viscosity.

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Einstein’s Equations

in the Radiation-dominated Era

•

Combine Einstein’s equations:

“non-adiabatic” pressure

25

Decompose the total

pressure perturbation into the total energy density perturbation and the rest.

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Einstein’s Equations

in the Radiation-dominated Era

•

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.

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Adiabatic Solution in the Radiation-dominated Era

•

Low-frequency limit (super-sound-horizon scales, qrs << 1)

•

^Φ^ADI -> –2ζ/3 = constant

•

High-frequency limit (sub-sound-horizon scales, qrs >> 1)

•

^Φ^ADI^{-> 2ζ}

ADI

where

27

Kodama & Sasaki (1986, 1987)

The potential decays -> The integrated Sachs-Wolfe Eﬀect

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Adiabatic Solution in the Radiation-dominated Era

•

Low-frequency limit (super-sound-horizon scales, qrs << 1)

•

^Φ^ADI -> –2ζ/3 = constant

•

High-frequency limit (sub-sound-horizon scales, qrs >> 1)

•

^Φ^ADI^{-> 2ζ}

ADI

where

Poisson Equation

& oscillation solution for radiation

28

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Sound Wave Solution in the Radiation-dominated Era

The solution is

where

Kodama & Sasaki (1986, 1987); Baumann, Green, Meyers & Wallisch (2016)

29

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Sound Wave Solution in the Radiation-dominated Era

The solution is

where

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Sound Wave Solution in the Radiation-dominated Era

The solution is

where

i.e., _ADI _ADI

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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!

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33

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Roles of viscosity

•

Neutrino viscosity: Gravitational Impact

•

Modify potentials:

•

Photon viscosity: Hydrodynamical Impact

•

Viscous photon-baryon fluid: damping of sound waves

Silk (1968) “Silk damping”

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Part III: Damping of the Sound Waves

35

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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

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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)

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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 obtain

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The 2nd-order

Tight-coupling Approximation

•

Eliminating d2 and using the fact that R is proportional to the scale factor, we obtain

•

^{Getting π}^γ 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)

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The 2nd-order

Tight-coupling Approximation

•

Eliminating d2 and using the fact that R is proportional to the scale factor, we obtain

•

^{Getting π}^γ 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)

(41)

Damped Oscillator

•

Using the energy conservation to replace δuγ with δργ/ργ, we obtain, for q >> aH,

New term, giving damping!

41

where

(42)

Damped Oscillator

•

Important for high frequencies (large multipoles)

42

New term, giving damping!

where

(43)

Damped Oscillator

•

43

New term, giving damping!

Exponential dampling!

The new solution is

⇡ exp q ² / _T n ¯ _e H

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(44)

Damped Oscillator

• New term, giving damping!

Exponential Silk dampling!

The new solution is

Silk

“diﬀusion length”

= length traveled by photon’s random walks

(45)

The Diffusion Length

Random walk

• The mean free path of the photon between scatterings is (σ

T

n

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

T

n

e

/H.

• Then, the length traveled by photons by random walks within the Hubble time is (σ

T

n

e

)

^–1

times √N

scatterings

• The diﬀusion length is thus (σ

T

n

e

)

^–1

times √N

scatterings

= (σ

T

n

e

H)

^–1/2

.

45

Silk

“diﬀusion length”

= length traveled by photon’s random walks

(46)

The Diffusion Damping

•

Diﬀusion mixes hot and cold photons -> Damping of anisotropies

by Wayne Hu

46

(47)

Planck Collaboration (2016)

Sachs-Wolfe Sound Wave

Silk Damping?

(48)

Additional Damping

fuzziness

( )

•

The power spectrum is

[

] ²

with q -> l/rL. The damping factor is thus exp

(

^–

2

^q²^/q^silk2

).

•

^q^silk^(t^L) = 0.139 Mpc^–1. This corresponds to a multipole of lsilk ~ qsilk rL/√2 = 1370. 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)

(49)

Sachs-Wolfe Sound Wave

Silk+Fuzziness Damping

Total damping:

q

D–2

= q

silk–2

+ q

fuzziness–2

q

D

~ 0.11 Mpc

^–1

, giving

l

D

~ q

D

r

L

/√2 ~ 1125

Planck Collaboration (2016)

(50)

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

(51)

Appendix: Neutrino Viscosity

51

(52)

High-frequency solution without neutrino viscosity

The solution is

where

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52

(53)

High-frequency solution with neutrino viscosity

The solution is

where

Chluba & Grin (2013)

non-zero value!

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53

(54)

High-frequency solution with neutrino viscosity

Using the formula for trigonometry, we write

where

(Hu & Sugiyama 1996)

(Bashinsky & Seljak 2004)

Phase shift!

54

(55)

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

55

(Bashinsky & Seljak 2004)

Lecture 7: Details of the Acoustic Oscillation