Lecture 6: Acoustic Oscillation

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Lecture 6: Acoustic Oscillation

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Part I: Hydrodynamics of Photon- baryon Fluid

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Creation of sound waves in the fireball Universe

Basic equations

• Conservation equations (energy and momentum)

• Equation of state, relating pressure to energy density of the α component

• General relativistic version of the “Poisson equation”, relating gravitational potential to energy density

• Evolution of the “anisotropic stress” (viscosity)

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P _↵ = P _↵ (⇢ _↵ )

This is still the Newtonian expression, which must be extended to GR.

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Energy Conservation

( )

velocity potential

anisotropic stress:

[or, viscosity]

v_↵ = 1

a r u_↵

α = baryon, photon, neutrino, dark matter

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˙

⇢ + ¯ ⇢ r ² u = 0

¯

⇢ u ˙ = P

•

Total energy conservation:

•

C.f., Total energy conservation [unperturbed]

•

C.f., Newtonian result

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¯

⇢

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Energy Conservation

•

Again, this is the eﬀect of locally-defined inhomogeneous scale factor, i.e.,

•

The spatial metric is given by

•

Thus, locally we can define a new scale factor:

ds

²

= a

²

(t) exp( 2 )dx

²

˜

a(t, x) = a(t) exp( )

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α = baryon, photon, neutrino, dark matter

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Energy Conservation

•

Momentum flux going outward (inward) -> reduction (increase) in the energy density

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α = baryon, photon, neutrino, dark matter

˙

⇢ + ¯ ⇢ r ² u = 0

¯

⇢ u ˙ = P

•

) (

¯

⇢

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Momentum Conservation

•

Total momentum conservation

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Cosmological redshift of the momentum

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Gravitational force given by potential gradient

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Force given by pressure gradient

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Force given by gradient of anisotropic stress

v_↵ = 1

a r u_↵

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˙

⇢ + ¯ ⇢ r ² u = 0

¯

⇢ u ˙ = P

•

) (

¯

⇢

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•

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

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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

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Perturbed pressure involves contributions from the

bulk viscosity

^:

Equation of State

P =

P _⌫ =

8

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•

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

^:

Equation of State

P =

P _⌫ =

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If you know a bit of GR:

The reason for this is that the trace of the stress-energy of relativistic species vanishes:

∑ μ=0,1,2,3 Τ μμ = 0

T

₀⁰

+

X

3

i=1

T

_iⁱ

= ⇢ + 3P + r

²

⇡ = 0

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Two remarks

Do we need to sum over α?

• In the standard scenario that we shall assume throughout this lecture,

• 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 & Coulomb scattering.

This must be taken into account when writing down separate momentum conservation equations.

• Next, we solve the conservation equations to derive the sound wave propagating in the fireball Universe.

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•

Fourier transformation replaces

Conservation Equations for Photons and Baryons

r

²

! q

²

momentum transfer via scattering

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• Conservation Equations for Photons and Baryons

r

²

! q

²

what about

photon’s viscosity?

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Formation of the “Photon-baryon Fluid”

Nobel Prize in Physics (2019)

• Photons are not an ideal fluid.

Photons free-stream at the speed of light.

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The energy and momentum conservation equations are not enough because we need to specify the evolution of viscosity.

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Solving for viscosity requires information of the phase-space distribution function of photons: Boltzmann equation.

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However, frequent scattering of photons with baryons(*) can make photons behave as a fluid: Photon-baryon fluid.

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Peebles & Yu (1970); Sunyaev & Zeldovich (1970)

(*)Photons scatter with electrons via Thomson scattering. Protons scatter with electrons via Coulomb scattering.

Thus we can say, eﬀectively, photons scatter with baryons

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https://www.nobelprize.org

Sound waves in the fireball Universe, predicted in 1970

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https://www.nobelprize.org/uploads/2019/10/fig2_fy_en_backgroundradiation.pdf

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At the ICGC2011 conference, Goa, India

¹⁶

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Sound waves in the fireball Universe, predicted in 1970

The Franklin Institute of Physics

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The predicted sound wave was found in 1999-2000.

No one (Peebles, Sunyaev, or Zeldovich) thought that this would ever be observed,

because the eﬀect seemed so tiny.

The golden lesson to learn

It does not matter how small the eﬀect would seem to you now. Publish your

calculation!

If the eﬀect is worth measuring, it will be measured.

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Part II: Tight-coupling approximation

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• Let’s solve them!

r

²

! q

²

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Tight-coupling Approximation

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

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Using the energy conservation to replace δuγ with δργ/ργ, we obtain

The wave equation, with the speed of sound of c

s2

= 1/3(1+R)!

Peebles & Yu (1970)

Tight-coupling Approximation

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¨

⇢_q + c²_s q² ⇢_q = 0

(c.f.)

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Sound Wave!

• To simplify the equation, let’s first look at the high-frequency solution

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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:

Peebles & Yu (1970); Sunyaev & Zeldovich (1970)

The sound wave solution!

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⇢_q(t) = A_q cos(qc_st) + B_q sin(qc_st)

(c.f.)

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Recap

Focus on physics!

• Photons are not a fluid; but Thomson scattering couples photons to baryons, forming a photon-baryon fluid.

•

The reduced sound speed, cs2=1/3(1+R), emerges automatically. Beautiful!

•

The relevant sound horizon is

•

^δρ^γ^/4ρ^γis 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

r

_s

=

Z

t 0

dt

⁰

a(t

⁰

) c

_s

(t

⁰

)

L

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Part III: Build a Universe!

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https://wmap.gsfc.nasa.gov/resources/camb_tool/index.html

Running this web tool requires Flash Player.

Enable it before using this tool.

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https://map.gsfc.nasa.gov/resources/camb_tool

You change these “cosmological parameters” to make the blue curve in the power spectrum figure

match the data points (and the red curve)

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Your Mission

“Fit” the data

1. Find the parameters that match the data points

2. Record the behaviour of the power spectrum, when you vary a parameter

• For example: What happens when you reduce the “Spectral Index”? What happens when you increase “Atoms”?

• Tip: Where to start? Start by varying one parameter away from the best-fitting parameter you found in (1)

• Explore the behaviours of as many parameters as you have time to explore 3. Document your findings in the shared note.

• I have not yet taught you how the power spectrum depends on the parameters.

So, collect data yourself now; it helps you understand physics later.

Lecture 6: Acoustic Oscillation