Lecture 2

Lecture 2

- Power spectrum of gravitational anisotropy

- Temperature anisotropy from sound waves

COBE 4-year Power Spectrum

Bennett et al. (1996)

The SW formula

allows us to determine the

3d power

spectrum of φ

^at

the last scattering surface from Cl.

But how?

•

But this is not exactly what we want. We want the statistical average of this quantity.

gives…

Sachs & Wolfe (1967) T (ˆn)

T₀ = 1

3 (t_L, rˆ_L)

SW Power Spectrum

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

where we defined

consequence of “statistical homogeneity”

φ

and used

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

SW Power Spectrum

•

Thus, the power spectrum of the CMB in the SW limit is

•

In the flat-sky approximation,

Perpendicular wavenumber, (qperp)²

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

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

n=1

full-sky correction

n=1 n=1.2 ± 0.3

(68%CL)

Bennett et al. (1996)

COBE 4-year Power Spectrum

Bennett et al. (1996)

WMAP 9-year Power Spectrum

Bennett et al. (2013)

Planck 29-mo Power Spectrum

Planck Collaboration (2016)

Planck 29-mo Power Spectrum

Planck Collaboration (2016)

Clearly, the SW

prediction does not fit!

Missing physics:

Hydrodynamics

(sound waves)

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

This is a viscous ^fluid,

in which the amplitude of

sound waves damps

at shorter wavelength

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

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

)

Comoving Sound ^Horizon

•

Then, we replace the speed of light with a time- dependent 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

qr s > 1

Sound Speed

•

Sound speed of an adiabatic fluid is given by

-

δP: pressure perturbation

-

δρ: density perturbation

•

For a baryon-photon system:

We can ignore the baryon pressure because it is much smaller than the photon pressure

Sound Speed

•

Using the adiabatic relationship between photons and baryons:

•

and pressure-density relation of a relativistic fluid, δPγ=δργ/3, We obtain

[i.e., the ratio of the number densities of baryons and photons is equal everywhere]

•

Or equivalently

where

sound speed is reduced!

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

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

For the last-scattering redshift of z

L

=1090

(or last-scattering temperature of T

L

=2974 K),

r s = 145.3 Mpc

We cannot ignore the effects of sound waves if qr

s

>1. Since l~qr

L

, this means

l > r L /r s = 96

where we used r

L

=13.95 Gpc

Creation of Sound Waves:

Basic Equations

1. Conservation equations (energy and momentum)

2. Equation of state, relating pressure to energy density 3. General relativistic version of the “Poisson equation”,

relating gravitational potential to energy density 4. Evolution of the “anisotropic stress” (viscosity)

P = P (⇢)

•

Total energy conservation:

•

C.f., Total energy conservation [unperturbed]

Energy Conservation

( )

velocity potential

anisotropic stress:

[or, viscosity]

v_↵ = 1

ar u_↵

Energy Conservation

•

Total 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( )

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

( )

Momentum Conservation

•

Total momentum conservation

•

Cosmological redshift of the momentum

•

Gravitational force given by potential gradient

•

Force given by pressure gradient

•

Force given by gradient of anisotropic stress

v_↵ = 1

ar u_↵

•

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 _⌫ =

•

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 _⌫ =

The reason for this is that

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

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

•

Fourier transformation replaces

Conservation Equations for Photons and Baryons

r

²

! q

²

momentum transfer via scattering

•

Fourier transformation replaces

Conservation Equations for Photons and Baryons

r

²

! q

²

what about photon’s viscosity?

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

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

•

Fourier transformation replaces

Let’s solve them!

r

²

! q

²

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.

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 c

s2

= 1/3(1+R)!

Sound Wave!

•

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

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

Solution: SOUND WAVE!

Recap

•

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

•

^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

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”

10 Gpc/h today 1 Gpc/h today 100 Mpc/h today

10 Mpc/h today 1 Mpc/h today

“enter the horizon”

Radiation Era

Last scattering

Matter Era

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

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:

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.,

Integrate

, then

integration constant

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 ζ

α

, i.e., ζ α =ζ

Bardeen, Steinhardt & Turner (1983); Weinberg (2003); Lyth, Malik & Sasaki (2005)

q EQ

•

Which fluctuation entered the horizon before the matter- radiation equality?

•

^{qEQ = aEQHEQ ~ 0.01 (ΩMh2/0.14) Mpc–1}

•

At the last scattering surface, this subtends the multipole of

l

EQ

= q

EQ

r

L

~ 140

Entered the horizon during the radiation era

What determines the

locations and heights of the peaks?

Does the sound-wave

solution explain it?

Peak Locations?

•

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

High-frequency solution, for q >> aH

[We will show it later.]

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

High-frequency solution, for q >> aH

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!

Better Solution in

Radiation-dominated Era

•

In the radiation-dominated era, R << 1

•

Change the independent variable from the time (t) to

Going back to the original tight-coupling equation..

Better Solution in

Radiation-dominated Era

•

In the radiation-dominated era, R << 1

•

Change the independent variable from the time (t) to

Then the equation simplifies to

where

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

)

Better Solution in

Radiation-dominated Era

Then the equation simplifies to

where

The solution is

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:

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:

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.

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.

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

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

We shall ignore this

Solution (Adiabatic)

in 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 -> 2ζ}

ADI

where

damp

Kodama & Sasaki (1986, 1987)

Solution (Adiabatic)

in 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 -> 2ζ}

ADI

where

damp

Poisson Equation

& oscillation solution for radiation

Solution (Adiabatic)

in 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 -> 2ζ}

ADI

where

damp

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

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 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)

Roles of viscosity

•

Neutrino viscosity

•

Modify potentials:

•

Photon viscosity

•

Viscous photon-baryon fluid: damping of sound waves

Silk (1968) “Silk damping”

High-frequency solution without neutrino viscosity

The solution is

where

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High-frequency solution with neutrino viscosity

The solution is

where

Chluba & Grin (2013)

non-zero value!

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High-frequency solution with neutrino viscosity

Using the formula for trigonometry, we write

where

Hu & Sugiyama (1996)

Bashinsky & Seljak (2004) Phase shift!

High-frequency solution with neutrino viscosity

The solution is

where

Hu & Sugiyama (1996)

Bashinsky & Seljak (2004) 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

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

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.

Today: Tight-coupling

Approximation (2nd-order)

•

When Thomson scattering is efficient, the relative velocity between photons and baryons is small. We write

[d2 is an arbitrary dimensionless variables]

•

And take .. We obtain

where

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

Kaiser (1983)

= 32 45

¯

⇢

T n ¯ _e @ _i @ _j u

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γ γ

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

Kaiser (1983)

= 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

Damped Oscillator

•

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

New term, giving damping!

where

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)

Damped Oscillator

•

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

New term, giving damping!

Exponential dampling!

SOLUTION:

⇡ 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 dampling!

SOLUTION:

Silk

Silk “diffusion length”

= length traveled by photon’s random walks

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 =

(σ T n e H) –1/2

Diffusion Damping

•

Diffusion mixes hot and cold photons -> Damping of anisotropies

by Wayne Hu

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 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)

Planck Collaboration (2016)

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

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