Lecture 4: Power Spectrum

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Lecture 4: Power Spectrum

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Outstanding Questions

• Where does anisotropy in CMB temperature come from?

• This is the origin of galaxies, stars, planets, and everything else we see around us, including

ourselves

• The leading idea: quantum fluctuations in

vacuum, stretched to cosmological length scales by a rapid exponential expansion of the universe

called “cosmic inflation” in the very early universe

How do we analyse the

data like this? ²

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Data Analysis

• Decompose temperature fluctuations in the sky into a set of waves with

various wavelengths

• Make a diagram showing the strength of each wavelength: Power Spectrum

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Long Wavelength Short Wavelength

180 degrees/(angle in the sky)

Amplitude of W aves [ μ K 2 ]

WMAP Collaboration

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Power Spectrum,

Explained

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Part I: Spherical Harmonics

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Fourier transform?

• The simplest way to decompose fluctuations into waves is Fourier transform.

• However, Fourier transform works only for plane waves in flat space.

• The sky is a sphere. How do we decompose fluctuations on a sphere into waves?

• The answer: Spherical Harmonics.

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Spherical harmonics

Wait, don’t run! It is not as bad as you may remember from the QM class…

• Dipole patterns (l=1)

8 (l,m)=(1,0) (l,m)=(1,1)

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a _` _m = ( 1) ^m a ^⇤ _`m : suﬃcient to consider only m>=0

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Dipole Temperature Anisotropy of the CMB

Due to the motion of Solar System with respect to the CMB rest frame

• Temperature anisotropy towards “+” is ΔT/T = v/c = 1.23 x 10 ^–3

• Thus, ΔT = 3.355 mK ⁹

in Galactic coordinates

The Solar System is moving towards this direction at 369 km/s.

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a ₁₀ = 5.124 mK str ^1/2 ,

a ₁₁ = 0.3384 3.215i mK str ^1/2 ,

a _{1 1} = a ^⇤ ₁₁

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(l,m)=(2,0) (l,m)=(2,1)

(l,m)=(2,2)

✓ = ⇡

`

For l=m ^{, a half-}

wavelength, λ θ /2,

corresponds to π/l.

Therefore, λ θ =2π/l

10

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(l,m)=(3,0) (l,m)=(3,1)

(l,m)=(3,2) (l,m)=(3,3)

✓ = ⇡

`

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[Values of Temperatures in the Sky Minus 2.725 K] / [Root Mean Square]

Fraction of the Number of Pixels Having Those T emperatur es

Histogram: WMAP Data Red Line: Gaussian

WMAP Collaboration

Variance of CMB Temperature

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• Values of a lm depend on

coordinates, but the squared amplitude, , does not depend on coordinates

Angular Power Spectrum

• The angular power spectrum, C l , quantifies how much

correlation power we have at a given angular separation.

• More precisely: it is l(2l+1)C l /4π that gives the fluctuation power at a given angular separation, ~π/l.

We can see this by computing variance:

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

What physics can we learn

from this

measurement?

14 Φ!!

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Gravitational Potential in 3D to Temperature in 2D

More generally: How is a plane wave in 3D projected on the sky?

• Take a single plane wave for the potential, going in the z direction:

15 φ=cos(qz)

r L

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T (ˆ n)

T ₀ = 1

3 (t _L , nr ˆ _L )

Let’s use the Sachs-Wolfe formula for the adiabatic initial condition:

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(t _L , x) / A(t _L ) cos(qz )

• A(t L ): Amplitude

• q: Wavenumber in 3D

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Gravitational Potential in 3D to Temperature in 2D

More generally: How is a plane wave in 3D projected on the sky?

16 φ=cos(qz)

r L

In the x-axis, the angle θ 1 subtends the half wavelength λ/2, with

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= 2⇡ /q

With trigonometry, we find

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tan ✓ ₁ ' ✓ ₁ = /2 r _L

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/2

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= ⇡

qr _L

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` ₁ ⇡ ⇡

✓ ₁ = qr _L

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tan ✓ ₂ ' ✓ ₂ > /2

r _L = ⇡

qr _L

Gravitational Potential in 3D to Temperature in 2D

More generally: How is a plane wave in 3D projected on the sky?

17 φ=cos(qz)

r L

In the z-axis, the angle θ 2 is subtends bigger than the half wavelength λ/2, with

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= 2⇡ /q

With trigonometry, we find

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` ₂ ⇡ ⇡

✓ ₂ < qr _L

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How do we understand the relationship between the 3D

wavenumber of the gravitational potential, Φ , and the 2D

wavenumber of the temperature anisotropy, l ?

18

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(t _L , x) =

Z d ³ q

(2⇡ ) ³ ^q (t _L ) exp(iq · x) ^?

t L : the time at the last scattering surface

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Part II: Flat-sky (Small-angle) Approximation

19

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Fourier transform?

• The simplest way to decompose fluctuations into waves is Fourier transform.

• However, Fourier transform works only for plane waves in flat space.

• The sky is a sphere. How do we decompose fluctuations on a sphere into waves?

• The answer: Spherical Harmonics.

20 • But, this seems too complicated for understanding the

relationship between the gravitational potential in 3D and the temperature anisotropy in 2D (i.e., sky).

• Alternative (approximate) approach?

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Fourier transform!

Approximately correct in a small region in the sky

• Take z-axis to anywhere we want in the sky. Then, treat a small area

around the z-axis as a “flat sky”.

• We then apply the usual 2D Fourier transform to analyse temperature

fluctuations, and relate it to the 3D Fourier transform of the potential Φ.

21 Focus here.

“Flat sky”, if θ is small

ˆ

n = (sin ✓ cos , sin ✓ sin , cos ✓ )

(22)

2D Fourier Transform

C.f.,

( )

22

(23)

• Take the inverse 2D Fourier transform of the Sachs-Wolfe formula for the adiabatic initial condition:

a(l) of the Sachs-Wolfe effect

23 T (ˆ n)

T ₀ = 1

3 (t _L , nr ˆ _L ) r L

1 [flat-sky approximation]

*q is the 3D Fourier wavenumber

• And Fourier transform Φ in 3D:

(t _L , x) =

Z d ³ q

(2⇡ ) ³ ^q (t _L ) exp(iq · x)

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Flat-sky Result

• ^{It is} now manifest that only the

perpendicular wavenumber contributes to l, i.e., l=q perp r L , giving l<qr L

i.e.,

r L

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` ₁ ⇡ ⇡

✓ ₁ = qr _L

24

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Flat-sky Result

• ^{It is} now manifest that only the

perpendicular wavenumber contributes to l, i.e., l=q perp r L , giving l<qr L

i.e.,

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` ₂ ⇡ ⇡

✓ ₂ < qr _L

r L

25

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The relationship between q and l Understood?

Let’s go to the full sky treatment.

26

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• Take the inverse spherical harmonics transform

of the Sachs-Wolfe formula for the adiabatic initial condition:

a lm of the Sachs-Wolfe effect

27 T (ˆ n)

T ₀ = 1

3 (t _L , nr ˆ _L )

r L

• And Fourier transform Φ in 3D:

*q is the 3D Fourier wavenumber

(t _L , x) =

Z d ³ q

(2⇡ ) ³ ^q (t _L ) exp(iq · x)

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Spherical wave decomposition of a plane wave

• This is the exact formula relating Φ in 3D at the last

scattering surface to a lm . How do we understand this?

• How to obtain a plane wave by combining spherical waves? The answer is

• which is called the “partial wave decomposition” or “Rayleigh’s formula”. Then we obtain

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q -> l projection

• A half wavelength, λ/2, at the last scattering surface

subtends an angle of λ/2r L . Since q=2π/λ, the angle is given by δθ=π/qr L . Comparing this with the relation δθ=π/l, we

obtain l=qr L . How can we see this?

• For l>>1, the spherical Bessel function, j l (qr L ), peaks at l~qr L and falls gradually toward qr L >l. Thus, a given q

mode contributes to large angular scales too. We learned this already from

the flat-sky approximation!

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Part III: Power Spectrum of the Sachs-Wolfe Effect

30

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Let’s compute the temperature power spectrum

Temperature C l

• ^{We use}

to compute

31

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Result

The power spectrum of the Sachs-Wolfe eﬀect?

• ^{We use}

to compute

32 • But this is not exactly what we want. We want the

statistical average of this quantity.

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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

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Power Spectrum of φ

• In addition, if depends only on the magnitude of the separation r and not on the directions, then

Lecture 4: Power Spectrum