Lecture 3: Gravitational Effects on Temperature Anisotropy

Lecture 3: Gravitational Effects on Temperature Anisotropy

1

Part I: Sachs-Wolfe Effect(s)

2

Evolution of photon’s energy

Sachs & Wolfe (1967)

•

Let’s find a (formal) solution for p by integrating this equation over time.

3

γⁱ is a unit vector of the direction of photon’s momentum:

Newtonian

gravitational potential

Scalar curvature

perturbation Tensor perturbation

= Gravitational wave

Evolution of photon’s energy

Sachs & Wolfe (1967)

•

Let’s find a (formal) solution for p by integrating this equation over time.

4

γⁱ is a unit vector of the direction of photon’s momentum:

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1 ap

d(ap)

dt =

Evolution of photon’s energy

Sachs & Wolfe (1967)

•

Let’s find a (formal) solution for p by integrating this equation over time.

5

γⁱ is a unit vector of the direction of photon’s momentum:

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1 ap

d(ap)

dt =

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d

dt + ˙

because

Formal Solution (Scalar)

or

Line-of-sight direction

“L” for “Last scattering surface”

Sachs & Wolfe (1967)

Present-day time

Comoving distance (r)

6

Formal Solution (Scalar)

Line-of-sight direction Initial Condition

Sachs & Wolfe (1967)

Comoving distance (r)

7

Formal Solution (Scalar)

Line-of-sight direction

Comoving distance (r) Gravitational Redshit

Sachs & Wolfe (1967)

8

Formal Solution (Scalar)

Line-of-sight direction

“integrated Sachs-Wolfe” (ISW) effect

Sachs & Wolfe (1967)

Comoving distance (r)

9

Part II: Initial Condition

10

Initial Condition

Only the data can tell us!

• Were photons hot, or cold, at the bottom of the potential well at the last scattering surface?

11

“Adiabatic Initial Condition”

The initial condition that fits the current data best

•

Definition: “Ratios of the number densities of all species are equal everywhere initially”

•

^{For ith and jth species, ni(x)/nj}(x) = constant

•

For a quantity X(t,x), let us define the fluctuation, δX, as

•

Then, the adiabatic initial condition is

12

n _i (t _initial , x)

¯

n _i (t _initial ) = n _j (t _initial , x)

¯

n _j (t _initial )

Example of the adiabatic initial condition

Thermal equilibrium

•

When photons and baryons were in thermal equilibrium in the past, then

•

^{nphoton ~ T3 and nbaryon ~ T3}

•

That is to say, thermal equilibrium naturally gives rise to the adiabatic initial condition, because nphoton / nbaryon = constant

•

This gives

13

• “B” for “Baryons”

• ρ is the mass density

A Big Question

•

How about dark matter?

•

If dark matter and photons were in thermal equilibrium in the past, then they should also obey the adiabatic initial condition

•

If not, there is no a priori reason to expect the adiabatic initial condition!

•

The current data are consistent with the adiabatic initial condition. This means something important for the nature of dark matter!

We shall assume the adiabatic initial condition throughout the lectures

14

Adiabatic solution

Was the temperature hot or cold at the bottom of potential?

•

At the last scattering surface, the temperature

fluctuation is given by the matter density fluctuation as

15

T (t _L , x)

T ¯ (t _L ) = 1 3

⇢ _M (t _L , x)

¯

⇢ _M (t _L )

Adiabatic solution

Was the temperature hot or cold at the bottom of potential?

•

On large scales, the matter density fluctuation during the matter-dominated era is given by

16

T (t _L , x)

T ¯ (t _L ) = 1 3

⇢ _M (t _L , x)

¯

⇢ _M (t _L ) = 2

3 (t _L , x)

Hot at the bottom of the potential well, but…

⇢

_M

/ ⇢ ¯

_M

= 2

Adiabatic solution

Was the temperature hot or cold at the bottom of potential?

•

^Therefore,

17

T (ˆ n)

T ₀ = 1

3 (t _L , r ˆ _L )

This is negative in an over-density region!

18

Part III: Gravitational Lensing

19

Equation of motion for photons

Evolution of the direction of photon’s momentum

•

Instead of the magnitude of photon’s momentum, write the equation of motion for photon’s momentum

20

in terms of the unit vector of the direction of photon’s momentum, γ

ⁱ

:

y

x

“u” labels photon’s path

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d

ⁱ

dt = 1 a

X

3

j=1

(

^{i j ij}

) @

@ x

^j

( + )

The sum of two potentials!

Einstein’s what could-have-been the biggest blunder

Φ or Φ+Ψ?

•

In 1911, Einstein calculated the deflection of light by Sun, and concluded that it would be 0.87 arcsec.

•

At that time, Einstein had not realised yet the role of spatial curvature (Ψ).

Thus, his metric was still ds42 = –(1+2Φ)dt² + dx². As a result, his prediction was a factor of two too small: the correct value is 1.75 arcsec.

•

In 1914, the expedition organised by Erwin Freundlich (Berliner Sternwarte) to detect the deflection of light by Sun during the total solar eclipse failed.

•

In 1916, Einstein predicted 1.75 arcsec by incorporating Ψ, which is equal to Φ.

•

In 1919, the expedition organised by Arthur Eddington (Cambridge Observatory) confirmed Einstein’s prediction.

21

What if Freundlich’s

expedition was successful?

Getting 1.75 arcsec

Let’s calculate!

22

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d

ⁱ

dt = 1 a

X

3

j=1

(

^{i j ij}

) @

@ x

^j

( + )

Sun

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= = GM/R

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d

²

dt = 2

²

X

j

j

@

@ x

^j

2 @

@ x

²

Look at i=2:

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= (2nd order) + 2GM b

[(x

¹

)

²

+ b

²

]

^3/2

Integrating over dt = dx

¹

, we obtain

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2

= 4GM b

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= 8.49 ⇥ 10

⁶

rad = 1.75 arcsec

^Yay!

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R = 6.96 ⇥ 10⁸ m M = 1.99 ⇥ 10³⁰ kg

(

Gravitational lensing effect on the CMB

What does it do to CMB?

•

The important fact: the gravitational lensing effect does not change the surface brightness.

•

This means that the value of CMB temperature does not change by lensing;

only the directions change.

•

You might be asked during your PhD exam: “Is the uniform CMB temperature affected by lensing?” The answer is no.

•

Only the anisotropy (and polarisation; Lecture 6) is affected:

23

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T _lensed (ˆ n) = T _unlensed (ˆ n + d)

Basak, Prunet & Benumbed (2008)

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T _unlensed (ˆ n)

²⁴

Basak, Prunet & Benumbed (2008)

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T _lensed (ˆ n) = T _unlensed (ˆ

²⁵

n + d)

Gravitational lensing effect on the CMB

Deflection angle and the “lens potential”

•

The vector “d” is called the deflection angle. For the scalar perturbation, we can write d as a gradient of a scalar potential (like the electric field):

with

26

<latexit sha1_base64="RuUn0A57rCXg5XeJy6WwBHj8dXw=">AAACMHicbVDLSgMxFM34tr6qLt0Ei9AilBlRdCOICrpUaFXolJLJ3GmDmcyQ3BHK0E9y46foRkERt36F6WNhrQcCh3PO5eaeIJXCoOu+OVPTM7Nz8wuLhaXlldW14vrGjUkyzaHOE5nou4AZkEJBHQVKuEs1sDiQcBvcn/X92wfQRiSqht0UmjFrKxEJztBKreKFfw4SGa21cl/HVIIyEPbKfochVRV6TMf9TI0ndnM/iGjYq7SKJbfqDkAniTciJTLCVav47IcJz2JQyCUzpuG5KTZzplFwCb2CnxlIGb9nbWhYqlgMppkPDu7RHauENEq0fQrpQP09kbPYmG4c2GTMsGP+en3xP6+RYXTUzIVKMwTFh4uiTFJMaL89GgoNHGXXEsa1sH+lvMM042g7LtgSvL8nT5Kbvap3UHWv90snp6M6FsgW2SZl4pFDckIuyRWpE04eyQt5Jx/Ok/PqfDpfw+iUM5rZJGNwvn8ACNyoXg==</latexit>

T _lensed (ˆ n) = T _unlensed (ˆ n + d)

<latexit sha1_base64="v2NgJs1BOK6i8R/kAH4pE/+eHn8=">AAACFnicbVBNS8MwGE79nPOr6tFLcAheHK0oehGGXjxOcB+wlpGm6RaWpiFJhVH6K7z4V7x4UMSrePPfmG4FdfOBwJPned83eZ9AMKq043xZC4tLyyurlbXq+sbm1ra9s9tWSSoxaeGEJbIbIEUY5aSlqWakKyRBccBIJxhdF37nnkhFE36nx4L4MRpwGlGMtJH69nHmBREMc3gJvUginHkCSU0R84Si+c9tiDTked+uOXVnAjhP3JLUQIlm3/70wgSnMeEaM6RUz3WE9rNiKGYkr3qpIgLhERqQnqEcxUT52WStHB4aJYRRIs3hGk7U3x0ZipUax4GpjJEeqlmvEP/zeqmOLvyMcpFqwvH0oShlUCewyAiGVBKs2dgQhCU1f4V4iEw62iRZNSG4syvPk/ZJ3T2rO7entcZVGUcF7IMDcARccA4a4AY0QQtg8ACewAt4tR6tZ+vNep+WLlhlzx74A+vjG96An9c=</latexit>

d = @

@ n ˆ

r

L

: the comoving distance from the observer to the last scattering surface

<latexit sha1_base64="BzhNW19xBpwkYTAN5eitTShBNfk=">AAACL3icbVDLSgMxFM34tr6qLt0Ei9CilhlRdCOIgrhwUcHaQqcOmTRjg5nMkNwRyjB+kRt/xY2IIm79CzNtF9p6IOTk3Hu4ucePBddg22/WxOTU9Mzs3HxhYXFpeaW4unajo0RRVqeRiFTTJ5oJLlkdOAjWjBUjoS9Yw78/y+uNB6Y0j+Q19GLWDsmd5AGnBIzkFc/dWPOy2yWAZeV41+USPPs2Vd5l1lGPbqAIzR+7KssvlZXdWpdvuzXNK2W1M/BhVfGKJbtq94HHiTMkJTREzSu+uJ2IJiGTQAXRuuXYMbRTooBTwbKCm2gWE3pP7ljLUElCpttpf98Mbxmlg4NImSMB99XfjpSEWvdC33SGBLp6tJaL/9VaCQRH7ZTLOAEm6WBQkAgMEc7Dwx2uGAXRM4RQxc1fMe0SkxGYiAsmBGd05XFys1d1Dqr21X7p5HQYxxzaQJuojBx0iE7QBaqhOqLoCb2gd/RhPVuv1qf1NWidsIaedfQH1vcP3bGoTg==</latexit>

(ˆ n) =

Z _r

_L

0

dr r _L r

r _L r ( + )(r, nr ˆ )

Part IV: Power Spectrum

27

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?

²⁸

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

29

Long Wavelength Short Wavelength

180 degrees/(angle in the sky)

Amplitude of W aves [ μ K

2

]

WMAP Collaboration

30

Power Spectrum, Explained

31

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.

32

Spherical harmonics

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

•

Dipole patterns (l=1)

33

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

<latexit sha1_base64="XLU8YyiTG6GvUNJlVpconh9498g=">AAACCnicbVDLSsNAFJ3UV62vqEs3o0WoQksiim6EohuXFewD2jRMppN26EwSZiZCCVm78VfcuFDErV/gzr9x2kbQ1gMXDufcy733eBGjUlnWl5FbWFxaXsmvFtbWNza3zO2dhgxjgUkdhywULQ9JwmhA6ooqRlqRIIh7jDS94fXYb94TIWkY3KlRRByO+gH1KUZKS665j9ykQxiDZZ7CS1gq20ddDn9EnnaPXbNoVawJ4DyxM1IEGWqu+dnphTjmJFCYISnbthUpJ0FCUcxIWujEkkQID1GftDUNECfSSSavpPBQKz3oh0JXoOBE/T2RIC7liHu6kyM1kLPeWPzPa8fKv3ASGkSxIgGeLvJjBlUIx7nAHhUEKzbSBGFB9a0QD5BAWOn0CjoEe/bledI4qdhnFev2tFi9yuLIgz1wAErABuegCm5ADdQBBg/gCbyAV+PReDbejPdpa87IZnbBHxgf3+vYmIE=</latexit>

a

_{` m}

= ( 1)

^m

a

^⇤_`m : sufficient to consider only m>=0

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

³⁴

in Galactic coordinates

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

<latexit sha1_base64="DzXFOjULeDw9KICe1jd04HINdHg=">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</latexit>

a

₁₀

= 5.124 mK str

^1/2

,

a

₁₁

= 0.3384 3.215i mK str

^1/2

,

a

_{1 1}

= a

^⇤₁₁

(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

35

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

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

✓ = ⇡

`

36

[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

37

•

Values of alm depend on

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

Angular Power Spectrum

•

The angular power spectrum, Cl, 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:

38

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

What physics can we learn

from this

measurement?

39

Φ!!

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:

40

φ=cos(qz)

r L

<latexit sha1_base64="ciE46FQAfLsnTsfc9vWVrW1JpLA=">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</latexit>

T (ˆ n)

T ₀ = 1

3 (t _L , nr ˆ _L )

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

<latexit sha1_base64="BlRi7QPiMbQzYaNAs1Xj9M8U1qE=">AAACEnicbVC7TsMwFHV4lvIKMLJYVEithKoEgWAssDAwFIk+pCaKHNdprTpxsB1EifoNLPwKCwMIsTKx8Tc4bQZoOZKlo3Pu1fU5fsyoVJb1bczNLywuLRdWiqtr6xub5tZ2U/JEYNLAnHHR9pEkjEakoahipB0LgkKfkZY/uMj81h0RkvLoRg1j4oaoF9GAYqS05JkVp96nZeVdHaSOH8D7UQU6seCx4hCeZXrFwVyWbx8qnlmyqtYYcJbYOSmBHHXP/HK6HCchiRRmSMqObcXKTZFQFDMyKjqJJDHCA9QjHU0jFBLppuNII7ivlS4MuNAvUnCs/t5IUSjlMPT1ZIhUX057mfif10lUcOqmNIoTRSI8ORQkDOrAWT+wSwXBig01QVhQ/VeI+0ggrHSLRV2CPR15ljQPq/Zx1bo+KtXO8zoKYBfsgTKwwQmogUtQBw2AwSN4Bq/gzXgyXox342MyOmfkOzvgD4zPH10inAQ=</latexit>

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

•A(tL): Amplitude

•q: Wavenumber in 3D

Gravitational Potential in 3D to Temperature in 2D

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

41

φ=cos(qz)

r L

In the x-axis, the angle θ1 subtends the half wavelength λ/2, with

<latexit sha1_base64="x3GKfoqJ977fd0YlXtya7cCrL94=">AAAB+XicbVDLSsNAFL3xWesr6tLNYBFc1aQouhGKblxWsA9oQplMJu3QySTOTAol9E/cuFDErX/izr9x2mahrQcGDuecy71zgpQzpR3n21pZXVvf2Cxtlbd3dvf27YPDlkoySWiTJDyRnQArypmgTc00p51UUhwHnLaD4d3Ub4+oVCwRj3qcUj/GfcEiRrA2Us+2PW7CIUY3qOal7PypZ1ecqjMDWiZuQSpQoNGzv7wwIVlMhSYcK9V1nVT7OZaaEU4nZS9TNMVkiPu0a6jAMVV+Prt8gk6NEqIokeYJjWbq74kcx0qN48AkY6wHatGbiv953UxH137ORJppKsh8UZRxpBM0rQGFTFKi+dgQTCQztyIywBITbcoqmxLcxS8vk1at6l5WnYeLSv22qKMEx3ACZ+DCFdThHhrQBAIjeIZXeLNy68V6tz7m0RWrmDmCP7A+fwDzyZKQ</latexit>

= 2⇡ /q

With trigonometry, we find

<latexit sha1_base64="X4EFaW9VhI0XU906xeHGIJoHoDU=">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</latexit>

tan ✓ ₁ ' ✓ ₁ = /2 r _L

<latexit sha1_base64="iCWO2yRii38MI98xt+EjA9lex0A=">AAAB8HicbVDLSgMxFL1TX7W+qi7dBIvgqs6Uii6LblxWsA9ph5LJZNrQJDMkGaEM/Qo3LhRx6+e4829M21lo64HA4Zxzyb0nSDjTxnW/ncLa+sbmVnG7tLO7t39QPjxq6zhVhLZIzGPVDbCmnEnaMsxw2k0UxSLgtBOMb2d+54kqzWL5YCYJ9QUeShYxgo2VHvvcRkN8URuUK27VnQOtEi8nFcjRHJS/+mFMUkGlIRxr3fPcxPgZVoYRTqelfqppgskYD2nPUokF1X42X3iKzqwSoihW9kmD5urviQwLrScisEmBzUgvezPxP6+Xmujaz5hMUkMlWXwUpRyZGM2uRyFTlBg+sQQTxeyuiIywwsTYjkq2BG/55FXSrlW9y6p7X680bvI6inACp3AOHlxBA+6gCS0gIOAZXuHNUc6L8+58LKIFJ585hj9wPn8AILWP9g==</latexit>

/2

<latexit sha1_base64="kioQwswNOG0n9Yz3t9xY5U1JAfk=">AAAB+nicbVDLSsNAFL2pr1pfqS7dDBbBVUlE0Y1QdOPCRQX7gCaEyXTSDp08nJkoJeZT3LhQxK1f4s6/cdpmoa0HLhzOuZd77/ETzqSyrG+jtLS8srpWXq9sbG5t75jV3baMU0Foi8Q8Fl0fS8pZRFuKKU67iaA49Dnt+KOrid95oEKyOLpT44S6IR5ELGAEKy15ZvXCCQQmmZOwPLsX3k3umTWrbk2BFoldkBoUaHrml9OPSRrSSBGOpezZVqLcDAvFCKd5xUklTTAZ4QHtaRrhkEo3m56eo0Ot9FEQC12RQlP190SGQynHoa87Q6yGct6biP95vVQF527GoiRVNCKzRUHKkYrRJAfUZ4ISxceaYCKYvhWRIdZRKJ1WRYdgz7+8SNrHdfu0bt2e1BqXRRxl2IcDOAIbzqAB19CEFhB4hGd4hTfjyXgx3o2PWWvJKGb24A+Mzx+mcpRB</latexit>

= ⇡

qr _L

<latexit sha1_base64="J7DM3WyxZs7rsnfW+7jb3zxK794=">AAACEHicbVC7SgNBFJ31GeMramkzGESrsCuKNkLQxsIignlANiyzk7tmcHZ3nLkrhiWfYOOv2FgoYmtp5984eRS+Dlw4nHMv994TKikMuu6nMzU9Mzs3X1goLi4tr6yW1tYbJs00hzpPZapbITMgRQJ1FCihpTSwOJTQDK9Ph37zFrQRaXKJfQWdmF0lIhKcoZWC0o4PUgaez5TS6R31I8147isxyH3sAbLAGxzf6OA8KJXdijsC/Uu8CSmTCWpB6cPvpjyLIUEumTFtz1XYyZlGwSUMin5mQDF+za6gbWnCYjCdfPTQgG5bpUujVNtKkI7U7xM5i43px6HtjBn2zG9vKP7ntTOMjjq5SFSGkPDxoiiTFFM6TId2hQaOsm8J41rYWynvMZsJ2gyLNgTv98t/SWOv4h1U3Iv9cvVkEkeBbJItsks8ckiq5IzUSJ1wck8eyTN5cR6cJ+fVeRu3TjmTmQ3yA877F0d9nV4=</latexit>

` ₁ ⇡ ⇡

✓ ₁ = qr _L

<latexit sha1_base64="KOT0HQYakgfjUS6ujnQfZaK9/tA=">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</latexit>

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?

42

φ=cos(qz)

r L

In the z-axis, the angle θ2 is subtends bigger than the half wavelength λ/2, with

<latexit sha1_base64="x3GKfoqJ977fd0YlXtya7cCrL94=">AAAB+XicbVDLSsNAFL3xWesr6tLNYBFc1aQouhGKblxWsA9oQplMJu3QySTOTAol9E/cuFDErX/izr9x2mahrQcGDuecy71zgpQzpR3n21pZXVvf2Cxtlbd3dvf27YPDlkoySWiTJDyRnQArypmgTc00p51UUhwHnLaD4d3Ub4+oVCwRj3qcUj/GfcEiRrA2Us+2PW7CIUY3qOal7PypZ1ecqjMDWiZuQSpQoNGzv7wwIVlMhSYcK9V1nVT7OZaaEU4nZS9TNMVkiPu0a6jAMVV+Prt8gk6NEqIokeYJjWbq74kcx0qN48AkY6wHatGbiv953UxH137ORJppKsh8UZRxpBM0rQGFTFKi+dgQTCQztyIywBITbcoqmxLcxS8vk1at6l5WnYeLSv22qKMEx3ACZ+DCFdThHhrQBAIjeIZXeLNy68V6tz7m0RWrmDmCP7A+fwDzyZKQ</latexit>

= 2⇡ /q

With trigonometry, we find <latexit sha1_base64="iCWO2yRii38MI98xt+EjA9lex0A=">AAAB8HicbVDLSgMxFL1TX7W+qi7dBIvgqs6Uii6LblxWsA9ph5LJZNrQJDMkGaEM/Qo3LhRx6+e4829M21lo64HA4Zxzyb0nSDjTxnW/ncLa+sbmVnG7tLO7t39QPjxq6zhVhLZIzGPVDbCmnEnaMsxw2k0UxSLgtBOMb2d+54kqzWL5YCYJ9QUeShYxgo2VHvvcRkN8URuUK27VnQOtEi8nFcjRHJS/+mFMUkGlIRxr3fPcxPgZVoYRTqelfqppgskYD2nPUokF1X42X3iKzqwSoihW9kmD5urviQwLrScisEmBzUgvezPxP6+Xmujaz5hMUkMlWXwUpRyZGM2uRyFTlBg+sQQTxeyuiIywwsTYjkq2BG/55FXSrlW9y6p7X680bvI6inACp3AOHlxBA+6gCS0gIOAZXuHNUc6L8+58LKIFJ585hj9wPn8AILWP9g==</latexit>

/2

<latexit sha1_base64="pLPl9qz7ZRR2RyFJAcSFcrhhuNg=">AAACD3icbVC7TsNAEDyHVwivACWNRQSiiuwIBAVFBA0FRZAIIMWRdb6sk1PO9nG3RkSW/4CGX6GhACFaWjr+hsujgMBIK41mdrW7E0jBNTrOl1WYmZ2bXygulpaWV1bXyusbVzpJFYMmS0SibgKqQfAYmshRwI1UQKNAwHXQPx3613egNE/iSxxIaEe0G/OQM4pG8su7Hgjh1zwqpUruvVBRlnmS55mHPUDq1/LjW+Wf++WKU3VGsP8Sd0IqZIKGX/70OglLI4iRCap1y3UktjOqkDMBeclLNUjK+rQLLUNjGoFuZ6N/cnvHKB07TJSpGO2R+nMio5HWgygwnRHFnp72huJ/XivF8Kid8VimCDEbLwpTYWNiD8OxO1wBQzEwhDLFza0261GTCZoISyYEd/rlv+SqVnUPqs7FfqV+MomjSLbINtkjLjkkdXJGGqRJGHkgT+SFvFqP1rP1Zr2PWwvWZGaT/IL18Q3rCJ01</latexit>

` ₂ ⇡ ⇡

✓ ₂ < qr _L

It is not simple! Let’s formulate this more precisely in the next lecture.

43