Lecture 3: Gravitational Effects on Temperature Anisotropy

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Lecture 3: Gravitational Effects on Temperature Anisotropy

1

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Part I: Sachs-Wolfe Effect(s)

2

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Evolution of photon’s energy

Sachs & Wolfe (1967)

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

3 γ ⁱ is a unit vector of the direction of photon’s momentum:

Newtonian

gravitational potential

Scalar curvature

perturbation Tensor perturbation

= Gravitational wave

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Evolution of photon’s energy

Sachs & Wolfe (1967)

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

4 γ ⁱ is a unit vector of the direction of photon’s momentum:

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1 ap

d(ap)

dt =

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Evolution of photon’s energy

Sachs & Wolfe (1967)

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

5 γ ⁱ 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

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Formal Solution (Scalar)

or

Line-of-sight direction

“L” for “Last scattering surface”

Sachs & Wolfe (1967)

Present-day time

Comoving distance (r)

6

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Formal Solution (Scalar)

Line-of-sight direction Initial Condition

Sachs & Wolfe (1967)

Comoving distance (r)

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Formal Solution (Scalar)

Line-of-sight direction

Comoving distance (r) Gravitational Redshit

Sachs & Wolfe (1967)

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Formal Solution (Scalar)

Line-of-sight direction

“integrated Sachs-Wolfe” (ISW) eﬀect

Sachs & Wolfe (1967)

Comoving distance (r)

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Part II: Initial Condition

10

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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

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“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 i} ^th ^{and j} ^th ^{species, n} ⁱ ^(x)/n ^j (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 )

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Example of the adiabatic initial condition

Thermal equilibrium

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

• ⁿ ^photon ^{~ T} ³ ^{and n} ^baryon ^{~ T} ³

• That is to say, thermal equilibrium naturally gives rise to the adiabatic initial condition, because n photon / n baryon = constant

• This gives

13 • “B” for “Baryons”

• ρ is the mass density

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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

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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 )

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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

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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!

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Part III: Gravitational Lensing

19

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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!

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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 ds 42 = –(1+2Φ)dt ² + dx ² . 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?

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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 ⇥ 10 ⁸ m M = 1.99 ⇥ 10 ³⁰ kg

(

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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 8) is aﬀected:

23

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

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Basak, Prunet & Benumbed (2008)

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T _unlensed (ˆ n) ²⁴

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Basak, Prunet & Benumbed (2008)

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

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

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d = @

@ n ˆ

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

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(ˆ n) =

Z _r _L

0 dr r _L r

r _L r ( + )(r, nr ˆ )

Lecture 3: Gravitational Effects on Temperature Anisotropy