The Linear Universe Matt Johnson Perimeter Institute/York - - PowerPoint PPT Presentation

The Linear Universe

Matt Johnson Perimeter Institute/York University

Thursday, 4 July, 13

• Most of cosmology is described by General Relativity and

Relativistic Hydrodynamics.

Modelling the Universe

Thursday, 4 July, 13

• Most of cosmology is described by General Relativity and

Relativistic Hydrodynamics.

Modelling the Universe

gµν(x, t) = ρi(x, t) = uµ(x, t) =

metric fluid densities fluid velocities

• The fundamental variables:

Thursday, 4 July, 13

• Most of cosmology is described by General Relativity and

Relativistic Hydrodynamics.

Modelling the Universe

Gµν = 8πGTµν

Tµν = X

i

(ρi + pi) uiµuiν + pigµν

rµT µν = 0

• The laws: Einstein and continuity equations

gµν(x, t) = ρi(x, t) = uµ(x, t) =

metric fluid densities fluid velocities

• The fundamental variables:

Thursday, 4 July, 13

• Most of cosmology is described by General Relativity and

Relativistic Hydrodynamics.

Modelling the Universe

Gµν = 8πGTµν

Tµν = X

i

(ρi + pi) uiµuiν + pigµν

rµT µν = 0

• The laws: Einstein and continuity equations

Set of coupled PDE’s -- need initial conditions!

gµν(x, t) = ρi(x, t) = uµ(x, t) =

metric fluid densities fluid velocities

• The fundamental variables:

Thursday, 4 July, 13

The Linear Universe

• For much of the history of the Universe:

gµν(x, t) = ¯ gµν(t) + δgµν(x, t) ρi(x, t) = ¯ ρi(t) + δρi(x, t) uµ(x, t) = ¯ uµ + uµ(x, t)

δ = small

Thursday, 4 July, 13

The Linear Universe

• For much of the history of the Universe:

gµν(x, t) = ¯ gµν(t) + δgµν(x, t) ρi(x, t) = ¯ ρi(t) + δρi(x, t) uµ(x, t) = ¯ uµ + uµ(x, t)

δ = small

• Why was the universe so nearly homogeneous?

Thursday, 4 July, 13

The Linear Universe

• For much of the history of the Universe:

gµν(x, t) = ¯ gµν(t) + δgµν(x, t) ρi(x, t) = ¯ ρi(t) + δρi(x, t) uµ(x, t) = ¯ uµ + uµ(x, t)

δ = small

• Why was the universe so nearly homogeneous?
• For today: this is an extraordinary convenience!

Thursday, 4 July, 13

The Linear Universe

• For much of the history of the Universe:

gµν(x, t) = ¯ gµν(t) + δgµν(x, t) ρi(x, t) = ¯ ρi(t) + δρi(x, t) uµ(x, t) = ¯ uµ + uµ(x, t)

δ = small

• Why was the universe so nearly homogeneous?
• For today: this is an extraordinary convenience!

Homogeneous Background Fluctuations Universe

Gµν = 8πGTµν ¯ Gµ⌫ = 8⇡G ¯ Tµ⌫ Gµ⌫ = 8⇡GTµ⌫

Thursday, 4 July, 13

Initial Conditions

• The homogeneous Universe:

pi = wiρi

Types of fluids

¯ ρi(t0)

Densities today

Thursday, 4 July, 13

Initial Conditions

• The homogeneous Universe:

pi = wiρi

Types of fluids

¯ ρi(t0)

Densities today

• The Linear Universe:

P [gµ⌫ (x, t = 0)] P [⇢i (x, t = 0)] P ⇥ uiµ (x, t = 0) ⇤

Characterize statistics of inhomogeneities! Assume our Universe is typical.

Thursday, 4 July, 13

τ

6 Parameter Model of the Universe

ΛCDM

3 parameters 2 parameters 1 parameter (for CMB)

P [δgµν (x, t = 0)]

Thursday, 4 July, 13

Modelling the universe

That’s it! The rest is details.

Initially small fluctuations collapse to form galaxies, stars, etc.

Thursday, 4 July, 13

• GR is highly non-linear - inferring the state of the

early universe would be like asking for the weather 100 million years ago based on the weather today.

• No general classification of metrics - how to

characterize initial conditions?

• Shock waves, singularities, oh my!

Giving Thanks

• The non-linear Universe

Thursday, 4 July, 13

• Simple evolution allows initial conditions to be

inferred.

• Background evolution and growth of structure can

be analyzed separately.

• Simple classification of initial conditions and metric

degrees of freedom.

• Physics on different scales evolves independently

(Fourier modes independent).

Giving Thanks

• The linear Universe

Thursday, 4 July, 13

The rest Now for some details....

Thursday, 4 July, 13

The homogeneous universe

Thursday, 4 July, 13

The homogeneous universe

?Big Bang? 9.1 Billion Years: our sun ignites. 13.7 Billion Years: the present. 100 million years galaxies and first stars form. 380,000 years: neutral atoms form.

10−6 seconds: protons and neutrons form.

1 second: atomic nuclei form.

Hot Dense Cool Diffuse

Thursday, 4 July, 13

The homogeneous universe

ds2 = dt2 + a2(t)δijdxidxj

• The metric in a flat, homogeneous, isotropic universe:

comoving coordinates

Thursday, 4 July, 13

The homogeneous universe

ds2 = dt2 + a2(t)δijdxidxj

• The metric in a flat, homogeneous, isotropic universe:

comoving coordinates

∆x

time

∆s

constant comoving distance = growing physical distance

Thursday, 4 July, 13

The homogeneous universe

• Conformal time:

ds2 = a2(η) ⇥ dη2 + δijdxidxj⇤ ⌘ = Z dt a(t)

Thursday, 4 July, 13

The homogeneous universe

• Conformal time:

ds2 = a2(η) ⇥ dη2 + δijdxidxj⇤

time space

The big bang: when a(t)=0

⌘ ⌘0 = ±(x x0) ⌘ = Z dt a(t)

Thursday, 4 July, 13

The homogeneous universe

• Conformal time:

time space

The big bang: when a(t)=0

ds2 = a2(η) ⇥ dη2 + δijdxidxj⇤ ⌘ = Z dt a(t)

particle horizon

∆x = Z dt a(t) = Z a0=1

a=0

dln(a) aH

comoving horizon

Thursday, 4 July, 13

The homogeneous universe

H2 ⌘ ✓ ˙ a a ◆ = 8πGρ 3 Gµν = 8πGTµν

• Equations of motion in a homogeneous universe:

Thursday, 4 July, 13

The homogeneous universe

H2 ⌘ ✓ ˙ a a ◆ = 8πGρ 3 Gµν = 8πGTµν

• Equations of motion in a homogeneous universe:

˙ ρ = 3H (ρ + p) p = wρ rµT µν = 0

Thursday, 4 July, 13

The homogeneous universe

H2 ⌘ ✓ ˙ a a ◆ = 8πGρ 3 Gµν = 8πGTµν

• Equations of motion in a homogeneous universe:

˙ ρ = 3H (ρ + p) p = wρ rµT µν = 0 ρ = ρ0a−3(1+w)

• Solutions:

a(t) = a0t

2 3(1+w)

different fluids gravitate differently!

Thursday, 4 July, 13

LogHtimeL LogHΡL

The homogeneous universe

radiation matter dark energy

w = 1/3, ⇢ / a−4 w = 0, ⇢ / a−3 w = 1, , ⇢ / const. ρ = ρ0a−3(1+w)

Thursday, 4 July, 13

0.2 0.4 0.6 0.8 1.0 1.2 1.4 t t0 0.5 1.0 1.5 a

The homogeneous universe

• Evolution of the scale factor:

matter dark energy radiation

Thursday, 4 July, 13

The homogeneous universe

• Energy budget:

✓ H H0 ◆2 = X

i

Ωia−3(1+wi) X

i

Ωi = 1

Thursday, 4 July, 13

The homogeneous universe

• Energy budget:

✓ H H0 ◆2 = X

i

Ωia−3(1+wi) X

i

Ωi = 1

• Ωr ⇠ 10−4

Thursday, 4 July, 13

The homogeneous universe

• Energy budget:

✓ H H0 ◆2 = X

i

Ωia−3(1+wi) X

i

Ωi = 1

• Ωr ⇠ 10−4

H0 = 100h km s Mpc = h 3000 Mpc

1 Mpc = 3 × 1023 meters

1 Mpc = 3.3 × 106 Lyr

Thursday, 4 July, 13

The homogeneous universe

z = λobs − λem λobs

• Redshift:

a = 1 1 + z a = 1, z = 0

Thursday, 4 July, 13

The homogeneous universe

z = λobs − λem λobs

• Redshift:

a = 1 1 + z a = 1, z = 0 zeq = 3400, z∗ = 1090, zre ⇠ 11, zgal ⇠ 11 12, zsurveys < ⇠ 1, zΛ = .28, zVirgo = .003

Thursday, 4 July, 13

• There is structure in the Universe:

ρ(x, t) = 1 V X ρ(k, t)ei~

k·~ x

• In Fourier space:

ρ(k, t)e ρ(k, 3000 ¯ ρ(t)f

The Inhomogeneous Universe

Thursday, 4 July, 13

• There is structure in the Universe:

ρ(x, t) = 1 V X ρ(k, t)ei~

k·~ x

• In Fourier space:

ρ(k, t)e ρ(k, 3000 ¯ ρ(t)f

The Inhomogeneous Universe

Thursday, 4 July, 13

• There is structure in the Universe:

ρ(x, t) = 1 V X ρ(k, t)ei~

k·~ x

• In Fourier space:

ρ(k, t)e ρ(k, 3000 ¯ ρ(t)f

The Inhomogeneous Universe

Thursday, 4 July, 13

• There is structure in the Universe:

ρ(x, t) = 1 V X ρ(k, t)ei~

k·~ x

• In Fourier space:

ρ(k, t)e ρ(k, 3000 ¯ ρ(t)f

large on small scales

The Inhomogeneous Universe

Thursday, 4 July, 13

The Inhomogeneous Universe

• There is structure in the Universe:

Structure: δρ

¯ ρ > ⇠ 1 O(106) O(103) O(1)

galaxies clusters superclusters

Thursday, 4 July, 13

The Inhomogeneous Universe

• There is structure in the Universe:

Structure: δρ

¯ ρ > ⇠ 1 O(106) O(103) O(1)

galaxies clusters superclusters k = .1 .01Mpc−1

• There is structure on all scales which have had a chance to

undergo gravitational collapse.

• The largest structures in the Universe define a scale above

which the fluctuations in density are linear:

wave number

λ ∼ 1% of observable Universe

Thursday, 4 July, 13

The Inhomogeneous Universe

• There is structure in the Universe:

Luminous: Baryons Photons Semi-Luminous: Neutrinos

Thursday, 4 July, 13

The Inhomogeneous Universe

• There is structure in the Universe:

Dark: Dark Matter Dark Energy Luminous: Baryons Photons Semi-Luminous: Neutrinos

Thursday, 4 July, 13

Gravitational Instability

• For non-relativistic matter in flat space:

@⇢ @t + r · (⇢~ v) = 0 @~ v @t + (~ v · r)~ v + rp ⇢ + r = 0 r2 = 4⇡G⇢

continuity Euler Poisson

Thursday, 4 July, 13

⇢ = ¯ ⇢ + ⇢ = ¯ + ~ v = ¯ ~ v + ~ v p = ¯ p + c2

s⇢

• Linearize:

c2

s = ∂p

∂ρ = w

Gravitational Instability

• For non-relativistic matter in flat space:

@⇢ @t + r · (⇢~ v) = 0 @~ v @t + (~ v · r)~ v + rp ⇢ + r = 0 r2 = 4⇡G⇢

continuity Euler Poisson

Thursday, 4 July, 13

Gravitational Instability

• Linearized equation of motion:

@2⇢ @t2 c2

sr2⇢ 4⇡G¯

⇢⇢ = 0

Thursday, 4 July, 13

Gravitational Instability

• Linearized equation of motion:

@2⇢ @t2 c2

sr2⇢ 4⇡G¯

⇢⇢ = 0

⇢(t, ~ x) = Z d3k (2⇡)3 ⇢(t, k)ei~

k·~ x

• Fourier transform:

Thursday, 4 July, 13

Gravitational Instability

• Linearized equation of motion:

@2⇢ @t2 c2

sr2⇢ 4⇡G¯

⇢⇢ = 0 @2⇢ @t2 +

• c2

sk2 4⇡G¯

⇢

• ⇢ = 0

⇢(t, ~ x) = Z d3k (2⇡)3 ⇢(t, k)ei~

k·~ x

• Fourier transform:

Thursday, 4 July, 13

Gravitational Instability

• Linearized equation of motion:

@2⇢ @t2 c2

sr2⇢ 4⇡G¯

⇢⇢ = 0 @2⇢ @t2 +

• c2

sk2 4⇡G¯

⇢

• ⇢ = 0

⇢(t, k) = A exp (i!(k)t) + B exp (i!(k)t) !(k) = p k2c2

s 4⇡G¯

⇢

⇢(t, ~ x) = Z d3k (2⇡)3 ⇢(t, k)ei~

k·~ x

• Fourier transform:

Thursday, 4 July, 13

!(k) = p k2c2

s 4⇡G¯

⇢

Gravitational Instability

Thursday, 4 July, 13

!(k) = p k2c2

s 4⇡G¯

⇢ J = 2⇡ kJ = cs ✓ ⇡ G¯ ⇢ ◆1/2

• Jeans scale: competition between pressure and gravity

Gravitational Instability

Thursday, 4 July, 13

!(k) = p k2c2

s 4⇡G¯

⇢ J = 2⇡ kJ = cs ✓ ⇡ G¯ ⇢ ◆1/2

• Jeans scale: competition between pressure and gravity

< J > J

sound waves gravitational collapse

Gravitational Instability

Thursday, 4 July, 13

• In an expanding universe waves are stretched:

ph = a(t)com

Gravitational Instability

∆x

time

∆s

Thursday, 4 July, 13

• In an expanding universe waves are stretched:

ph = a(t)com

Gravitational Instability

∆x

time

∆s

• Only gravitationally bound (non-linear) structures separate

from the Hubble flow.

• Expansion inhibits collapse.

Thursday, 4 July, 13

Gravitational Instability

tcoll ⇠ (4πG¯ ρ)−1/2 ⇠ H−1

Expansion will be relevant!

Thursday, 4 July, 13

Gravitational Instability

tcoll ⇠ (4πG¯ ρ)−1/2 ⇠ H−1

Expansion will be relevant!

 d2 dt2 + 2H d dt + ✓ c2

s

k2

com

a2 4⇡G¯ ⇢ ◆ ⇢(t, kcom) ¯ ⇢(t) = 0

• Including expansion (on small scales):

Thursday, 4 July, 13

Gravitational Instability

tcoll ⇠ (4πG¯ ρ)−1/2 ⇠ H−1

Expansion will be relevant!

 d2 dt2 + 2H d dt + ✓ c2

s

k2

com

a2 4⇡G¯ ⇢ ◆ ⇢(t, kcom) ¯ ⇢(t) = 0

• Including expansion (on small scales):

⇢ ¯ ⇢ / a ⇢ ¯ ⇢ / const. c2

s = 0

radiation matter dark energy

⇢ ¯ ⇢ / log(a)

Thursday, 4 July, 13

Gravitational Instability

tcoll ⇠ (4πG¯ ρ)−1/2 ⇠ H−1

Expansion will be relevant!

 d2 dt2 + 2H d dt + ✓ c2

s

k2

com

a2 4⇡G¯ ⇢ ◆ ⇢(t, kcom) ¯ ⇢(t) = 0

• Including expansion (on small scales):

⇢ ¯ ⇢ / a ⇢ ¯ ⇢ / const. c2

s = 0

radiation matter dark energy

⇢ ¯ ⇢ / log(a)

To have structure, need:

δρ ¯ ρ (teq) > ⇠ 3 ⇥ 10−4

Thursday, 4 July, 13

Gravitational Instability

• An important scale: comoving horizon

1 aH

• Horizon crossing: k = aH

comoving scale conformal time η

k 1 aH = η

superhorizon subhorizon horizon crossing

Thursday, 4 July, 13

Gravitational Instability

• An important scale: comoving horizon

1 aH

• Horizon crossing: k = aH

comoving scale conformal time η

1 aH = η

no gravitational collapse gravitational collapse horizon crossing GR necessary

Thursday, 4 July, 13

Gravitational Instability

• An important scale: comoving horizon

1 aH

• Horizon crossing: k = aH

comoving scale conformal time η

1 aH = η

smaller scales collapse first

Thursday, 4 July, 13

Gravitational Instability

• An important scale: comoving horizon

1 aH

• Horizon crossing: k = aH

comoving scale conformal time η

1 aH = η

extrapolating, all scales start out superhorizon

Thursday, 4 July, 13

Gravitational Instability

• An important scale: comoving horizon

1 aH

• Horizon crossing: k = aH

comoving scale conformal time η

Thursday, 4 July, 13

Gravitational Instability

• An important scale: comoving horizon

1 aH

• Horizon crossing: k = aH

comoving scale conformal time η

Perturbations generated here? causal !Ruled out by data! e.g. topological defects

Thursday, 4 July, 13

Gravitational Instability

• An important scale: comoving horizon

1 aH

• Horizon crossing: k = aH

comoving scale conformal time η

Perturbations generated here? acausal !Agrees with data! Inflation: fixes causality Perturbations generated here? causal !Ruled out by data! e.g. topological defects

Thursday, 4 July, 13

Gravitational Instability

• An important scale: comoving horizon

1 aH

• Horizon crossing: k = aH

comoving scale conformal time η

superhorizon subhorizon subhorizon Inflation and alternatives

Thursday, 4 July, 13

Relativistic Perturbations

• The full model:

rµT µ⌫ = 0 Gµ⌫ = 8⇡GTµ⌫ ) + δgµν(x, t) ) + δρi(x, t) + uµ(x, t)

Thursday, 4 July, 13

Relativistic Perturbations

• The full model:

rµT µ⌫ = 0 Gµ⌫ = 8⇡GTµ⌫ ) + δgµν(x, t) ) + δρi(x, t) + uµ(x, t)

• Metric fluctuations

) + δgµν(x, t)

4 scalar 4 vector 2 tensor

) + δρi(x, t)

gravity waves decay, not major player

Thursday, 4 July, 13

Relativistic Perturbations

• The full model:

rµT µ⌫ = 0 Gµ⌫ = 8⇡GTµ⌫ ) + δgµν(x, t) ) + δρi(x, t) + uµ(x, t)

• Metric fluctuations

) + δgµν(x, t)

4 scalar 4 vector 2 tensor

) + δρi(x, t)

gravity waves decay, not major player

• Gauge choice: only 6 DOF.

ds2 = a(⌘)2 ⇥ (1 + 2Ψ) d⌘2 + (1 + 2Φ) ijdxidxj⇤

“Conformal Newtonian Gauge”

Thursday, 4 July, 13

Important events

equality

Thursday, 4 July, 13

Important events

equality

photons tightly coupled to baryons imperfect fluid with dark matter perturbations begin to grow

c2

s ' 1/2

Thursday, 4 July, 13

Important events

recombination equality

photons tightly coupled to baryons imperfect fluid with dark matter perturbations begin to grow

c2

s ' 1/2

Thursday, 4 July, 13

Important events

recombination equality

photons and baryons decouple: CMB is released! baryons begin to collapse into dark matter halos neutral atoms form photons tightly coupled to baryons imperfect fluid with dark matter perturbations begin to grow

c2

s ' 1/2

Thursday, 4 July, 13

Important events

recombination equality

photons and baryons decouple: CMB is released! baryons begin to collapse into dark matter halos neutral atoms form

reionization

photons tightly coupled to baryons imperfect fluid with dark matter perturbations begin to grow

c2

s ' 1/2

Thursday, 4 July, 13

Important events

recombination equality

photons and baryons decouple: CMB is released! baryons begin to collapse into dark matter halos neutral atoms form

reionization

first non-linear structure some CMB photons re- scatter stars reionize the Universe hierarchical structure formation photons tightly coupled to baryons imperfect fluid with dark matter perturbations begin to grow

c2

s ' 1/2

Thursday, 4 July, 13

Baryon Acoustic Oscillations

• Before recombination, photons and baryons are coupled:

 d2 dt2 + 2H d dt + ✓ c2

s

k2

com

a2 4⇡G¯ ⇢ ◆ ⇢(t, kcom) ¯ ⇢(t) = 0

effective pressure from coupling

Thursday, 4 July, 13

Baryon Acoustic Oscillations

• Before recombination, photons and baryons are coupled:

 d2 dt2 + 2H d dt + ✓ c2

s

k2

com

a2 4⇡G¯ ⇢ ◆ ⇢(t, kcom) ¯ ⇢(t) = 0

effective pressure from coupling

• Sound horizon:

rs = Z ⌘ dη cs(η)

frequency of oscillation for sound waves

Standard ruler!

δρ ¯ ρ ∝ cos [krs(η)]

Thursday, 4 July, 13

Baryon Acoustic Oscillations

• Before recombination, photons and baryons are coupled:

 d2 dt2 + 2H d dt + ✓ c2

s

k2

com

a2 4⇡G¯ ⇢ ◆ ⇢(t, kcom) ¯ ⇢(t) = 0

effective pressure from coupling Maximum at recombination Minimum at recombination

kn = nπ rs , n = 1, 2, 3, . . .

Thursday, 4 July, 13

The CMB

• The Universe is filled with a gas of photons.

dN(t, ~ x) = f(t, ~ x, p)d3xd3p (2⇡)3 f = 1 exp h

p T (t)

i 1

Bose-Einstein distribution

Thursday, 4 July, 13

The CMB

• The Universe is filled with a gas of photons.

dN(t, ~ x) = f(t, ~ x, p)d3xd3p (2⇡)3 f = 1 exp h

p T (t)

i 1

Bose-Einstein distribution

T(t)

• The Universe’s most perfect blackbody.
• The temperature today is 2.73 K.
• - all photons stretched equally

Thursday, 4 July, 13

The CMB

• Perturbations are characterized by:

f = 1 exp h

p T (t)(1+Θ(~ x,t,ˆ p))

i 1

Θ = T T

Thursday, 4 July, 13

The CMB

• Perturbations are characterized by:

f = 1 exp h

p T (t)(1+Θ(~ x,t,ˆ p))

i 1

Θ = T T

Observers at each position see an anisotropic distribution of photons

dN(tnow, ~ xhere) = Z f(tnow, ~ x, p)(~ xhere)d3xd3p (2⇡)3

Thursday, 4 July, 13

The CMB

✓⇢ ¯ ⇢ ◆

dm

, vdm, ✓⇢ ¯ ⇢ ◆

b

, vb, Θ, Φ, Ψ

Full set of coupled variables go into finding the temperature anisotropy

Thursday, 4 July, 13

The CMB

✓⇢ ¯ ⇢ ◆

dm

, vdm, ✓⇢ ¯ ⇢ ◆

b

, vb, Θ, Φ, Ψ

Full set of coupled variables go into finding the temperature anisotropy

• Sachs-Wolfe -- valid on largest scales

✓∆T T ◆

fin

= ✓∆T T ◆

init

Φinit = Φinit 3

Intrinsic temperature variations gravitational redshift

Thursday, 4 July, 13

The CMB

• Integrated Sachs-Wolfe -- time dependence of

potentials.

✓⇢ ¯ ⇢ ◆

dm

, vdm, ✓⇢ ¯ ⇢ ◆

b

, vb, Θ, Φ, Ψ

Full set of coupled variables go into finding the temperature anisotropy

• Sachs-Wolfe -- valid on largest scales

✓∆T T ◆

fin

= ✓∆T T ◆

init

Φinit = Φinit 3

Intrinsic temperature variations gravitational redshift

Thursday, 4 July, 13

The CMB

• In linear theory, can sum up the contribution from each

fourier mode separately:

x y

Thursday, 4 July, 13

The CMB

• In linear theory, can sum up the contribution from each

fourier mode separately:

x y

Thursday, 4 July, 13

The CMB

• In linear theory, can sum up the contribution from each

fourier mode separately:

x y

Thursday, 4 July, 13

The CMB

• Convenient to perform spherical harmonic transform:

Θ(t, ~ x, ˆ p) = X

` `

X

m=−`

a`mY`m(ˆ p)

Thursday, 4 July, 13

The CMB

• Convenient to perform spherical harmonic transform:

Θ(t, ~ x, ˆ p) = X

` `

X

m=−`

a`mY`m(ˆ p)

• The transfer function:

a`m = Z d3k (2π)3 ∆`(k)Φinit(k)Y`m(ˆ k) Φinit(k) ! a`m

projection, evolution, ISW, etc.

Thursday, 4 July, 13

The CMB

• Convenient to perform spherical harmonic transform:

Θ(t, ~ x, ˆ p) = X

` `

X

m=−`

a`mY`m(ˆ p)

• The transfer function:

a`m = Z d3k (2π)3 ∆`(k)Φinit(k)Y`m(ˆ k) Φinit(k) ! a`m

projection, evolution, ISW, etc.

• Computed numerically: CAMB, CMBFast, etc.

Thursday, 4 July, 13

The Power Spectrum

• Fluctuations are characterized statistically:

hΦ(k)Φ(k0)i = δ3(k k0)P(k)

ha`ma`0m0i = ``0mm0C` C` = 2 ⇡ Z dk k2∆2

`(k)P(k)

Relates statistics of primordial fluctuations to the statistics of fluctuations in the CMB

hΦ(k)Φ(k0)Φ(k00)i = 0

gaussian: .......... ..........

Thursday, 4 July, 13

The Power Spectrum

P(k) = Akns1

C`

Acoustic Peaks Sachs-Wolfe Damping tail

Thursday, 4 July, 13

Comments

• In a statistically homogeneous and isotropic universe with

gaussian fluctuations, the power spectrum is all the information there is.

• There is power on scales of order the size of the
• bservable universe -- superhorizon fluctuations.
• The structure of the acoustic peaks is determined by the

contents of the universe as well as the initial conditions.

Thursday, 4 July, 13

Other cool things in CMB

• Lensing of the CMB: information on intervening structure
• Polarization of the CMB: primordial gravitational waves
• Sunyaev-Zeldovich effect: shadows of galaxy clusters in the

CMB

• Combined mass and number of neutrinos.

Thursday, 4 July, 13

6 Parameter Model of the Universe

ΛCDM

Planck Planck+lensing Planck+WP Parameter Best fit 68% limits Best fit 68% limits Best fit 68% limits Ωbh2 . . . . . . . . . . 0.022068 0.02207 ± 0.00033 0.022242 0.02217 ± 0.00033 0.022032 0.02205 ± 0.00028 Ωch2 . . . . . . . . . . 0.12029 0.1196 ± 0.0031 0.11805 0.1186 ± 0.0031 0.12038 0.1199 ± 0.0027 100θMC . . . . . . . . 1.04122 1.04132 ± 0.00068 1.04150 1.04141 ± 0.00067 1.04119 1.04131 ± 0.00063 τ . . . . . . . . . . . . 0.0925 0.097 ± 0.038 0.0949 0.089 ± 0.032 0.0925 0.089+0.012

−0.014

ns . . . . . . . . . . . 0.9624 0.9616 ± 0.0094 0.9675 0.9635 ± 0.0094 0.9619 0.9603 ± 0.0073 ln(1010As) . . . . . . . 3.098 3.103 ± 0.072 3.098 3.085 ± 0.057 3.0980 3.089+0.024

−0.027

Thursday, 4 July, 13

6 Parameter Model of the Universe

ΛCDM

Planck Planck+lensing Planck+WP Parameter Best fit 68% limits Best fit 68% limits Best fit 68% limits Ωbh2 . . . . . . . . . . 0.022068 0.02207 ± 0.00033 0.022242 0.02217 ± 0.00033 0.022032 0.02205 ± 0.00028 Ωch2 . . . . . . . . . . 0.12029 0.1196 ± 0.0031 0.11805 0.1186 ± 0.0031 0.12038 0.1199 ± 0.0027 100θMC . . . . . . . . 1.04122 1.04132 ± 0.00068 1.04150 1.04141 ± 0.00067 1.04119 1.04131 ± 0.00063 τ . . . . . . . . . . . . 0.0925 0.097 ± 0.038 0.0949 0.089 ± 0.032 0.0925 0.089+0.012

−0.014

ns . . . . . . . . . . . 0.9624 0.9616 ± 0.0094 0.9675 0.9635 ± 0.0094 0.9619 0.9603 ± 0.0073 ln(1010As) . . . . . . . 3.098 3.103 ± 0.072 3.098 3.085 ± 0.057 3.0980 3.089+0.024

−0.027

Thursday, 4 July, 13