[PPT] - Cosmology & CMB Set5:Initial Conditions and Inflation Davide PowerPoint Presentation

SLIDE 1

Cosmology & CMB

Set5:Initial Conditions and Inflation Davide Maino Universit` a degli Studi di Milano ISAPP 2011 Ph.D. School

SLIDE 2

Einstein Equations - I

For perturbation Θ we need Θ(0) and similarly Φ and Ψ and

their time derivatives

Einstein equations describe the evolution. Take the time-time

component G0

0 = g00

R00 − 1

2g00R

= (−1 + 2Ψ)R00 − R

2 and taking the perturbed part and working in Fourier space δG0

0 = −6H∂tΨ + 6ΨH2 − 2Φk2

a2 = 8πGδT0 where T0

0 = −ργ(1 + 4Θ0). The same is true also for other

species with density contrast δi, where i = b, cdm

SLIDE 3

Einstein Equations - II

Moving to conformal time

k2Φ+3 ˙ a a

˙

Φ − ˙ a aΨ

= 4πGa2(ρcdmδcdm+ρbδb+4ργΘ0+4ρνN0)

where N0 is monopole term for neutrinos

This is Poisson equation taking into account expansion.
Taking the space-space equation

k2(Φ + Ψ) = 32ρπa2G(ργΘ2 + ρνN2) where Θ2 and N2 are quadrupole moment for photons and neutrinos.

Scattering is effective, only monopole and dipole are important

Φ = −Ψ

SLIDE 4

Initial Conditions - I

BE for photons, baryons and dark matter

˙ Θ + ikµΘ = − ˙ Φ − ikµΨ − ˙ τ[Θ0 − Θ + µvb] ˙ δb(cdm) + ikvb(cdm) = −3 ˙ Φ

At early times all scales of interest are outside the horizon

kη ≪ 1 ˙ Θ ≃ Θ η → ˙ Θ kΘ ≃ Θ ηkΘ ≫ 1 → ˙ Θ ≫ Θ we neglect all Θ and terms with k ˙ Θ0 + ˙ Φ = 0 ˙ δb(cdm) = −3 ˙ Φ

SLIDE 5

Initial Conditions - II

With this approximation and taking contribution only from

radiation, the time-time equation reads 3 ˙ a a

˙

Φ − ˙ a aΨ

= 16πGa2ργΘ0
For RD a ∝ t1/2 ∝ η and ˙

a/a ∝ 1/η ˙ Φ η − Ψ η2 = 28πGa2 3 ργΘ0 = 2 ˙ a a 2 ργ ρ Θ0 ≃ 2 η2 Θ0 using Friedmann equation in conformal time

SLIDE 6

Initial Conditions - III

Take the conformal time derivative and using ˙

Θ0 + ˙ Φ = 0 ˙ Φ + η¨ Φ − ˙ Ψ = −2 ˙ Φ η¨ Φ + 4 ˙ Φ =

Search solution of the form Φ = ηp. Two solutions with p = 0

(time independent) and p = −3 (rapidly decreasing). Φ = 2Θ0

For density contrast

˙ δ = 3 ˙ Θ0 → δ = 3Θ + constant and perturbations are classified according to value for the

constant. If zero perturbations are called adiabatic otherwise

iso-curvature

SLIDE 7

Inflation - I

Why Φ = 0 and how is possible Φ = 2Θ0?
Conformal time η is maximum distance a photon can travels

from the beginning: particle horizon

Scales λ ∝ k−1 with kη ≪ 1 are larger than particle horizon

hence no causal contact. Only when re-enter horizon physics (i.e. Compton Scattering) plays its game

However CMB is highly isotropic all over the sky: how is this

possible?

SLIDE 8

Inflation - I

SLIDE 9

Inflation - II

The comoving horizon

η = dt a = da a 1 aH logarithmic integral of the comoving Hubble radius (aH)−1, the maximum distance a particle can travel in a unit expansion time

Particles separated now by a distance > (aH)−1 cannot talk now

but if the Hubble radius were larger in the past they can talk each

ther
But H−1 ∝ a2 in RD or H−1 ∝ a3/2 in MD

SLIDE 10

Inflation - II

SLIDE 11

Inflation - III

(aH)−1 has to decrease i.e. (aH) has to increase

d dt

ada/dt

a

= d2a

dt2 > 0 which is an accelerated expansion usually called inflation

With H almost constant during inflation

da a = Hdt → a(t) = aeexp[H(t − te)] t < te

SLIDE 12

Inflation - III

SLIDE 13

Inflation - III

Accelerated expansion

¨ a a = −4πG 3 (ρ + 3p) which is < 0 for both radiation and matter

¨

a > 0 ↔ ρ + 3p < 0 ↔ p < −ρ/3

This new ingredient has to dominate the Universe energy content

SLIDE 14

Inflation and horizon problem

During de Sitter phase horizon scale H−1 is almost constant
If inflation lasts long enough, all interesting scales are outside

the horizon

e-foldings

N = ln[HI(te − ti)]

To solve horizon we require that present horizon scale H−1

was reduced during inflation to a scale λH0(ti) < H−1

I

λH0(ti) = H−1 ate at0 ati ate

= H−1

T0 Tf

e−N ≤ H−1

I

In units of h = c = G = 1 Hubble constant H0 = 9.6 × 10−62

and T0 = 1.9 × 10−32 N ≥ ln T0 H0

≃ 67 + ln

Te HI

SLIDE 15

Inflation and flatness problem

During de Sitter phase H−1 is constant

Ω − 1 = k a2H2 ∝ 1 a2

The fine-tuning required to have Ω0 ≃ 1 today is

|Ω − 1| ∼ 10−60 at the end of inflation |Ω − 1|t=te |Ω − 1|t=ti = ai ae 2 = e−2N

Taking |Ω − 1|t=ti ≃ 1 the e-foldings required are ≃ 70
Inflation ⇒ Ω0 = 1 but does not change geometry but magnify

curvature radius so that locally the Universe appears flat

SLIDE 16

Inflation with a scalar field - I

Scalar field: next step of complication after simple constant (Φ,

Ψ, gµν)

In QFT each particle has a quantum field associated but none of

the known particle has a field able to do inflation, not even Higgs boson

The field

φ(x, t) = φ0(t) + δφ(x, t) φ0 is classical part of the field, δφ are quantum fluctuations

Stress-energy tensor for a scalar field

Tα

β = gα ν ∂νφ∂βφ − gα β

1 2gµν∂µ∂νφ + V(φ)

SLIDE 17

Inflation with a scalar field - II

Take the classical part, depends only on t

Tα

β = −gα 0 g0 β( ˙

φ0)2 + gα

β

1 2( ˙ φ0)2 − V(φ)

T0

0 = −ρ and Ti i = p

−ρ = − ˙ φ2

0 + 1

2 ˙ φ2

0 − V(φ0) ⇒ ρ = 1

2 ˙ φ2

0+V(φ0)

p = 1 2 ˙ φ2

0−V(φ0)

If potential energy dominates p = −ρ and inflation may occur

SLIDE 18

Equation of motion

˙ a a 2 = 8πG 3 ρ = 8πG 3 ˙ φ2 2 + V(φ0)

Take time derivative

2da/dt a

¨

a a − ˙ a a 2 = 8πG 3

˙

φ0 ¨ φ0 + V′ ˙ φ0

2da/dt

a

−4πG

ρ 3 + p

− 8πG

3 ρ

=

8πG 3

˙

φ0 ¨ φ0 + V′ ˙ φ0

−8πGH ˙

φ2 = 8πG 3

˙

φ0 ¨ φ0 + V′ ˙ φ0

¨

φ0 + 3H ˙ φ0 + V′ = 0

SLIDE 19

Slow-roll condition

To have inflation we require ˙

φ2

0 ≪ V(φ) so potential φ0 is slowly

rolling down its potential i.e. potential is very flat

We can neglect ¨

φ0 and Friedmann becomes H2 ≃ 8πG 3 V(φ0) dynamic is dominated by potential energy of the scalar field

Slow-roll requires

˙ φ0

2 ≪ V(φ0)

⇒ (V′)2 V ≪ H2 ¨ φ0 ≪ 3H ˙ φ0 ⇒ V′′ ≪ H2

SLIDE 20

Slow-roll parameters

It is useful to define

ǫ = − ˙ H H2 = 4πG ˙ φ2 H2 = 1 16πG V′ V 2 η = 1 8πG V′′ V

= 1

3 V′′ H2 δ = η − ǫ = − ¨ φ0 H ˙ φ0

ǫ quantifies how much H changes during inflation

¨ a a = ˙ H + H2 = (1 − ǫ)H2 inflation is possible only if ǫ < 1.

In general slow-roll is attained when ǫ ≪ 1 and |η| ≪ 1

SLIDE 21

Slow-roll parameters

Take Friedmann equations

H2 = 4πG ˙ φ2 ǫ ¨ a a = (1 − ǫ)H2 take time derivative of the first and divide by (˙ a/a)2 −2Hǫ = 2 ¨ φ0 ˙ φ0 + ˙ ǫ ǫ

Replace ¨

φ0 with δ ˙ ǫ = −2Hǫ (δ + ǫ)

Evolution of ǫ is second-order in parameters i.e. almost constant

during inflation

SLIDE 22

Choose a Frame

Gravity acts on all components: small fluctuations in φ are

related to metric perturbation giving rise to perturbations of the curvature Ψ

Comiving slicing: orthogonal to comoving observers world-line.

Free-falling observed for which expansion is isotropic δφcom = 0

Scalar field transforms

˜ δφ = δφ − ˙ φT and ˜ δφ = 0 with T = δφ/ ˙ φ Ψ → Ψcom = ψ + HT = Ψ + H δφ ˙ φ

Comoving curvature perturbation

R = Ψ + H δφ ˙ φ ≃ H δφ ˙ φ

SLIDE 23

Perturbations

Consider now also the perturbation term δφ and linearize the

perturbation ¨ δφ + 2 ˙ a a ˙ δφ + k2δφ + a2V′′δφ = 0 and in slow-roll inflation V′′ is negligible

For scales λ ≪ H−1 we have k ≫ aH and the second term can

be neglected ¨ δφ + k2δφ = 0 harmonic oscillator with frequency k2. Fluctuations do not grow

For scales λ ≫ H−1 above the horizon, k ≪ aH neglect the last

term ¨ δφ + 2 ˙ a a ˙ δφ = 0 fluctuations are constant

SLIDE 24

Slow-Roll Evolution

Rewrite u ≡ aδφ to remove expansion damping

¨ u +

k2 − 2

˙ a a 2 u = 0

for conformal time measured from the end of inflation

˜ η = η − ηend ˜ η = a

aend

da Ha2 ≈ − 1 aH

More compact slow-roll equation

¨ u +

k2 − 2

˜ η

u = 0

SLIDE 25

Slow-Roll limit

Slow-roll equation has exact solution

u = A

k ± i

˜ η

e∓ik˜

η

For |k˜

η| ≫ 1 (early times, inside the Hubble length) behaves as free oscillator lim

|k˜ η|→∞ u = Ake∓ik˜ η

Normalization A is set by the origin of quantum fluctuations of φ

SLIDE 26

Slow-Roll limit

For |k˜

η| ≪ 1 (late times, outside Hubble length) fluctuations freeze in lim

|k˜ η|→0 u

= ∓ i ˜ ηA = ∓iHaA δφ = ∓iHA R = ±iHA ˙ a a 1 ˙ φ

From Slow-Roll

˙ a a 2 1 ˙ φ2 = 4πG ǫ

Curvature Power Spectrum

∆2

R ≡ k3|R|2

2π2 = 8πG H2 (2π)2ǫk3A2

SLIDE 27

Quantum Fluctuations

Simple harmonic oscillator ≪ Hubble length

¨ u + k2u = 0

Quantize the oscillator. Transform into operators (give them hat)

ˆ u = u(k, η)ˆ a + u⋆(k, η)ˆ a† where u(k, η) satisfies the classical equation of motion

Creation and annihilation operators

[ˆ a, ˆ a†] ≡ ˆ aˆ a† − ˆ a†ˆ a = 1 [ˆ a, ˆ a] = [ˆ a†, ˆ a†] = 0

Normlization of wave-function

u(k, η) = 1 √ 2k e−ik˜

η

SLIDE 28

Quantum Fluctuations

Zero point fluctuations of ground state

u2 = 0|ˆ u†ˆ u|0 = 0|(u⋆ˆ a† + uˆ a)(uˆ a + u⋆ˆ a†)|0 = 0|ˆ aˆ a†|0|u(k, η)|2 = 0|[ˆ a, ˆ a†] + ˆ a†ˆ a|0|u(k, η)|2 = |u(k, η)|2 = 1 2k

So A = (2k3)−1/2 and curvature power spectrum

∆2

R = 8πG

2 H2 (2π)2ǫ, ∆2

δφ =

H2 (2π)2

As long as H is constant the spectrum is scale invariant

SLIDE 29

Tilt

Scalar spectral index quantifies the scale invariant nature of the

spectrum dln∆2

R

dlnk ≡ ns − 1 = 2dlnH dlnk − dlnǫ dlnk

Evaluate at horizon crossing where fluctuations freeze

dlnH dlnk |−k˜

η=1

= k H dH d˜ η |−k˜

η=1

d˜ η dk |−k˜

η=1

= k H (−aH2ǫ)|−k˜

η=1

1 k2 = −ǫ where aH = −1/˜ η = k

SLIDE 30

Tilt

Evolution of ǫ

dlnǫ dlnk = − dlnǫ dln˜ η = −2a(δ + ǫ)H˜ η = 2(δ + ǫ)

Tilt in slow-roll approximation

ns = 1 − 4ǫ − 2δ

Typical inflationary models has ns ≈ 1 yielding scale invariant

power spectrum

SLIDE 31

Inflationary Zoology

Despite the simplicity of the paradigm there are lot of

inflationary models differing in the potential and for the underlying particle physics theory

Three groups: small fields, large fields and hybrid models
ccupying different regions in the (ǫ, η) plane
Large Field models: 0 < η < 2ǫ; V(φ) = Λ4(φ/µ)p and

V(φ) = Λ4exp(φ/µ)

Small Field models: η < −ǫ; V(φ) = Λ4[1 − (φ/µ)p]
Hybrid models: 0 < 2ǫ < η; V(φ) = Λ4[1 + (φ/µ)p]
Different model predicts different amounts of, e.g., gravitational

waves and/or non-gaussianity

SLIDE 32

Large Field Models

Large field models include monomial potential V(φ) = Aφn for

which ǫ ≈ n2 16πGφ2 δ ≈ ǫ − n(n − 1) 8πGφ2

Slow-roll requires large fields values φ > (8πG)−1/2 = Mpl
Thus ǫ ∼ |δ| and observing ns = 1 indicates finite ǫ
WMAP tilt still support some large fields model, potentially

making gravitational waves observables

SLIDE 33

Small Field Models

If φ is near the maximum of the potential

V(φ) = V0 − 1 2µ2φ2

Inflation takes places if V0 dominates

ǫ ≈ 1 16πG µ4φ2 V2 δ ≈ ǫ + 1 8πG µ2 V0 → δ ǫ = V0 µ2φ2 ≫ 1

Tilt reflects δ: ns ≈ 1 − 2δ and ǫ is much smaller

ǫ = r 16 = 8πG 2 dφ dN 2

The more φ rolls, the larger tensor modes.
Small field does not roll enough to develop detectable GW

SLIDE 34

Hybrid Models

The field φ is rolling down the potential

V(φ) = V0 + 1 2m2φ2

Slow-roll parameters are similar to small field

ǫ ≈ 1 16πG m4φ2 V2 δ ≈ ǫ − 1 8πG m2 V0

V0 domination: ǫ small, δ < 0 and nS > 1 - blue tilt
m2 domination, monomial-like

SLIDE 35

Current Status - WMAP 7yr

SLIDE 36

Non-Gaussianity

In single field slow-roll inflation, inflaton φ is a free-field - modes

do not interact and fluctuations are Gaussian to a high degree

Non-gaussianities are at least second order effect and with

fluctuations at 10−5 level, this is a 10−10 effect!

Second order fluctuations are in general further reduced by order

ǫ,δ R(2) =

R(1)2

O(ǫ, δ)

In general fluctuations in Newtonian gauge on Φ are in the form

Φ(x) = Φ(1)(x) + fNL

Φ(1)(x)

2 − Φ(1)(x)2

and general prediction fNL = O(1)

SLIDE 37

Non-Gaussinity

Decompose in harmonics

φ(k) =

d3xe−ik·xΦ(x) = Φ(1)(k)

+ fNL

d3xe−ik·x
d3k1

(2π)3 eik1·xΦ(1)(k1)

d3k2

(2π)3 eik2·xΦ(1)(k2) and since

d3xei(k−k′)·x = (2π)3δ(k − k′) yields

Φ(k) = Φ(1)(k) + fNL

d3k1

(2π)3 Φ(1)(k1)Φ(1)(k − k′)

The first order term is a Gaussian field represented by the power

spectrum Φ(1)(k)Φ(1)(k′) = (2π)3δ(k + k′)PΦ(k) we get a bi-spectrum contribution

SLIDE 38

Non-Gaussianity

The bi-spectrum is proportional to the product of power spectra

Φ(k1)Φ(k2)Φ(k3) = (2π)3δ(k1 + k2 + k3) [2fNLPΦ(k1)PΦ(k2) + perm]

Bi-spectrum contains most of the information on local

Non-Gaussinity.

Higher order correlation functions do exist and can be measured

with good S/N but they have larger sample variance and do not help on fNL

From CMB total intensity anisotropies fNL < 80 with 95% CL
Planck is expected to reach fNL ∼ few

SLIDE 39

Curvaton

Single field inflation has almost null non-gaussianity but beyond

this larger fNL can be generated

Suppose beside inflaton there is another scalar field σ called

curvaton during inflation that has quantum fluctuations ∆2

σ ≈

H 2π 2

After inflation curvaton oscillates around its minimum and

decays leaving its field fluctuations as density fluctuations

SLIDE 40

Curvaton

The energy density of the field is the square of field amplitude

ρσ = m2

σσ2

Fractional density fluctuations are related to fractional field

fluctuations δρσ ρσ ≈ 2δσ σ and so power spectrum gets a boost by 1/σ2

⋆ (at horizon crossing)

One can tune to make it dominate near the end of inflation
Dominating energy density means curvature perturbation like the

usual inflaton field