[PPT] - Inflationary non-Gaussianity: theoretical predictions and PowerPoint Presentation

SLIDE 1

Inflationary non-Gaussianity: theoretical predictions and observational consequences

Gabriel Jung, thesis defense Supervisor: Bartjan van Tent 22 May 2018, at the Laboratoire de Physique Th´ eorique

Gabriel Jung

LPT Orsay

22 May 2018 1 / 23

SLIDE 2

Inflation

History of

ur

universe

Source: ESA

Inflation (Starobinsky, 1980; Guth, 1981)

Period of fast and accelerated expansion in the very early universe
Energy content of the universe dominated by a scalar field
Solves several issues of the standard Big Bang theory

Gabriel Jung

LPT Orsay

22 May 2018 2 / 23

SLIDE 3

Cosmic Microwave Background (CMB)

CMB (observed in 1964 by Penzias and Wilson)

Relic radiation emitted 380000 years after the Big Bang
Blackbody spectrum at a temperature of 2.725 ± 0.001 K
Coming from all parts of the sky with a uniform temperature
Observational window on the primordial universe

Gabriel Jung

LPT Orsay

22 May 2018 3 / 23

SLIDE 4

Cosmic Microwave Background (CMB)

CMB (observed in 1964 by Penzias and Wilson)

Relic radiation emitted 380000 years after the Big Bang
Blackbody spectrum at a temperature of 2.725 ± 0.001 K
Coming from all parts of the sky with a uniform temperature
Observational window on the primordial universe
Precision cosmology era:

(WMAP + Planck) Measurements

f

relative temperature fluctuations of

rder 10−5

CMB seen by Planck

Gabriel Jung

LPT Orsay

22 May 2018 3 / 23

SLIDE 5

Perturbations

Primordial fluctuations → Large-scale structures present now
Explained by inflation: scalar field quantum fluctuations + expansion
Single-field inflation → perturbations are frozen on super-Hubble

scales → simple link between inflationary perturbations and CMB anisotropies

Inflation After Inflation Perturbations Hubble length

horizon crossing horizon re-entry

frozen Time Comoving scales

Gabriel Jung

LPT Orsay

22 May 2018 4 / 23

SLIDE 6

Perturbations

First-order perturbation theory:

Many inflation models predict an almost Gaussian and almost

scale-invariant distribution of the perturbations ⇒ We only need the power spectrum (Fourier transform of the two-point correlator): Ps ∝ kns−1 with ns = 0.968 ± 0.006 (Planck, 1502.02114)

Gabriel Jung

LPT Orsay

22 May 2018 4 / 23

SLIDE 7

Perturbations

First-order perturbation theory:

Many inflation models predict an almost Gaussian and almost

scale-invariant distribution of the perturbations ⇒ We only need the power spectrum (Fourier transform of the two-point correlator): Ps ∝ kns−1 with ns = 0.968 ± 0.006 (Planck, 1502.02114) Second-order perturbation theory: Deviations from Gaussianity ⇒ Three point-correlation function ↔ bispectrum Parametrized by the amplitude parameter fNL Different sources:

At late times: extra-galactic and galactic foregrounds
Primordial ⇒ important to discriminate inflation models

⇒ Many different shapes to look for in the observational data

Gabriel Jung

LPT Orsay

22 May 2018 4 / 23

SLIDE 8

Plan

1. Non-Gaussianity in two-field inflation

Jung & Van Tent (1611.09233)

Perturbations
Background description (slow-roll parameters)
Observables
The slow-roll approximation
Sum potentials
Monomial potentials
2. Non-Gaussianity in CMB observations

Jung, Racine & Van Tent (to be published)

Binned bispectrum estimator
Galactic foregrounds: dust
CMB data analyses

Gabriel Jung

LPT Orsay

22 May 2018 5 / 23

SLIDE 9

Non-Gaussianity in two-field inflation

SLIDE 10

Local non-Gaussianity

Gravitational potential:

Φ = ΦL + f local

NL

(Φ2

L − ΦL2)

ΦL ∝ ˆ a + ˆ a† (annihilation and creation operators)

ΦΦΦ ≈ f local

NL

Φ2

L2

Peaks in the squeezed configuration:
ne small ℓ and two large ℓ’s
Planck (1502.01592): f local

NL

= 0.8 ± 5.0

Gabriel Jung

LPT Orsay

22 May 2018 6 / 23

SLIDE 11

Local non-Gaussianity

Gravitational potential:

Φ = ΦL + f local

NL

(Φ2

L − ΦL2)

ΦL ∝ ˆ a + ˆ a† (annihilation and creation operators)

ΦΦΦ ≈ f local

NL

Φ2

L2

Peaks in the squeezed configuration:
ne small ℓ and two large ℓ’s
Planck (1502.01592): f local

NL

= 0.8 ± 5.0

Can be produced during multiple-field inflation!

Two kinds of perturbations which interact on super-Hubble scales:

Adiabatic: perturbations in the total energy density
Isocurvature: relative fluctuations between the different components
Long-wavelength formalism (Rigopoulos, Shellard and van Tent:

astro-ph/0504508, Tzavara and van Tent: 1012.6027)

Gabriel Jung

LPT Orsay

22 May 2018 6 / 23

SLIDE 12

Local non-Gaussianity

Gravitational potential:

Φ = ΦL + f local

NL

(Φ2

L − ΦL2)

ΦL ∝ ˆ a + ˆ a† (annihilation and creation operators)

ΦΦΦ ≈ f local

NL

Φ2

L2

Peaks in the squeezed configuration:
ne small ℓ and two large ℓ’s
Planck (1502.01592): f local

NL

= 0.8 ± 5.0

Can be produced during multiple-field inflation!

Two kinds of perturbations which interact on super-Hubble scales:

Adiabatic: perturbations in the total energy density
Isocurvature: relative fluctuations between the different components
Long-wavelength formalism (Rigopoulos, Shellard and van Tent:

astro-ph/0504508, Tzavara and van Tent: 1012.6027) Green’s functions:

¯

v22: isocurvature mode

¯

v12: isocurvature contribution to adiabatic Important hypothesis: ¯ v22 vanishes at the end of inflation

Gabriel Jung

LPT Orsay

22 May 2018 6 / 23

SLIDE 13

Multiple-field inflation

Main motivation: many scalar fields in high-energy theories

ϕ σ

Gabriel Jung

LPT Orsay

22 May 2018 7 / 23

SLIDE 14

Multiple-field inflation

Main motivation: many scalar fields in high-energy theories

Two-field models: W (φ, σ)

⇒ Sufficient to study new effects

Nice description of two-field infla-

tion ⇒ Trajectory in field space

Time coordinate: number of e-

folds Orthonormal basis e1 = (e1φ, e1σ) , e2 = (e1σ, −e1φ)

Groot Nibbelink & van Tent: hep-ph/0011325 & 0107272

e1 e1ϕ e1σ ϕ σ

e1φ =

˙ φ

√

˙ φ2+ ˙ σ2

e1σ =

˙ σ

√

˙ φ2+ ˙ σ2

Gabriel Jung

LPT Orsay

22 May 2018 7 / 23

SLIDE 15

Multiple-field inflation

Main motivation: many scalar fields in high-energy theories

Two-field models: W (φ, σ)

⇒ Sufficient to study new effects

Nice description of two-field infla-

tion ⇒ Trajectory in field space

Time coordinate: number of e-

folds Orthonormal basis e1 = (e1φ, e1σ) , e2 = (e1σ, −e1φ)

Groot Nibbelink & van Tent: hep-ph/0011325 & 0107272

e1 e1ϕ e1σ ϕ σ

e1φ =

˙ φ

√

˙ φ2+ ˙ σ2

e1σ =

˙ σ

√

˙ φ2+ ˙ σ2

Slow-roll parameters ǫ = −

˙ H H = ˙ φ2+ ˙ σ2 2

η =

¨ φ ˙ φ+¨ σ ˙ σ ˙ φ2+ ˙ σ2 − ǫ

η⊥ =

¨ φ ˙ σ+¨ σ ˙ φ ˙ φ2+ ˙ σ2

ǫ and η, similar to the usual ǫ and η of single-field inflation
η⊥: perpendicular (e2) acceleration, purely multiple-field effect

Link adiabatic/isocurvature perturbations: ˙ ¯ v12 = 2η⊥¯ v22

Gabriel Jung

LPT Orsay

22 May 2018 7 / 23

SLIDE 16

Observables

Spectral index: single-field formula + multiple-field correction ns − 1 = −4ǫ∗ − 2η

∗ −

¯ v12 1 + (¯ v12)2

4η⊥

∗ − 2¯

v12(ǫ∗ + η

∗ + ˜

W22∗)

Subscript ∗: evaluated at horizon-crossing

Gabriel Jung

LPT Orsay

22 May 2018 8 / 23 fNL = 0.8 ± 5.0 ns = 0.968 ± 0.006

SLIDE 17

Observables

Spectral index: single-field formula + multiple-field correction ns − 1 = −4ǫ∗ − 2η

∗ −

¯ v12 1 + (¯ v12)2

4η⊥

∗ − 2¯

v12(ǫ∗ + η

∗ + ˜

W22∗)

Subscript ∗: evaluated at horizon-crossing

Non-Gaussianity −6 5fNL = −2(¯ v12)2 [1 + (¯ v12)2]2 (giso + gsr + gint) giso = (ǫ + η)(¯ v22)2 + ¯ v22 ˙ ¯ v22 gsr = −ǫ∗ + η

∗

2¯ v 2

12

+ η⊥

∗ ¯

v12 2 − 3 2

ǫ∗ + η

∗ − χ∗ + η⊥ ∗

¯ v12

gint = −

t

t∗

dt′ 2(η⊥)2(¯ v22)2 + (ǫ + η)¯ v22 ˙ ¯ v22 + (˙ ¯ v22)2 −G13(t, t′)¯ v22(Ξ¯ v22 + 9η⊥ ˙ ¯ v22)

Gabriel Jung
LPT Orsay

22 May 2018 8 / 23 fNL = 0.8 ± 5.0 ns = 0.968 ± 0.006

SLIDE 18

Observables

Spectral index: single-field formula + multiple-field correction ns − 1 = −4ǫ∗ − 2η

∗ −

¯ v12 1 + (¯ v12)2

4η⊥

∗ − 2¯

v12(ǫ∗ + η

∗ + ˜

W22∗)

Subscript ∗: evaluated at horizon-crossing

Non-Gaussianity −6 5fNL = −2(¯ v12)2 [1 + (¯ v12)2]2 (giso + gsr + gint) giso: pure isocurvature term, set to 0 at the end of inflation gsr = −ǫ∗ + η

∗

2¯ v 2

12

+ η⊥

∗ ¯

v12 2 − 3 2

ǫ∗ + η

∗ − χ∗ + η⊥ ∗

¯ v12

gint = −

t

t∗

dt′ 2(η⊥)2(¯ v22)2 + (ǫ + η)¯ v22 ˙ ¯ v22 + (˙ ¯ v22)2 −G13(t, t′)¯ v22(Ξ¯ v22 + 9η⊥ ˙ ¯ v22)

Gabriel Jung
LPT Orsay

22 May 2018 8 / 23 fNL = 0.8 ± 5.0 ns = 0.968 ± 0.006

SLIDE 19

Observables

Spectral index: single-field formula + multiple-field correction ns − 1 = −4ǫ∗ − 2η

∗ −

¯ v12 1 + (¯ v12)2

4η⊥

∗ − 2¯

v12(ǫ∗ + η

∗ + ˜

W22∗)

Subscript ∗: evaluated at horizon-crossing

Non-Gaussianity −6 5fNL = −2(¯ v12)2 [1 + (¯ v12)2]2 (giso + gsr + gint) giso: pure isocurvature term, set to 0 at the end of inflation gsr = O(10−2): generalization of single-field fNL (unobservable) gint = − t

t∗

dt′ 2(η⊥)2(¯ v22)2 + (ǫ + η)¯ v22 ˙ ¯ v22 + (˙ ¯ v22)2 −G13(t, t′)¯ v22(Ξ¯ v22 + 9η⊥ ˙ ¯ v22)

Gabriel Jung
LPT Orsay

22 May 2018 8 / 23 fNL = 0.8 ± 5.0 ns = 0.968 ± 0.006

SLIDE 20

Observables

Spectral index: single-field formula + multiple-field correction ns − 1 = −4ǫ∗ − 2η

∗ −

¯ v12 1 + (¯ v12)2

4η⊥

∗ − 2¯

v12(ǫ∗ + η

∗ + ˜

W22∗)

Subscript ∗: evaluated at horizon-crossing

Non-Gaussianity −6 5fNL = −2(¯ v12)2 [1 + (¯ v12)2]2 (giso + gsr + gint) giso: pure isocurvature term, set to 0 at the end of inflation gsr = O(10−2): generalization of single-field fNL (unobservable) gint : complicated integral we need to solve and only large contribution to fNL at the end of inflation!

Gabriel Jung

LPT Orsay

22 May 2018 8 / 23 fNL = 0.8 ± 5.0 ns = 0.968 ± 0.006

SLIDE 21

Two approximations

Slow-roll approximation Fields are rolling down slowly on an almost flat-potential

Approximation:

ǫ and η ≪ 1

ǫ ≪ 1 → quasi-exponential expansion

η ≪ 1 → ¨ φ ≪ ˙ φ

Additional hypothesis:

η⊥ ≪ 1

ns − 1 ≪ 1 ⇒ slow-roll at horizon-

crossing

ϕ  ϕ W(ϕ)

Gabriel Jung

LPT Orsay

22 May 2018 9 / 23

SLIDE 22

Two approximations

Slow-roll approximation Fields are rolling down slowly on an almost flat-potential

Approximation:

ǫ and η ≪ 1

ǫ ≪ 1 → quasi-exponential expansion

η ≪ 1 → ¨ φ ≪ ˙ φ

Additional hypothesis:

η⊥ ≪ 1

ns − 1 ≪ 1 ⇒ slow-roll at horizon-

crossing

ϕ  ϕ W(ϕ)

Sum potential: W (φ, σ) = U(φ) + V (σ) gint = ǫ¯ v 2

22 − ǫ∗−

e2

1φ∗ ˜

Vσσ∗ − e2

1σ∗ ˜

Uφφ∗ 2e1φ∗e1σ∗ ¯ v12 with ˜ Uφφ =

1 3H2 ∂2W ∂φ∂φ and ˜

Vσσ =

1 3H2 ∂2W ∂σ∂σ

Gabriel Jung

LPT Orsay

22 May 2018 9 / 23

SLIDE 23

Non-Gaussianity

−6 5fNL = e2

1φ∗ ˜

Vσσ∗ − e2

1σ∗ ˜

Uφφ∗ e1φ∗e1σ∗ ¯ v123 (1 + ¯ v122)2 + O(10−2) ¯ v12: contribution of the isocurvature mode to the adiabatic mode

 

Gabriel Jung

LPT Orsay

22 May 2018 10 / 23

SLIDE 24

Non-Gaussianity

−6 5fNL = e2

1φ∗ ˜

Vσσ∗ − e2

1σ∗ ˜

Uφφ∗ e1φ∗e1σ∗ ¯ v123 (1 + ¯ v122)2 + O(10−2) ¯ v12: contribution of the isocurvature mode to the adiabatic mode

v 123 1+v

_ 1222

1 v 12

0.325 2 4 6 8 10 0.0 0.1 0.2 0.3 0.4 0.5 0.6 v 12

Two inequalities (1):

¯

v12

3

(1+¯ v122)2

< 0.325

(2):

¯

v12

3

(1+¯ v122)2

<
1

¯ v12

, (equality for ”large” ¯

v12)

Gabriel Jung

LPT Orsay

22 May 2018 10 / 23

SLIDE 25

Non-Gaussianity

−6 5fNL = e2

1φ∗ ˜

Vσσ∗ − e2

1σ∗ ˜

Uφφ∗ e1φ∗e1σ∗ ¯ v123 (1 + ¯ v122)2 + O(10−2) ¯ v12: contribution of the isocurvature mode to the adiabatic mode

v 123 1+v

_ 1222

1 v 12

0.325 2 4 6 8 10 0.0 0.1 0.2 0.3 0.4 0.5 0.6 v 12

Two inequalities (1):

¯

v12

3

(1+¯ v122)2

< 0.325

(2):

¯

v12

3

(1+¯ v122)2

<
1

¯ v12

, (equality for ”large” ¯

v12)

(1) ⇒

− 6

5fNL

< 0.325
e2

1φ∗ ˜

Vσσ∗−e2

1σ∗ ˜

Uφφ∗ e1φ∗e1σ∗

e2

1φ∗ ≈ 1 and e2 1σ∗ ≪ 1 , necessary for large fNL

Gabriel Jung

LPT Orsay

22 May 2018 10 / 23

SLIDE 26

Non-Gaussianity

−6 5fNL = e2

1φ∗ ˜

Vσσ∗ − e2

1σ∗ ˜

Uφφ∗ e1φ∗e1σ∗ ¯ v123 (1 + ¯ v122)2 + O(10−2) ¯ v12: contribution of the isocurvature mode to the adiabatic mode

v 123 1+v

_ 1222

1 v 12

0.325 2 4 6 8 10 0.0 0.1 0.2 0.3 0.4 0.5 0.6 v 12

Two inequalities (1):

¯

v12

3

(1+¯ v122)2

< 0.325

(2):

¯

v12

3

(1+¯ v122)2

<
1

¯ v12

, (equality for ”large” ¯

v12)

(1) ⇒

− 6

5fNL

< 0.325
e2

1φ∗ ˜

Vσσ∗−e2

1σ∗ ˜

Uφφ∗ e1φ∗e1σ∗

e2

1φ∗ ≈ 1 and e2 1σ∗ ≪ 1 , necessary for large fNL

Within this limit, at the end of inflation: ¯ v12 = −V∗e1φ∗

W∗e1σ∗

(2) ⇒

−6

5fNL

<
−Vσσ∗

V∗

Gabriel Jung
LPT Orsay

22 May 2018 10 / 23

SLIDE 27

Adding the constraints on the spectral index

Observables in the limit e2

1φ∗ ≈ 1 and e2 1σ∗ ≪ 1

ns − 1 = −2ǫ∗ + 2 ˜ Vσσ∗ − 6

5fNL = − Vσσ∗ V∗

Correct spectral index ns: ˜ Vσσ∗ = Vσσ∗

W∗ ⇒ Vσσ∗ ≪ W∗

Gabriel Jung

LPT Orsay

22 May 2018 11 / 23

fNL = 0.8 ± 5.0 ns = 0.968 ± 0.006

SLIDE 28

Adding the constraints on the spectral index

Observables in the limit e2

1φ∗ ≈ 1 and e2 1σ∗ ≪ 1

ns − 1 = −2ǫ∗ + 2 ˜ Vσσ∗ − 6

5fNL = − Vσσ∗ V∗

Correct spectral index ns: ˜ Vσσ∗ = Vσσ∗

W∗ ⇒ Vσσ∗ ≪ W∗

But fNL large ⇒ Vσσ∗ is not negligible Combination of the constraints V∗ <

∼ Vσσ∗ ≪ W∗ + e2 1σ∗ ≪ 1

Gabriel Jung

LPT Orsay

22 May 2018 11 / 23

fNL = 0.8 ± 5.0 ns = 0.968 ± 0.006

SLIDE 29

Adding the constraints on the spectral index

Observables in the limit e2

1φ∗ ≈ 1 and e2 1σ∗ ≪ 1

ns − 1 = −2ǫ∗ + 2 ˜ Vσσ∗ − 6

5fNL = − Vσσ∗ V∗

Correct spectral index ns: ˜ Vσσ∗ = Vσσ∗

W∗ ⇒ Vσσ∗ ≪ W∗

But fNL large ⇒ Vσσ∗ is not negligible Combination of the constraints V∗ <

∼ Vσσ∗ ≪ W∗ + e2 1σ∗ ≪ 1

Summary

Almost single-field ⇒ Far from the turn of the field trajectory

(before the turn ¯ v12 negligible, same for fNL)

Slow-roll ⇒ Everything is evolving slowly
Time is limited between horizon-crossing and the end of inflation:

∼ 60 e-folds at most Is it possible to have the turn of the field trajectory without breaking the slow-roll approximation?

Gabriel Jung

LPT Orsay

22 May 2018 11 / 23

fNL = 0.8 ± 5.0 ns = 0.968 ± 0.006

SLIDE 30

Monomial potentials

Turn starting before the end of slow-roll

W (φ, σ) = αφ2 + C + βσ4 (+λσ6)

Turn starting ”too late”

W (φ, σ) = αφ2 + C + βσ2 (+λσ4)

Gabriel Jung

LPT Orsay

22 May 2018 12 / 23

SLIDE 31

Monomial potentials

Turn starting before the end of slow-roll

W (φ, σ) = αφ2 + C + βσ4 (+λσ6)

Turn starting ”too late”

W (φ, σ) = αφ2 + C + βσ2 (+λσ4) Monomial potential U(φ) = αφn and V (σ) = βσm + C Quasi-single field approximation ⇒ Analytical expressions of the slow-roll functions using m, n and only four parameters ǫ, Nφ, ns and fNL

ǫ indicates the end of slow-roll at ǫ = 0.1
Nφ: expansion due to φ after horizon-crossing Nφ <

∼ 60 e-folds

Observational constraint: ns = 0.968 ± 0.006
Choice: fNL = O(1) (large non-Gaussianity)

Gabriel Jung

LPT Orsay

22 May 2018 12 / 23

SLIDE 32

Constraining the potential parameter space: fNL

+ +

|- 6

5fNL|>0.2

|- 6

5fNL|>1

|- 6

5fNL|>5

1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 n m

Parameter space of m and n

Testing different values of fNL up to the Planck constraints

Gabriel Jung

LPT Orsay

22 May 2018 13 / 23

SLIDE 33

Constraining the potential parameter space: fNL

+ +

|- 6

5fNL|>0.2

|- 6

5fNL|>1

|- 6

5fNL|>5

1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 n m

Parameter space of m and n

Testing different values of fNL up to the Planck constraints Results:

Blank region is

excluded: most of the parameter space.

Only possibility: n < 4

with m > n

Two-field quadratic

inflation is excluded

Gabriel Jung

LPT Orsay

22 May 2018 13 / 23

SLIDE 34

Constraining the potential parameter space: ns

+ +

ΔnS=0.012 ΔnS=0.006 ΔnS=0.0015 ΔnS=0.

1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 n m

Parameter space of m and n

Effect

f

a reasonable improvement

n

the mea- surement of ns

Gabriel Jung

LPT Orsay

22 May 2018 14 / 23

SLIDE 35

Constraining the potential parameter space: ns

+ +

ΔnS=0.012 ΔnS=0.006 ΔnS=0.0015 ΔnS=0.

1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 n m

Parameter space of m and n

Effect

f

a reasonable improvement

n

the mea- surement of ns

CORE-like

experiment: ∆ns ≈ 0.0015 (Planck/4)

Actually it is the lower

bound on ns which is the most interesting

Increasing this lower bound

invalidates most m and n

Gabriel Jung

LPT Orsay

22 May 2018 14 / 23

SLIDE 36

Summary

Sum potential

Simple expressions different from single-field inflation

ns − 1 = −2ǫ∗ + 2 ˜ Vσσ∗ − 6

5fNL = − Vσσ∗ V∗

Large fNL ⇒ Quasi single-field inflation and fine tuning
Observations of ns constrain also fNL (depending on the model)

Monomial potential

Large fNL is possible but most of parameter space is excluded
φ has to dominate almost the whole time after horizon-crossing
The turn can only occur near the end of the slow-roll regime

⇒ η and η⊥ are O(10−1) at the start of the turn. ⇒ Then O(1) during the turn is easy, slow-roll is broken ⇒ In that case, slow-roll predictions cannot be trusted

Gabriel Jung

LPT Orsay

22 May 2018 15 / 23

SLIDE 37

Summary

Sum potential

Simple expressions different from single-field inflation

ns − 1 = −2ǫ∗ + 2 ˜ Vσσ∗ − 6

5fNL = − Vσσ∗ V∗

Large fNL ⇒ Quasi single-field inflation and fine tuning
Observations of ns constrain also fNL (depending on the model)

Monomial potential

Large fNL is possible but most of parameter space is excluded
φ has to dominate almost the whole time after horizon-crossing
The turn can only occur near the end of the slow-roll regime

⇒ η and η⊥ are O(10−1) at the start of the turn ⇒ Then O(1) during the turn is easy, slow-roll is broken ⇒ In that case, slow-roll predictions cannot be trusted ⇒ We need to go beyond the slow-roll approximation!!

Gabriel Jung

LPT Orsay

22 May 2018 15 / 23

SLIDE 38

Beyond the slow-roll approximation

Differentiating three times the integral form of gint:

(η⊥)2 ... g int + η⊥ 3η⊥ − ǫη⊥ + 6ηη⊥ − 2ξ⊥ ¨ gint + (η⊥)2 −12ǫ + 6χ + 6(η)2 + 6(η⊥)2 + 4ξ + η⊥ 3 ˜ W211 − 8ηξ⊥ + 2(ξ⊥)2 ˙ gint = −18(η⊥)2χ2 + 2(η)2(η⊥)2ξ − 6ηη⊥ξ⊥χ + 6(η)2(η⊥)2χ − 6(η⊥)2χξ − 2(η⊥)4ξ − 6(η⊥)4χ − 18ǫη(η⊥)2χ + 12ǫ2η(η⊥)2 + 12ǫ(η)2(η⊥)2 − 6ǫη⊥ξ⊥χ − 12ǫ2(η⊥)2χ − 12ǫ(η⊥)4 − 3η(η⊥)2 ˜ W111 − 3ǫ(η⊥)2 ˜ W111 + 3(η⊥)3 ˜ W211 + 3η(η⊥)2 ˜ W221 + 3ǫ(η⊥)2 ˜ W221 − 3(η⊥)3 ˜ W222 − 2ηη⊥ξξ⊥ + 6ǫη(η⊥)2ξ − 12ǫ2ηη⊥ξ⊥ + 20ǫ3η(η⊥)2 + 28ǫ2(η)2(η⊥)2 − 12ǫη(η⊥)4 + 12ǫ(η)3(η⊥)2 − 2ǫη⊥ξξ⊥ −8ǫ(η)2η⊥ξ⊥ + 4ǫ2(η⊥)2ξ − 4ǫ3η⊥ξ⊥ − 4ǫ(η⊥)3ξ⊥ + 4ǫ4(η⊥)2 − 12ǫ2(η⊥)4 (¯ v22)2 + −36(η⊥)2χ − 6ǫη(η⊥)2 − 12ǫ2(η⊥)2 − 6(η⊥)4 − 6ǫ(η⊥)2χ + 6(η)2(η⊥)2 − 6ηη⊥ξ⊥ + 6η(η⊥)2χ − 6(η⊥)2ξ − 6ǫη⊥ξ⊥ − 12η⊥χξ⊥ − 3(η⊥)2 ˜ W111 − 3ηη⊥ ˜ W211 − 3η⊥ǫ ˜ W211 + 6(η⊥)2 ˜ W221 − 2η⊥ξξ⊥ − 2η(ξ⊥)2 + 2(η)2η⊥ξ⊥ + 2η(η⊥)2ξ − 2(η⊥)3ξ⊥ − 8ǫηη⊥ξ⊥ + 18ǫ(η)2(η⊥)2 + 24ǫ2η(η⊥)2 + 4ǫ(η⊥)2ξ − 6ǫ2η⊥ξ⊥ −6(η⊥)4ǫ + 6ǫ3(η⊥)2 − 2ǫ(ξ⊥)2 ¯ v22¯ v32 + −18(η⊥)2 − 6ǫ(η⊥)2 + 6η(η⊥)2 − 12η⊥ξ⊥ − 3η⊥ ˜ W211 + 6ǫη(η⊥)2 + 2ǫ2(η⊥)2 +2ηη⊥ξ⊥ − 2ǫη⊥ξ⊥ − 2(ξ⊥)2 (¯ v32)2 Gabriel Jung

LPT Orsay

22 May 2018 16 / 23

SLIDE 39

Beyond the slow-roll approximation

Differentiating three times the integral form of gint:

(η⊥)2 ... g int + η⊥ 3η⊥ − ǫη⊥ + 6ηη⊥ − 2ξ⊥ ¨ gint +

η⊥ (...) + 2(ξ⊥)2

˙ gint = K22(¯ v22)2 + K23¯ v22 ˙ ¯ v22 + K33(˙ ¯ v22)2,

Using slow-roll at horizon-crossing:

gint(t) = A∗ ¯ v12(t) + t

t∗

dt′P(t′)

Gabriel Jung

LPT Orsay

22 May 2018 16 / 23

SLIDE 40

Beyond the slow-roll approximation

Differentiating three times the integral form of gint:

(η⊥)2 ... g int + η⊥ 3η⊥ − ǫη⊥ + 6ηη⊥ − 2ξ⊥ ¨ gint +

η⊥ (...) + 2(ξ⊥)2

˙ gint = K22(¯ v22)2 + K23¯ v22 ˙ ¯ v22 + K33(˙ ¯ v22)2,

Using slow-roll at horizon-crossing:

gint(t) = A∗ ¯ v12(t) + t

t∗

dt′P(t′) The slow-roll solution

gint = ǫ¯ v2

22 − ǫ∗−

e2

1φ∗ ˜

Vσσ∗ − e2

1σ∗ ˜

Uφφ∗ 2e1φ∗e1σ∗ ¯ v12

works beyond the slow-roll approxi- mation if ǫ is small during the turn because t

t∗ dt′P(t′) is negligible!

Gabriel Jung

LPT Orsay

22 May 2018 16 / 23

SLIDE 41

Non-Gaussianity in CMB

bservations

SLIDE 42

Non-Gaussianity in observational data

Angle-averaged bispectrum: Bℓ1ℓ2ℓ3 =

S2 dˆ

Ω Tℓ1(ˆ Ω)Tℓ2(ˆ Ω)Tℓ3(ˆ Ω)

where Tℓ(ˆ

Ω) =

ℓ

m=−ℓ

aℓmYℓm(ˆ Ω) fNL estimator ˆ fNL = Bobs

ℓ1ℓ2ℓ3

Bth

ℓ1ℓ2ℓ3

, but it only uses one triplet (ℓ1, ℓ2, ℓ3)

Gabriel Jung

LPT Orsay

22 May 2018 17 / 23

SLIDE 43

Non-Gaussianity in observational data

Angle-averaged bispectrum: Bℓ1ℓ2ℓ3 =

S2 dˆ

Ω Tℓ1(ˆ Ω)Tℓ2(ˆ Ω)Tℓ3(ˆ Ω)

where Tℓ(ˆ

Ω) =

ℓ

m=−ℓ

aℓmYℓm(ˆ Ω) fNL estimator ˆ fNL = 1 N

ℓ1≤ℓ2≤ℓ3

1 Var(Bobs

ℓ1ℓ2ℓ3/Bth ℓ1ℓ2ℓ3)

Bobs

ℓ1ℓ2ℓ3

Bth

ℓ1ℓ2ℓ3

, weighted sum

Gabriel Jung

LPT Orsay

22 May 2018 17 / 23

SLIDE 44

Non-Gaussianity in observational data

Angle-averaged bispectrum: Bℓ1ℓ2ℓ3 =

S2 dˆ

Ω Tℓ1(ˆ Ω)Tℓ2(ˆ Ω)Tℓ3(ˆ Ω)

where Tℓ(ˆ

Ω) =

ℓ

m=−ℓ

aℓmYℓm(ˆ Ω) fNL estimator ˆ fNL = 1 N

ℓ1≤ℓ2≤ℓ3
Bth

ℓ1ℓ2ℓ3

2 Var(Bobs

ℓ1ℓ2ℓ3)

Bobs

ℓ1ℓ2ℓ3

Bth

ℓ1ℓ2ℓ3

, weighted sum

Gabriel Jung

LPT Orsay

22 May 2018 17 / 23

SLIDE 45

Non-Gaussianity in observational data

Angle-averaged bispectrum: Bℓ1ℓ2ℓ3 =

S2 dˆ

Ω Tℓ1(ˆ Ω)Tℓ2(ˆ Ω)Tℓ3(ˆ Ω)

where Tℓ(ˆ

Ω) =

ℓ

m=−ℓ

aℓmYℓm(ˆ Ω) fNL estimator ˆ fNL = 1 N

ℓ1≤ℓ2≤ℓ3

Bth

ℓ1ℓ2ℓ3Bobs ℓ1ℓ2ℓ3

Var(Bobs

ℓ1ℓ2ℓ3) ,

N: normalization factor

Gabriel Jung

LPT Orsay

22 May 2018 17 / 23

SLIDE 46

Non-Gaussianity in observational data

Angle-averaged bispectrum: Bℓ1ℓ2ℓ3 =

S2 dˆ

Ω Tℓ1(ˆ Ω)Tℓ2(ˆ Ω)Tℓ3(ˆ Ω)

where Tℓ(ˆ

Ω) =

ℓ

m=−ℓ

aℓmYℓm(ˆ Ω) fNL estimator ˆ fNL = 1 N

ℓ1≤ℓ2≤ℓ3

Bth

ℓ1ℓ2ℓ3Bobs ℓ1ℓ2ℓ3

Var(Bobs

ℓ1ℓ2ℓ3) ,

N: normalization factor Different sources of variance:

Cosmic Variance: inherent statistical uncertainty due to the fact we
nly have one realization of the sky
Experiment: noisy modes have a larger variance, observing only a

part of the sky increases the variance...

Gabriel Jung

LPT Orsay

22 May 2018 17 / 23

SLIDE 47

Binned bispectrum estimator

Issue: O(109) triplets in the case of Planck

Gabriel Jung

LPT Orsay

22 May 2018 18 / 23

SLIDE 48

Binned bispectrum estimator

Issue: O(109) triplets in the case of Planck Different solutions:

Separable bispectrum:
ℓ1≤ℓ2≤ℓ3

→

ℓ1
×
ℓ2
×
ℓ3
Several different implementations: KSW, modal estimators

Gabriel Jung

LPT Orsay

22 May 2018 18 / 23

SLIDE 49

Binned bispectrum estimator

Issue: O(109) triplets in the case of Planck Different solutions:

Separable bispectrum:
ℓ1≤ℓ2≤ℓ3

→

ℓ1
×
ℓ2
×
ℓ3
Several different implementations: KSW, modal estimators
Binned bispectrum: broader filters Ti =
ℓ∈bin i

Tℓ

Bucher, Van Tent & Carvalho (0911.1642) and Bucher, Racine & Van Tent (1509.08107)

⇒ Estimator: ˆ fNL = 1 N

i1≤i2≤i3
ℓ1,ℓ2,ℓ3∈bins

Bth

ℓ1ℓ2ℓ3

ℓ1,ℓ2,ℓ3∈bins

Bobs

ℓ1ℓ2ℓ3

ℓ1,ℓ2,ℓ3∈bins

Var(Bobs

ℓ1ℓ2ℓ3)

Gabriel Jung
LPT Orsay

22 May 2018 18 / 23

SLIDE 50

Binned bispectrum estimator

Issue: O(109) triplets in the case of Planck Different solutions:

Separable bispectrum:
ℓ1≤ℓ2≤ℓ3

→

ℓ1
×
ℓ2
×
ℓ3
Several different implementations: KSW, modal estimators
Binned bispectrum: broader filters Ti =
ℓ∈bin i

Tℓ

Bucher, Van Tent & Carvalho (0911.1642) and Bucher, Racine & Van Tent (1509.08107)

⇒ Estimator: ˆ fNL = 1 N

i1≤i2≤i3
ℓ1,ℓ2,ℓ3∈bins

Bth

ℓ1ℓ2ℓ3

ℓ1,ℓ2,ℓ3∈bins

Bobs

ℓ1ℓ2ℓ3

ℓ1,ℓ2,ℓ3∈bins

Var(Bobs

ℓ1ℓ2ℓ3)

New implementation:

The binned bispectrum of a map is computed explicitly ⇒ Use the binned bispectrum of a map as a theoretical template

Gabriel Jung

LPT Orsay

22 May 2018 18 / 23

SLIDE 51

Galactic foregrounds: dust

Dust (Planck 2015: 1502.01588)

Small grains ( <

∼ 1 µm) in the interstellar medium

Strongly non-Gaussian
Dominant signal in the galactic plane

Gabriel Jung

LPT Orsay

22 May 2018 19 / 23

SLIDE 52

Galactic foregrounds: dust

Dust (Planck 2015: 1502.01588)

Small grains ( <

∼ 1 µm) in the interstellar medium

Strongly non-Gaussian
Dominant signal in the galactic plane
Use a mask!

Gabriel Jung

LPT Orsay

22 May 2018 19 / 23

SLIDE 53

Bispectrum of the dust

Even with the mask, galactic dust → strong contamination
Several techniques to clean the data (Commander, Smica... )

Gabriel Jung

LPT Orsay

22 May 2018 20 / 23

SLIDE 54

Bispectrum of the dust

Even with the mask, galactic dust → strong contamination
Several techniques to clean the data (Commander, Smica...)

Bispectrum signal over noise: ⇒ Strongly non-Gaussian, peaks in the squeezed configuration → Correlations between small and large scales

Gabriel Jung

LPT Orsay

22 May 2018 20 / 23

SLIDE 55

Bispectrum of the dust

Even with the mask, galactic dust → strong contamination
Several techniques to clean the data (Commander, Smica...)

Weights in multipole space:

500 1000 1500 2000 2 100 200 300 400 500 1

dust

500 1000 1500 2000 2 100 200 300 400 500 1

local

⇒ Dust and local shapes are correlated (∼ −50%)

Gabriel Jung

LPT Orsay

22 May 2018 20 / 23

SLIDE 56

Analyses

Tests with the dust numerical template: 100 CMB Gaussian simulations + known amount of dust

2015 Planck analysis → 57 bins Correlation dust/local: ∼ 50% Local Dust Dust 100% (expected f dust

NL

= 1) Indep −91 ± 15 0.83 ± 0.19 Joint −28 ± 16 0.77 ± 0.21 Difficult to differentiate the two shapes

Gabriel Jung

LPT Orsay

22 May 2018 21 / 23

SLIDE 57

Analyses

Tests with the dust numerical template: 100 CMB Gaussian simulations + known amount of dust

2015 Planck analysis → 57 bins Correlation dust/local: ∼ 50% Local Dust Dust 100% (expected f dust

NL

= 1) Indep −91 ± 15 0.83 ± 0.19 Joint −28 ± 16 0.77 ± 0.21 Difficult to differentiate the two shapes Add bins (low ℓ)

= = = = = = = = = ⇒

70 bins (correlation dust/local: ∼ 40%) Local Dust Dust 100% (expected f dust

NL

= 1) Indep −72 ± 11 0.97 ± 0.20 Joint −3 ± 15 0.97 ± 0.23 Dust 0% (expected f dust

NL

= 0) Indep −0.1 ± 0.5 0.001 ± 0.004 Joint −0.4 ± 0.8 0.000 ± 0.004

Gabriel Jung

LPT Orsay

22 May 2018 21 / 23

SLIDE 58

Analyses

Tests with the dust numerical template: 100 CMB Gaussian simulations + known amount of dust

2015 Planck analysis → 57 bins Correlation dust/local: ∼ 50% Local Dust Dust 100% (expected f dust

NL

= 1) Indep −91 ± 15 0.83 ± 0.19 Joint −28 ± 16 0.77 ± 0.21 Difficult to differentiate the two shapes Add bins (low ℓ)

= = = = = = = = = ⇒

70 bins (correlation dust/local: ∼ 40%) Local Dust Dust 100% (expected f dust

NL

= 1) Indep −72 ± 11 0.97 ± 0.20 Joint −3 ± 15 0.97 ± 0.23 Dust 0% (expected f dust

NL

= 0) Indep −0.1 ± 0.5 0.001 ± 0.004 Joint −0.4 ± 0.8 0.000 ± 0.004

Analyses with the dust numerical template: CMB cleaned map

Local Dust (expected f dust

NL

= 0) Indep 2.1 ± 5.4 −0.05 ± 0.06 Joint −7 ± 11 −0.08 ± 0.08

Gabriel Jung

LPT Orsay

22 May 2018 21 / 23

SLIDE 59

Analyses

Tests with the dust numerical template: 100 CMB Gaussian simulations + known amount of dust

2015 Planck analysis → 57 bins Correlation dust/local: ∼ 50% Local Dust Dust 100% (expected f dust

NL

= 1) Indep −91 ± 15 0.83 ± 0.19 Joint −28 ± 16 0.77 ± 0.21 Difficult to differentiate the two shapes Add bins (low ℓ)

= = = = = = = = = ⇒

70 bins (correlation dust/local: ∼ 40%) Local Dust Dust 100% (expected f dust

NL

= 1) Indep −72 ± 11 0.97 ± 0.20 Joint −3 ± 15 0.97 ± 0.23 Dust 0% (expected f dust

NL

= 0) Indep −0.1 ± 0.5 0.001 ± 0.004 Joint −0.4 ± 0.8 0.000 ± 0.004

Analyses with the dust numerical template: Raw sky map at 143 GHz

Local Dust (expected f dust

NL

= 1) Indep −65 ± 14 0.84 ± 0.28 Joint −12 ± 21 0.78 ± 0.32

Gabriel Jung

LPT Orsay

22 May 2018 21 / 23

SLIDE 60

Summary

New implementation in the binned bispectrum code

⇒ use the binned bispectrum of any map as a theoretical template

Method tested with several galactic foregrounds → dust (here)

⇒ We detected the expected amount in both simulations and

bservational data

⇒ Foregrounds correlated to the local shape (similar squeezed configuration) ⇒ Choice of bins at low ℓ is important

Gabriel Jung

LPT Orsay

22 May 2018 22 / 23

SLIDE 61

Conclusion

SLIDE 62

Conclusion and outlook

New feature of the binned bispectrum estimator: numerical

theoretical templates determined from any map ⇒ Tested with the galactic foregrounds which are strongly non-Gaussian and correlated to the local shape (large contamination)

Method ready to analyze E-polarization (new Planck release)

Gabriel Jung

LPT Orsay

22 May 2018 23 / 23

SLIDE 63

Conclusion and outlook

New feature of the binned bispectrum estimator: numerical

theoretical templates determined from any map ⇒ Tested with the galactic foregrounds which are strongly non-Gaussian and correlated to the local shape (large contamination)

Method ready to analyze E-polarization (new Planck release)
Conditions for large fNL for two-field sum potentials

W (φ, σ) = U(φ) + V (σ) in and beyond the slow-roll approximation:

Quasi single-field inflation at horizon-crossing
The ratio Vσσ∗

V∗

has to be O(1)

However, Vσσ∗

W∗

is constrained by observations of the spectral index. When combined with spectral index observational constraints ⇒ Hard to have fNL ∼ O(1), even impossible for most of the parameter space with monomial potentials

Is the situation similar for more complicated models?
Next generation of CMB experiments can improve a lot the

constraints on non-Gaussianity, especially if the spectral index is in the higher values of the current bounds

Gabriel Jung

LPT Orsay

22 May 2018 23 / 23

SLIDE 64

Thank you for your attention!

Gabriel Jung

LPT Orsay

22 May 2018 23 / 23