CMB Non-Gaussianity as a probe of the physics of …phd.fisica.unimi.it/assets/Seminario-Troja.pdfCMB Non-Gaussianity as a probe of the physics of the primordial Universe Antonino

CMB Non-Gaussianity as a probe of the physics of theprimordial Universe

Antonino Troja

Universita degli Studi di MilanoFacolta di Scienze e Tecnologie

November 17, 2014

Antonino Troja (UniMi) CMB non-Gaussianity November 17, 2014 1 / 23

CMB: Cosmic Microwave Background


CMB: Temperature Anisotropies

PLANCK Collaboration (2013)


Anisotropies and Inflation

Inflation −→ CMB Anisotropies


Inflation Scalar Field

INFLATON φ(x, t) = φ0(t) + δφ(x, t)

Spatial average of the field

Vacuum Fluctuations

VacuumFluctuations

PrimordialGravitationalFluctuations

MatterFluctuations

CMBAnisotropy

Inflation After Inflation





Vacuum Fluctuations

VacuumFluctuations


MatterFluctuations

CMBAnisotropy






Vacuum Fluctuations

VacuumFluctuations


MatterFluctuations

CMBAnisotropy






Vacuum Fluctuations

VacuumFluctuations


MatterFluctuations

CMBAnisotropy



Inflationary Models

There is NOT only one Inflation model.

Single-Field

Standard Scenario

k-inflation

DBI inflation

...

Multi-Fields

Curvaton Scenario

Ghost inflation

D-cceleration scenario

...

Which is the correct one?


Inflationary Models

There is NOT only one Inflation model.

Single-Field

Standard Scenario

k-inflation

DBI inflation

...

Multi-Fields

Curvaton Scenario

Ghost inflation

D-cceleration scenario

...

Which is the correct one?


Inflationary Foresight: Non-Gaussianity

Primordial Non-Gaussianity is one of the foresight of the Inflationarymodels.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 1 2 3 4 5 6

Maxwell-Boltzmann




0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 1 2 3 4 5 6

Maxwell-Boltzmann




0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 1 2 3 4 5 6

Maxwell-BoltzmannGaussian


Foresight of the models

Single-Field Models

The primordial fluctuation distribution is GAUSSIAN.

Multi-Fields Models

The primordial fluctuation distribution is NON-GAUSSIAN.

The amounts of non-Gaussianity depends on the model.


Primordial non-Gaussianity

Primordial Non-Gaussianity in the gravitational field Φ(x) is parametrizedby the so-called Bardeen potential:

Φ(x) = ΦL(x) + fNL(ΦL(x)2 − 〈ΦL(x)2〉) + gNL[ΦL(x)3]

Gaussian component

non-Gaussian component

AnisotropiesNG

PrimordialNG





Gaussian component


AnisotropiesNG

PrimordialNG





Gaussian component


AnisotropiesNG

PrimordialNG


Non-Gaussianity: Geometrical Interpretation

fNL3rd-order:Skewness

gNL4th-order:Kurtosis


fNL: Skewness

Skewness: asymmetry of the distribution.

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0 0.2 0.4 0.6 0.8 1 1.2 1.4

Right side of MB

Left side of MB

fNL < 0


gNL: Kurtosis

Kurtosis: height of the tails of the distribution.

0

0.1

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0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

Left side of MB, gNL > 0

Right side of MB, gNL < 0

Gaussian distribution


Gaussian Distribution

The Gaussian distribution is fully described by the lowest moments: mean(µ) and variance (σ):

G (x) = 1√2πσ

e−(x−µ)2

2σ2

Skewness = Kurtosis = 0 −→ fNL = gNL = 0


Gaussian Distribution

The Gaussian distribution is fully described by the lowest moments: mean(µ) and variance (σ):

G (x) = 1√2πσ

e−(x−µ)2

2σ2

Skewness = Kurtosis = 0 −→ fNL = gNL = 0


Evaluation of fNL and gNL

In order to evaluate fNL and gNL we use the high-order correlationfunctions. In particular their harmonic counterpart.

3-point correlation function (Fourier space):

〈Φ(k1)Φ(k2)Φ(k3)〉 = (2π)3δ(3)(k1 + k2 + k3)BΦ(k1, k2, k3),

BispectrumBΦ(k1, k2, k3) = fNLF (k1, k2, k3)



In order to evaluate fNL and gNL we use the high-order correlationfunctions. In particular their harmonic counterpart.


〈Φ(k1)Φ(k2)Φ(k3)〉 = (2π)3δ(3)(k1 + k2 + k3)BΦ(k1, k2, k3),

BispectrumBΦ(k1, k2, k3) = fNLF (k1, k2, k3)




〈Φ(k1)Φ(k2)Φ(k3)Φ(k4)〉 =

= (2π)4δ(4)(k1 + k2 + k3 + k4)TΦ(k1, k2, k3, k4),

Trispectrum

TΦ(k1, k2, k3, k4) = gNL[PΦ(k1)PΦ(k2)PΦ(k3) + (3perm.)]


State of the Art

Optimal Estimator : Unbiased estimator with the lowest variance amongthe other ones.

Planck Collaboration (2013):

fNL = 2.7 ± 5.8 (1σ)

Single-field models are preferred.


State of the Art

Sekiguchi and Sugiyama, WMAP 9yrs (2013)

gNL = (−3.3 ± 2.2) × 105 (1σ)

gNL doesn’t have any statistical significance!


State of the Art

Sekiguchi and Sugiyama, WMAP 9yrs (2013)

gNL = (−3.3 ± 2.2) × 105 (1σ)

gNL doesn’t have any statistical significance!


Spherical Needlet System

I propose an optimal estimator using the Spherical Needlets system:

ψjk(x) :=√λjk∑l

b

(l

B j

) l∑m=−l

Ylm(ξjk)Y lm(x)

It is possible to write the Trispectrum Estimator using the Needletcoefficients.


Needlet Coefficients

Reconstruction Formula:

T (x) =∑j ,k

βjkψjk(x)


Needlet Coefficients

Reconstruction Formula:

T (x) =∑j ,k

βjkψjk(x)


State of the art and Future perspective

Spherical Needlet Estimator:

J j1j2j3j4=

√4π

Nj4

∑k4

βj1k4βj2k4βj3k4βj4k4

σj1σj2σj3σj4+ C (βj1k4 , βj2k4 , βj3k4 , βj4k4)

Application on CMB temperature data;

Application on CMB polarization data;

Application to other dataset, like LSS data;

The main goal is to evaluate gNL with a precision never achieved before.


Thanks for your attention.


Documents

CMB Non-Gaussianity as a probe of the physics of …phd.fisica.unimi.it/assets/Seminario-Troja.pdfCMB Non-Gaussianity as a probe of the physics of the primordial Universe Antonino