Trispectrum Estimator of Primordial Perturbation in Equilateral Type Non-Gaussian Models

Ｋｅｉｓｕｋｅ　Ｉｚｕｍｉ　（泉　圭介）Collaboration with Shuntaro Mizuno

Kazuya Koyama

InflationThe problem of Big Bang cosmology

Flatness problemHorizon problem

Inflation can solve these problems by the exponentially expansion.

Additional advantage of inflation

Primordial fluctuations are created quantum mechanically. These fluctuations become the seed of the structure of Universe.

However, there are O(100) inflation models.　　 Identification of inflation model is one of important tasks.

How?

More accurate observation of primordial fluctuations (CMB)Observation of gravitational wave (tensor fluctuations)

Cosmic Microwave Background (CMB)What we observe?

In early universe, the energy density is highand photon can not propagate freely.

Last scattering surfaceSince universe expands, at some time photon can propagate freely.We see this surface and measure the temperature.

http://map.gsfc.nasa.gov/

WMAP 7year

The perturbation produced in inflation era

Almost the same temperature about 3000K (2.7K now)

There is small fluctuation ΔT/T～ 10^-5

Gravitational perturbation

Temperature perturbation

Statistics of CMB fluctuation

Origin is quantum fluctuation in inflation era.

(Almost) Gaussian

Almost de Sitter expansion.

(Almost) scale invariant

interaction

Non-Gaussianity

3-point function -> bispectrum4-point function -> trispectrum

Scale dependence

Other direction

WMAP à 10 < f NL < 74 (95% CL)polarization

Bispectrum

hð(k1)ð(k2)ð(k3)i =(2ù)3î 3(k1 + k2 + k3)Bð(k1;k2;k3)Definition of bispectrum of curvature perturbation Bð(k1;k2;k3)

Bð(k1;k2;k3)Because of isotropy and homogeneity, depends only on amplitude of momenta k1;k2;k3

Assuming scale invariance , depends on two parameters Bð(k1;k2;k3) k2=k1;k3=k1

Shape of Bispectrumlocal

k2=k1k3=k1

0

1 0.5

1

equilateral orthogonal

Local shape

B localð = 2(kà 3

1 kà 32 + (2perm:))

k2=k1k3=k1

0

1 0.5

1

k1

k2

k3

Small scale

Small scaleLarge scale

Definition of local shape (k1k2k3)2B localð

Maximum

k2=k1 = 1k3=k1 = 0

Pð(k1)

Large scale

Local limit of bispectrum can be interpreted as powerspectrum on background modulated by large scale perturbation

ð(x) = ðgauss(x) + 53f NL(ð2

gauss(x) à hð2gauss(x)i)

Derivation of local shape

Equilateral and orthogonal shape

k2=k1 k3=k1

0

1 0.5

1

k1

k2

k3

(k1k2k3)2Bequilð

Bequilð = à 6(kà 3

1 kà 32 +(2perm:))

Definition of equilateral shape

+ 6(kà 11 kà 2

2 kà 33 + (5perm:)) à 12kà 2

1 kà 22 kà 2

3

Maximum: equilateral shape

k2=k1 = k3=k1 = 1

k2=k1 k3=k1

01

(k1k2k3)2Borthð

Definition of orthogonal shape

Bequilð = à 18(kà 3

1 kà 32 + (2perm:))

à 24(kà 11 kà 2

2 kà 33 + (5perm:))

+ 18kà 21 kà 2

2 kà 23

In single field inflation model, all bispectra can be written as linear combination of local, equilateral and orthogonal shapes.

Non-GaussianityBispectrum : Leading order non-Gaussianity

à 10 < f NL < 74 (95% CL)WMAP

PLANCK

jf NLj < O(1)If , it can be observed.

advantageEase of calculation and data analysis.

disadvantageOnly see a part of full informationFor instance, it is difficult to distinguish between DBI inflation and ghost inflation .

Trispectrum : Next order non-Gaussianityadvantage

Complication of calculation and data analysis.disadvantage

More informationsIn Trispectrum, can we see difference between DBI inflation and ghost inflation?

6 parameters

jgNLj < 105 à 106WMAP

PLANCKjgNLj < 560If ,

it can be observed.

Defining inner product of Trispectrum shapes, we quantify similarity between two shapes.

Komatsu et al. 2010

Regan et al. 2010

Kogo, Komatsu 2006

PLANCK homepage http://www.sciops.esa.int/index.php?project=PLANCK

Inner Product and correlator

hð(k1)ð(k2)ð(k3)ð(k4)i =(2ù)3î 3(k1 + k2 + k3 + k4)Tð(k1;k2;k3;k4;k12;ò4)Definition of bispectrum of curvature perturbation Tð(k1;k2;k3;k4;k12;ò4)

depends on 6 parametersShape function

ST(k1;k2;k3;k4;k12;ò4) = (k1k2k3k4)2k12Tð(k1;k2;k3;k4;k12;ò4)Inner product

F[ST;ST0] =Rdk1dk2dk3dk4dk12dò4 STST0w

correlator

C[ST;ST0] = F[ST;ST0]= F[ST;ST]F[ST0;ST0]q

Non-gaussianity parameter

gequilNL = F[ST;Sequil

T ]=F[SequilT ;Sequil

T ]

SequilT = 3

64(2ù2Pð)3ò

(P i=14 ki)5

k12Q

i=14 ki + 11perms:

ó

Resultcorrelator

gequilNL

Highly correlated

Low correlation

Some models can be discriminated by trispectrum

Summary

Analysis of non-Gaussianity of primordial perturbation is one of way to discriminate inflation models.

We can distinguish some of models by trispectrum.

We also see non-gaussianity parameter in some models. gequilNL

Thank you for your attention.