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.