Analytic Minkowski Functionals of the CMBnlg.koubo/2010_3/Matsubara.pdfAnalytic Minkowski...

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Analytic Minkowski Functionals of the CMB Taka Matsubara (Nagoya U.) @Yukawa Hall 2010/3/26 Phys. Rev. D in press, arXiv:1001.2321 2010326日金曜日

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Page 1: Analytic Minkowski Functionals of the CMBnlg.koubo/2010_3/Matsubara.pdfAnalytic Minkowski Functionals of the CMB Taka Matsubara (Nagoya U.) @Yukawa Hall 2010/3/26 Phys. Rev. D in press,

Analytic Minkowski Functionals of the CMB

Taka Matsubara (Nagoya U.)

@Yukawa Hall2010/3/26

Phys. Rev. D in press, arXiv:1001.2321

2010年3月26日金曜日

Page 2: Analytic Minkowski Functionals of the CMBnlg.koubo/2010_3/Matsubara.pdfAnalytic Minkowski Functionals of the CMB Taka Matsubara (Nagoya U.) @Yukawa Hall 2010/3/26 Phys. Rev. D in press,

Minkowski Functionals can

simultaneously constrain

(and other models)

2010年3月26日金曜日

Page 3: Analytic Minkowski Functionals of the CMBnlg.koubo/2010_3/Matsubara.pdfAnalytic Minkowski Functionals of the CMB Taka Matsubara (Nagoya U.) @Yukawa Hall 2010/3/26 Phys. Rev. D in press,

Non-Gaussian fluctuations

• The power spectrum cannot distinguish the non-Gaussianity

Chingangbam & Park 2009

gNL = +106

gNL = -106

2010年3月26日金曜日

Page 4: Analytic Minkowski Functionals of the CMBnlg.koubo/2010_3/Matsubara.pdfAnalytic Minkowski Functionals of the CMB Taka Matsubara (Nagoya U.) @Yukawa Hall 2010/3/26 Phys. Rev. D in press,

Higher-order polyspectra• Non-Gaussianity is basically characterized by higher-

order statistics beyond the power spectrum

• Polyspectra : multipole expansions of the higher-order correlation function

• B: bispectrum, T: trispectrum

• 3j-symbol appears due to rotational symmetry

2010年3月26日金曜日

Page 5: Analytic Minkowski Functionals of the CMBnlg.koubo/2010_3/Matsubara.pdfAnalytic Minkowski Functionals of the CMB Taka Matsubara (Nagoya U.) @Yukawa Hall 2010/3/26 Phys. Rev. D in press,

Detecting the non-Gaussianity in the CMB

• Direct measurements of polyspectra

• Too complex due to many arguments

• Optimal weighting method

• Optimal estimators

• Geometrical analysis of patterns in CMB anisotropy

• Minkowski functions etc.

2010年3月26日金曜日

Page 6: Analytic Minkowski Functionals of the CMBnlg.koubo/2010_3/Matsubara.pdfAnalytic Minkowski Functionals of the CMB Taka Matsubara (Nagoya U.) @Yukawa Hall 2010/3/26 Phys. Rev. D in press,

Geometrical analysis of patterns in CMB anisotropy

• To characterize the geometrical patterns, the isotemperature contours in a smoothed CMB map is considered

2010年3月26日金曜日

Page 7: Analytic Minkowski Functionals of the CMBnlg.koubo/2010_3/Matsubara.pdfAnalytic Minkowski Functionals of the CMB Taka Matsubara (Nagoya U.) @Yukawa Hall 2010/3/26 Phys. Rev. D in press,

Geometrical analysis of patterns in CMB anisotropy

• To characterize the geometrical patterns, the isotemperature contours in a smoothed CMB map is considered

2010年3月26日金曜日

Page 8: Analytic Minkowski Functionals of the CMBnlg.koubo/2010_3/Matsubara.pdfAnalytic Minkowski Functionals of the CMB Taka Matsubara (Nagoya U.) @Yukawa Hall 2010/3/26 Phys. Rev. D in press,

Geometrical analysis of patterns in CMB anisotropy

• To characterize the geometrical patterns, the isotemperature contours in a smoothed CMB map is considered

2010年3月26日金曜日

Page 9: Analytic Minkowski Functionals of the CMBnlg.koubo/2010_3/Matsubara.pdfAnalytic Minkowski Functionals of the CMB Taka Matsubara (Nagoya U.) @Yukawa Hall 2010/3/26 Phys. Rev. D in press,

Geometrical analysis of patterns in CMB anisotropy

• To characterize the geometrical patterns, the isotemperature contours in a smoothed CMB map is considered

2010年3月26日金曜日

Page 10: Analytic Minkowski Functionals of the CMBnlg.koubo/2010_3/Matsubara.pdfAnalytic Minkowski Functionals of the CMB Taka Matsubara (Nagoya U.) @Yukawa Hall 2010/3/26 Phys. Rev. D in press,

Geometrical analysis of patterns in CMB anisotropy

• To characterize the geometrical patterns, the isotemperature contours in a smoothed CMB map is considered

2010年3月26日金曜日

Page 11: Analytic Minkowski Functionals of the CMBnlg.koubo/2010_3/Matsubara.pdfAnalytic Minkowski Functionals of the CMB Taka Matsubara (Nagoya U.) @Yukawa Hall 2010/3/26 Phys. Rev. D in press,

Geometrical analysis of patterns in CMB anisotropy

• To characterize the geometrical patterns, the isotemperature contours in a smoothed CMB map is considered

2010年3月26日金曜日

Page 12: Analytic Minkowski Functionals of the CMBnlg.koubo/2010_3/Matsubara.pdfAnalytic Minkowski Functionals of the CMB Taka Matsubara (Nagoya U.) @Yukawa Hall 2010/3/26 Phys. Rev. D in press,

Geometrical analysis of patterns in CMB anisotropy

• To characterize the geometrical patterns, the isotemperature contours in a smoothed CMB map is considered

2010年3月26日金曜日

Page 13: Analytic Minkowski Functionals of the CMBnlg.koubo/2010_3/Matsubara.pdfAnalytic Minkowski Functionals of the CMB Taka Matsubara (Nagoya U.) @Yukawa Hall 2010/3/26 Phys. Rev. D in press,

Minkowski Functionals• The Minkowski functionals: statistical measures of

geometrical properties in the isotemperature contours (functions of threshold temperature)

• Area of high-temp. regions

• Length of iso-temp. contours

• [No. of high-temp. regions] - [No. of low-temp. regions] ( Euler number)

ex.)

2010年3月26日金曜日

Page 14: Analytic Minkowski Functionals of the CMBnlg.koubo/2010_3/Matsubara.pdfAnalytic Minkowski Functionals of the CMB Taka Matsubara (Nagoya U.) @Yukawa Hall 2010/3/26 Phys. Rev. D in press,

Minkowski Functionals

2010年3月26日金曜日

Page 15: Analytic Minkowski Functionals of the CMBnlg.koubo/2010_3/Matsubara.pdfAnalytic Minkowski Functionals of the CMB Taka Matsubara (Nagoya U.) @Yukawa Hall 2010/3/26 Phys. Rev. D in press,

Minkowski Functionals• Analytic formulas for Gaussian fields are well

known

• Tomita’s formula (Tomita 1986)

2010年3月26日金曜日

Page 16: Analytic Minkowski Functionals of the CMBnlg.koubo/2010_3/Matsubara.pdfAnalytic Minkowski Functionals of the CMB Taka Matsubara (Nagoya U.) @Yukawa Hall 2010/3/26 Phys. Rev. D in press,

Primordial non-Gaussianity and Minkowski Functionals

• Minkowski functionals, as functions of the threshold, have universal shapes for Gaussian fields

-4-2 0 2 4

-4 -3 -2 -1 0 1 2 3 4

V2

0 0.2 0.4 0.6 0.8 1 1.2 1.4

V1

0 0.2 0.4 0.6 0.8

1

V0

Different shape: non-Gaussian signature

2010年3月26日金曜日

Page 17: Analytic Minkowski Functionals of the CMBnlg.koubo/2010_3/Matsubara.pdfAnalytic Minkowski Functionals of the CMB Taka Matsubara (Nagoya U.) @Yukawa Hall 2010/3/26 Phys. Rev. D in press,

Primordial non-Gaussianity and Minkowski Functionals

• Analysis by the WMAP team

• Minkowski functionals in the WMAP5 data and differences from the Gaussian predictions are plotted

• To date, Gaussian fluctuations are consistent with the data within (correlated) error bars

Komatsu et al. (2009)

2010年3月26日金曜日

Page 18: Analytic Minkowski Functionals of the CMBnlg.koubo/2010_3/Matsubara.pdfAnalytic Minkowski Functionals of the CMB Taka Matsubara (Nagoya U.) @Yukawa Hall 2010/3/26 Phys. Rev. D in press,

Primordial non-Gaussianity and Minkowski Functionals

• It was common to use numerical simulations to give theoretical predictions for MFs of non-Gaussian fields, model by model

• Large computational cost, but results are not general

• We need general formulas for non-Gaussian fields

• However, there are infinite types of non-Gaussian fields

• When the non-Gaussianity is weak, an expansion in terms of the non-Gaussianity is possible. General formulas are found.

• TM (1994); TM (2004); Hikage et al. (2008); TM (2010)

2010年3月26日金曜日

Page 19: Analytic Minkowski Functionals of the CMBnlg.koubo/2010_3/Matsubara.pdfAnalytic Minkowski Functionals of the CMB Taka Matsubara (Nagoya U.) @Yukawa Hall 2010/3/26 Phys. Rev. D in press,

Minkowski Functionals in non-Gaussian Fields

• Results:

2010年3月26日金曜日

Page 20: Analytic Minkowski Functionals of the CMBnlg.koubo/2010_3/Matsubara.pdfAnalytic Minkowski Functionals of the CMB Taka Matsubara (Nagoya U.) @Yukawa Hall 2010/3/26 Phys. Rev. D in press,

Minkowski Functionals in non-Gaussian Fields

• The expansion is very good for the CMB, since

• Coefficients appeared

• calculated from moments of temperature and its derivatives

• skewness, kurtosis and their derivatives

2010年3月26日金曜日

Page 21: Analytic Minkowski Functionals of the CMBnlg.koubo/2010_3/Matsubara.pdfAnalytic Minkowski Functionals of the CMB Taka Matsubara (Nagoya U.) @Yukawa Hall 2010/3/26 Phys. Rev. D in press,

Minkowski Functionals in non-Gaussian Fields

• Those coefficients are given by weighted sums of the bispectrum and trispectrum

2010年3月26日金曜日

Page 22: Analytic Minkowski Functionals of the CMBnlg.koubo/2010_3/Matsubara.pdfAnalytic Minkowski Functionals of the CMB Taka Matsubara (Nagoya U.) @Yukawa Hall 2010/3/26 Phys. Rev. D in press,

Minkowski Functionals in non-Gaussian Fields

• From the above, analytic formulas can be evaluated for a given set of bispectrum and trispectrum

• This result is model-independent, no matter what kind of primordial non-Gaussianity is assumed

• (as long as the expansion scheme is good)

• Promising method to constrain models of the early universe which predict the primordial non-Gaussianity

2010年3月26日金曜日

Page 23: Analytic Minkowski Functionals of the CMBnlg.koubo/2010_3/Matsubara.pdfAnalytic Minkowski Functionals of the CMB Taka Matsubara (Nagoya U.) @Yukawa Hall 2010/3/26 Phys. Rev. D in press,

Application to the Local-type non-Gaussianity

• Application of the general formulas to the Local-type non-Gaussianity

• Newtonian potential has the form:

• The form of bispectrum and trispectrum in this case is well known [Komatsu & Spergel (2001); Okamoto & Hu (2002)]

• bispectrum

• trispectrum

• Therefore, the analytic formulas of the MFs are also linear combinations of these parameters

2010年3月26日金曜日

Page 24: Analytic Minkowski Functionals of the CMBnlg.koubo/2010_3/Matsubara.pdfAnalytic Minkowski Functionals of the CMB Taka Matsubara (Nagoya U.) @Yukawa Hall 2010/3/26 Phys. Rev. D in press,

Application to the Local-type non-Gaussianity

• Comparison with numerical simulations (Sachs-Wolfe limit)

-0.001

0

0.001

0.002

-4 -3 -2 -1 0 1 2 3 4

(V0 -

V0G

) / V

0G,m

ax

0 0.2 0.4 0.6 0.8

1

V0

fNL = 102, gNL = 106

-0.004-0.002

0 0.002 0.004 0.006

-4 -3 -2 -1 0 1 2 3 4

(V1 -

V1G

) / V

1G,m

ax

0 0.2 0.4 0.6 0.8

1 1.2 1.4

V1

fNL = 102

gNL = 106

-0.02-0.015-0.01

-0.005 0

0.005

-4 -3 -2 -1 0 1 2 3 4

(V2 -

V2G

) / V

2G,m

ax

-4-2 0 2 4

V2

fNL = 102, gNL = 106

upper panels: values of Minkowski Functionalslower panels: difference from Gaussian

data points: averages of 100,000 non-Gaussian realizations

2010年3月26日金曜日

Page 25: Analytic Minkowski Functionals of the CMBnlg.koubo/2010_3/Matsubara.pdfAnalytic Minkowski Functionals of the CMB Taka Matsubara (Nagoya U.) @Yukawa Hall 2010/3/26 Phys. Rev. D in press,

Non-degeneracy of parameters• Analytic MFs: linear combination of

• The coefficients are orthogonal functions of nu

• Therefore those parameters can be independently determined by observations: non-degeneracy

colored lines: contributions from fNL and gNL

(contributions from tauNL is negligibly small in this example)

fNL

gNL

-0.02-0.015

-0.01-0.005

0 0.005

-4 -3 -2 -1 0 1 2 3 4

(V2 -

V2G

) / V

2G,m

ax

-4-2 0 2 4

V2

fNL = 102, gNL = 106

2010年3月26日金曜日

Page 26: Analytic Minkowski Functionals of the CMBnlg.koubo/2010_3/Matsubara.pdfAnalytic Minkowski Functionals of the CMB Taka Matsubara (Nagoya U.) @Yukawa Hall 2010/3/26 Phys. Rev. D in press,

Summary• Minkowski Functionals:

• Statistics of geometric patterns, sensitive to the non-Gaussianity

• Analytic formulas up to cubic order in general nG field

• Independent determination of fNL, gNL, tauNL possible

• Future:

• Constrain gNL and tauNL by WMAP data (with Hikage)

• Constrain non-local-type models from WMAP data

• Applications to the polarization map (straight)

2010年3月26日金曜日

Page 27: Analytic Minkowski Functionals of the CMBnlg.koubo/2010_3/Matsubara.pdfAnalytic Minkowski Functionals of the CMB Taka Matsubara (Nagoya U.) @Yukawa Hall 2010/3/26 Phys. Rev. D in press,

Comparisons with WMAP Simulations

• Hikage et al. (2008): fNL

• Simulated map with observational noises

• Analytic MFs can actually constrain NG

2010年3月26日金曜日

Page 28: Analytic Minkowski Functionals of the CMBnlg.koubo/2010_3/Matsubara.pdfAnalytic Minkowski Functionals of the CMB Taka Matsubara (Nagoya U.) @Yukawa Hall 2010/3/26 Phys. Rev. D in press,

Analysis of WMAP data• Hikage et al. (2008) : fNL

• WMAP3

Similar analysis on gNL, tauNL is going on

(with Hikage)

2010年3月26日金曜日