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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
Minkowski Functionals can
simultaneously constrain
(and other models)
2010年3月26日金曜日
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
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
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
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
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
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
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
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
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
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
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
Minkowski Functionals
2010年3月26日金曜日
Minkowski Functionals• Analytic formulas for Gaussian fields are well
known
• Tomita’s formula (Tomita 1986)
2010年3月26日金曜日
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
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
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
Minkowski Functionals in non-Gaussian Fields
• Results:
2010年3月26日金曜日
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
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
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
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
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
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
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
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
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
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