angular polyspectra in cosmology: lss bispectrum and cmb ... · 1) ( x n)i is de ned as the excess...
TRANSCRIPT
Angular Polyspectra in Cosmology:LSS Bispectrum and CMB Trispectrum
Antonino Troja
Universita degli Studi di Milano
22 June 2017
Outline
I Polyspectra: Definition and Statistical Meaning;
I Inflation Overview;
I LSS Bispectrum;
I CMB Trispectrum;
I Conclusion.
Outline
I Polyspectra: Definition and Statistical Meaning;
I Inflation Overview;
I LSS Bispectrum;
I CMB Trispectrum;
I Conclusion.
Polyspectra: Definition
Given a random field Φ(x), we define the n-point correlationfunction:
〈Φ(x1) · · ·Φ(xn)〉
is defined as the excess probability, compared to a randomdistribution, of finding n points in a certain configuration.
The polyspectrum of order n-1 is the Fouriercounterpart of the n-point correlation function.
Polyspectra: Definition
Given a random field Φ(x), we define the n-point correlationfunction:
〈Φ(x1) · · ·Φ(xn)〉
is defined as the excess probability, compared to a randomdistribution, of finding n points in a certain configuration.
The polyspectrum of order n-1 is the Fouriercounterpart of the n-point correlation function.
Power Spectrum
Given a field Φ(x) and its Fourier transform
Φ(k) =
∫d3xΦ(x)e ik·x
Power Spectrum:
〈Φ(k1)Φ(k2)〉 = (2π)3δ(3)(k1 + k2)PΦ(k1)
k1
k2
Bispectrum
Bispectrum:
〈Φ(k1)Φ(k2)Φ(k3)〉 = (2π)3δ(3)(k1 + k2 + k3)BΦ(k1, k2, k3)
k1
k2
k3
Trispectrum
Trispectrum:
〈Φ(k1)Φ(k2)Φ(k3)Φ(k4)〉 = (2π)3δ(3)(k1+k2+k3+k4)TΦ(k1, k2, k3, k4)
k1
k2
k3
k4
Polyspectra: Statistical definition
In statistics, the polyspectra are related to the momenta of adistribution.
P(k) −→ variance
B(k1, k2, k3) −→ skewness
T (k1, k2, k3) −→ kurtosis
Gaussian distribution
A 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
Gaussian vs Non-Gaussian
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
Gaussian vs Non-Gaussian
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
fNL: SkewnessSkewness: asymmetry of the distribution.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
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: KurtosisKurtosis: height of the tails of the distribution.
0
0.1
0.2
0.3
0.4
0.5
0.6
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
Polyspectra on the Sphere
If the field is defined on the sphere, the Fourier transform isperformed using the Spherical Harmonics, i.e. the basis for thefunctions defined on S2.
Φ(x) =
∫d3xΦ(k)e−ik·x −→
∞∑l=0
l∑m=−l
almYlm(x)
Φ(k) −→ alm
P(k) −→ Cl
B(k1, k2, k3) −→ Bl1l2l3
T (k1, k2, k3, k4) −→ Tl1l2l3l4
Polyspectra on the Sphere
If the field is defined on the sphere, the Fourier transform isperformed using the Spherical Harmonics, i.e. the basis for thefunctions defined on S2.
Φ(x) =
∫d3xΦ(k)e−ik·x −→
∞∑l=0
l∑m=−l
almYlm(x)
Φ(k) −→ alm
P(k) −→ Cl
B(k1, k2, k3) −→ Bl1l2l3
T (k1, k2, k3, k4) −→ Tl1l2l3l4
Outline
I Polyspectra: Definition and Statistical Meaning;
I Inflation Overview;
I LSS Bispectrum;
I CMB Trispectrum;
I Conclusion.
What do we study?
What do they have in common?
Inflationary seeds
Both CMB and LSS were seeded by Inflaton vacuum perturbation!
Inflation
Inflaton and Inflation
INFLATON φ(x, t) = φ0(t) + δφ(x, t)
I Spatial average of the field
I Vacuum Fluctuations
VacuumFluctuations
PrimordialGravitationalFluctuations
MatterFluctuations
CMBAnisotropy
Inflation After Inflation
Inflaton and Inflation
INFLATON φ(x, t) = φ0(t) + δφ(x, t)
I Spatial average of the field
I Vacuum Fluctuations
VacuumFluctuations
PrimordialGravitationalFluctuations
MatterFluctuations
CMBAnisotropy
Inflation After Inflation
Inflaton and Inflation
INFLATON φ(x, t) = φ0(t) + δφ(x, t)
I Spatial average of the field
I Vacuum Fluctuations
VacuumFluctuations
PrimordialGravitationalFluctuations
MatterFluctuations
CMBAnisotropy
Inflation After Inflation
Inflaton and Inflation
INFLATON φ(x, t) = φ0(t) + δφ(x, t)
I Spatial average of the field
I Vacuum Fluctuations
VacuumFluctuations
PrimordialGravitationalFluctuations
MatterFluctuations
CMBAnisotropy
Inflation After Inflation
Primordial Gravitational Field
We can write the primordial gravitational field Φ(x) by mean of theso-called Bardeen potential:
Φ(x) = ΦL(x) + fNL(ΦL(x)2 − 〈ΦL(x)2〉) + gNL[ΦL(x)3]
Gaussian component
non-Gaussian component
Primordial Gravitational Field
We can write the primordial gravitational field Φ(x) by mean of theso-called Bardeen potential:
Φ(x) = ΦL(x) + fNL(ΦL(x)2 − 〈ΦL(x)2〉) + gNL[ΦL(x)3]
Gaussian component
non-Gaussian component
Non-Gaussianity
The amplitude of fNL and gNL affect both the photon and thematter distribution.
CMB: the distribution of the anisotropies inherits the primordialnon-gaussianity.
LSS: fNL and gNL affect the way the gravitational collapse happens.
Non-Gaussianity
The amplitude of fNL and gNL affect both the photon and thematter distribution.
CMB: the distribution of the anisotropies inherits the primordialnon-gaussianity.
LSS: fNL and gNL affect the way the gravitational collapse happens.
Non-Gaussianity
The amplitude of fNL and gNL affect both the photon and thematter distribution.
CMB: the distribution of the anisotropies inherits the primordialnon-gaussianity.
LSS: fNL and gNL affect the way the gravitational collapse happens.
Last Remark
There is NOT only one Inflation model.
Single-Field
I Standard Scenario
I k-inflation
I DBI inflation
I ...
Multi-Fields
I Curvaton Scenario
I Ghost inflation
I D-cceleration scenario
I ...
Which is the correct one?
Last Remark
There is NOT only one Inflation model.
Single-Field
I Standard Scenario
I k-inflation
I DBI inflation
I ...
Multi-Fields
I Curvaton Scenario
I Ghost inflation
I D-cceleration scenario
I ...
Which is the correct one?
Last Remark
Standard ModelThe primordial fluctuation distribution is GAUSSIAN.
Non-Standard ModelsThe primordial fluctuation distribution is NON-GAUSSIAN.
The amounts of non-Gaussianity depends on the model.
Outline
I Polyspectra: Definition and Statistical Meaning;
I Inflation Overview;
I LSS Bispectrum;
I CMB Trispectrum;
I Conclusion.
Large Scale Structure of the Universe
Matter Bispectrum
LSS bispectrum is composed by two terms:
B(k1, k2, k3) = BI (k1, k2, k3) + BG (k1, k2, k3),
where BI is the primordial bispectrum, parametrized by fNL,and BG is the gravitational evolution bispectrum.
Matter Bispectrum
k1
k2
k3k1
k2
k3θ k1
k2
k3
3D triangle configuration
k1
k2
k3
Photometric Samples
Photometric Filters
Photometric Error
φ(z) =dNg
dz
∫dzp P(z |zp)W (zp)
DatasetMeasurements performed on 125 LSS mocks covering 1/8 of skyat z = 0.5 from MICE catalogue (Fosalba et al. 2008, Crocce etal. 2010).
0 187
Number of galaxies
Results
Smallphoto-zerror
Largephoto-zerror
small binsize large binsize
−0.2
0.0
0.2
0.4
0.6
0.8
1.0
L1L
2L
3b L
1L
2L
3
×10−3
σz = 0.03
∆z(1+z)
= 0.03
Nbody
GMTLSC
95 119 143 167 191 215 239L1
−0.5
0.0
0.5
1.0
1.5
2.0
b L1L
2L
3/bth L
1L
2L
3
Non-Triangular Bins
−0.2
0.0
0.2
0.4
0.6
0.8
1.0
L1L
2L
3b L
1L
2L
3
×10−3
σz = 0.03
∆z(1+z)
= 0.05
95 119 143 167 191 215 239L1
−0.5
0.0
0.5
1.0
1.5
2.0
b L1L
2L
3/bth L
1L
2L
3
−0.2
0.0
0.2
0.4
0.6
0.8
1.0
L1L
2L
3b L
1L
2L
3
×10−3
σz = 0.06
∆z(1+z)
= 0.03
95 119 143 167 191 215 239L1
−0.5
0.0
0.5
1.0
1.5
2.0
b L1L
2L
3/bth L
1L
2L
3
−0.2
0.0
0.2
0.4
0.6
0.8
1.0
L1L
2L
3b L
1L
2L
3
×10−3
σz = 0.06
∆z(1+z)
= 0.05
95 119 143 167 191 215 239L1
−0.5
0.0
0.5
1.0
1.5
2.0
b L1L
2L
3/bth L
1L
2L
3
Outline
I Polyspectra: Definition and Statistical Meaning;
I Inflation Overview;
I LSS Bispectrum;
I CMB Trispectrum;
I Conclusion.
CMB Non-Gaussianity
PrimordialNG
AnisotropiesNG
Evaluate NG in the CMB allows to constrain the inflationarymodels.
CMB Non-Gaussianity
PrimordialNG
AnisotropiesNG
Evaluate NG in the CMB allows to constrain the inflationarymodels.
NG Estimators
fNL
3rd-ordermomentum:
Skewness
Bispectrum
gNL
4th-ordermomentum:
Kurtosis
Trispectrum
State of the Art
Optimal Estimator : Unbiased estimator with the lowest varianceamong the other ones. Planck Collaboration (2015):
fNL = 2.5 ± 5.7 (1σ)
Single-field models are preferred.
State of the Art
Planck Collaboration (2015):
gNL = (−9.0 ± 7.7) × 104 (1σ)
gNL has a very low statistical significance!
State of the Art
Planck Collaboration (2015):
gNL = (−9.0 ± 7.7) × 104 (1σ)
gNL has a very low statistical significance!
Spherical Needlet System
ψjk(x) :=√λjk∑l
b
(l
B j
) l∑m=−l
Ylm(ξjk)Y lm(x)
100 101 102 103 104
`
0.0
0.2
0.4
0.6
0.8
1.0
b( l B
j
)
Scale 1Scale 2Scale 3Scale 4Scale 5Scale 6Scale 7Scale 8Scale 9Scale 10Scale 11Scale 12
Localization Property
Localization property in real space means raise of the statisticalsignificance in presence of incomplete sky coverage (i.e. ALWAYS).
Needlet Coefficients
Reconstruction Formula:
T (x) =∑l ,m
almYlm(x) =∑j ,k
βjkψjk(x)
Needlet Deconvolution
-439.221 501.718 -44.9847 42.5258
-71.5606 68.7117 -216.269 215.141
gNL Estimation
−600000 −400000 −200000 0 200000 400000 600000gNL
0
20
40
60
80
100
120
140
Nocc
σgNL Estimation
20 40 60 80 100 120 140 160 180 200N covset
1.22
1.24
1.26
1.28
1.30
1.32σg N
Lfit 5ptfit 3 of 5ptfit 7ptfit 4 of 7ptDataData add
Future Perspectives
I The angular bispectrum of LSS seems to work well whencompared with predictions, now it is time to constrainparameters with it;
I The Needlet trispectrum of CMB is computational feasibleand ready to constrain gNL from PLANCK data.