angular polyspectra in cosmology: lss bispectrum and cmb ... · 1) ( x n)i is de ned as the excess...

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.

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

Outline



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

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.

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 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 LSS Bispectrum;

I CMB Trispectrum;

I Conclusion.

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!

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.

angular polyspectra in cosmology: lss bispectrum and cmb ... · 1) ( x n)i is de ned as the excess...

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