3 years in a nutshell:
Development and application of polyspectra to
modern Cosmology
Antonino Troja
Università degli Studi di Milano
Facoltà di Scienze e Tecnologie
October 26, 2016
PhD Goals
CMB
LSS
I
Needlet Trispectrum
Estimator;
I
Angular Bispectrum
Prediction;
I
gNL estimation.
I
Estimator.
Cosmological Observables
Polyspectra: Why
CMB
Temeprature
Anisotropy
LSS
Galaxy
Posistion
The best way to analize a distribution is to study its polyspectra.
Polyspectra: Definition
Given a random field Φ(x), we define the n-point correlation
function:
hΦ(x1 ) · · · Φ(xn )i
is defined as the excess probability, compared to a random
distribution, of finding n points in a certain configuration.
Polyspectra: Definition
Given a random field Φ(x), we define the n-point correlation
function:
hΦ(x1 ) · · · Φ(xn )i
is defined as the excess probability, compared to a random
distribution, of finding n points in a certain configuration.
The Angular polyspectrum of order n-1 is the
Fourier counterpart of the n-point correlation function.
Power Spectrum
Given a spherical field Φ(x) and its Fourier transform
Z
Φ(k) = d 3 xΦ(x)e ik·x
Power Spectrum:
hΦ(k1 )Φ(k2 )i = (2π)3 δ (3) (k1 + k2 )PΦ (k1 )
k2
k1
Bispectrum
Bispectrum:
hΦ(k1 )Φ(k2 )Φ(k3 )i = (2π)3 δ (3) (k1 + k2 + k3 )BΦ (k1 , k2 , k3 )
k3
k2
k1
Trispectrum
Trispectrum:
hΦ(k1 )Φ(k2 )Φ(k3 )Φ(k4 )i = (2π)3 δ (3) (k1 +k2 +k3 +k4 )TΦ (k1 , k2 , k3 , k4 )
k3
k2
k4
k1
Polyspectra: Statistical definition
In statistics, the polyspectra are defined as the higher order
momentum of a distribution.
P(k) −→ variance
B(k1 , k2 , k3 ) −→ skewness
T (k1 , k2 , k3 ) −→ kurtosis
fNL : Skewness
Skewness: asymmetry of the distribution.
1.8
1.6
fNL < 0
1.4
1.2
Right side of MB
1
Left side of MB
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
gNL : Kurtosis
Kurtosis: height of the tails of the distribution.
0.6
0.5
Right side of MB, gNL < 0
0.4
0.3
Gaussian distribution
0.2
0.1
0
Left side of MB, gNL > 0
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
Gaussian distribution
The Gaussian distribution is fully described by the lowest moments:
mean (µ) and variance (σ):
G (x) =
√1 e
2πσ
−(x−µ)2
2σ 2
Polyspectra help in discern Gaussian from non-Gaussian
distribution.
Polyspectra on the Sphere
If the field is defined on the sphere, the Fourier transform is
performed using the Spherical Harmonics, i.e. the basis for the
functions defined on S 2 .
Z
Φ(x) =
d 3 xΦ(k)e −ik·x −→
∞ X
l
X
l=0 m=−l
Φ(k) −→ alm
P(k) −→ Cl
B(k1 , k2 , k3 ) −→ Bl1 l2 l3
T (k1 , k2 , k3 , k4 ) −→ Tl1 l2 l3 l4
alm Ylm (x)
Cosmic Microwave Background:
Needlet Trispectrum
CMB Non-Gaussianity
There is NOT only one Inflation model.
Single-Field
Multi-Fields
I
Standard Scenario
I
Curvaton Scenario
I
k-inflation
I
Ghost inflation
I
...
I
...
Each model provides a differrent NG track in the CMB anisotropies
distribution.
CMB Non-Gaussianity
There is NOT only one Inflation model.
Single-Field
Multi-Fields
I
Standard Scenario
I
Curvaton Scenario
I
k-inflation
I
Ghost inflation
I
...
I
...
Each model provides a differrent NG track in the CMB anisotropies
distribution.
Evaluate NG in the CMB allows to
constrain the inflationary models.
State of the Art: Planck 2015
fNL = 2.5 ± 5.7 (1σ)
gNL = (−9.0 ± 7.7) × 104 (1σ)
State of the Art: Planck 2015
fNL = 2.5 ± 5.7 (1σ)
gNL = (−9.0 ± 7.7) × 104 (1σ)
Single-field models are preferred. gNL has a very low statistical
significance!
Spherical Needlet System
I proposed an optimal estimator using the Spherical Needlets
system:
X
l
p X
l
Ylm (ξjk )Y lm (x)
ψjk (x) := λjk
b
Bj
l
m=−l
Localization property in real space means raise of the statistical
significance in presence of incomplete sky coverage (i.e. ALWAYS).
Needlet Coefficients
Reconstruction Formula:
X
X
T (x) =
βjk ψjk (x) ≡
alm Ylm (x)
j,k
l,m
Needlet Trispectrum
Spherical Needlet Optimal Estimator:
s
4π X βj1 k4 βj2 k4 βj3 k4 βj4 k4
Jjj31jj42 =
+ C (βj1 k4 , βj2 k4 , βj3 k4 , βj4 k4 )
Nj4
σj1 σj2 σj3 σj4
k4
j1 ≤ j2 ≤ j3 ≤ j4
Consistency Checks
I tested the estimator using simulation at WMAP (Nside = 512)
and PLANCK (Nside = 2048) resolution.
I used Gaussian simulation and non-Gaussian simulation
input
(Elsner&Wandelt (2009)) with gNL
= 3.0 · 105 .
Resolution
512
512
512
2048
2048
2048
nj
12
14
18
12
16
20
Gaussian sim.
0.0 ± 3.6
0.0 ± 3.6
0.0 ± 3.6
0.1 ± 1.3
0.0 ± 1.4
0.0 ± 1.6
Non-Gaussian sim.
3.0 ± 4.1
3.0 ± 4.0
2.9 ± 4.2
3.2 ± 2.3
3.2 ± 2.5
3.1 ± 2.5
Fisher
2.2
2.2
2.2
0.7
0.7
0.7
Consistency Checks
I tested the estimator using simulation at WMAP (Nside = 512)
and PLANCK (Nside = 2048) resolution.
I used Gaussian simulation and non-Gaussian simulation
input
(Elsner&Wandelt (2009)) with gNL
= 3.0 · 105 .
Resolution
512
512
512
2048
2048
2048
nj
12
14
18
12
16
20
Gaussian sim. Non-Gaussian sim.
0.0 ± 3.6
3.0 ± 4.1
0.0 ± 3.6
3.0 ± 4.0
0.0 ± 3.6
2.9 ± 4.2
0.1 ± 1.3
3.2 ± 2.3
0.0 ± 1.4
3.2 ± 2.5
0.0 ± 1.6
3.1 ± 2.5
FAILED!
Fisher
2.2
2.2
2.2
0.7
0.7
0.7
Large Scale Structure:
Angular Bispectrum
LSS
Galaxy Bias
We measure the posistion of the galaxy in space. We don’t know
anything about the distribution of dark matter.
We assume that galaxies trace the matter distribution writing:
1
δg (x) ' b1 δ(x) + b2 δ 2 (x)
2
where:
I
δg (x) is the galaxy density field;
I
δ(x) is the matter density field;
I
b1 is the linear bias parameter;
I
b2 is the non-linear bias parameter.
Constrain b1 and b2 is one of the main goal of modern Cosmology.
3D Matter Bispectrum
Since the number of configuration in 3D space is greater that than
in 2D space, the bias parameter b1 and b2 have been constrained
using the 3D bispectrum so far.
Angular Bispectrum
My goal is to understand if the Angular Bispectrum can add any
information to the 3D bispectrum.
What I do is to integrate the galaxy distribution along the line of
sight, convoluting with a window function, in order to consider a
slice of sky in the redshift axis.
Angular Bispectrum: Predictions
bl1 l2 l3
3 Z
2
dz1 dz2 dz3 φ(z1 )φ(z2 )φ(z3 )
=
π
Z
× dk1 dk2 dk3 k12 k22 k32 B(k1 , k2 .k3 , z1 , z2 , z3 )
× jl1 (k1 r (z1 ))jl2 (k2 r (z2 ))jl3 (k3 r (z3 ))
Z
× dxx 2 jl1 (k1 x)jl2 (k2 x)jl3 (k3 x)
Angular Bispectrum: Predictions
Z
bl1 l2 l3 =
1
B
dzφ(z)3
4
r (z1 ) |r (z)0 |2
l1 + 12 l2 + 12 l3 + 12
,
,
,z
r (z1 ) r (z2 ) r (z3 )
!
Angular Bispectrum: Measurements
Measurements performed on 125 LSS mocks covering 1/8 of sky
(MICE catalogue).
The estimator used is the “classic” one:
b̂l1 l2 l3 =
1
X
hl1 l2 l3
m1 m2 m3
al1 m1 al2 m2 al3 m3
l1
l2
l3
m1 m2 m3
hl1 l2 l3 =
p
Ntrip (l1 , l2 , l3
Angular Bispectrum: All configurations
All-Shapes Bispectrum ∆` = 24
0.007
Limber Linear Prediction
Limber Scoccimarro-Couchman Prediction
Limber Gil-Marin Prediction
Measurements
0.006
0.005
`31 b`1 `2 `3
0.004
0.003
0.002
0.001
0.000
3.0
0
50
100
0
50
100
150
200
250
150
200
250
b`1 `2 `3 /bth
`1 `2 `3
2.5
2.0
1.5
1.0
0.5
0.0
`1
Angular Bispectrum: Equilateral configurations
Equilateral Bispectrum ∆` = 24
0.0007
Limber Linear Prediction
Limber Scoccimarro-Couchman Prediction
Limber Gil-Marin Prediction
Measurements
0.0006
0.0005
`3 b```
0.0004
0.0003
0.0002
0.0001
0.0000
0
50
100
0
50
100
150
200
250
150
200
250
b``` /bth
```
2.5
2.0
1.5
1.0
0.5
0.0
`
Conclusion and Goals
CMB
I
Introduce a new selection alghoritm for the Trispectrum
estimator;
I
Solve the NG issue by testing more deeply the NG simulations;
LSS
I
Costrain bias parameters b1 and b2 using Angular Bispectrum;
I
Introduce non-local bias;
Thanks for your Attention!