View
2
Download
0
Category
Preview:
Citation preview
Cold and hot spots at large scales
Airam Marcos-Caballero
June 17, 2015
Meeting on Fundamental Cosmology, Santander
Simulating spots on the sphereTheoretical profiles
Covariance of the ProfilesCold Spot Analysis
Outline
1 Simulating spots on the sphere
2 Theoretical profiles
3 Covariance of the Profiles
4 Cold Spot Analysis
A. Marcos-Caballero Cold and hot spots 2 / 22
Simulating spots on the sphereTheoretical profiles
Covariance of the ProfilesCold Spot Analysis
Outline
1 Simulating spots on the sphere
2 Theoretical profiles
3 Covariance of the Profiles
4 Cold Spot Analysis
A. Marcos-Caballero Cold and hot spots 3 / 22
Simulating spots on the sphereTheoretical profiles
Covariance of the ProfilesCold Spot Analysis
We define a spot in the field ψ through its derivatives up to secondorder:
ψ(n) = ψ + ∂iψ θi +1
2∂i∂jψ θiθj . . .
A field in the sphere is given in terms of the spherical harmonics:
ψ(n) =
∞∑l=0
l∑m=−l
a`mY`m(n)
How to express these derivatives in terms of the sphericalharmonics coefficients?
A. Marcos-Caballero Cold and hot spots 4 / 22
Simulating spots on the sphereTheoretical profiles
Covariance of the ProfilesCold Spot Analysis
Peak height (m = 0)
ψ =
∞∑`=0
√2`+ 1
4πa`0
First derivatives (m = 1)
D†ψ =
∞∑`=0
√2`+ 1
4π
√(`+ 1)!
(`− 1)!a`1
Second derivatives (m = 0 and m = 2)
∇2ψ = −∞∑`=0
√2`+ 1
4π
(`+ 1)!
(`− 1)!a`0 Scalar (m = 0)
D2ψ = −∞∑`=0
√2`+ 1
4π
√(`+ 2)!
(`− 2)!a`2 Spinor (m = 2)
A. Marcos-Caballero Cold and hot spots 5 / 22
Simulating spots on the sphereTheoretical profiles
Covariance of the ProfilesCold Spot Analysis
Quantities with different spin are uncorrelated due to rotationalinvariance.
Scalar: ψ, ∇2ψ
Vector: D†ψ
Tensor: D2ψ
Peak variables
Peak height ν = ψσ(ψ)
Mean curvature κ = ∇2ψσ(∇2ψ)
Ellipticity ε = D2ψσ(D2ψ)
κ =1
a2+
1
b2
ε =1
b2
(1− b2
a2
)ei2α
P (ν, κ, ε) = P (ν)P (κ|ν)P (ε)
A. Marcos-Caballero Cold and hot spots 6 / 22
Simulating spots on the sphereTheoretical profiles
Covariance of the ProfilesCold Spot Analysis
We separate the peak degrees of freedom and the rest of the field.
a`m, e`m=⇒
ν, κ, ε: peak degrees of freedom.
a`m, e`m: degrees of freedomuncorrelated with the peak.
We orthogonalize the covarianve matrix:
C =
(ν, κ, ε 0
0 a`m, e`m
)Once the peak variables are uncorrelated is very easy to simulatemaps with the spots and calculate expectation values.
A. Marcos-Caballero Cold and hot spots 7 / 22
Simulating spots on the sphereTheoretical profiles
Covariance of the ProfilesCold Spot Analysis
Simulations
Maximum with ν = 5
Elliptical spot
A. Marcos-Caballero Cold and hot spots 8 / 22
Simulating spots on the sphereTheoretical profiles
Covariance of the ProfilesCold Spot Analysis
Outline
1 Simulating spots on the sphere
2 Theoretical profiles
3 Covariance of the Profiles
4 Cold Spot Analysis
A. Marcos-Caballero Cold and hot spots 9 / 22
Simulating spots on the sphereTheoretical profiles
Covariance of the ProfilesCold Spot Analysis
In order to get the profile we calculate the expectation value of thefield with some peak contraints.
There are two ways of averaging the peak degrees of freedom:
With the density of peaks −→ Stacking
With the probability density −→ Single spot
n(ν, κ, ε) ∝ |H(κ, ε)|P (ν, κ, ε)
A. Marcos-Caballero Cold and hot spots 10 / 22
Simulating spots on the sphereTheoretical profiles
Covariance of the ProfilesCold Spot Analysis
ν fixed and min/max selection
p(θ) =ν
σC(θ) +
∆κ(ν)
(1− ρ2)σ2
(∇2 − ρσ2
σ
)C(θ)
where ∆κ(ν) = 〈κ〉ν − ρν is the shift of the curvature mean due tothe min/max selection.
-60
-40
-20
0
20
40
60
80
100
120
0 0.5 1 1.5 2 2.5 3
T(θ
) [µ
K]
θ
Peak heightCurvature
Total
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
-4 -2 0 2 4
∆κ(ν
)
ν
Single spotStacking
Wavelet 5°
A. Marcos-Caballero Cold and hot spots 11 / 22
Simulating spots on the sphereTheoretical profiles
Covariance of the ProfilesCold Spot Analysis
Polarization
Cartesian coordinates
Q, U
Polar coordinates
Qr, Ur
A. Marcos-Caballero Cold and hot spots 12 / 22
Simulating spots on the sphereTheoretical profiles
Covariance of the ProfilesCold Spot Analysis
T (θ) = Ψ†CTT (θ)
E(θ) = Ψ†CTE(θ)
B(θ) = Ψ†CTB(θ)
Ψ(θ) =
(ψ0
∇2ψ0
)
Cij(θ) = Σ−1(
Cij(θ′)∇2Cij(θ′)
)
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
0 0.5 1 1.5 2 2.5 3 3.5
Qr(
θ) [µ
K]
θ
ν = 1ν = 2ν = 3ν = 4
E(θ)←→ Qr(θ)
B(θ)←→ Ur(θ)
A. Marcos-Caballero Cold and hot spots 13 / 22
Simulating spots on the sphereTheoretical profiles
Covariance of the ProfilesCold Spot Analysis
Outline
1 Simulating spots on the sphere
2 Theoretical profiles
3 Covariance of the Profiles
4 Cold Spot Analysis
A. Marcos-Caballero Cold and hot spots 14 / 22
Simulating spots on the sphereTheoretical profiles
Covariance of the ProfilesCold Spot Analysis
Profiles covariance
C(θ, θ′) =1
2πδ(cos θ − cos θ′)⊗ C(θ)
C(θ, θ′) =
∞∑`=0
2`+ 1
4πC` P`(cos θ)P`(cos θ′)
C` = const. =⇒ C(θ, θ′) =1
2πδ(cos θ − cos θ′)
A. Marcos-Caballero Cold and hot spots 15 / 22
Simulating spots on the sphereTheoretical profiles
Covariance of the ProfilesCold Spot Analysis
Covariance bias due to peak selection
The covariance of ψ and ∇2ψ is modified when peak constraintsare imposed
Σ −→ Σ + ∆Σ
CTT (θ, θ′) = CTTrings(θ, θ′)︸ ︷︷ ︸
Intrinsic
+ CTT (θ)†∆ΣCTT (θ′)︸ ︷︷ ︸Peak selection
CEE(θ, θ′) = CEErings(θ, θ′) + CTE(θ)†∆ΣCTE(θ′)
CTE(θ, θ′) = CTErings(θ, θ′) + CTT (θ)†∆ΣCTE(θ′)
C(θ) = Σ−1(
C(θ′)∇2C(θ′)
)A. Marcos-Caballero Cold and hot spots 16 / 22
Simulating spots on the sphereTheoretical profiles
Covariance of the ProfilesCold Spot Analysis
Outline
1 Simulating spots on the sphere
2 Theoretical profiles
3 Covariance of the Profiles
4 Cold Spot Analysis
A. Marcos-Caballero Cold and hot spots 17 / 22
Simulating spots on the sphereTheoretical profiles
Covariance of the ProfilesCold Spot Analysis
Peak selection on Wavelet space
We select the CS interms of thewavelet coefficient.
C(θ) −→ w(θ)⊗C(θ)
C` −→ w`C`
p(θ) −→ w(θ)⊗p(θ) -300
-250
-200
-150
-100
-50
0
50
0 5 10 15 20 25
T(θ
) [µ
K]
θ
A. Marcos-Caballero Cold and hot spots 18 / 22
Simulating spots on the sphereTheoretical profiles
Covariance of the ProfilesCold Spot Analysis
Cold Spot simulations
Real space
Wavelet space (R = 5◦)
A. Marcos-Caballero Cold and hot spots 19 / 22
Conclusions
We have developed the methodology to simulate spots on thesphere (without the flat approximation).
Previous results on peaks are recovered.
We differenciate between density (stacking) and probability(single spot) average.
Theoretical calculations of the covariance matrix between theprofiles T and Qr.
Higher order moments of the profile can be calculated usingthis scheme.
It is possible to generalized this procedure to spots withellipticity (oriented stacking).
Applications to the Cold Spot.
Simulating spots on the sphereTheoretical profiles
Covariance of the ProfilesCold Spot Analysis
Thank you
A. Marcos-Caballero Cold and hot spots 21 / 22
Cold and hot spots at large scales
Airam Marcos-Caballero
June 17, 2015
Meeting on Fundamental Cosmology, Santander
Recommended