Airam Marcos-Caballero June 17, 2015 Meeting on ...max.ifca.unican.es/MFC2015/talks/Number02_MarcosCaballero.pdf · Airam Marcos-Caballero June 17, 2015 Meeting on Fundamental Cosmology,

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



Outline








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?




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)




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)




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.




Simulations

Maximum with ν = 5

Elliptical spot




Outline








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)




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°




Polarization

Cartesian coordinates

Q, U

Polar coordinates

Qr, Ur




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)




Outline








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′)




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



Outline








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




Cold Spot simulations

Real space

Wavelet space (R = 5◦)


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.



Thank you


Cold and hot spots at large scales

Airam Marcos-Caballero

June 17, 2015

Meeting on Fundamental Cosmology, Santander

Documents

Airam Marcos-Caballero June 17, 2015 Meeting on ...max.ifca.unican.es/MFC2015/talks/Number02_MarcosCaballero.pdf · Airam Marcos-Caballero June 17, 2015 Meeting on Fundamental Cosmology,