**Patricio Vielva**

**Patricio Vielva**

**Astrophysics Department (IFCA, Santander)****Wiaux, Vielva, Martínez-González & Vandergheynst, 2006, PRL**

**Bernard’s **

** Motivation**

** Our approach: the CMB structures can "watch"**

** Steerable wavelets**

** Application to the WMAP data**
** What next?**

**Conclusions**

**Bernard’s **

**(from Hansen et al. 2004)**
** Motivation**

** Our approach: the **
**CMB structures can **

**"watch"**

** Steerable wavelets**

** Application to the **
**WMAP data**

** What next?**

** Conclusions**

**Motivation**

Many works have analysed the WMAP data for studying whether it is consistent with the isotropy principle.

Some works based on wavelets have detected a very large and very cold spot in the southern hemisphere showing a significant deviation from an isotropic GRF (Vielva et al. 2004, Cruz et al. 2005, 2006).

Certain analyses find a strong evidence for a north-south asymmetry
*maximized in a coordinate system with the north pole *close to the north
ecliptic pole (e.g. Eriksen et al. 2004, Hansen et al. 2004, Land &

Magueijo 2005a).

Other works find an anomalous alignment between the low multipoles of the CMB, suggesting a preferred direction near the ecliptic plane and close to the axis of the dipole (e.g. Copi et al. 2004,2005, Schwarz et al.

2004, de Oliveira-Costa et al., Land & Magueijo 2005b).

Some authors have not found any strong evidence for the isotropy violation (e.g. Hajian et al. 2005)

**Bernard’s **

Considering the previous results, we propose an alternative method for probing the statistical isotropy of the CMB

it relies on the analysis of the alignment of structures on the CMB

preferred directions in the universe are defined as the directions towards which local features of the CMB are mostly oriented

the number of times a direction is “watched” represents a signal on the sphere, D(), allowing also for the analysis of its corresponding angular power spectrum

the analysis is feasible thanks to the steerable wavelet decomposition of the data, which also allows to probe different scales of the features

** Motivation**

** Our approach: the **
**CMB structures can **

**"watch"**

** Steerable wavelets**

** Application to the **
**WMAP data**

** What next?**

** Conclusions**

**Bernard’s **

**Great circle**
**Great circle**

**Seen twice**

** Motivation**

** Our approach: the **
**CMB structures can **

**"watch"**

** Steerable wavelets**

** Application to the **
**WMAP data**

** What next?**

** Conclusions**

**Our approach: the CMB structures can "watch"**

**An even signal is obtained**

**Bernard’s **

Modified from the Max Tegmark web site

The wavelet transform of a signal gives us information about:

the scale of the structures presented in the signal the position in which those structures are located

and (for the continuous case) it is obtained by convolving the signal with the wavelet.

Steerable wavelets are the natural extension of isotropic and directional wavelets, which have been successfully and extensively applied to many different topics within the CMB data analysis.

The steerable wavelets were introduced by Freeman & Adelson at the fall of the 80’s and the rise of the 90’s. They have been recently extended to the sphere by Wiaux et al. 2005.

They appear as a solution for the multi-directional image analysis:

any direction can be explored through a linear combination of a
*given basis. *

** Motivation**

** Our approach: the **
**CMB structures can **

**"watch"**

** Steerable wavelets**

** Application to the **
**WMAP data**

** What next?**

** Conclusions**

**Bernard’s **

The wavelet coefficient at a given *position (x, y), at a given *
*scale (R) and at a particular *orientation , can be expressed
as:

Their properties allows to explore possibilities that are (in practice) unfeasible by using standard techniques, like the one proposed in this work: the anisotropy analysis by studying the alignment of the CMB structures.

Typical examples of steerable wavelets are the Nth-directional derivatives of a Gaussian function

###

###

^{M}*m*

*m*

*m*

*x* *R*

*R* *x*

1

### , )

### ( ,

### ,

###

###

** Motivation**

** Our approach: the **
**CMB structures can **

**"watch"**

** Steerable wavelets**

** Application to the **
**WMAP data**

** What next?**

** Conclusions**

**Steerable wavelets**

**Bernard’s **

### ^{xx}

^{xx}

### _{xx}

_{xx}

### _{xx}

_{xx}

### _{xx}

_{xx}

### _{xx}

_{xx}

### _{xx}

_{xx}

### ^{yy}

^{yy}

###

### *xy*

cos2 sin^{2} sin2

** Motivation**

** Our approach: the **
**CMB structures can **

**"watch"**

** Steerable wavelets**

** Application to the **
**WMAP data**

** What next?**

** Conclusions**

**Bernard’s **

### ^{} ^{} ^{} ^{} ^{} ^{}

###

_{}

*x*,

*R*

_{xx}*x*,

*R*cos

^{2}

_{yy}*x*,

*R*sin

^{2}

_{xy}*x*,

*R*sin2

###

###

* 0

###

###

**Orientation of the maximum value of the wavelet coefficients**

**Orientation of the feature**

### 2 2 2

### tan

_{0}

###

_{*}

###

_{0}

###

###

###

###

###

*xy*
*xx*

*xy*

**Orientation given the maximum of the wavelet coefficient:**

**By computing three wavelet coefficient maps (for each **
**scale), the local orientation that better matches that of the **
**CMB features can be obtained**

** Motivation**

** Our approach: the **
**CMB structures can **

**"watch"**

** Steerable wavelets**

** Application to the **
**WMAP data**

** What next?**

** Conclusions**

**Steerable wavelets**

**Bernard’s **

**{ ** ^{ } **, , ...}**

^{ }

###

** Motivation**

** Our approach: the **
**CMB structures can **

**"watch"**

** Steerable wavelets**

** Application to the **
**WMAP data**

** What next?**

** Conclusions**

**Bernard’s **

**Wavelet coefficients ****matching the features ****orientations **

**Number of times that a ****position is “watched” **

**by the features**

###

###

###

^{(}

^{)}

1

### 1 , )

### (

###

###

^{N}

^{cros}*c*

*c*

*R*

*W* *R*

*D* *A*

** Motivation**

** Our approach: the **
**CMB structures can **

**"watch"**

** Steerable wavelets**

** Application to the **
**WMAP data**

** What next?**

** Conclusions**

**Steerable wavelets**

**Bernard’s **

** Motivation**

** Our approach: the **
**CMB structures can **

**"watch"**

** Steerable wavelets**

** Application to the **
**WMAP data**

** What next?**

** Conclusions**

**Bernard’s **

WMAP web site

**The analysed map is the one proposed by Komatsu et al. **

**2003 (NG paper of WMAP-1st yr). The map is degraded down **
**to N**_{side}**=32. The Kp0 mask is applied. Scales from 5 to 30 **
**degrees are explored.**

** Motivation**

** Our approach: the **
**CMB structures can **

**"watch"**

** Steerable wavelets**

** Application to the **
**WMAP data**

** What next?**

** Conclusions**

**Application to the WMAP data**

**Bernard’s **

** D**_{R}**() map for the co-added WMAP map @ 8.3º angular size given ****in terms of the 1 – p-value (estimated from 10000 simulations)**

** Motivation**

** Our approach: the **
**CMB structures can **

**"watch"**

** Steerable wavelets**

** Application to the **
**WMAP data**

** What next?**

** Conclusions**

**Bernard’s **

**There are 20 anomalous directions in the sky, that have been **

**“watched” more times than any of the analysed simulations. **

**The probability of having, at least, this number of so “watched” **

**positions is 0.01 %**

Best fit of a great circle passing by the anomalous directions

) 48 ,

264

(*l* ^{o}*b* ^{o}

Dipole direction

) 56 ,

331

(*l* ^{o}*b* ^{o}

Normal direction to the detected plane

) 29 ,

91

(*l* ^{o}*b* * ^{o}*
)

30 ,

96

(*l* ^{o}*b* ^{o}

NEP

**This results synthesises previous anomalies !!**

** Motivation**

** Our approach: the **
**CMB structures can **

**"watch"**

** Steerable wavelets**

** Application to the **
**WMAP data**

** What next?**

** Conclusions**

**Application to the WMAP data**

**Bernard’s **

**Q band** **V band**

**W band**

** Motivation**

** Our approach: the **
**CMB structures can **

**"watch"**

** Steerable wavelets**

** Application to the **
**WMAP data**

** What next?**

** Conclusions**

**Bernard’s **

** D**_{R}**() map for the co-added WMAP map @ 8.3º angular size given ****in terms of the # (estimated from 10000 simulations)**

** Motivation**

** Our approach: the **
**CMB structures can **

**"watch"**

** Steerable wavelets**

** Application to the **
**WMAP data**

** What next?**

** Conclusions**

**Application to the WMAP data**

**Bernard’s **

### Are those anomalous directions due to an specific

### “watching” of certain structures located in specific places?

### Or, on the contrary, they come from positions on the sky

### homogenously distributed?

** Motivation**

** Our approach: the **
**CMB structures can **

**"watch"**

** Steerable wavelets**

** Application to the **
**WMAP data**

** What next?**

** Conclusions**

### 1yr

### 3yr

**Bernard’s **

### Same structures are found, but signification decreases: from

### 0.01% to 0.42%

Current work is in progress to find the possible source of this anisotropy detection:

the coincidence of the preferred directions detected with the EP and dipole axes suggest possible unknown systematics

the angular size in which the preferred directions appear is compatible with topological defects (like textures) or secondary anisotropies due to the Rees-Sciama effect

although the detected anisotropy seems to be the same at all the frequencies, it must be also considered that foregrounds could generate aligned structures of several degrees

the analysis of the *angular power spectrum of D() could help *
to study the multipole distribution of the anomalies

** Motivation**

** Our approach: the **
**CMB structures can **

**"watch"**

** Steerable wavelets**

** Application to the **
**WMAP data**

** What next?**

** Conclusions**

**What next?**

**Bernard’s **

A new approach based on the alignment of the CMB structures has been proposed for studying the isotropic principle.

The method relies in the application of steerable wavelets, that allows to identified the local orientation of the features at each scale.

The application to the 1st yr-WMAP data shows that, at scales of 8.3º (multipole range between l=11 and l=27), 20 anomalous directions are detected.

Those directions identified, first, a plane which perpendicular direction is close to the dipole axis, and second, a most prominent position on that plane that is extremely close to the NEP.

This result has been confirmed with the second WMAP release.

Further analysis is needed to identify the possible source of this anisotropy.

Steerable wavelets open a door to fast oriented multi-scale analyses of the CMB

** Motivation**

** Our approach: the **
**CMB structures can **

**"watch"**

** Steerable wavelets**

** Application to the **
**WMAP data**

** What next?**

** Conclusions**

**Bernard’s **