# A Comprehensive Overview of the Cold Spot

Authors: Patricio Vielva
Journal: Advances in Astronomy
 (2010)

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## Abstract

The report of a significant deviation of the CMB temperature anisotropies distribution from Gaussianity (soon after the public release of the WMAP data in 2003) has become one of the most solid WMAPanomalies. This detection grounds on an excess of the kurtosis of the Spherical Mexican HatWavelet coefficients at scales of around
10 degrees. At these scales, a prominent feature—located in the southern Galactic hemisphere—was highlighted
from the rest of the SMHW coefficients: the Cold Spot. This paper presents a comprehensive overview related
to the study of the Cold Spot, paying attention to the non-Gaussianity detection methods, the morphological
characteristics of the Cold Spot, and the possible sources studied in the literature to explain its nature. Special
emphasis is made on the Cold Spot compatibility with a cosmic texture, commenting on future tests that would
help to give support or discard this hypothesis.


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## Body

## 1. Introduction

Besides the great success of the NASA WMAP satellite on providing a detailed knowledge of the cosmological parameters that define the physical properties of the Universe (e.g.,  [1, 2]), some unexpected results have attracted the attention of the cosmological community soon after the first release of the WMAP data: the so-called WMAP anomalies. Some of these anomalies are related to hemispherical asymmetries (e.g.,  [3–11]), an anomalous alignment of the quadrupole and octopole components (e.g.,  [12–21]), significantly low variance of the CMB temperature fluctuations (e.g.,  [22–25]), or anomalous alignment of the CMB features toward the Ecliptic poles (e.g.,  [26, 27]). Some of these aspects are addressed in this special issue. In addition to the previous findings, the prominent cold spot (hereinafter, the  Cold Spot) detected in the southern hemisphere by [28] became one of the most studied anomalies of the WMAP data. The  Cold Spot was identified after testing that the Spherical Mexican Hat Wavelet (SMHW) coefficients of the WMAP data presented an excess of kurtosis (at scales of around 10∘ in the sky), as compared to the distribution derived from isotropic and Gaussian CMB simulations. This paper presents a complete review on the detection and characterization of this nonstandard signature, and a description of the different attempts made so far in understanding what could be the cause behind such departure from the standard inflationary paradigm. It is organized as follows: in Section 2, the author justifies the use of wavelets as a natural tool for probing the Gaussianity of the CMB temperature fluctuations. the author also presents the different statistics applied in the wavelet space that had led to point out the WMAP data incompatibility with the standard model. In Section 3, the author briefly describes the morphological characteristics of the  Cold Spot. The important question of the actual significance of the detection of the  Cold Spot and the aspects associated with a posteriori interpretations are addressed in Section 4. Some of the different sources that have been considered in the literature to explain the  Cold Spot feature are discussed in Section 5. In Section 6, the author explains in detail a plausible hypothesis to accommodate the existence of the  Cold Spot together with the standard cosmological model: a cosmic texture. In addition, the author also describe possible followup tests that could help to confirm or discard such hypothesis. Finally, the author’s conclusions are given in Section 7.


## 2. The Non-Gaussianity Detection

The  Cold Spot was firstly identified through ablind Gaussianity test of the WMAP first-year data [28]. This test was designed to probe the isotropic and Gaussian nature of the CMB, as predicted by the standard inflationary model (see, e.g., [29]), and it was based in a multiresolution analysis performed with the Spherical Mexican Hat Wavelet (SMHW). In this section, the author summarizes the non-Gaussianity detection that led to the identification of the  Cold Spot. The author starts by justifying why an analysis based on wavelets was proposed and presenting the main characteristics of the wavelet used in the analysis: the SMHW. Afterwards, the author explains which statistics (all of them based on the SMHW coefficients) reported the original deviations from Gaussianity of the WMAP data.


### 2.1. Why a Wavelet?

Nowadays, the CMB scientific community is already very familiar with the application of wavelets (and other members of the-lets zoo, like curvelets, ridglets or needlets) to data analysis. However, it is worth recalling that this is a relatively new custom. Although wavelet applications in cosmology shyly started already in mid-80s, it was not until 1997 that the first application to CMB was presented [30], precisely in an exercise devoted to probe the Gaussianity of the CMB, and it was just a year afterwards that the first CMB data analyses with wavelets came to light [31, 32], in particular, with two applications to COBE [33] data. During the last decade, the application of wavelets to extract cosmological information from CMB data has growth considerably, in many different branches: compact source detection (e.g., [34–39]), Gaussianity (e.g., [28, 40–51]), cross-correlation with large scale structure (e.g., [52–55]), decomposition of the coupled E/B signals [56], probing isotropy (e.g., [9, 26, 27]), cosmic string detection [57], microwave sky recovery (e.g., [58–61]), CMB denoising (e.g., [34, 62]), CMB power-spectrum determination [63], and primordial power-spectrum recovery (e.g., [64]) are some of the application fields. The author refers the reader to [65, 66] for some reviews on the wavelets applications to CMB data, with a particular emphasis on data analysis on the sphere.The wavelet transform (e.g., [67, 68]) has become very popular for a major reason: they offer a unique opportunity to probe scale-dependent phenomena, but keeping, at the same time, information about spatial localization. This is a clear advantage—for many purposes—over classical Fourier or harmonic transforms: physical processes typically exhibit a clear scale-dependent behaviour, and, often, such behaviour differs enough from one phenomenon to another (e.g.,  in a microwave image, the localized emission due to cluster of galaxies has very different properties as compared to the large-scale signal produced by the Galactic components).The capability of emphasising or amplifying some features (at a particular scale) makes wavelets unique to probe the Gaussianity of the CMB: there are many different physical processes that might introduce non-Gaussian signatures into a CMB signal, at a very particular scale range (e.g.,  primordial non-Gaussianity due to nonstandard inflationary scenarios, cosmic defects like strings or textures, secondary anisotropies, foreground emissions, etc.).In this sense, when a Gaussianity analysis of the CMB is performed in wavelet space, we are in a very adequate framework to probe, almost separately, any potential non-Gaussian signatures present in the data. The ability of wavelets to amplify a given signature is explained because they can be seen as compensated filters (i.e.,∫ℜnΨ=0, where ℜn is the space where the wavelet Ψ is defined). This property is not satisfied by other standard filters, as the top-hat or the Gaussian functions. As mentioned above, wavelets do not only provide us with the capability of selecting a given scale range, but they also allow one to keep the information about spatial localization: we do not just study, for instance, the compatibility with Gaussianity, but we are also able to identify where in the data a deviation might be spatially localized. This intrinsic property of the wavelet transform (spatial localization) is also unique to explore not only Gaussianity, but also isotropy, since the statistical properties can be studied (almost independently) from one region in the data to another, in a self-consistent way. Hereinafter, the author will focus on the use of the two-dimensional (2D) continuous wavelet transform (CWT, see, e.g., Chapter 2 in [68]) and, in particular, on the 2D CWT defined on the sphere. There are different ways to define wavelets on the sphere (e.g., [69–72]); most of the works related to the  Cold Spot have been performed using the definition of the isotropic Spherical Mexican Hat Wavelet (SMHW) proposed by [45] which is a stereographic projection of the Mexican Hat Wavelet, as proposed by [69](1)Ψ(θ;R)=12πRNR[1+(y2)2]2[2-(yR)2]e-y2/2R2,
where y≡2tan(θ/2) is the stereographic projection variable, θ∈[0,π) is the colatitude, and the constant NR≡1+(R2/2)+(R4/4) is chosen such as the square of the wavelet function Ψ(θ;R) is normalized to unity. In Figure 1, the radial profile of the SMHW, for different wavelet scales R, is shown. For a signal T(θ,ϕ) defined on the sphere, its spherical harmonic coefficients tℓm are defined as(2)tℓm=∫dΩYℓm*(θ,ϕ)T(θ,ϕ),
where dΩ=dθsinθdϕ, the spherical coordinates are, as mentioned above, the colatitude θ (related to the latitude b as b=π/2-θ), and ϕ∈[0,2π) is the longitude. The function Yℓm(θ,ϕ) is the spherical harmonic of order ℓ and m, and * denotes complex conjugation. For an isotropic 2D CWT on the sphere, the wavelet coefficients w(θ,ϕ;R) are obtained as(3)w(θ,ϕ;R)=∑ℓ=0ℓmax∑m=-ℓℓtℓmΨℓ(R)Yℓm(θ,ϕ)T(θ,ϕ),
where Ψℓ(R) is the window function associated with the wavelet function (e.g.,  Ψ(θ;R) in (1)), and ℓmax represents the maximum multipole associated with a given resolution of the signal T(θ,ϕ), typically limited by the size of the pixel adopted to represent such signal on the sphere.Figure 1
Figure shows the profile on the SMHW as a function of the angular distanceθ for different wavelet scales R, ranging from 1∘ to 45∘. Notice that the amplitude of the SMHW at θ≡0 has been fixed to 1, to allow for an easier comparison.


### 2.2. The Statistics

The  Cold Spot was firstly detected via a positive deviation of the kurtosis of the SMHW coefficients at scales of aroundR≈300∘. The inspection of the map of the SMHW coefficients at these scales revealed the presence of a very large and cold spot in the southern hemisphere. The comparison of the amplitude and the area of this cold spot as compared with Gaussian simulations showed that it was particularly anomalous. Finally, a higher criticism test of the SMHW coefficients also indicated a deviation, at around the same wavelet scales, also showing that the major source for such deviation was located in the position of the  Cold Spot. In the following subsections, the author describes briefly these statistics: the kurtosis, the amplitude, the area and the higher criticism.


#### 2.2.1. The Kurtosis

The two most obvious indicators for a possible deviation from Gaussianity of a given data setz={z1,z2,…,zN} are the third (μ3) and fourth (μ4) central moments. The n-central moment of a random set of N numbers is defined as(4)μn=1N∑i=1N(zi-z¯)n,
where z¯=(1/N)∑i=1Nzi is the mean value of the data set. For a Gaussian random data set of zero mean (i.e., μ1=z¯≡0) and dispersion σ (i.e., μ2≡σ2), it is trivial to prove that higher order central moments are either zero (in the case of μ2n+1, for n≥1) or a given function of the dispersion (in the case of μ2n, for n≥1). Usually, it is much more convenient to work with normalized central moments νn, such as(5)νn=μnσn.
The normalized central moments are more convenient than central moments, since they are referred to the intrinsic fluctuations of the random data set (represented by the dispersion). This helps to absorb into νn possible uncertainties on the knowledge on the amplitude of the fluctuations of the random sample. The normalized central moment ν3 is normally referred to as the skewness (S) whereas K≡ν4-3 is called kurtosis. The reason for the subtraction of the number 3 in the previous expression comes from the fact that for a Gaussian random variable, μ4=3σ4 and, therefore, ν4=3. The previous definition assures that as it happens for the skewness (S), the kurtosis (K) of a Gaussian field is zero. Applying these concepts to the SMHW coefficients w(θ,ϕ;R), the skewness SR and the kurtosis KR of the wavelet coefficients, as a function of the wavelet scale R, can be defined as follows:(6)SR=1σR31Npix(R)∑i=1Npix(R)w(θi,ϕi;R)3,KR=1σR41Npix(R)∑i=1Npix(R)w(θi,ϕi;R)4-3,
where at each scale, it is assumed that the coefficients w(θ,ϕ;R) have zero mean. σR is the dispersion of the wavelet coefficients at the scale R(7)σR=[1Npix(R)∑i=1Npix(R)w(θi,ϕi;R)2]1/2.
In the previous expressions, Npix(R) represents the number of wavelet coefficients at a given scale R. Notice that very often, CMB data is not available in the full celestial sphere (e.g., because strong contamination from astrophysical foregrounds has to be masked). In most of the works in the CMB field, the HEALPix tessellation [73] is adopted. In this scheme, the resolution of a given image represented on the sphere is given by the  NSIDE parameter, which indicates how many divisions of the 12 basic pixels are required to achieve such resolution. The NSIDE parameter is related with the number of the pixels (Npix) required to fill the sphere at that resolution as: Npix=12NSIDE2. In Figure 2, the major result of the seminal work [28] on the  Cold Spot is shown. It represents the kurtosis of the wavelet coefficients, as a function of the scale, for the first release of the WMAP data (blue stars). The solid yellow line represents the mean value obtained from 10,000 CMB Gaussian random simulations, taking into account the instrumental properties of the analyzed data, and generated from an angular power spectrum derived from the best-fit cosmological model. The coloured regions (red, green, and magenta) represent the acceptance intervals at 32%, 5%, and 1%, respectively. Notice that at R scales of 250 and 300 arcmin, the kurtosis of the WMAP data was above the 1% acceptance interval. In detail, the excess of kurtosis is given by a p-value of ≈4 ×10-3 for the two scales. No significant deviations were found related to the skewness of the wavelet coefficients. The analysis was repeated in the two Galactic hemispheres separately. This was motivated by previous findings of asymmetries related to the genus [74] and the N-point correlation function [3]. It was found that whereas there was not deviation on the northern hemisphere, the excess of kurtosis was even more remarkable in the southern region. In particular, at the scale of 250 arcmin, the kurtosis of the SMHW coefficients was associated with a p-value of ≈2×10-3. Again, no deviation on the skewness was noticed. Hence, these analyses indicated that the source for the deviation on the kurtosis of the wavelet coefficient was related to feature/s with a typical length of around 10∘ in the sky and located in the southern hemisphere. Thanks to the frequency/spatial properties of the SMHW decomposition, it was possible to study in detail such features. In the left panel of Figure 3, the cleaned CMB map obtained from the fourth release of the WMAP data is given. In particular, the author plot the optimal combination for cosmological analysis (hereinafter, the QVW map), obtained as a noise-weighted linear combination of the Q-, V-, and W-bands, previously cleaned via a template fitting (e.g., [75]). Regions highly contaminated by Galactic foregrounds, nearby clusters, and extragalactic point sources have been masked. In the right panel, the SMHW coefficients of the previous map are represented, at the scale R=250 arcmin. It is evident the presence of the  Cold Spot in the southwest side of the image. In particular, the centre of the Cold Spot is estimated to be θ=147∘ and ϕ=209∘. The study of the  Cold Spot, through the application of follow-up tests, provided further evidences for its anomalous nature. The author reviews these tests in the following subsections.Figure 2
Figure showing the positive deviation of the kurtosis of the wavelet coefficients at scalesR of 250 and 300 arcmin found by [28]. Red, green and magenta regions represent the acceptance intervals at 32%, 5%, and 1%, respectively.(a) CMB cleaned map derived from the 4th release of the WMAP data, obtained via the template fitting technique described in [75]. (b) wavelet coefficients of the previous map, obtained after the SMHW convolution at a scale of R=250 arcmin. The location of the  Cold Spot is indicated in both panels by the circle. The centre of the  Cold Spot is θ=147∘ and ϕ=210∘.
(a)(b)


#### 2.2.2. The Amplitude

One of the most trivial statistics to study extreme values (as the  Cold Spot) in a random sample is the largest/smallest observation. In [28], it was established that the temperature of the  Cold Spot was −4.57 times the dispersion of the SMHW coefficients at R=250 arcmin. This cold value represented a p-value of.01 (relative to Monte Carlo simulations). A more robust statistic related to the extreme values is the MAX statistic, understood as the largest observation (in absolute value). For the particular case of the SMHW coefficients, MAX is defined, at scale R, as(8)MAXR=max{|w(θi,ϕi;R)|}.
The MAX statistic is more robust than selecting the coldest of the extrema, since the selection of the lowest values could be seen as an a posteriori selection. This statistic was studied in [76], showing that the  Cold Spot was always the maximum absolute observation of the WMAP data at scales around 300 arcmin, representing an upper tail probability of 0.38% (relative to Monte Carlo simulations). This value was less significant than the one mentioned in the previous subsection. The reason for this change is, as commented, that the MAX statistic is more robust than simply selecting the smallest values of the observations.


#### 2.2.3. The Area

The area above or below a given threshold is one of the most common statistics used to characterize the properties of a random field. In particular, the area is the most commonly Minkowski functional used in the literature (see, e.g., [77–79]). (For 2D images, there are three Minkowski functionals, namely, the contour or length, the area, and the genus. These three quantities are defined above/below a given threshold.) Generalizing this concept to the case of the wavelet coefficients, we can define cold (AR-ν) and hot (AR+ν) areas, at a given threshold ν and a given scale R, as(9)AR-ν=#{w(θi,ϕi;R)<-ν},AR+ν=#{w(θi,ϕi;R)≥+ν},
where the number operator #{conditioni} indicates how many times conditioni is satisfied, for i ranging from 1 to Npix(R). The cold and hot areas of the WMAP data were analyzed by [46]. It was reported that whereas the hot area was consistent with the expected behaviour for the standard Gaussian model (at all the scales R and thresholds ν), the cold area was not compatible. In particular, deviations from Gaussianity were found, again, at SMHW scales of R≈300∘. The deviation took place for thresholds equal or smaller than -3σR (see Figure 4). The analysis per different regions of the sky confirmed that the anomaly on the cold area was localized in the southern-west Galactic quadrant of the sky, and that the  Cold Spot was responsible for this anomaly. In particular, the cold area of the WMAP data (at the mentioned scales, and below a threshold of -3σR) was found anomalous with a probability of ≈99.7% whereas it became fully compatible with Gaussian simulations, once the  Cold Spot was not considered in the analysis. As for the case of the MAX statistic, a more conservative estimator (i.e., less dependent from the fact that the Cold Spot is negative) can be considered, just by selecting the maximum value of the previous cold and hot areas(10)ARν=max{AR-ν,AR+ν}.
This new statistic was used by [76], finding, again, that the WMAP data was anomalous about thresholds larger than |3σR|, for scales of the SMHW of around 300 arcmin.(a) cold area of the SMHW coefficients (atR=300 arcmin), as a function of the threshold (ν). (b) cold area (AR-ν) of the SMHW coefficients (at ν=3σR), as a function of the scale (R). As in Figure 2, the red, green, and magenta regions represent the 32%, 5%, and 1% acceptance intervals, respectively. These plots correspond to the analysis done by [46].
(a)(b)


#### 2.2.4. The Higher Criticism

Higher criticism (HC) is a relatively new statistic introduced in 2004 by [80], and firstly applied to the context of probing the Gaussianity of the CMB only a year after by [81]. Although there is not a unique definition for the HC, all the forms proposed in the literature satisfy the same key concept: HC is a measurement of the distance between a given sample of n elements to a Gaussian probability density distribution, established by means of the difference between the p-value pi of a given observation Xi—assuming it comes from a N(0,1)—, and its cardinal position on the sorted list (in increasing order) of p-values pi (i.e., pi-1<pi<pi+1,foralli=1,…,n). The HC associated with the sample is just defined as the largest value of such differences.This concept can be applied to the SMHW coefficients of a given signal (e.g.,  the QVW map) at a given scaleR. This was the analysis proposed by [81]. Let us adopt the following definition for the HC associated with Npix(R) wavelet coefficients w(θi,ϕi;R), at scale R:(11)HCNpix(R)=max{HCNpixi(R)},
where the maximization is made over the quantity HCNpixi(R), that provides the difference between the experimental probability of the wavelet coefficients w(θi,ϕi;R) at scale R and the corresponding theoreticalpvalue. Such quantity reads as(12)HCNpixi(R)=Npix|(i/Npix(R))-pi(R)|pi(R)(1-pi(R)),
where the p-value is given by pi(R)=P{|N(0,1)|>|ŵ(θi,ϕi;R)|}. The ̂ operator indicates that the Npix(R) SMHW coefficients at the scale R have been transformed into a zero mean and unit variance sample. The author remarks that the p-values have been sorted in increasing order.As explained in [81], the HC represents, under certain conditions, some advantages with respect to more traditional statistics designed to study the Gaussianity of a given sample. In particular, HC seems to be a better estimator than the MAX statistic: whereas the latter is designed to capture Gaussian deviations caused by very large values of the distribution, the HC is also sensitive to anomalies produced by moderate values. In addition, the HC can identify which values (in a given sample) are the ones that differ from the theoretical Gaussian distribution.In [81], it was reported that the HCNpix(R) was above the 1% acceptance interval, again, at the SMHW scale R=300 arcmin. They found that, actually, all the SMHW coefficients associated with HCNpixi(R) values above the 1% acceptance interval set by CMB Gaussian simulations, where localized in the position of the  Cold Spot. This extra test was an additional support to the anomalous nature of the WMAP data, and of the  Cold Spot in particular. Results were confirmed by [76] for the analysis of the second WMAP data release, reporting an upper tail probability even lower than for the 1-year data.As an illustration of the HC statistic, in Figure5, the author represents (in solid red) the HCNpixi values obtained from the QVW map in the real space, and (in dot-dash blue) the corresponding curve for R=300 arcmin. These quantities are normalized to their minimum values, for a better comparison. They are represented in an increasing order. It is remarkable that, for the case of the analyses performed on the SMHW coefficients, there is a tail of very large values of HCNpixi, that are not present for the real space case. This is due to the ability of the SMHW transform to enhance features of a given scale and shape. In Figure 6, these values are represented on the celestial sphere (left panel for the real space case, and right panel for the SMHW coefficients). The figure indicates that there are not particular signatures in the real space whereas the SMHW coefficients at R=300 arcmin allow us for a clear identification of the features causing the anomalous values of the HCNpix(R). In particular, the key role played by the  Cold Spot is highlighted.Figure 5
TheHCNpixi values obtained from the analysis of the QVW map. The solid red line corresponds to the application of (12) to the CMB map in the real space (i.e., R≡0) whereas the dot-dash blue line corresponds to the analysis of the SMHW coefficients at scale R=300 arcmin. HCNpixi curves are normalized by their respective minimum values.Maps ofHCNpixi obtained from the analysis of the QVW map derived from the WMAP data. Left panel corresponds to the study of the real space case, while right panel shows the outcome of the analysis of the SMHW coefficients at R=300 arcmin. Whereas for the former there are not particular signatures in the map, the wavelet analysis shows some prominent features, being the Cold Spot the most pronounced one.
(a)(b)


## 2.1. Why a Wavelet?

Nowadays, the CMB scientific community is already very familiar with the application of wavelets (and other members of the-lets zoo, like curvelets, ridglets or needlets) to data analysis. However, it is worth recalling that this is a relatively new custom. Although wavelet applications in cosmology shyly started already in mid-80s, it was not until 1997 that the first application to CMB was presented [30], precisely in an exercise devoted to probe the Gaussianity of the CMB, and it was just a year afterwards that the first CMB data analyses with wavelets came to light [31, 32], in particular, with two applications to COBE [33] data. During the last decade, the application of wavelets to extract cosmological information from CMB data has growth considerably, in many different branches: compact source detection (e.g., [34–39]), Gaussianity (e.g., [28, 40–51]), cross-correlation with large scale structure (e.g., [52–55]), decomposition of the coupled E/B signals [56], probing isotropy (e.g., [9, 26, 27]), cosmic string detection [57], microwave sky recovery (e.g., [58–61]), CMB denoising (e.g., [34, 62]), CMB power-spectrum determination [63], and primordial power-spectrum recovery (e.g., [64]) are some of the application fields. The author refers the reader to [65, 66] for some reviews on the wavelets applications to CMB data, with a particular emphasis on data analysis on the sphere.The wavelet transform (e.g., [67, 68]) has become very popular for a major reason: they offer a unique opportunity to probe scale-dependent phenomena, but keeping, at the same time, information about spatial localization. This is a clear advantage—for many purposes—over classical Fourier or harmonic transforms: physical processes typically exhibit a clear scale-dependent behaviour, and, often, such behaviour differs enough from one phenomenon to another (e.g.,  in a microwave image, the localized emission due to cluster of galaxies has very different properties as compared to the large-scale signal produced by the Galactic components).The capability of emphasising or amplifying some features (at a particular scale) makes wavelets unique to probe the Gaussianity of the CMB: there are many different physical processes that might introduce non-Gaussian signatures into a CMB signal, at a very particular scale range (e.g.,  primordial non-Gaussianity due to nonstandard inflationary scenarios, cosmic defects like strings or textures, secondary anisotropies, foreground emissions, etc.).In this sense, when a Gaussianity analysis of the CMB is performed in wavelet space, we are in a very adequate framework to probe, almost separately, any potential non-Gaussian signatures present in the data. The ability of wavelets to amplify a given signature is explained because they can be seen as compensated filters (i.e.,∫ℜnΨ=0, where ℜn is the space where the wavelet Ψ is defined). This property is not satisfied by other standard filters, as the top-hat or the Gaussian functions. As mentioned above, wavelets do not only provide us with the capability of selecting a given scale range, but they also allow one to keep the information about spatial localization: we do not just study, for instance, the compatibility with Gaussianity, but we are also able to identify where in the data a deviation might be spatially localized. This intrinsic property of the wavelet transform (spatial localization) is also unique to explore not only Gaussianity, but also isotropy, since the statistical properties can be studied (almost independently) from one region in the data to another, in a self-consistent way. Hereinafter, the author will focus on the use of the two-dimensional (2D) continuous wavelet transform (CWT, see, e.g., Chapter 2 in [68]) and, in particular, on the 2D CWT defined on the sphere. There are different ways to define wavelets on the sphere (e.g., [69–72]); most of the works related to the  Cold Spot have been performed using the definition of the isotropic Spherical Mexican Hat Wavelet (SMHW) proposed by [45] which is a stereographic projection of the Mexican Hat Wavelet, as proposed by [69](1)Ψ(θ;R)=12πRNR[1+(y2)2]2[2-(yR)2]e-y2/2R2,
where y≡2tan(θ/2) is the stereographic projection variable, θ∈[0,π) is the colatitude, and the constant NR≡1+(R2/2)+(R4/4) is chosen such as the square of the wavelet function Ψ(θ;R) is normalized to unity. In Figure 1, the radial profile of the SMHW, for different wavelet scales R, is shown. For a signal T(θ,ϕ) defined on the sphere, its spherical harmonic coefficients tℓm are defined as(2)tℓm=∫dΩYℓm*(θ,ϕ)T(θ,ϕ),
where dΩ=dθsinθdϕ, the spherical coordinates are, as mentioned above, the colatitude θ (related to the latitude b as b=π/2-θ), and ϕ∈[0,2π) is the longitude. The function Yℓm(θ,ϕ) is the spherical harmonic of order ℓ and m, and * denotes complex conjugation. For an isotropic 2D CWT on the sphere, the wavelet coefficients w(θ,ϕ;R) are obtained as(3)w(θ,ϕ;R)=∑ℓ=0ℓmax∑m=-ℓℓtℓmΨℓ(R)Yℓm(θ,ϕ)T(θ,ϕ),
where Ψℓ(R) is the window function associated with the wavelet function (e.g.,  Ψ(θ;R) in (1)), and ℓmax represents the maximum multipole associated with a given resolution of the signal T(θ,ϕ), typically limited by the size of the pixel adopted to represent such signal on the sphere.Figure 1
Figure shows the profile on the SMHW as a function of the angular distanceθ for different wavelet scales R, ranging from 1∘ to 45∘. Notice that the amplitude of the SMHW at θ≡0 has been fixed to 1, to allow for an easier comparison.


## 2.2. The Statistics

The  Cold Spot was firstly detected via a positive deviation of the kurtosis of the SMHW coefficients at scales of aroundR≈300∘. The inspection of the map of the SMHW coefficients at these scales revealed the presence of a very large and cold spot in the southern hemisphere. The comparison of the amplitude and the area of this cold spot as compared with Gaussian simulations showed that it was particularly anomalous. Finally, a higher criticism test of the SMHW coefficients also indicated a deviation, at around the same wavelet scales, also showing that the major source for such deviation was located in the position of the  Cold Spot. In the following subsections, the author describes briefly these statistics: the kurtosis, the amplitude, the area and the higher criticism.


### 2.2.1. The Kurtosis

The two most obvious indicators for a possible deviation from Gaussianity of a given data setz={z1,z2,…,zN} are the third (μ3) and fourth (μ4) central moments. The n-central moment of a random set of N numbers is defined as(4)μn=1N∑i=1N(zi-z¯)n,
where z¯=(1/N)∑i=1Nzi is the mean value of the data set. For a Gaussian random data set of zero mean (i.e., μ1=z¯≡0) and dispersion σ (i.e., μ2≡σ2), it is trivial to prove that higher order central moments are either zero (in the case of μ2n+1, for n≥1) or a given function of the dispersion (in the case of μ2n, for n≥1). Usually, it is much more convenient to work with normalized central moments νn, such as(5)νn=μnσn.
The normalized central moments are more convenient than central moments, since they are referred to the intrinsic fluctuations of the random data set (represented by the dispersion). This helps to absorb into νn possible uncertainties on the knowledge on the amplitude of the fluctuations of the random sample. The normalized central moment ν3 is normally referred to as the skewness (S) whereas K≡ν4-3 is called kurtosis. The reason for the subtraction of the number 3 in the previous expression comes from the fact that for a Gaussian random variable, μ4=3σ4 and, therefore, ν4=3. The previous definition assures that as it happens for the skewness (S), the kurtosis (K) of a Gaussian field is zero. Applying these concepts to the SMHW coefficients w(θ,ϕ;R), the skewness SR and the kurtosis KR of the wavelet coefficients, as a function of the wavelet scale R, can be defined as follows:(6)SR=1σR31Npix(R)∑i=1Npix(R)w(θi,ϕi;R)3,KR=1σR41Npix(R)∑i=1Npix(R)w(θi,ϕi;R)4-3,
where at each scale, it is assumed that the coefficients w(θ,ϕ;R) have zero mean. σR is the dispersion of the wavelet coefficients at the scale R(7)σR=[1Npix(R)∑i=1Npix(R)w(θi,ϕi;R)2]1/2.
In the previous expressions, Npix(R) represents the number of wavelet coefficients at a given scale R. Notice that very often, CMB data is not available in the full celestial sphere (e.g., because strong contamination from astrophysical foregrounds has to be masked). In most of the works in the CMB field, the HEALPix tessellation [73] is adopted. In this scheme, the resolution of a given image represented on the sphere is given by the  NSIDE parameter, which indicates how many divisions of the 12 basic pixels are required to achieve such resolution. The NSIDE parameter is related with the number of the pixels (Npix) required to fill the sphere at that resolution as: Npix=12NSIDE2. In Figure 2, the major result of the seminal work [28] on the  Cold Spot is shown. It represents the kurtosis of the wavelet coefficients, as a function of the scale, for the first release of the WMAP data (blue stars). The solid yellow line represents the mean value obtained from 10,000 CMB Gaussian random simulations, taking into account the instrumental properties of the analyzed data, and generated from an angular power spectrum derived from the best-fit cosmological model. The coloured regions (red, green, and magenta) represent the acceptance intervals at 32%, 5%, and 1%, respectively. Notice that at R scales of 250 and 300 arcmin, the kurtosis of the WMAP data was above the 1% acceptance interval. In detail, the excess of kurtosis is given by a p-value of ≈4 ×10-3 for the two scales. No significant deviations were found related to the skewness of the wavelet coefficients. The analysis was repeated in the two Galactic hemispheres separately. This was motivated by previous findings of asymmetries related to the genus [74] and the N-point correlation function [3]. It was found that whereas there was not deviation on the northern hemisphere, the excess of kurtosis was even more remarkable in the southern region. In particular, at the scale of 250 arcmin, the kurtosis of the SMHW coefficients was associated with a p-value of ≈2×10-3. Again, no deviation on the skewness was noticed. Hence, these analyses indicated that the source for the deviation on the kurtosis of the wavelet coefficient was related to feature/s with a typical length of around 10∘ in the sky and located in the southern hemisphere. Thanks to the frequency/spatial properties of the SMHW decomposition, it was possible to study in detail such features. In the left panel of Figure 3, the cleaned CMB map obtained from the fourth release of the WMAP data is given. In particular, the author plot the optimal combination for cosmological analysis (hereinafter, the QVW map), obtained as a noise-weighted linear combination of the Q-, V-, and W-bands, previously cleaned via a template fitting (e.g., [75]). Regions highly contaminated by Galactic foregrounds, nearby clusters, and extragalactic point sources have been masked. In the right panel, the SMHW coefficients of the previous map are represented, at the scale R=250 arcmin. It is evident the presence of the  Cold Spot in the southwest side of the image. In particular, the centre of the Cold Spot is estimated to be θ=147∘ and ϕ=209∘. The study of the  Cold Spot, through the application of follow-up tests, provided further evidences for its anomalous nature. The author reviews these tests in the following subsections.Figure 2
Figure showing the positive deviation of the kurtosis of the wavelet coefficients at scalesR of 250 and 300 arcmin found by [28]. Red, green and magenta regions represent the acceptance intervals at 32%, 5%, and 1%, respectively.(a) CMB cleaned map derived from the 4th release of the WMAP data, obtained via the template fitting technique described in [75]. (b) wavelet coefficients of the previous map, obtained after the SMHW convolution at a scale of R=250 arcmin. The location of the  Cold Spot is indicated in both panels by the circle. The centre of the  Cold Spot is θ=147∘ and ϕ=210∘.
(a)(b)


### 2.2.2. The Amplitude

One of the most trivial statistics to study extreme values (as the  Cold Spot) in a random sample is the largest/smallest observation. In [28], it was established that the temperature of the  Cold Spot was −4.57 times the dispersion of the SMHW coefficients at R=250 arcmin. This cold value represented a p-value of.01 (relative to Monte Carlo simulations). A more robust statistic related to the extreme values is the MAX statistic, understood as the largest observation (in absolute value). For the particular case of the SMHW coefficients, MAX is defined, at scale R, as(8)MAXR=max{|w(θi,ϕi;R)|}.
The MAX statistic is more robust than selecting the coldest of the extrema, since the selection of the lowest values could be seen as an a posteriori selection. This statistic was studied in [76], showing that the  Cold Spot was always the maximum absolute observation of the WMAP data at scales around 300 arcmin, representing an upper tail probability of 0.38% (relative to Monte Carlo simulations). This value was less significant than the one mentioned in the previous subsection. The reason for this change is, as commented, that the MAX statistic is more robust than simply selecting the smallest values of the observations.


### 2.2.3. The Area

The area above or below a given threshold is one of the most common statistics used to characterize the properties of a random field. In particular, the area is the most commonly Minkowski functional used in the literature (see, e.g., [77–79]). (For 2D images, there are three Minkowski functionals, namely, the contour or length, the area, and the genus. These three quantities are defined above/below a given threshold.) Generalizing this concept to the case of the wavelet coefficients, we can define cold (AR-ν) and hot (AR+ν) areas, at a given threshold ν and a given scale R, as(9)AR-ν=#{w(θi,ϕi;R)<-ν},AR+ν=#{w(θi,ϕi;R)≥+ν},
where the number operator #{conditioni} indicates how many times conditioni is satisfied, for i ranging from 1 to Npix(R). The cold and hot areas of the WMAP data were analyzed by [46]. It was reported that whereas the hot area was consistent with the expected behaviour for the standard Gaussian model (at all the scales R and thresholds ν), the cold area was not compatible. In particular, deviations from Gaussianity were found, again, at SMHW scales of R≈300∘. The deviation took place for thresholds equal or smaller than -3σR (see Figure 4). The analysis per different regions of the sky confirmed that the anomaly on the cold area was localized in the southern-west Galactic quadrant of the sky, and that the  Cold Spot was responsible for this anomaly. In particular, the cold area of the WMAP data (at the mentioned scales, and below a threshold of -3σR) was found anomalous with a probability of ≈99.7% whereas it became fully compatible with Gaussian simulations, once the  Cold Spot was not considered in the analysis. As for the case of the MAX statistic, a more conservative estimator (i.e., less dependent from the fact that the Cold Spot is negative) can be considered, just by selecting the maximum value of the previous cold and hot areas(10)ARν=max{AR-ν,AR+ν}.
This new statistic was used by [76], finding, again, that the WMAP data was anomalous about thresholds larger than |3σR|, for scales of the SMHW of around 300 arcmin.(a) cold area of the SMHW coefficients (atR=300 arcmin), as a function of the threshold (ν). (b) cold area (AR-ν) of the SMHW coefficients (at ν=3σR), as a function of the scale (R). As in Figure 2, the red, green, and magenta regions represent the 32%, 5%, and 1% acceptance intervals, respectively. These plots correspond to the analysis done by [46].
(a)(b)


### 2.2.4. The Higher Criticism

Higher criticism (HC) is a relatively new statistic introduced in 2004 by [80], and firstly applied to the context of probing the Gaussianity of the CMB only a year after by [81]. Although there is not a unique definition for the HC, all the forms proposed in the literature satisfy the same key concept: HC is a measurement of the distance between a given sample of n elements to a Gaussian probability density distribution, established by means of the difference between the p-value pi of a given observation Xi—assuming it comes from a N(0,1)—, and its cardinal position on the sorted list (in increasing order) of p-values pi (i.e., pi-1<pi<pi+1,foralli=1,…,n). The HC associated with the sample is just defined as the largest value of such differences.This concept can be applied to the SMHW coefficients of a given signal (e.g.,  the QVW map) at a given scaleR. This was the analysis proposed by [81]. Let us adopt the following definition for the HC associated with Npix(R) wavelet coefficients w(θi,ϕi;R), at scale R:(11)HCNpix(R)=max{HCNpixi(R)},
where the maximization is made over the quantity HCNpixi(R), that provides the difference between the experimental probability of the wavelet coefficients w(θi,ϕi;R) at scale R and the corresponding theoreticalpvalue. Such quantity reads as(12)HCNpixi(R)=Npix|(i/Npix(R))-pi(R)|pi(R)(1-pi(R)),
where the p-value is given by pi(R)=P{|N(0,1)|>|ŵ(θi,ϕi;R)|}. The ̂ operator indicates that the Npix(R) SMHW coefficients at the scale R have been transformed into a zero mean and unit variance sample. The author remarks that the p-values have been sorted in increasing order.As explained in [81], the HC represents, under certain conditions, some advantages with respect to more traditional statistics designed to study the Gaussianity of a given sample. In particular, HC seems to be a better estimator than the MAX statistic: whereas the latter is designed to capture Gaussian deviations caused by very large values of the distribution, the HC is also sensitive to anomalies produced by moderate values. In addition, the HC can identify which values (in a given sample) are the ones that differ from the theoretical Gaussian distribution.In [81], it was reported that the HCNpix(R) was above the 1% acceptance interval, again, at the SMHW scale R=300 arcmin. They found that, actually, all the SMHW coefficients associated with HCNpixi(R) values above the 1% acceptance interval set by CMB Gaussian simulations, where localized in the position of the  Cold Spot. This extra test was an additional support to the anomalous nature of the WMAP data, and of the  Cold Spot in particular. Results were confirmed by [76] for the analysis of the second WMAP data release, reporting an upper tail probability even lower than for the 1-year data.As an illustration of the HC statistic, in Figure5, the author represents (in solid red) the HCNpixi values obtained from the QVW map in the real space, and (in dot-dash blue) the corresponding curve for R=300 arcmin. These quantities are normalized to their minimum values, for a better comparison. They are represented in an increasing order. It is remarkable that, for the case of the analyses performed on the SMHW coefficients, there is a tail of very large values of HCNpixi, that are not present for the real space case. This is due to the ability of the SMHW transform to enhance features of a given scale and shape. In Figure 6, these values are represented on the celestial sphere (left panel for the real space case, and right panel for the SMHW coefficients). The figure indicates that there are not particular signatures in the real space whereas the SMHW coefficients at R=300 arcmin allow us for a clear identification of the features causing the anomalous values of the HCNpix(R). In particular, the key role played by the  Cold Spot is highlighted.Figure 5
TheHCNpixi values obtained from the analysis of the QVW map. The solid red line corresponds to the application of (12) to the CMB map in the real space (i.e., R≡0) whereas the dot-dash blue line corresponds to the analysis of the SMHW coefficients at scale R=300 arcmin. HCNpixi curves are normalized by their respective minimum values.Maps ofHCNpixi obtained from the analysis of the QVW map derived from the WMAP data. Left panel corresponds to the study of the real space case, while right panel shows the outcome of the analysis of the SMHW coefficients at R=300 arcmin. Whereas for the former there are not particular signatures in the map, the wavelet analysis shows some prominent features, being the Cold Spot the most pronounced one.
(a)(b)


## 2.2.1. The Kurtosis

The two most obvious indicators for a possible deviation from Gaussianity of a given data setz={z1,z2,…,zN} are the third (μ3) and fourth (μ4) central moments. The n-central moment of a random set of N numbers is defined as(4)μn=1N∑i=1N(zi-z¯)n,
where z¯=(1/N)∑i=1Nzi is the mean value of the data set. For a Gaussian random data set of zero mean (i.e., μ1=z¯≡0) and dispersion σ (i.e., μ2≡σ2), it is trivial to prove that higher order central moments are either zero (in the case of μ2n+1, for n≥1) or a given function of the dispersion (in the case of μ2n, for n≥1). Usually, it is much more convenient to work with normalized central moments νn, such as(5)νn=μnσn.
The normalized central moments are more convenient than central moments, since they are referred to the intrinsic fluctuations of the random data set (represented by the dispersion). This helps to absorb into νn possible uncertainties on the knowledge on the amplitude of the fluctuations of the random sample. The normalized central moment ν3 is normally referred to as the skewness (S) whereas K≡ν4-3 is called kurtosis. The reason for the subtraction of the number 3 in the previous expression comes from the fact that for a Gaussian random variable, μ4=3σ4 and, therefore, ν4=3. The previous definition assures that as it happens for the skewness (S), the kurtosis (K) of a Gaussian field is zero. Applying these concepts to the SMHW coefficients w(θ,ϕ;R), the skewness SR and the kurtosis KR of the wavelet coefficients, as a function of the wavelet scale R, can be defined as follows:(6)SR=1σR31Npix(R)∑i=1Npix(R)w(θi,ϕi;R)3,KR=1σR41Npix(R)∑i=1Npix(R)w(θi,ϕi;R)4-3,
where at each scale, it is assumed that the coefficients w(θ,ϕ;R) have zero mean. σR is the dispersion of the wavelet coefficients at the scale R(7)σR=[1Npix(R)∑i=1Npix(R)w(θi,ϕi;R)2]1/2.
In the previous expressions, Npix(R) represents the number of wavelet coefficients at a given scale R. Notice that very often, CMB data is not available in the full celestial sphere (e.g., because strong contamination from astrophysical foregrounds has to be masked). In most of the works in the CMB field, the HEALPix tessellation [73] is adopted. In this scheme, the resolution of a given image represented on the sphere is given by the  NSIDE parameter, which indicates how many divisions of the 12 basic pixels are required to achieve such resolution. The NSIDE parameter is related with the number of the pixels (Npix) required to fill the sphere at that resolution as: Npix=12NSIDE2. In Figure 2, the major result of the seminal work [28] on the  Cold Spot is shown. It represents the kurtosis of the wavelet coefficients, as a function of the scale, for the first release of the WMAP data (blue stars). The solid yellow line represents the mean value obtained from 10,000 CMB Gaussian random simulations, taking into account the instrumental properties of the analyzed data, and generated from an angular power spectrum derived from the best-fit cosmological model. The coloured regions (red, green, and magenta) represent the acceptance intervals at 32%, 5%, and 1%, respectively. Notice that at R scales of 250 and 300 arcmin, the kurtosis of the WMAP data was above the 1% acceptance interval. In detail, the excess of kurtosis is given by a p-value of ≈4 ×10-3 for the two scales. No significant deviations were found related to the skewness of the wavelet coefficients. The analysis was repeated in the two Galactic hemispheres separately. This was motivated by previous findings of asymmetries related to the genus [74] and the N-point correlation function [3]. It was found that whereas there was not deviation on the northern hemisphere, the excess of kurtosis was even more remarkable in the southern region. In particular, at the scale of 250 arcmin, the kurtosis of the SMHW coefficients was associated with a p-value of ≈2×10-3. Again, no deviation on the skewness was noticed. Hence, these analyses indicated that the source for the deviation on the kurtosis of the wavelet coefficient was related to feature/s with a typical length of around 10∘ in the sky and located in the southern hemisphere. Thanks to the frequency/spatial properties of the SMHW decomposition, it was possible to study in detail such features. In the left panel of Figure 3, the cleaned CMB map obtained from the fourth release of the WMAP data is given. In particular, the author plot the optimal combination for cosmological analysis (hereinafter, the QVW map), obtained as a noise-weighted linear combination of the Q-, V-, and W-bands, previously cleaned via a template fitting (e.g., [75]). Regions highly contaminated by Galactic foregrounds, nearby clusters, and extragalactic point sources have been masked. In the right panel, the SMHW coefficients of the previous map are represented, at the scale R=250 arcmin. It is evident the presence of the  Cold Spot in the southwest side of the image. In particular, the centre of the Cold Spot is estimated to be θ=147∘ and ϕ=209∘. The study of the  Cold Spot, through the application of follow-up tests, provided further evidences for its anomalous nature. The author reviews these tests in the following subsections.Figure 2
Figure showing the positive deviation of the kurtosis of the wavelet coefficients at scalesR of 250 and 300 arcmin found by [28]. Red, green and magenta regions represent the acceptance intervals at 32%, 5%, and 1%, respectively.(a) CMB cleaned map derived from the 4th release of the WMAP data, obtained via the template fitting technique described in [75]. (b) wavelet coefficients of the previous map, obtained after the SMHW convolution at a scale of R=250 arcmin. The location of the  Cold Spot is indicated in both panels by the circle. The centre of the  Cold Spot is θ=147∘ and ϕ=210∘.
(a)(b)


## 2.2.2. The Amplitude

One of the most trivial statistics to study extreme values (as the  Cold Spot) in a random sample is the largest/smallest observation. In [28], it was established that the temperature of the  Cold Spot was −4.57 times the dispersion of the SMHW coefficients at R=250 arcmin. This cold value represented a p-value of.01 (relative to Monte Carlo simulations). A more robust statistic related to the extreme values is the MAX statistic, understood as the largest observation (in absolute value). For the particular case of the SMHW coefficients, MAX is defined, at scale R, as(8)MAXR=max{|w(θi,ϕi;R)|}.
The MAX statistic is more robust than selecting the coldest of the extrema, since the selection of the lowest values could be seen as an a posteriori selection. This statistic was studied in [76], showing that the  Cold Spot was always the maximum absolute observation of the WMAP data at scales around 300 arcmin, representing an upper tail probability of 0.38% (relative to Monte Carlo simulations). This value was less significant than the one mentioned in the previous subsection. The reason for this change is, as commented, that the MAX statistic is more robust than simply selecting the smallest values of the observations.


## 2.2.3. The Area

The area above or below a given threshold is one of the most common statistics used to characterize the properties of a random field. In particular, the area is the most commonly Minkowski functional used in the literature (see, e.g., [77–79]). (For 2D images, there are three Minkowski functionals, namely, the contour or length, the area, and the genus. These three quantities are defined above/below a given threshold.) Generalizing this concept to the case of the wavelet coefficients, we can define cold (AR-ν) and hot (AR+ν) areas, at a given threshold ν and a given scale R, as(9)AR-ν=#{w(θi,ϕi;R)<-ν},AR+ν=#{w(θi,ϕi;R)≥+ν},
where the number operator #{conditioni} indicates how many times conditioni is satisfied, for i ranging from 1 to Npix(R). The cold and hot areas of the WMAP data were analyzed by [46]. It was reported that whereas the hot area was consistent with the expected behaviour for the standard Gaussian model (at all the scales R and thresholds ν), the cold area was not compatible. In particular, deviations from Gaussianity were found, again, at SMHW scales of R≈300∘. The deviation took place for thresholds equal or smaller than -3σR (see Figure 4). The analysis per different regions of the sky confirmed that the anomaly on the cold area was localized in the southern-west Galactic quadrant of the sky, and that the  Cold Spot was responsible for this anomaly. In particular, the cold area of the WMAP data (at the mentioned scales, and below a threshold of -3σR) was found anomalous with a probability of ≈99.7% whereas it became fully compatible with Gaussian simulations, once the  Cold Spot was not considered in the analysis. As for the case of the MAX statistic, a more conservative estimator (i.e., less dependent from the fact that the Cold Spot is negative) can be considered, just by selecting the maximum value of the previous cold and hot areas(10)ARν=max{AR-ν,AR+ν}.
This new statistic was used by [76], finding, again, that the WMAP data was anomalous about thresholds larger than |3σR|, for scales of the SMHW of around 300 arcmin.(a) cold area of the SMHW coefficients (atR=300 arcmin), as a function of the threshold (ν). (b) cold area (AR-ν) of the SMHW coefficients (at ν=3σR), as a function of the scale (R). As in Figure 2, the red, green, and magenta regions represent the 32%, 5%, and 1% acceptance intervals, respectively. These plots correspond to the analysis done by [46].
(a)(b)


## 2.2.4. The Higher Criticism

Higher criticism (HC) is a relatively new statistic introduced in 2004 by [80], and firstly applied to the context of probing the Gaussianity of the CMB only a year after by [81]. Although there is not a unique definition for the HC, all the forms proposed in the literature satisfy the same key concept: HC is a measurement of the distance between a given sample of n elements to a Gaussian probability density distribution, established by means of the difference between the p-value pi of a given observation Xi—assuming it comes from a N(0,1)—, and its cardinal position on the sorted list (in increasing order) of p-values pi (i.e., pi-1<pi<pi+1,foralli=1,…,n). The HC associated with the sample is just defined as the largest value of such differences.This concept can be applied to the SMHW coefficients of a given signal (e.g.,  the QVW map) at a given scaleR. This was the analysis proposed by [81]. Let us adopt the following definition for the HC associated with Npix(R) wavelet coefficients w(θi,ϕi;R), at scale R:(11)HCNpix(R)=max{HCNpixi(R)},
where the maximization is made over the quantity HCNpixi(R), that provides the difference between the experimental probability of the wavelet coefficients w(θi,ϕi;R) at scale R and the corresponding theoreticalpvalue. Such quantity reads as(12)HCNpixi(R)=Npix|(i/Npix(R))-pi(R)|pi(R)(1-pi(R)),
where the p-value is given by pi(R)=P{|N(0,1)|>|ŵ(θi,ϕi;R)|}. The ̂ operator indicates that the Npix(R) SMHW coefficients at the scale R have been transformed into a zero mean and unit variance sample. The author remarks that the p-values have been sorted in increasing order.As explained in [81], the HC represents, under certain conditions, some advantages with respect to more traditional statistics designed to study the Gaussianity of a given sample. In particular, HC seems to be a better estimator than the MAX statistic: whereas the latter is designed to capture Gaussian deviations caused by very large values of the distribution, the HC is also sensitive to anomalies produced by moderate values. In addition, the HC can identify which values (in a given sample) are the ones that differ from the theoretical Gaussian distribution.In [81], it was reported that the HCNpix(R) was above the 1% acceptance interval, again, at the SMHW scale R=300 arcmin. They found that, actually, all the SMHW coefficients associated with HCNpixi(R) values above the 1% acceptance interval set by CMB Gaussian simulations, where localized in the position of the  Cold Spot. This extra test was an additional support to the anomalous nature of the WMAP data, and of the  Cold Spot in particular. Results were confirmed by [76] for the analysis of the second WMAP data release, reporting an upper tail probability even lower than for the 1-year data.As an illustration of the HC statistic, in Figure5, the author represents (in solid red) the HCNpixi values obtained from the QVW map in the real space, and (in dot-dash blue) the corresponding curve for R=300 arcmin. These quantities are normalized to their minimum values, for a better comparison. They are represented in an increasing order. It is remarkable that, for the case of the analyses performed on the SMHW coefficients, there is a tail of very large values of HCNpixi, that are not present for the real space case. This is due to the ability of the SMHW transform to enhance features of a given scale and shape. In Figure 6, these values are represented on the celestial sphere (left panel for the real space case, and right panel for the SMHW coefficients). The figure indicates that there are not particular signatures in the real space whereas the SMHW coefficients at R=300 arcmin allow us for a clear identification of the features causing the anomalous values of the HCNpix(R). In particular, the key role played by the  Cold Spot is highlighted.Figure 5
TheHCNpixi values obtained from the analysis of the QVW map. The solid red line corresponds to the application of (12) to the CMB map in the real space (i.e., R≡0) whereas the dot-dash blue line corresponds to the analysis of the SMHW coefficients at scale R=300 arcmin. HCNpixi curves are normalized by their respective minimum values.Maps ofHCNpixi obtained from the analysis of the QVW map derived from the WMAP data. Left panel corresponds to the study of the real space case, while right panel shows the outcome of the analysis of the SMHW coefficients at R=300 arcmin. Whereas for the former there are not particular signatures in the map, the wavelet analysis shows some prominent features, being the Cold Spot the most pronounced one.
(a)(b)


## 3. The Characteristics of the Cold Spot

In this section, the author summarizes briefly some of the most important properties of the  Cold Spot. The author will focus in two major aspects: its morphology and its frequency dependence.The morphological properties of the  Cold Spot are different depending whether we do the analysis in the real or wavelet space. As it was pointed out in [46], the region associated with the  Cold Spot, in the real space, appears as formed by several small cold spots. The amplitude of the most prominent of these spots is ≲−350 μK with a size of ≈1∘. None of these structures is particularly anomalous. The image of the  Cold Spot in the real space is shown in the left panel of Figure 7. It is, however, in the wavelet space where the  Cold Spot appears more interesting. In the right panel of Figure 7, the author presents a close view of the  Cold Spot after convolution with the SMHW at a scale R=250 arcmin. Besides all the anomalous characteristics previously discussed (i.e., area, HC, and MAX), the  Cold Spot appears as a very symmetric feature. However, this effect could be biased since, after all, the SMHW is an isotropic filter and, therefore, the symmetric features of the  Cold Spot could be amplified, erasing any possible intrinsic anisotropy. This issue was studied in detail by [82]. Instead of applying an isotropic wavelet, the anisotropic Elliptical Spherical Mexican Hat Wavelet (ESMHW) was adopted. The cleaned CMB map derived from the WMAP was transformed into ESMHW coefficients (at the scales for which the WMAP data appeared as anomalous), for different ratios ζ between the smallest and the largest axes of the ESMHW, and for different orientations. This work proved that the maximum matching between the  Cold Spot and the ESMHW took place when ζ∈[0.875,1] and, hence, indicated that the  Cold Spot structure was quite close to be isotropic (assuming that the ratio of the ESMHW axes mimics, somehow, the ratio between the  Cold Spot axes).Image of the  Cold Spot (θ=-147∘,ϕ=209∘) in the real (a) and wavelet (b) spaces. The homogenously filled circles correspond to positions where known extragalactic point sources have been masked.
(a)(b)The frequency dependence of the  Cold Spot has been a matter of study soon after its discovery. Already in [28], the SMHW coefficients of the cleaned WMAP frequency channels (namely, Q, V, and W bands) were computed, and the mean value of the coefficients associated with the  Cold Spot at the scale R=250 arcmin was estimated. No obvious frequency dependence of this mean value of the wavelet coefficients was found, hence, being fully consistent with the expected behaviour for the CMB emission  and, therefore, quite different to the typical frequency dependence of the Galactic foregrounds. (The analyzed WMAP frequency channels were in thermodynamical temperature and, therefore, the CMB appears as a frequency independent emitter. Notice that, since the SMHW transform is a linear operation, the same behaviour is expected for the wavelet coefficients.)


## 4. The Significance of the Detection: The A Posteriori Issue

One of the most questioned aspects of theWMAP anomalies in general, and of the  Cold Spot in particular, is the issue of the actual significance of the detection. This is a very important point that is intimately linked to the blind nature of all the Gaussianity/isotropy tests that led to the report of such anomalies.The author reviews where this problem comes from: if many tests are performed in a given data set, it is not strange that some of them report some deviation from the null hypothesis. It is quite usual to face the following situation: a set ofblind tests (i.e., tests that just challenge the compatibility of the data with a given null hypothesis, H0, and not confronting such hypothesis with an alternative one, H1) claim a given incompatibility of the WMAP data. A subsequent test is performed, taking into account the previous finding and, usually, in such a way that the initial reported deviation is now found at higher significance. In this procedure, there are two weak points: the first one, already mentioned, is to assess the probability of finding a deviation as the one claimed during the first step, taking into account all the possible tests that were performed. The second one is the credibility of the probability for the follow-up test, where a particularity was studied in greater detail.As mentioned above, this is a common situation for theWMAP anomalies works and, therefore, the  Cold Spot is not an exception. Several tests were made in the first work by [28], namely, the estimation of the skewness and the kurtosis at several scales of the SMHW. A particular deviation was highlighted: the excess of kurtosis at several scales around R=250 arcmin. After that, the  Cold Spot was identified as a prominent feature, and further tests (the MAX, the cold area, the HC) were applied. The author believes that there is not a unique and clear way to solve these ambiguities and, to the author’s view, this point is not usually addressed in the literature. However, whereas for the latter aspect (i.e., the significance for the follow-up tests) the solution is hard, for the former there could be some possible getaway, at least depending on the complexity of the preliminary analysis. Actually, the  Cold Spot is one of the few WMAP anomalies where this particular aspect has been considered with deeper interest. In fact, it was the matter of several papers [47, 82] and, in particular, of [76].In this last work, the significance of the first detection was addressed, focusing in thea posteriori selection of the statistic (the kurtosis) and the scale range (≈250 arcmin). A conservative procedure to establish the p-value of the non-Gaussian detection, based on the characteristics of the analysis, was proposed. More specifically, since 30 statistics were applied to the QVW map (i.e., the skewness and the kurtosis of the SMHW coefficients at 15 scales), and only 3 out of these 30 statistics were found as anomalous (i.e., the kurtosis of the SMHW at scales R=200,250, and 300 arcmin were outside the 1% acceptance interval—see [76] for details), then, it was decided to estimate the significance of the non-Gaussianity detection by exploring in how many out of 10,000 CMB Gaussian simulations it was observed that the skewness or the kurtosis of the SMHW coefficients were outside the 1% acceptance interval, at least, at three scales. The p-value obtained in this manner was 0.0185. This p-value can serve, as explained in [76], as a conservative probability related to the non-Gaussianity associated with the SMHW analysis.In this spirit, the following up tests (e.g.,  the amplitude, the area, or the HC) can be just seen as additional probes to explore/understand the previous deviation, rather than as independent sources for establishing a proper significance level for the detection.Recently, [83] has questioned the non-Gaussianity found by [28], since the excess of the kurtosis was clearly found with the SMHW, but it was not the case with other analyzing kernels (proposed in [83]), as the top-hat and the Gaussian filters. The authors argued that these tools are more natural than a wavelet like the SMHW and that, therefore, the selection of the SMHW is somehow a posteriori. Contrary to this reasoning, the author found that the results obtained by [83] imply a different conclusion: the lack of detection when analyzing with the top-hat and the Gaussian filter is a proof of the issue discussed in Section 2.1, namely, that any filtering kernel is not necessary suitable for the detection of any non-Gaussian feature. It is clear that some features (like point sources, cosmic strings, or textures) are much better detected after applying optimal or targeted filters, rather than general ones (like the top-hat or the Gaussian functions).The reason why a compensate filter as the SMHW gets a much larger amplification as compared to uncompensated kernels as the previous ones is that it is much more efficient to remove the background fluctuations above and below a given scale interval. Even more, it can be shown (e.g.,  [84]) that the SMHW is close to the optimal or Matched filter to detect objects with a Gaussian-like profile embedded in CMB-dominated background (as it is the case in the region of the  Cold Spot). The form of a matched filter designed for a given situation is defined, not only by the shape of the feature to be detected, but also by the statistical properties of the background. In particular, the matched filter (defined in the Harmonic space) is proportional to the shape of the feature and inversely proportional to the angular power spectrum of the background. Therefore, bearing in mind that (at degree scales) the CMB is well described by an angular power spectrum close to Cℓ∝ℓ-2, it is trivial to show that the SMHW is near to be an optimal tool for detecting a feature described by a Gaussian-like profile and embedded in such background.The author remarks that as it was said in Section2.1, the selection of wavelets as a suitable tool for non-Gaussian analysis, as it was shown by several authors in the past (e.g.,  [40, 45, 85]), cannot be considered as an a posteriori choice, but, rather, as a natural option for studying scale-dependent phomena. In particular, the compensation property satisfied by wavelets makes them extremely good analyzing kernels to amplify certain features, precisely because it assures a strong suppression of the large-scale fluctuations of the background.Finally, it is worth recalling that the  Cold Spot has been identified as an anomalous feature by other tools different from the SMHW: by directional wavelets [47], scalar indices [86, 87], steerable wavelets [27], needlets [48], and the Kolmogorov stochastic parameter [88].


## 5. Some Possible Sources to Explain the  Cold Spot

To find an explanation for the non-Gaussianity deviation associated with the  Cold Spot is the next step, once its anomalous nature (i.e., noncompatible with the standard inflationary scenario) is accepted. With this aim, many efforts have been done in the last years, considering different sources for the observed anomaly. The possible causes addressed so far account forsystematics effects, mostly due to instrumental aspects that are not well understood/modelled, such as spurious emissions due to foregrounds or contaminants of the cosmological signal, nonaccounted secondary anisotropies induced on the CMB photons, as the interaction with the ionized medium (e.g.,  the Sunyaev-Zeldovich effect) or the nonlinear evolution of the gravitational potential (e.g.,  the Rees-Sciama effect), and, of course, alternative (or complementary) models to the standard inflationary scenario (as cosmic defects). In the following subsection, and also in Section 6, the author addresses these possibilities, starting from those hypotheses that are less dramatic from the point of view of strong implications for the standard cosmological model.


### 5.1. Systematics

To test the influence of unknown/unmodeled systematics on the non-Gaussianity deviation is, as one could imagine, a very hard task. The analyses that can be carried out to probe such sources are, basically of two types.One of these analyses areconsistency tests. As in most of the CMB experiments, WMAP satellite can provided us with cleaned CMB maps (e.g., following the template fitting approach described by [75]) for several detectors. Therefore, focusing in the non-Gaussian deviation associated with the  Cold Spot, an obvious procedure would be to check whether the application of the different statistical tools reveals that such a feature is associated with only one detector, or a smaller set of detectors. If this were the case that would be a clear indication for a lack of consistency and, therefore, that the non-Gaussianity detection is associated with a given instrumental feature. This was done by [28, 46] for the kurtosis (KR) and the area (ARν) of the SMHW wavelet coefficients. No inconsistency was found: the excess of kurtosis and of area was found to be the same for every difference assembly. As an illustration, in 8, the kurtosis of the SMHW coefficients (as a function of the scale KR) for 4 different CMB maps is presented. In particular, results for the Q-, V-, and W-band cleaned CMB maps are shown, together with the curve obtained from the analysis of the optimal QVW-map. These curves are quite similar, which indicates that the non-Gaussian signal is presented in all the WMAP detectors, at a similar level.The second type of analyses arenull tests. The cosmological frequencies of the WMAP satellite (i.e., Q-band at 41 GHz, V-band at 61 GHz, and W-band at 94 GHz) are made from more than one difference assembly. Hence, just subtracting difference assemblies at the corresponding band can produce noise maps per frequency. Neglecting small differences from the optical beams and the band-pass widths, the CMB and foreground emissions have been cancelled out in this new map. Therefore, the application of the statistical tools to these difference maps helps to check whether the non-Gaussian signal is a noisy artifact (if such signal is still present) or not (if consistency with Monte Carlo simulations is found). These was done by [28, 46], again, for the kurtosis and the area of the SMHW coefficients, respectively. As for the previous type of systematics probe, there was a clear indication that the non-Gaussian signal was not related to any instrumental signature. As an example, in Figure 9, the result obtained by [28] for KR is presented. On the left panel, the author represents the variation of the SMHW wavelet coefficients for the difference map constructed at 41 GHz as Q1-Q2. Similarly, the results for the V1-V2 map at 61 GHz are provided in the middle panel. Finally, in the right panel, the author gives the output for the difference map obtained at the 94 GHz band as the combination W1-W2+W3-W4. Results for the WMAP data is given as blue asterisks whereas, as for previous figures, the red, green, and magenta regions provide the 32%, 5%, and 1% acceptance intervals, respectively.Summarizing,consitency and null tests do not reveal the presence of systematics behind the non-Gaussianity associated with the  Cold Spot. Besides these test, it is important to remark that the angular size associated with this feature is ≈10∘ in the sky. It is not trivial to think in a systematic effect affecting at this scale, and providing such localized feature in the sky as the  Cold Spot.Finally, the subsequent releases of the WMAP data (where the modelling of the instrumental properties have been improving with time) have shown that there are not changes in the non-Gaussianity deviation, except for a slight increasing on its significance, which reflects the higher signal-to-noise ratio that WMAP data is getting as observational time increases.


### 5.2. Foregrounds

Astrophysical contaminants orforegrounds are the next possible origin for the non-Gaussianity associated with the  Cold Spot. It is well known that foregrounds are highly non-Gaussian signals. It is worth commenting that although the  Cold Spot is negative, it is still possible to think in an additive source (as foregrounds are) as a feasible explanation. To understand this point, it is important to recall that the QVW map (that, as the author said before, is commonly adopted in the literature for cosmological analyses) is obtained as a noise-weighted linear combination of cleaned CMB maps at different frequencies. These maps (at 41, 61, and 94 GHz) are produced, as mentioned previously, via a template fitting (see, e.g., [75]). Therefore, any oversubtraction of foregrounds templates could cause a foreground residual in the form of a cold emission. (In some works, a VW map is adopted, that is, a map built as a noise-weighted linear combination of the cleaned V- and W-bands.)Since, as mentioned before, the size of the  Cold Spot is of several degrees, it is really hard to believe that this feature could be associated with residuals from point sources (i.e., from radio and infrared galaxies). Therefore, only galactic emissions (as synchrotron, free-free, and thermal and spinning dust) could be responsible for a large feature as the  Cold Spot. However, notice that the  Cold Spot is located at57∘ from the Galactic plane, and at a longitude of 209∘. In other words, the  Cold Spot is placed in a region of low Galactic contamination. According to the previous reasoning, it is already hard to make compatible the presence of the  Cold Spot with a given Galactic emission. Even though, of course, the issue has been a matter of discussion. The author reviews here some of these analyses.The most obvious test is, of course, to check whether there is or not any frequency dependence of the statistical estimators that indicated the Gaussian deviation. For instance, in [28, 82], the kurtosis of the SMHW coefficients (KR) was studied for the different CMB-cleaned maps obtained at the Q-, V-, and W-bands.The results obtained for this study are given in Figure8. The kurtosis KR is presented for the 41, 61, and 94 GHz channels, and for the noise-weighted lineal combination (the QVW map). It is remarkable the high similarity of the curves. The pattern of the kurtosis, as a function of the SMHW scale, is the same for all the maps. The same is observed for its normalization. An equivalent test can be done for the area of the SMHW coefficients (ARν), as proposed by [46, 82]. Results, at R=300 arcmin, are presented in Figure 10. Notice that the agreement of the area of the SMHW coefficients (above threshold ν= 3, 3.5, 4σR) among different frequency bands (Q, V, and W) and the combined QVW map (represented by the solid lines in the figure) is very high. This kind of tests shows that there is not any evident frequency dependence of the statistics associated with the non-Gaussian deviation and, therefore, that such anomaly is fully consistent with the expected behaviour for a CMB feature.Figure 8
Frequency dependence of the excess of the kurtosis of the SMHW coefficients. The plot shows the variation of the kurtosis as a function of the scaleKR for cleaned CMB maps obtained from the WMAP data at three cosmological frequencies (Q-, V-, and W-bands), and for the optimal CMB cleaned map provided by WMAP, as noise-weighted combinations of the previous maps (the QVW map). The coloured regions represent, as for other figures, the 32%, 5% and 1% acceptance intervals provided by simulations.Null tests performed on difference maps, free of any foreground and cosmological signal. (a) represents the kurtosis of the SMHW coefficients, as a function of the scale R, for the Q1-Q2 map, that is, for a map built as the difference of the difference assemblies observations at 41 GHz. (b) is as the previous one, but for the V1-V2 difference map, that is, at 61 GHz. Finally, (c) provides the variation of the SMHW coefficients as a function of the wavelet scale, but for the difference map obtained with the 4 difference assemblies of WMAP at 90 GHz, that is, W1-W2+W3-W4. The blue asterisks represent values for the data whereas red, green and magenta regions represented the acceptance intervals at 32%, 5%, and 1%, respectively, determined by simulations.
(a)(b)(c)Figure 10
Frequency dependence of area of the SMHW coefficients. The plot shows the area of the wavelet coefficients atR=300 arcmin for different thresholds (from top to bottom: 3, 3.5, and 4σR), as a function of the frequency. The circles represent the values obtained for cleaned CMB maps obtained from the WMAP data at three cosmological frequencies (Q-, V-, and W-bands). The solid lines represent the areas, at different thresholds, for the noise-weighted QVW map.Additional tests supporting this idea have been proposed in the literature. First [28], the kurtosis of the SMHW coefficients was analyzed for a CMB-free map, constructed as the combination of the 4 difference assemblies at the W-band minus the sum of the 4 ones at Q- and V-band (i.e., W1+W2+W3+W4-Q1-Q2-V1-V2). This kind of map could have a contribution of the Galactic contaminants (outside a given observing mask), since foreground emissions are not frequency independent. This analysis did not show any significant deviation from the expected behaviour from Gaussian simulations, and, therefore, discarding the presence of significant foreground residuals.Second, as suggested by [82], different CMB recoveries from the WMAP data (where independent and alternative component separation approaches were followed) could be analyzed. In particular, the cleaned CMB maps obtained by [89] were probed. The kurtosis, the area, and the MAX of the SMHW coefficients do not change significantly from the different CMB maps.Reference [82] explored a more complicate scenario: a situation in which combinations of different foreground emissions could mimic, in the region of the  Cold Spot, the behaviour associated with the CMB, that is, a frequency independent global emission. To check this possibility, several templates were used as tracers of the Galactic foregrounds, namely, the Rodhes/HartRa0 2326 MHz [90] radio survey for synchrotron, the Hα by [91] for the free-free, and the thermal dust model by [92]. Authors studied the expected contribution of foregrounds in the region of the  Cold Spot, and they found that taking into account the uncertainties in the extrapolation of these templates from their original observations to the WMAP frequency range, it was possible to find a global Galactic emission that was nearly frequency-independent from 41 to 94 GHz (i.e., from Q- to W-bands). However, it was found that the emission was at a level of one order of magnitude below the Cold Spot temperature. It was checked that even accounting twice for that hypothetical foreground emission, it was not possible to reconcile observations with the Gaussian model.All these tests on the impact of the foregrounds indicated that the non-Gaussian signal associated with the Cold Spot was fully consistent with a CMB like frequency dependence, and that the role played by astrophysical contaminants was negligible.


### 5.3. The Sunyaev-Zeldovich Effect

After checking that the possible impact of systematics and foregrounds on the non-Gaussian detection is very low, the next step is to study whether secondary anisotropies of the CMB could explain the anomalous nature of the Cold Spot.The Sunyaev-Zeldovich effect (SZ) could be a potential candidate to explain the anomaly (it is produced by the inverse Compton interaction of the CMB photons, as they cross the hot electron gas that is found in clusters of galaxies). Two major reasons support this possibility: first, the size of the Cold Spot is nearly compatible with the fluctuations caused by the nearest clusters of galaxies, and second, this fluctuations  (in the frequency range covered by WMAP) produce cold spots [93].There is no evidence for the existence of any large cluster in the direction of the  Cold Spot. However, [94] reported the presence of a large concentration of galaxies in that direction—referred to as the Eridanus super-group—that could account for a total mass of ≈1014M⊙ (see [95] for details).This scenario implies, therefore, that the SZ could explain the nature of the  Cold Spot, at least, partially. In other words, it could be possible that a combination of a SZ contribution plus a large (but not anomalous) CMB fluctuation could account for the  Cold Spot emission. This was studied by [82], and the results are summarized in Figure 11. In this figure, the values of the SMHW coefficients w(θ,ϕ;R), at the position of the  Cold Spot and at R=300 arcmin, are given for the eight CMB cleaned maps at the Q-, V-, and W-bands. The corresponding error bars were computed from CMB plus noise simulations, corresponding to the instrumental properties of these detectors, and convolved with the appropriate SMHW kernel.Figure 11
Best-fit of the WMAP data—across the 8 difference assemblies for theQ-, V-, and W-bands—to a CMB (solid line) and a SZ spectra (dashed line). The fit is performed to the minimum amplitude of the Cold Spot in wavelet space, at a scale of R=300 arcmin. Error bars are obtained from Gaussian simulations.Three different fits to the data were explored. First, a pure CMB spectrum was used, giving a very good fit with a reducedχ2=1.00. Such a fit is represented by the solid line in Figure 11. Second, a pure SZ spectrum was tested, obtaining a very poor fit (dashed line) with a reduced χ2=9.12. Finally, a joint fit to a CMB plus SZ spectra was explored, obtaining an amplitude for the SZ spectrum consistent with zero  and a reduced χ2 quite similar to the first case. These results confirm that the frequency dependence of the Cold Spot is consistent with a CMB-like spectrum. They also rule out the possibility that the SZ is playing any significant role.


### 5.4. The Late Evolution of the Large-Scale Structure

Another secondary anisotropy that could explain the anomalous nature of the  Cold Spot is the one due to the nonlinear evolution of the gravitational field: the so-called Rees-Sciama effect (RS). In particular, it is known [96, 97] that voids in the large-scale structure could induce a negative nonlinear anisotropy in the CMB photons. The size of such secondary anisotropies depends on the proper size of the void and its redshift. Therefore, as for the SZ, the RS is another potential candidate to explain the anomalous nature of the Cold Spot.Extra support for this hypothesis came from two different paths. On the one hand, theoretical works as [98, 99] proposed that a very large void (of ≈300h-1 Mpc) and located at low redshift (z≪1) could produce large negative CMB fluctuations such as the  Cold Spot, even with modest density contrast values (i.e., in a quasilinear regime). On the other hand, it was suggested [100] that the NVSS catalogue [101] seems to show, at the position of the  Cold Spot, a lack in the number count of radio galaxies.Against these ideas, some criticisms can be made. First, this kind of voids are not observed and, even more, according to currentN-body simulations [102] they are extremely rare events (≈13σ, i.e., much more rare than the  Cold Spot deviation itself, that was a 98.15% event!) in the standard cosmological framework. Second, the claim made on the NVSS data has been recently questioned by [103], suggesting that such finding was an artefact caused by possible systematics related with the NVSS data processing, the statistical procedure and the a posteriori selection of the  Cold Spot position.In addition, observational campaigns on the region of the  Cold Spot were made by [104] with the MegaCam on the Canada-France-Hawaii Telescope and by [105] with the VIMOS spectrograph on the very large telescope. Both works have reported that the large-scale structure in that direction, and up to z≈1, is fully consistent with the standard model and that there is no evidence of such large voids as those required by the nonlinear evolution of the gravitational potential hypothesis.


## 5.1. Systematics

To test the influence of unknown/unmodeled systematics on the non-Gaussianity deviation is, as one could imagine, a very hard task. The analyses that can be carried out to probe such sources are, basically of two types.One of these analyses areconsistency tests. As in most of the CMB experiments, WMAP satellite can provided us with cleaned CMB maps (e.g., following the template fitting approach described by [75]) for several detectors. Therefore, focusing in the non-Gaussian deviation associated with the  Cold Spot, an obvious procedure would be to check whether the application of the different statistical tools reveals that such a feature is associated with only one detector, or a smaller set of detectors. If this were the case that would be a clear indication for a lack of consistency and, therefore, that the non-Gaussianity detection is associated with a given instrumental feature. This was done by [28, 46] for the kurtosis (KR) and the area (ARν) of the SMHW wavelet coefficients. No inconsistency was found: the excess of kurtosis and of area was found to be the same for every difference assembly. As an illustration, in 8, the kurtosis of the SMHW coefficients (as a function of the scale KR) for 4 different CMB maps is presented. In particular, results for the Q-, V-, and W-band cleaned CMB maps are shown, together with the curve obtained from the analysis of the optimal QVW-map. These curves are quite similar, which indicates that the non-Gaussian signal is presented in all the WMAP detectors, at a similar level.The second type of analyses arenull tests. The cosmological frequencies of the WMAP satellite (i.e., Q-band at 41 GHz, V-band at 61 GHz, and W-band at 94 GHz) are made from more than one difference assembly. Hence, just subtracting difference assemblies at the corresponding band can produce noise maps per frequency. Neglecting small differences from the optical beams and the band-pass widths, the CMB and foreground emissions have been cancelled out in this new map. Therefore, the application of the statistical tools to these difference maps helps to check whether the non-Gaussian signal is a noisy artifact (if such signal is still present) or not (if consistency with Monte Carlo simulations is found). These was done by [28, 46], again, for the kurtosis and the area of the SMHW coefficients, respectively. As for the previous type of systematics probe, there was a clear indication that the non-Gaussian signal was not related to any instrumental signature. As an example, in Figure 9, the result obtained by [28] for KR is presented. On the left panel, the author represents the variation of the SMHW wavelet coefficients for the difference map constructed at 41 GHz as Q1-Q2. Similarly, the results for the V1-V2 map at 61 GHz are provided in the middle panel. Finally, in the right panel, the author gives the output for the difference map obtained at the 94 GHz band as the combination W1-W2+W3-W4. Results for the WMAP data is given as blue asterisks whereas, as for previous figures, the red, green, and magenta regions provide the 32%, 5%, and 1% acceptance intervals, respectively.Summarizing,consitency and null tests do not reveal the presence of systematics behind the non-Gaussianity associated with the  Cold Spot. Besides these test, it is important to remark that the angular size associated with this feature is ≈10∘ in the sky. It is not trivial to think in a systematic effect affecting at this scale, and providing such localized feature in the sky as the  Cold Spot.Finally, the subsequent releases of the WMAP data (where the modelling of the instrumental properties have been improving with time) have shown that there are not changes in the non-Gaussianity deviation, except for a slight increasing on its significance, which reflects the higher signal-to-noise ratio that WMAP data is getting as observational time increases.


## 5.2. Foregrounds

Astrophysical contaminants orforegrounds are the next possible origin for the non-Gaussianity associated with the  Cold Spot. It is well known that foregrounds are highly non-Gaussian signals. It is worth commenting that although the  Cold Spot is negative, it is still possible to think in an additive source (as foregrounds are) as a feasible explanation. To understand this point, it is important to recall that the QVW map (that, as the author said before, is commonly adopted in the literature for cosmological analyses) is obtained as a noise-weighted linear combination of cleaned CMB maps at different frequencies. These maps (at 41, 61, and 94 GHz) are produced, as mentioned previously, via a template fitting (see, e.g., [75]). Therefore, any oversubtraction of foregrounds templates could cause a foreground residual in the form of a cold emission. (In some works, a VW map is adopted, that is, a map built as a noise-weighted linear combination of the cleaned V- and W-bands.)Since, as mentioned before, the size of the  Cold Spot is of several degrees, it is really hard to believe that this feature could be associated with residuals from point sources (i.e., from radio and infrared galaxies). Therefore, only galactic emissions (as synchrotron, free-free, and thermal and spinning dust) could be responsible for a large feature as the  Cold Spot. However, notice that the  Cold Spot is located at57∘ from the Galactic plane, and at a longitude of 209∘. In other words, the  Cold Spot is placed in a region of low Galactic contamination. According to the previous reasoning, it is already hard to make compatible the presence of the  Cold Spot with a given Galactic emission. Even though, of course, the issue has been a matter of discussion. The author reviews here some of these analyses.The most obvious test is, of course, to check whether there is or not any frequency dependence of the statistical estimators that indicated the Gaussian deviation. For instance, in [28, 82], the kurtosis of the SMHW coefficients (KR) was studied for the different CMB-cleaned maps obtained at the Q-, V-, and W-bands.The results obtained for this study are given in Figure8. The kurtosis KR is presented for the 41, 61, and 94 GHz channels, and for the noise-weighted lineal combination (the QVW map). It is remarkable the high similarity of the curves. The pattern of the kurtosis, as a function of the SMHW scale, is the same for all the maps. The same is observed for its normalization. An equivalent test can be done for the area of the SMHW coefficients (ARν), as proposed by [46, 82]. Results, at R=300 arcmin, are presented in Figure 10. Notice that the agreement of the area of the SMHW coefficients (above threshold ν= 3, 3.5, 4σR) among different frequency bands (Q, V, and W) and the combined QVW map (represented by the solid lines in the figure) is very high. This kind of tests shows that there is not any evident frequency dependence of the statistics associated with the non-Gaussian deviation and, therefore, that such anomaly is fully consistent with the expected behaviour for a CMB feature.Figure 8
Frequency dependence of the excess of the kurtosis of the SMHW coefficients. The plot shows the variation of the kurtosis as a function of the scaleKR for cleaned CMB maps obtained from the WMAP data at three cosmological frequencies (Q-, V-, and W-bands), and for the optimal CMB cleaned map provided by WMAP, as noise-weighted combinations of the previous maps (the QVW map). The coloured regions represent, as for other figures, the 32%, 5% and 1% acceptance intervals provided by simulations.Null tests performed on difference maps, free of any foreground and cosmological signal. (a) represents the kurtosis of the SMHW coefficients, as a function of the scale R, for the Q1-Q2 map, that is, for a map built as the difference of the difference assemblies observations at 41 GHz. (b) is as the previous one, but for the V1-V2 difference map, that is, at 61 GHz. Finally, (c) provides the variation of the SMHW coefficients as a function of the wavelet scale, but for the difference map obtained with the 4 difference assemblies of WMAP at 90 GHz, that is, W1-W2+W3-W4. The blue asterisks represent values for the data whereas red, green and magenta regions represented the acceptance intervals at 32%, 5%, and 1%, respectively, determined by simulations.
(a)(b)(c)Figure 10
Frequency dependence of area of the SMHW coefficients. The plot shows the area of the wavelet coefficients atR=300 arcmin for different thresholds (from top to bottom: 3, 3.5, and 4σR), as a function of the frequency. The circles represent the values obtained for cleaned CMB maps obtained from the WMAP data at three cosmological frequencies (Q-, V-, and W-bands). The solid lines represent the areas, at different thresholds, for the noise-weighted QVW map.Additional tests supporting this idea have been proposed in the literature. First [28], the kurtosis of the SMHW coefficients was analyzed for a CMB-free map, constructed as the combination of the 4 difference assemblies at the W-band minus the sum of the 4 ones at Q- and V-band (i.e., W1+W2+W3+W4-Q1-Q2-V1-V2). This kind of map could have a contribution of the Galactic contaminants (outside a given observing mask), since foreground emissions are not frequency independent. This analysis did not show any significant deviation from the expected behaviour from Gaussian simulations, and, therefore, discarding the presence of significant foreground residuals.Second, as suggested by [82], different CMB recoveries from the WMAP data (where independent and alternative component separation approaches were followed) could be analyzed. In particular, the cleaned CMB maps obtained by [89] were probed. The kurtosis, the area, and the MAX of the SMHW coefficients do not change significantly from the different CMB maps.Reference [82] explored a more complicate scenario: a situation in which combinations of different foreground emissions could mimic, in the region of the  Cold Spot, the behaviour associated with the CMB, that is, a frequency independent global emission. To check this possibility, several templates were used as tracers of the Galactic foregrounds, namely, the Rodhes/HartRa0 2326 MHz [90] radio survey for synchrotron, the Hα by [91] for the free-free, and the thermal dust model by [92]. Authors studied the expected contribution of foregrounds in the region of the  Cold Spot, and they found that taking into account the uncertainties in the extrapolation of these templates from their original observations to the WMAP frequency range, it was possible to find a global Galactic emission that was nearly frequency-independent from 41 to 94 GHz (i.e., from Q- to W-bands). However, it was found that the emission was at a level of one order of magnitude below the Cold Spot temperature. It was checked that even accounting twice for that hypothetical foreground emission, it was not possible to reconcile observations with the Gaussian model.All these tests on the impact of the foregrounds indicated that the non-Gaussian signal associated with the Cold Spot was fully consistent with a CMB like frequency dependence, and that the role played by astrophysical contaminants was negligible.


## 5.3. The Sunyaev-Zeldovich Effect

After checking that the possible impact of systematics and foregrounds on the non-Gaussian detection is very low, the next step is to study whether secondary anisotropies of the CMB could explain the anomalous nature of the Cold Spot.The Sunyaev-Zeldovich effect (SZ) could be a potential candidate to explain the anomaly (it is produced by the inverse Compton interaction of the CMB photons, as they cross the hot electron gas that is found in clusters of galaxies). Two major reasons support this possibility: first, the size of the Cold Spot is nearly compatible with the fluctuations caused by the nearest clusters of galaxies, and second, this fluctuations  (in the frequency range covered by WMAP) produce cold spots [93].There is no evidence for the existence of any large cluster in the direction of the  Cold Spot. However, [94] reported the presence of a large concentration of galaxies in that direction—referred to as the Eridanus super-group—that could account for a total mass of ≈1014M⊙ (see [95] for details).This scenario implies, therefore, that the SZ could explain the nature of the  Cold Spot, at least, partially. In other words, it could be possible that a combination of a SZ contribution plus a large (but not anomalous) CMB fluctuation could account for the  Cold Spot emission. This was studied by [82], and the results are summarized in Figure 11. In this figure, the values of the SMHW coefficients w(θ,ϕ;R), at the position of the  Cold Spot and at R=300 arcmin, are given for the eight CMB cleaned maps at the Q-, V-, and W-bands. The corresponding error bars were computed from CMB plus noise simulations, corresponding to the instrumental properties of these detectors, and convolved with the appropriate SMHW kernel.Figure 11
Best-fit of the WMAP data—across the 8 difference assemblies for theQ-, V-, and W-bands—to a CMB (solid line) and a SZ spectra (dashed line). The fit is performed to the minimum amplitude of the Cold Spot in wavelet space, at a scale of R=300 arcmin. Error bars are obtained from Gaussian simulations.Three different fits to the data were explored. First, a pure CMB spectrum was used, giving a very good fit with a reducedχ2=1.00. Such a fit is represented by the solid line in Figure 11. Second, a pure SZ spectrum was tested, obtaining a very poor fit (dashed line) with a reduced χ2=9.12. Finally, a joint fit to a CMB plus SZ spectra was explored, obtaining an amplitude for the SZ spectrum consistent with zero  and a reduced χ2 quite similar to the first case. These results confirm that the frequency dependence of the Cold Spot is consistent with a CMB-like spectrum. They also rule out the possibility that the SZ is playing any significant role.


## 5.4. The Late Evolution of the Large-Scale Structure

Another secondary anisotropy that could explain the anomalous nature of the  Cold Spot is the one due to the nonlinear evolution of the gravitational field: the so-called Rees-Sciama effect (RS). In particular, it is known [96, 97] that voids in the large-scale structure could induce a negative nonlinear anisotropy in the CMB photons. The size of such secondary anisotropies depends on the proper size of the void and its redshift. Therefore, as for the SZ, the RS is another potential candidate to explain the anomalous nature of the Cold Spot.Extra support for this hypothesis came from two different paths. On the one hand, theoretical works as [98, 99] proposed that a very large void (of ≈300h-1 Mpc) and located at low redshift (z≪1) could produce large negative CMB fluctuations such as the  Cold Spot, even with modest density contrast values (i.e., in a quasilinear regime). On the other hand, it was suggested [100] that the NVSS catalogue [101] seems to show, at the position of the  Cold Spot, a lack in the number count of radio galaxies.Against these ideas, some criticisms can be made. First, this kind of voids are not observed and, even more, according to currentN-body simulations [102] they are extremely rare events (≈13σ, i.e., much more rare than the  Cold Spot deviation itself, that was a 98.15% event!) in the standard cosmological framework. Second, the claim made on the NVSS data has been recently questioned by [103], suggesting that such finding was an artefact caused by possible systematics related with the NVSS data processing, the statistical procedure and the a posteriori selection of the  Cold Spot position.In addition, observational campaigns on the region of the  Cold Spot were made by [104] with the MegaCam on the Canada-France-Hawaii Telescope and by [105] with the VIMOS spectrograph on the very large telescope. Both works have reported that the large-scale structure in that direction, and up to z≈1, is fully consistent with the standard model and that there is no evidence of such large voids as those required by the nonlinear evolution of the gravitational potential hypothesis.


## 6. A Plausible Explanation: Cosmic Textures

In the previous section, the author has presented an overview of the several works carried out to establish whether the non-Gaussianity detection associated with the  Cold Spot could be explained in terms of systematics, foregrounds, and secondary anisotropies as the Sunyaev-Zeldovich and the Rees-Sciama effects. None of these possibilities seem to provide a satisfactory explanation, and, therefore, other sources should be investigated.In this section, the author pays attention to the suggestion made by [106]: the  Cold Spot could be caused by a cosmic texture. Cosmic textures [107] are a type of cosmic defects. They are supposed to be generated at some stage of the early Universe, associated with the symmetry-braking phase transitions that are predicted by certain theoretical models of high energy physics (see, e.g., [108] and references therein for a much more detailed explanation). In short, defects can be understood as space regions of a given phase state, surrounded by a space already in a new phase. In some cases, as for textures, these regions could collapse and, therefore, left an imprint on the CMB photons.Among the different types of cosmic defects, textures are the most plausible candidate to explain the anomalous nature of the  Cold Spot, since the interaction of the CMB photons with the time variation of the gravitational potential, associated with an eventual collapse of the texture, produces spots in the CMB fluctuations [109]. Even more, cosmic textures are expected to be a possible source of kurtosis deviation whereas the expected level of skewness is almost negligible (at least for values of the symmetry-breaking energy scale compatible with current observations). This is caused by the even probability of textures producing cold and hot spots and, therefore, providing a nearly symmetric distribution of temperature fluctuations. In fact, the equilateral fNL expected from textures goes as fNL≈1.5×10-100ψ06 [110], where ψ0 is the symmetry-breaking energy scale, measured in Gev. For typical limits in ψ0 imposed by the CMB angular power spectrum analysis (e.g., [111, 112]), the expected equilateral fNL is ≈10-9, that is, a tiny value well below the current constraints [113].The isotropic shape of the temperature fluctuations related to these spots can be approximated, at least at small angular distancesϑ, as [114](13)ΔTT(ϑ)=±ε11+4(ϑ/ϑc)2,
where ϑ represents the angular distance from the centre of the spot, and ϑc is a characteristic scale parameter of the spot—that is related to the redshift of the spot and the dynamics of the Universe [106]. The amplitude ε is proportional to the symmetry-breaking energy scale ψ0: ε=8π2Gψ02. It is worth remarking that, according to cosmic texture models, the amplitude ε is the same for every single spot generated by the collapsing defects.As mentioned above, the profile proposed in (13) is only valid up to small relative distances (e.g.,  ϑ≈ϑc) (the author knowledge, there is not any analytical or numerical solution for the full profile of a given cosmic texture yet).In order to have a profile valid at larger angular distances, [106] proposed to extend the profile given by (13) from its half-maximum, following a Gaussian function. The extension is done by imposing continuity, both, of the profile itself and of its first derivative.Taking into account these conditions, the assumed profile for the CMB temperature fluctuation caused by a collapsing cosmic texture would be given by(14)ΔTT=±{ε1+4(ϑ/ϑc)2,ifϑ≤ϑ*ε2e-(1/2ϑc2)(ϑ2+ϑ*2),ifϑ>ϑ*,
where ϑ*=3/2ϑc.In Figure12, the author shows the radial section of the above profile (solid line). The red and blue parts of the solid line correspond to the Gaussian function and to the profile of (13), respectively. The dashed lines represent the truncated regions for both curves. The cosmic texture parameters used in this profile are ϑ=5∘ (i.e., similar to the SMHW scale at which the  Cold Spot appears as anomalous) and ε=10-4. This value corresponds to a symmetry-breaking energy scale of ψ0=1.13×1016 GeV  that corresponds to a conservative upper limit imposed by CMB measurements (e.g., [111, 115]).Figure 12
Radial profile (solid blue/red lines) of a cosmic texture, as given in (14). The parameters defining this profile are ε=10-4 and ϑc=5∘. The dashed lines represent the truncated values of the profiles given by (13) (blue) and the Gaussian function (red). See text for details.The author remarks that the results obtained with a profile as the one given in (14), and that are reviewed in the next section, do not depend very much with the specific function adopted for the extrapolation. Similar results are obtained, for instance, when an exponential function or a SMHW-like kernel are used.


### 6.1. The Bayesian Framework

Attending to the issues discussed in the previous subsection, a cosmic texture could be a strong candidate to explain the  Cold Spot: textures produce spots on the CMB temperature fluctuations, they are non-Gaussian signals, and, depending on their amplitude (or the symmetry-breaking scale), they could be compatible with current constraints on the role played by cosmic defects on the structure formation and evolution of the Universe.Reference [106] proposed to make use of the texture profile of (14) to perform a hypothesis test to decide whether the WMAP data (in the position of the  Cold Spot) is more likely to be described by a large (but not anomalous) CMB spot (i.e., the nullH0 hypothesis) or by a cosmic texture of amplitude ε and size ϑ added to a random Gaussian and isotropic CMB field (i.e., the alternativeH1 hypothesis).The optimal way of performing such hypotheses test is within the Bayesian framework. Bayes' theorem states that, given a data setD and some unknown parameters Θ (defining a given model in the context of a given hypothesis Hi), the posterior probability of the parameters/model given the data P(Θ∣D,Hi), is related to the likelihood P(D∣Θ,Hi) (i.e., the probability of the data given the parameters/model) as(15)P(Θ∣D,Hi)=P(D∣Θ,Hi)P(Θ∣Hi)P(D∣Hi),
where P(Θ∣Hi) is a measurement of our a priori knowledge about the parameters/model (i.e., the prior), and P(D∣Hi) is a constant (i.e., it does not depend on the parameters/model) called Bayesian evidence (BE). The BE is nothing but the average likelihood with respect to the prior(16)P(D∣Hi)=∫P(D∣Θ,Hi)P(Θ∣Hi)dΘ,
and it is a largely used mechanism to perform hypotheses test. In particular, its role on different cosmology fields has been quite remarkable during the last years (e.g.,  [116] for dark energy studies, [117] for anisotropic models of the Universe expansion, [118] for studying different re-ionization models, [119] for point source detection, and [120] for exploring nonstandard inflationary models).The importance of BE for hypotheses test is clear. First, it is obvious that the quantity that we would like to obtain is a measurement of the probability of a given hypothesisH0, given the data, that is, P(H0∣D). This probability can be written, attending to the probability multiplication rule, as(17)P(H0∣D)=P(D∣H0)P(H0)P(D),
that is, it is proportional to the BE and to the probability of the hypothesis and inversely proportional to the probability of the data. Under certain circumstances, the probability of the hypothesis could be known, but, however, it is not the case for the probability of the data. In other words, we only can learn about the probability of the hypothesis H0 up to a factor. Therefore, what we can extract is a relative measurement of the probability of two hypotheses (H0 and H1), given the same data set D. This relative measurement is called the posterior probability ratio, ρ, and reads(18)ρ≡P(H1∣D)P(H0∣D)=E1E0P(H1)P(H0),
where, for simplicity, the author rewrite the BE, P(D∣Hi), as Ei. Hence, if ρ>1, we can conclude that the hypothesis H1 is favoured by the data with respect to H0. In some cases, there is not a clear choice for the probability of the hypotheses. In this case, empirical rules for the ratio of evidences—as the Jeffreys' scale [121]—are usually adopted.Therefore, the procedure required to explore whether the  Cold Spot is more likely to be explained in terms of a cosmic texture (H1) rather than by a Gaussian CMB fluctuation (H0) is clear: first, the likelihood is computed for both hypotheses; second, the BE is estimated, taking into account the adequate priors (16); finally, the posterior probability ratio (18) is evaluated, making use of suitable a priori probabilities for H0 and H1.This was the procedure followed in [106]. It is straightforward to show that since the noise term is caused by standard CMB Gaussian fluctuations and instrumental Gaussian noise, the likelihood function reads as(19)P(D∣Θ,Hi)∝exp(-χ22),
where χ2=(D-T(Θ))C-1(D-T(Θ))T. The correlation matrix C accounts for the full Gaussian CMB and noise correlations—that is, C=S+N, where sij∝∑CℓPℓ(cosθij) and nij=σi2δij, being Pℓ the Legendre polynomials, θij the angular distance between the pixels i and j, σi2 the instrumental noise contribution to pixel i, and δij represents the Kronecker delta. D represents the data (i.e., the QVW map), and the function T(Θ) represents the model behind the hypotheses—that is, (14) for H1, and ≡0 for H0. Finally, T denotes standard matrix transpose.The priors adopted by [106] for the parameters Θ≡(ε,ϑc) were chosen attending to observational constraints and cosmic texture simulations. In particular, the prior on the amplitude was |ε|≤10-4 whereas the prior of the size 1∘≤ϑc≤15∘ was assumed. The amplitude prior was uniformly distributed, and, as mentioned above, it is a conservative constraint imposed from the contribution of cosmic defects to the CMB angular power spectrum. The size ϑc follows a scale-invariant law, and the limits come from texture simulations. Textures below 1∘ should be smeared out by photon diffusion, and, in addition, they would be related to collapsing events above redshift ≈1000, which would not affect the CMB image. The upper limit is due to the unlikely probability of generating such large textures in the finite celestial sphere. Even so, [106] tested that results were not specially sensitivity to the prior selection, since the likelihood was clearly peaked, within a region of the parameter space clearly allowed by observations and texture models.The marginalization of the posterior probability in (15), led to the determination of the texture parameters, obtaining ε=7.3-3.6+2.5×10-5 and ϑc=4.9∘-2.4∘+2.8∘ at 95% confidence. The BE ratio was 150, which, in terms of the empirical rules [121], is a strong indication that the texture hypothesis for the  Cold Spot is favoured over the isotropic and Gaussian CMB fluctuation option. Adopting a ratio for the probability of the hypotheses given by the fraction of the sky that is covered by a cosmic texture as large as the one required for the  Cold Spot (≈0.017), the posterior probability ratio was ρ=2.5, which also favours the texture hypothesis.It is worth mentioning that the estimated value for the texture amplitude could be affected by selection bias. In [106], it is established that such bias could provide an overestimate of the texture amplitude by a factor of 2. This bias is caused because the texture amplitude is estimated in a low signal-to-noise regime, where the features placed in large background fluctuations are more easily detected. Even so, the estimated value for the texture amplitude, ε=7.3×10-5, would imply a symmetry-breaking energy scale of ψ0=8.7×1015 GeV, which, on the one hand, is fully compatible with more recent constraints imposed from the analysis of the CMB temperature and polarization power spectra (e.g., [112]), and, on the other hand, is in agreement with the predictions of most of the models for particle physics. In addition, by relating the angular size of the CMB texture profile to the cosmological parameters defining the geometry and evolution of the Universe, it was possible to establish that the texture collapse (that generated the CMB profile in (14)) occurred at redshift z≈6.Figure13 shows the effect of correcting the QVW map from the texture emission. In the left panel, the region of the sky where the  Cold Spot was identified is shown. The middle panel presents the best fit of the texture profile, according to the parameters previously mentioned. On the right panel, the author presents the resulting map after the subtraction of the estimated cosmic texture contribution. The  Cold Spot is noticeably reduced. More quantitative measurements were made by [106]. It was proved that if cosmic textures (adequate to the parameters fixed by the  Cold Spot analysis) were added to isotropic and Gaussian simulations of CMB signals, as seen in the QVW WMAP map, then the kurtosis of the SMHW became compatible. Even more, [122] showed that the WMAP map (corrected from the cosmic texture contribution in the location of the Cold Spot) was not compatible with anisotropic patterns for nonstandard expansions of the Universe (in particular, for Bianchi VIIh models), as it was previously the case for the uncorrected data (e.g., [117, 123]).(a) CMB temperature fluctuations obtained from WMAP data, centred in the position of the  Cold Spot. (b) best fit of a texture profile to the data, according to (14). (c) CMB map, at the position of the  Cold Spot, after the subtraction of the best-fit texture model.
(a)(b)(c)Finally, the author remarks that [95] proposed a similar approach as the one described in this section to study (attending to spatial templates) whether SZ and RS could provided more suitable hypotheses than the standard isotropic and Gaussian model. This Bayesian analysis indicated that neither of these hypotheses is favoured.


### 6.2. Followup Tests

The studies described in Section5 and in the previous subsection, clearly indicates that among all the realistic sources that could explain the anomalous nature of the  Cold Spot, only the cosmic texture hypothesis remains as a feasible option. The results obtained by [106], and reviewed in the previous subsection, have to be understood as a (clear) indication that the cosmic texture is plausible. However, before accepting it as the final explanation, it should be confirmed by additional tests. In particular, if the cosmic texture hypothesis would be the right one, then there are some clear predictions that could be tested (at least in the near feature, once ongoing/upcoming experiments as SPT, ACT, QUIJOTE, and ALMA are fully operative).In this subsection, the author comments on the three most obvious followup tests that could help to discard, or accept, the cosmic texture. These follow-up tests are: the searching of more textures, the local polarization of the CMB, and the local CMB lensing. These foreseen effects were firstly pointed out by [106].


#### 6.2.1. Looking for More Textures

If a texture were found in the location of the  Cold Spot, then, attending to cosmic texture models, there should be more cold and hot spots randomly distributed across the sky. In fact, the distribution of CMB spots caused by cosmic textures follows a scale-invariant law(20)Nsp(>ϑc)=4πνκ33ϑc2,
that is, the number of cold/hot spots with a scale equal or larger than ϑc, is inversely proportional to ϑc2 (see [124] for the grounds and [106] for a derivation). The ν and κ parameters are associated with the physics of the cosmic texture models, and their specific meaning is out of the scope of this paper. The author remarks here that according to state-of-the-art simulations, these parameters are well consistent with ν≈2 and κ≈0.1 values, when ϑc is expressed in radians. Before continuing the discussion, the author stresses that the number of expected cold/hot spots with a scale ϑc equal or larger than 5∘ (i.e., as the  Cold Spot scale) is ≈1. In other words, the fact that we only found a  Cold Spot of a scale similar or larger than the  Cold Spot size is fully consistent with the cosmic texture scenario. This fact, of course, is an extra support for the texture hypothesis causing the  Cold Spot emission.Let us come back to (20). A straightforward calculation tells as that if the cosmic texture hypothesis is correct, then the CMB temperature fluctuations should contain ≈28 spots with a scale ϑc≥1∘, and ≈7 spots with a scale ϑc≥2∘. Poisson errors can be safely assumed for these numbers. Therefore, the number of cold/hot spots expected in the WMAP data are 1 ≲Nsp(>2°) ≲ 13 at the 95% confidence level. Current work is in progress to check this prediction, by using a fast cluster nesting sampling algorithm—MULTINEST [125]—to explore the posterior probability ratio. There are well-founded hopes to find new textures. In particular, some non-Gaussian analyses as [27, 48, 88] reported some hot/cold spots (in addition to the  Cold Spot) as potentially anomalous.


#### 6.2.2. The Polarization of the CMB

The effect of a collapsing texture on the passing by CMB photons is nothing but a secondary anisotropy of the CMB fluctuations, whose origin is merely gravitational. Hence, the effect of such gravitational phenomenon on the E-mode polarization is almost negligible (only vector modes would be affected, which are well below the scalar mode contribution).Strictly speaking, this lack of polarization is not a unique signature produced by cosmic textures. As the author said, any secondary anisotropy of gravitational origin would cause it. However, these other effects (as huge voids) are quite implausible explanations (see Section5.4). For that reason, this effect is a valid follow-up test to probe the texture hypothesis.The procedure is simple: to compare the E-mode polarization in the position of a temperature spot (as large and extreme as the  Cold Spot is), under two different hypotheses, thenull or H0 one (i.e., the temperature spot is caused by a Gaussian fluctuation) and the alternative or H1 option (i.e., the temperature spot is a secondary anisotropy caused by the collapse of an evolving texture). In fact, as proposed by [126], the best discriminating measurement is the T-E correlation, rather than simply the E signal. This cross-correlation is expected to be close to zero for the H1 hypothesis. The approach suggested in this work was to estimate the correlation of the T and E profiles around the position of the spot temperature signal. This statistic was computed for many simulations according to the H0 and H1 hypotheses, and a hypothesis test was performed, via the definition of an optimal Fisher discriminant statistic (e.g., [127]).The method was applied to probe the capabilities of current and upcoming CMB experiments for discriminating between the two hypotheses. In particular, the cases of WMAP, Planck [128], and QUIJOTE [129] were considered.As the major conclusion, it can be established that the discrimination power of theT-E correlation is not very high. In fact, for an ideal noise-free experiment, at a power of the test of 0.5, the significance level is (up most) 0.8%. The reason for this limitation is that the characteristic size of the  Cold Spot is ≈10∘, which, roughly, corresponds to multipoles of ℓ≈40. It happens that at this multipole scales, the T-E angular cross-power spectrum is very close to zero, already for H0 and, therefore, it is hard to discriminate it from H1.The results are graphically summarized in Figure14. The author represent the significance level or p-value (for a power of the test of 0.5) as a function of the instrumental noise sensitivity for the E-mode polarization. The three experiments previously mentioned are indicated as vertical lines. Notice that the WMAP instrumental characteristics do not allow for any significant discrimination between hypotheses. With Planck, it would be possible to reach a modest 7% significance level. Finally, the upcoming QUIJOTE experiment would allow for a 1.4% detection. However, if the significance of the  Cold Spot obtained via the first non-Gaussianity wavelet analysis (i.e., 1.85%, as mentioned in Section 4) is also considered, [126] claims that a joint T and T-E significance detections of 0.025% and 0.12% could be imposed on the texture hypothesis, for the QUIJOTE and Planck experiments, respectively.Figure 14
The curve indicates the probability (thep-value) for rejecting the texture hypothesis, attending to the T-E correlation, as a function of the instrumental noise sensitivity in polarization. From left to right, the vertical lines indicate the noise level associated with QUIJOTE, Planck, and WMAP experiments.


#### 6.2.3. Gravitational Lensing

Besides the lack of polarization discussed in the previous subsection, the lensing of the CMB photons would be another foreseen effect caused by the gravitational field generated by the collapsing texture. This point was recently addressed by [130]. This work studied the capabilities of small-scale CMB experiments as ACT to detect a possible lensing effect occurring at the position of the  Cold Spot, caused by the gravitational field of a texture placed at z=6, and with a typical scale of 5∘, associated with a symmetry-breaking energy scale of ψ0=4.5×1015 GeV (i.e., comparable to the parameters determined by [106]).As for the polarization test, the power of the lensing analysis to probe the existence of the texture is relatively modest. For instance, detection is made at the 3σ level after 1000 minutes of integration time. In other words, this test would require of a dedicated observational campaign.


## 6.1. The Bayesian Framework

Attending to the issues discussed in the previous subsection, a cosmic texture could be a strong candidate to explain the  Cold Spot: textures produce spots on the CMB temperature fluctuations, they are non-Gaussian signals, and, depending on their amplitude (or the symmetry-breaking scale), they could be compatible with current constraints on the role played by cosmic defects on the structure formation and evolution of the Universe.Reference [106] proposed to make use of the texture profile of (14) to perform a hypothesis test to decide whether the WMAP data (in the position of the  Cold Spot) is more likely to be described by a large (but not anomalous) CMB spot (i.e., the nullH0 hypothesis) or by a cosmic texture of amplitude ε and size ϑ added to a random Gaussian and isotropic CMB field (i.e., the alternativeH1 hypothesis).The optimal way of performing such hypotheses test is within the Bayesian framework. Bayes' theorem states that, given a data setD and some unknown parameters Θ (defining a given model in the context of a given hypothesis Hi), the posterior probability of the parameters/model given the data P(Θ∣D,Hi), is related to the likelihood P(D∣Θ,Hi) (i.e., the probability of the data given the parameters/model) as(15)P(Θ∣D,Hi)=P(D∣Θ,Hi)P(Θ∣Hi)P(D∣Hi),
where P(Θ∣Hi) is a measurement of our a priori knowledge about the parameters/model (i.e., the prior), and P(D∣Hi) is a constant (i.e., it does not depend on the parameters/model) called Bayesian evidence (BE). The BE is nothing but the average likelihood with respect to the prior(16)P(D∣Hi)=∫P(D∣Θ,Hi)P(Θ∣Hi)dΘ,
and it is a largely used mechanism to perform hypotheses test. In particular, its role on different cosmology fields has been quite remarkable during the last years (e.g.,  [116] for dark energy studies, [117] for anisotropic models of the Universe expansion, [118] for studying different re-ionization models, [119] for point source detection, and [120] for exploring nonstandard inflationary models).The importance of BE for hypotheses test is clear. First, it is obvious that the quantity that we would like to obtain is a measurement of the probability of a given hypothesisH0, given the data, that is, P(H0∣D). This probability can be written, attending to the probability multiplication rule, as(17)P(H0∣D)=P(D∣H0)P(H0)P(D),
that is, it is proportional to the BE and to the probability of the hypothesis and inversely proportional to the probability of the data. Under certain circumstances, the probability of the hypothesis could be known, but, however, it is not the case for the probability of the data. In other words, we only can learn about the probability of the hypothesis H0 up to a factor. Therefore, what we can extract is a relative measurement of the probability of two hypotheses (H0 and H1), given the same data set D. This relative measurement is called the posterior probability ratio, ρ, and reads(18)ρ≡P(H1∣D)P(H0∣D)=E1E0P(H1)P(H0),
where, for simplicity, the author rewrite the BE, P(D∣Hi), as Ei. Hence, if ρ>1, we can conclude that the hypothesis H1 is favoured by the data with respect to H0. In some cases, there is not a clear choice for the probability of the hypotheses. In this case, empirical rules for the ratio of evidences—as the Jeffreys' scale [121]—are usually adopted.Therefore, the procedure required to explore whether the  Cold Spot is more likely to be explained in terms of a cosmic texture (H1) rather than by a Gaussian CMB fluctuation (H0) is clear: first, the likelihood is computed for both hypotheses; second, the BE is estimated, taking into account the adequate priors (16); finally, the posterior probability ratio (18) is evaluated, making use of suitable a priori probabilities for H0 and H1.This was the procedure followed in [106]. It is straightforward to show that since the noise term is caused by standard CMB Gaussian fluctuations and instrumental Gaussian noise, the likelihood function reads as(19)P(D∣Θ,Hi)∝exp(-χ22),
where χ2=(D-T(Θ))C-1(D-T(Θ))T. The correlation matrix C accounts for the full Gaussian CMB and noise correlations—that is, C=S+N, where sij∝∑CℓPℓ(cosθij) and nij=σi2δij, being Pℓ the Legendre polynomials, θij the angular distance between the pixels i and j, σi2 the instrumental noise contribution to pixel i, and δij represents the Kronecker delta. D represents the data (i.e., the QVW map), and the function T(Θ) represents the model behind the hypotheses—that is, (14) for H1, and ≡0 for H0. Finally, T denotes standard matrix transpose.The priors adopted by [106] for the parameters Θ≡(ε,ϑc) were chosen attending to observational constraints and cosmic texture simulations. In particular, the prior on the amplitude was |ε|≤10-4 whereas the prior of the size 1∘≤ϑc≤15∘ was assumed. The amplitude prior was uniformly distributed, and, as mentioned above, it is a conservative constraint imposed from the contribution of cosmic defects to the CMB angular power spectrum. The size ϑc follows a scale-invariant law, and the limits come from texture simulations. Textures below 1∘ should be smeared out by photon diffusion, and, in addition, they would be related to collapsing events above redshift ≈1000, which would not affect the CMB image. The upper limit is due to the unlikely probability of generating such large textures in the finite celestial sphere. Even so, [106] tested that results were not specially sensitivity to the prior selection, since the likelihood was clearly peaked, within a region of the parameter space clearly allowed by observations and texture models.The marginalization of the posterior probability in (15), led to the determination of the texture parameters, obtaining ε=7.3-3.6+2.5×10-5 and ϑc=4.9∘-2.4∘+2.8∘ at 95% confidence. The BE ratio was 150, which, in terms of the empirical rules [121], is a strong indication that the texture hypothesis for the  Cold Spot is favoured over the isotropic and Gaussian CMB fluctuation option. Adopting a ratio for the probability of the hypotheses given by the fraction of the sky that is covered by a cosmic texture as large as the one required for the  Cold Spot (≈0.017), the posterior probability ratio was ρ=2.5, which also favours the texture hypothesis.It is worth mentioning that the estimated value for the texture amplitude could be affected by selection bias. In [106], it is established that such bias could provide an overestimate of the texture amplitude by a factor of 2. This bias is caused because the texture amplitude is estimated in a low signal-to-noise regime, where the features placed in large background fluctuations are more easily detected. Even so, the estimated value for the texture amplitude, ε=7.3×10-5, would imply a symmetry-breaking energy scale of ψ0=8.7×1015 GeV, which, on the one hand, is fully compatible with more recent constraints imposed from the analysis of the CMB temperature and polarization power spectra (e.g., [112]), and, on the other hand, is in agreement with the predictions of most of the models for particle physics. In addition, by relating the angular size of the CMB texture profile to the cosmological parameters defining the geometry and evolution of the Universe, it was possible to establish that the texture collapse (that generated the CMB profile in (14)) occurred at redshift z≈6.Figure13 shows the effect of correcting the QVW map from the texture emission. In the left panel, the region of the sky where the  Cold Spot was identified is shown. The middle panel presents the best fit of the texture profile, according to the parameters previously mentioned. On the right panel, the author presents the resulting map after the subtraction of the estimated cosmic texture contribution. The  Cold Spot is noticeably reduced. More quantitative measurements were made by [106]. It was proved that if cosmic textures (adequate to the parameters fixed by the  Cold Spot analysis) were added to isotropic and Gaussian simulations of CMB signals, as seen in the QVW WMAP map, then the kurtosis of the SMHW became compatible. Even more, [122] showed that the WMAP map (corrected from the cosmic texture contribution in the location of the Cold Spot) was not compatible with anisotropic patterns for nonstandard expansions of the Universe (in particular, for Bianchi VIIh models), as it was previously the case for the uncorrected data (e.g., [117, 123]).(a) CMB temperature fluctuations obtained from WMAP data, centred in the position of the  Cold Spot. (b) best fit of a texture profile to the data, according to (14). (c) CMB map, at the position of the  Cold Spot, after the subtraction of the best-fit texture model.
(a)(b)(c)Finally, the author remarks that [95] proposed a similar approach as the one described in this section to study (attending to spatial templates) whether SZ and RS could provided more suitable hypotheses than the standard isotropic and Gaussian model. This Bayesian analysis indicated that neither of these hypotheses is favoured.


## 6.2. Followup Tests

The studies described in Section5 and in the previous subsection, clearly indicates that among all the realistic sources that could explain the anomalous nature of the  Cold Spot, only the cosmic texture hypothesis remains as a feasible option. The results obtained by [106], and reviewed in the previous subsection, have to be understood as a (clear) indication that the cosmic texture is plausible. However, before accepting it as the final explanation, it should be confirmed by additional tests. In particular, if the cosmic texture hypothesis would be the right one, then there are some clear predictions that could be tested (at least in the near feature, once ongoing/upcoming experiments as SPT, ACT, QUIJOTE, and ALMA are fully operative).In this subsection, the author comments on the three most obvious followup tests that could help to discard, or accept, the cosmic texture. These follow-up tests are: the searching of more textures, the local polarization of the CMB, and the local CMB lensing. These foreseen effects were firstly pointed out by [106].


### 6.2.1. Looking for More Textures

If a texture were found in the location of the  Cold Spot, then, attending to cosmic texture models, there should be more cold and hot spots randomly distributed across the sky. In fact, the distribution of CMB spots caused by cosmic textures follows a scale-invariant law(20)Nsp(>ϑc)=4πνκ33ϑc2,
that is, the number of cold/hot spots with a scale equal or larger than ϑc, is inversely proportional to ϑc2 (see [124] for the grounds and [106] for a derivation). The ν and κ parameters are associated with the physics of the cosmic texture models, and their specific meaning is out of the scope of this paper. The author remarks here that according to state-of-the-art simulations, these parameters are well consistent with ν≈2 and κ≈0.1 values, when ϑc is expressed in radians. Before continuing the discussion, the author stresses that the number of expected cold/hot spots with a scale ϑc equal or larger than 5∘ (i.e., as the  Cold Spot scale) is ≈1. In other words, the fact that we only found a  Cold Spot of a scale similar or larger than the  Cold Spot size is fully consistent with the cosmic texture scenario. This fact, of course, is an extra support for the texture hypothesis causing the  Cold Spot emission.Let us come back to (20). A straightforward calculation tells as that if the cosmic texture hypothesis is correct, then the CMB temperature fluctuations should contain ≈28 spots with a scale ϑc≥1∘, and ≈7 spots with a scale ϑc≥2∘. Poisson errors can be safely assumed for these numbers. Therefore, the number of cold/hot spots expected in the WMAP data are 1 ≲Nsp(>2°) ≲ 13 at the 95% confidence level. Current work is in progress to check this prediction, by using a fast cluster nesting sampling algorithm—MULTINEST [125]—to explore the posterior probability ratio. There are well-founded hopes to find new textures. In particular, some non-Gaussian analyses as [27, 48, 88] reported some hot/cold spots (in addition to the  Cold Spot) as potentially anomalous.


### 6.2.2. The Polarization of the CMB

The effect of a collapsing texture on the passing by CMB photons is nothing but a secondary anisotropy of the CMB fluctuations, whose origin is merely gravitational. Hence, the effect of such gravitational phenomenon on the E-mode polarization is almost negligible (only vector modes would be affected, which are well below the scalar mode contribution).Strictly speaking, this lack of polarization is not a unique signature produced by cosmic textures. As the author said, any secondary anisotropy of gravitational origin would cause it. However, these other effects (as huge voids) are quite implausible explanations (see Section5.4). For that reason, this effect is a valid follow-up test to probe the texture hypothesis.The procedure is simple: to compare the E-mode polarization in the position of a temperature spot (as large and extreme as the  Cold Spot is), under two different hypotheses, thenull or H0 one (i.e., the temperature spot is caused by a Gaussian fluctuation) and the alternative or H1 option (i.e., the temperature spot is a secondary anisotropy caused by the collapse of an evolving texture). In fact, as proposed by [126], the best discriminating measurement is the T-E correlation, rather than simply the E signal. This cross-correlation is expected to be close to zero for the H1 hypothesis. The approach suggested in this work was to estimate the correlation of the T and E profiles around the position of the spot temperature signal. This statistic was computed for many simulations according to the H0 and H1 hypotheses, and a hypothesis test was performed, via the definition of an optimal Fisher discriminant statistic (e.g., [127]).The method was applied to probe the capabilities of current and upcoming CMB experiments for discriminating between the two hypotheses. In particular, the cases of WMAP, Planck [128], and QUIJOTE [129] were considered.As the major conclusion, it can be established that the discrimination power of theT-E correlation is not very high. In fact, for an ideal noise-free experiment, at a power of the test of 0.5, the significance level is (up most) 0.8%. The reason for this limitation is that the characteristic size of the  Cold Spot is ≈10∘, which, roughly, corresponds to multipoles of ℓ≈40. It happens that at this multipole scales, the T-E angular cross-power spectrum is very close to zero, already for H0 and, therefore, it is hard to discriminate it from H1.The results are graphically summarized in Figure14. The author represent the significance level or p-value (for a power of the test of 0.5) as a function of the instrumental noise sensitivity for the E-mode polarization. The three experiments previously mentioned are indicated as vertical lines. Notice that the WMAP instrumental characteristics do not allow for any significant discrimination between hypotheses. With Planck, it would be possible to reach a modest 7% significance level. Finally, the upcoming QUIJOTE experiment would allow for a 1.4% detection. However, if the significance of the  Cold Spot obtained via the first non-Gaussianity wavelet analysis (i.e., 1.85%, as mentioned in Section 4) is also considered, [126] claims that a joint T and T-E significance detections of 0.025% and 0.12% could be imposed on the texture hypothesis, for the QUIJOTE and Planck experiments, respectively.Figure 14
The curve indicates the probability (thep-value) for rejecting the texture hypothesis, attending to the T-E correlation, as a function of the instrumental noise sensitivity in polarization. From left to right, the vertical lines indicate the noise level associated with QUIJOTE, Planck, and WMAP experiments.


### 6.2.3. Gravitational Lensing

Besides the lack of polarization discussed in the previous subsection, the lensing of the CMB photons would be another foreseen effect caused by the gravitational field generated by the collapsing texture. This point was recently addressed by [130]. This work studied the capabilities of small-scale CMB experiments as ACT to detect a possible lensing effect occurring at the position of the  Cold Spot, caused by the gravitational field of a texture placed at z=6, and with a typical scale of 5∘, associated with a symmetry-breaking energy scale of ψ0=4.5×1015 GeV (i.e., comparable to the parameters determined by [106]).As for the polarization test, the power of the lensing analysis to probe the existence of the texture is relatively modest. For instance, detection is made at the 3σ level after 1000 minutes of integration time. In other words, this test would require of a dedicated observational campaign.


## 6.2.1. Looking for More Textures

If a texture were found in the location of the  Cold Spot, then, attending to cosmic texture models, there should be more cold and hot spots randomly distributed across the sky. In fact, the distribution of CMB spots caused by cosmic textures follows a scale-invariant law(20)Nsp(>ϑc)=4πνκ33ϑc2,
that is, the number of cold/hot spots with a scale equal or larger than ϑc, is inversely proportional to ϑc2 (see [124] for the grounds and [106] for a derivation). The ν and κ parameters are associated with the physics of the cosmic texture models, and their specific meaning is out of the scope of this paper. The author remarks here that according to state-of-the-art simulations, these parameters are well consistent with ν≈2 and κ≈0.1 values, when ϑc is expressed in radians. Before continuing the discussion, the author stresses that the number of expected cold/hot spots with a scale ϑc equal or larger than 5∘ (i.e., as the  Cold Spot scale) is ≈1. In other words, the fact that we only found a  Cold Spot of a scale similar or larger than the  Cold Spot size is fully consistent with the cosmic texture scenario. This fact, of course, is an extra support for the texture hypothesis causing the  Cold Spot emission.Let us come back to (20). A straightforward calculation tells as that if the cosmic texture hypothesis is correct, then the CMB temperature fluctuations should contain ≈28 spots with a scale ϑc≥1∘, and ≈7 spots with a scale ϑc≥2∘. Poisson errors can be safely assumed for these numbers. Therefore, the number of cold/hot spots expected in the WMAP data are 1 ≲Nsp(>2°) ≲ 13 at the 95% confidence level. Current work is in progress to check this prediction, by using a fast cluster nesting sampling algorithm—MULTINEST [125]—to explore the posterior probability ratio. There are well-founded hopes to find new textures. In particular, some non-Gaussian analyses as [27, 48, 88] reported some hot/cold spots (in addition to the  Cold Spot) as potentially anomalous.


## 6.2.2. The Polarization of the CMB

The effect of a collapsing texture on the passing by CMB photons is nothing but a secondary anisotropy of the CMB fluctuations, whose origin is merely gravitational. Hence, the effect of such gravitational phenomenon on the E-mode polarization is almost negligible (only vector modes would be affected, which are well below the scalar mode contribution).Strictly speaking, this lack of polarization is not a unique signature produced by cosmic textures. As the author said, any secondary anisotropy of gravitational origin would cause it. However, these other effects (as huge voids) are quite implausible explanations (see Section5.4). For that reason, this effect is a valid follow-up test to probe the texture hypothesis.The procedure is simple: to compare the E-mode polarization in the position of a temperature spot (as large and extreme as the  Cold Spot is), under two different hypotheses, thenull or H0 one (i.e., the temperature spot is caused by a Gaussian fluctuation) and the alternative or H1 option (i.e., the temperature spot is a secondary anisotropy caused by the collapse of an evolving texture). In fact, as proposed by [126], the best discriminating measurement is the T-E correlation, rather than simply the E signal. This cross-correlation is expected to be close to zero for the H1 hypothesis. The approach suggested in this work was to estimate the correlation of the T and E profiles around the position of the spot temperature signal. This statistic was computed for many simulations according to the H0 and H1 hypotheses, and a hypothesis test was performed, via the definition of an optimal Fisher discriminant statistic (e.g., [127]).The method was applied to probe the capabilities of current and upcoming CMB experiments for discriminating between the two hypotheses. In particular, the cases of WMAP, Planck [128], and QUIJOTE [129] were considered.As the major conclusion, it can be established that the discrimination power of theT-E correlation is not very high. In fact, for an ideal noise-free experiment, at a power of the test of 0.5, the significance level is (up most) 0.8%. The reason for this limitation is that the characteristic size of the  Cold Spot is ≈10∘, which, roughly, corresponds to multipoles of ℓ≈40. It happens that at this multipole scales, the T-E angular cross-power spectrum is very close to zero, already for H0 and, therefore, it is hard to discriminate it from H1.The results are graphically summarized in Figure14. The author represent the significance level or p-value (for a power of the test of 0.5) as a function of the instrumental noise sensitivity for the E-mode polarization. The three experiments previously mentioned are indicated as vertical lines. Notice that the WMAP instrumental characteristics do not allow for any significant discrimination between hypotheses. With Planck, it would be possible to reach a modest 7% significance level. Finally, the upcoming QUIJOTE experiment would allow for a 1.4% detection. However, if the significance of the  Cold Spot obtained via the first non-Gaussianity wavelet analysis (i.e., 1.85%, as mentioned in Section 4) is also considered, [126] claims that a joint T and T-E significance detections of 0.025% and 0.12% could be imposed on the texture hypothesis, for the QUIJOTE and Planck experiments, respectively.Figure 14
The curve indicates the probability (thep-value) for rejecting the texture hypothesis, attending to the T-E correlation, as a function of the instrumental noise sensitivity in polarization. From left to right, the vertical lines indicate the noise level associated with QUIJOTE, Planck, and WMAP experiments.


## 6.2.3. Gravitational Lensing

Besides the lack of polarization discussed in the previous subsection, the lensing of the CMB photons would be another foreseen effect caused by the gravitational field generated by the collapsing texture. This point was recently addressed by [130]. This work studied the capabilities of small-scale CMB experiments as ACT to detect a possible lensing effect occurring at the position of the  Cold Spot, caused by the gravitational field of a texture placed at z=6, and with a typical scale of 5∘, associated with a symmetry-breaking energy scale of ψ0=4.5×1015 GeV (i.e., comparable to the parameters determined by [106]).As for the polarization test, the power of the lensing analysis to probe the existence of the texture is relatively modest. For instance, detection is made at the 3σ level after 1000 minutes of integration time. In other words, this test would require of a dedicated observational campaign.


## 7. Conclusions

In this paper the author has presented a comprehensive overview of the Cold Spot. Since its detection in 2003 by [28], this feature proved to be one of the most intriguing anomalies found in the WMAP data.The  Cold Spot was detected after performing a non-Gaussianity test on a cleaned CMB (obtained via a template fitting of the WMAP difference assemblies atQ-, V-, and W-bands). The non-Gaussian analysis was performed by comparing the values obtained for the skewness and the kurtosis of the SMHW coefficients (at several scales) for such cleaned CMB map to the distribution expected from isotropic and Gaussian CMB realizations. This analysis indicated an excess of the kurtosis at SMHW scales R of around 300 arcmin. Subsequent analysis of the SMHW coefficients, based on the area above/below a given threshold, the Higher Criticism, and the maximum value, agreed in detecting the non-Gaussian deviation occurring at the same scales, and confirmed the peculiar role played by the  Cold Spot. Avoiding any possible a posteriori choice of statistics, a conservative significance detection level of 1.85% was placed by [76].The Cold Spot was found to be highly isotropic [82], and the impact of possible systematics and residual foreground contamination was discarded [28, 46, 76, 82]. Equivalently, some secondary anisotropies (potentially responsible for the  Cold Spot emission) as the Sunyaev-Zeldovich and the Rees-Sciama effects, that, in principle, could be accepted as valid solutions, were show to be unlikely explanations [82, 95].A plausible explanation in terms of a cosmic defect was addressed by [106]. It would imply the presence of a collapsing cosmic texture at redshift z≈6, with a typical scale of ≈5∘, corresponding to a symmetry-breaking energy scale of ψ0≈9×1015. In addition, some follow-up tests were proposed to confirm or discard the cosmic texture hypothesis. In particular, one should expect to have more spots generated by evolving textures in CMB observations (e.g.,  ≈7 with scales ϑ≳2∘), a lack of E-mode polarization is predicted, as compared to the values associated with spots derived from a Gaussian CMB field; and, finally, a lensing of the CMB photons is potentially detectable with future small scale CMB experiments as ACT or SPT.Summarizing, the study of the  Cold Spot has provided a wealth of information, and the upcoming high-quality CMB data, as the one expected from Planck, guarantee that still more knowledge will come from the analysis of this very interesting feature.



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*Source: 592094-2010-11-04.xml*