Alignments of parity even/odd-only multipoles in CMB

TL;DR

This study analyzes parity asymmetries in the CMB multipoles from PLANCK data, finding significant alignments in odd multipoles but not in even ones, suggesting potential anisotropy features.

Contribution

It introduces directional statistics to compare parity even and odd multipoles and assesses their alignments, providing new insights into CMB anisotropy patterns.

Findings

No significant alignments in even multipoles.

Evidence for alignments in odd multipoles at ~2 sigma significance.

Alignment axes are consistent across different galactic cuts.

Abstract

We compare the statistics of parity even and odd multipoles of the cosmic microwave background (CMB) sky from PLANCK full mission temperature measurements. An excess power in odd multipoles compared to even multipoles has previously been found on large angular scales. Motivated by this apparent parity asymmetry, we evaluate directional statistics associated with even compared to odd multipoles, along with their significances. Primary tools are the \emph{Power Tensor} and \emph{Alignment Tensor} statistics. We limit our analysis to the first sixty multipoles i.e., $l=[2,61]$. We find no evidence for statistically unusual alignments of even parity multipoles. More than one independent statistic finds evidence for alignments of anisotropy axes of odd multipoles, with a significance equivalent to $\sim 2 \sigma$ or more. The robustness of alignment axes is tested by making galactic cuts and varying the multipole range. Very interestingly, the region spanned by the (a)symmetry axes is found to broadly contain other parity (a)symmetry axes previously observed in the literature.

Tables

Table 1. Table 1: The directions corresponding to the PEVs of l = 2 , 3 𝑙 2 3 l=2,3 modes obtained from Power tensor statistic are listed here. These axes are headless and the quoted direction is from the upper galactic hemisphere. These broadly point towards the CMB kinetic dipole direction ( ℓ , b ) = ( 264 ∘ , 48 ∘ ) ℓ 𝑏 superscript 264 superscript 48 (\ell,b)=(264^{\circ},48^{\circ}) as shown in subsequent plots. They are aligned at a mere separation of ≈ 6.2 ∘ absent superscript 6.2 \approx 6.2^{\circ} .

$l$	$(\ell ,b)$ (degrees)
2	(${239.8}^{\circ }$,${57.2}^{\circ }$)
3	(${244.3}^{\circ }$,${63.0}^{\circ }$)

Equations

$$\Delta T(\hat{n})=\sum _{l=2}^{\infty }\sum _{m=-l}^{+l}{a}_{lm}{Y}_{lm}(\hat{n}).$$

$$a_{lm}=⟨l,m|a(l)⟩,$$

$$|a(l){⟩}^{\prime }=|a(l)⟩+|\delta a(l)⟩,$$

$$\frac{\partial }{\partial {\Theta }_i\partial {\Theta }_j}⟨\delta a(l)|\delta a(l)⟩=⟨a(l)|J^i{J}^j|a(l)⟩\equiv A_{ij}.$$

$$A_{ij}(l)=\frac{1}{l(l+1)(2l+1)}\sum _{m{m}^{\prime }{m}^{\prime \prime }}{a}_{lm}{J}_{m{m}^{\prime }}^i{J}_{m^{\prime }{m}^{\prime \prime }}^j{a}_{l{m}^{\prime \prime }}^{\ast }.$$

$$S=-\sum _{\alpha =1}^3{\lambda }_{\alpha }\ln ({\lambda }_{\alpha }),$$

$$X_{ij}(l_{min},l_{max})=\sum _{l=l_{min}}^{l_{max}}{\tilde{e}}_l^i{\tilde{e}}_l^j,$$

$$S_X=-\sum _{\alpha =1}^3{\zeta }_{\alpha }\ln ({\zeta }_{\alpha }).$$

$$B_{l{l}^{\prime }}=Tr\{{\Pi }_l^{†}{\Pi }_{l^{\prime }}\}=⟨{\tilde{e}}_l|{\tilde{e}}_{l^{\prime }}{⟩}^2.$$

$$AD$$

$$f(x)=\frac{dN}{dx}=2\frac{dN}{d\cos \theta }|\frac{d\cos \theta }{dx}|=\frac{2}{2}\frac{d\sqrt{x}}{dx}=\frac{1}{2\sqrt{x}}.$$

Full text

††pubyear: 2017††pagerange: Alignments of parity even/odd-only multipoles in CMB–B

1 Introduction

Many tests of symmetry of the cosmic microwave background (CMB) sky have revealed unexplained anomalies on large angular scales, namely among low multipoles. Many low multipoles are plagued with anomalous features, associated with a breakdown of isotropy, with significances that varied between different data releases (de Oliveira-Costa et al., 2004; Ralston & Jain, 2004; Copi, Huterer & Starkman, 2004; Schwarz et al., 2004; Eriksen et al., 2004; Akrami et al., 2014; Vielva et al., 2004; Land & Magueijo, 2005a; Kim & Naselsky, 2010; Finelli et al., 2012). In some cases the anomalies have been attributed to statistical flukes. However, they have received significant interest from the cosmology community by way of alternate or independent analyses towards understanding these peculiarities (see for example Hajian & Souradeep (2003); Slosar & Seljak (2005); Land & Magueijo (2005b); Bielewicz et al. (2005); de Oliveira-Costa & Tegmark (2006); Copi et al. (2006); Wiaux et al. (2006); Abramo et al. (2006); Bernui et al. (2007); Gruppuso & Burigana (2009); Sarkar et al. (2011); Cruz et al. (2011); Rassat & Starck (2013); Rassat et al. (2014); Polastri, Gruppuso & Natoli (2015); Copi et al. (2015); Schwarz et al. (2016); Gruppuso et al. (2011); Hansen et al. (2011); Kim & Naselsky (2011); Maris et al. (2011); Aluri & Jain (2012); Naselsky et al. (2013); Eriksen et al. (2007); Bernui (2008); Lew (2008); Hansen et al. (2009); Bernui (2009); Paci et al. (2010); Santos, Villela & Wuensche (2012); Flender & Hotchkiss (2013); Rath & Jain (2013); Bernui, Oliveira & Pereira (2014); Quartin & Notari (2015); Aiola et al. (2015); Gurzadyan et al. (2009); Naselsky, Hansen & Kim (2011); Ben-David, Kovetz & Itzhaki (2012); Cruz et al. (2006, 2007); Nadathur et al. (2014)). Regardless of interpretation, the large scale anomalies have persisted from WMAP to PLANCK mission data, where the science teams pursued them with no final conclusion (Bennett et al., 2011, 2013; Planck Collaboration XXIII, 2014; Planck Collaboration XVI, 2016).

In this paper we uncover yet another peculiarity associated with low multipole CMB data. We compare the alignments of parity-even and parity-odd multipoles separately to explore any preferred directions associated with each. The significances of these directions point to a particular parity preference present in the data, and possible clues about their relation to other large angle CMB anomalies.

In Kim & Naselsky (2010), an anomalous point (inversion) parity asymmetry was reported to be present in CMB data at low$-l$. Odd multipoles of the CMB were found to have significantly more power compared to the even multipoles in the angular power spectrum from WMAP seven year data, following an earlier analysis that used WMAP first year data (Land & Magueijo, 2005a). Let $P^{+}=\langle D_{l}\rangle_{even-l}$ and $P^{-}=\langle D_{l}\rangle_{odd-l}$ denote mean power in even and odd multipoles, respectively, up to a chosen $l_{max}$ in the multipole range $l=[2,l_{max}]$. Here $D_{l}=l(l+1)C_{l}/2\pi$, and $C_{l}$ is the CMB angular power spectrum. Since the power $l(l+1)C_{l}\sim$ constant, at low multipoles, the ratio $R(l_{max})=P^{+}/P^{-}$ is expected to fluctuate about ‘$1$’. However it was found to be significantly lower than ‘$1$’ with a probability-to-exceed the observed value in data reaching a minimum of $\sim 3\sigma$ at $l_{max}=22$.

This was followed by other studies confirming the anomalous nature of this parity asymmetry (Gruppuso et al., 2011; Aluri & Jain, 2012). In the PLANCK 2015 analysis (Planck Collaboration XVI, 2016), the $p-$value of this asymmetry was evaluated to be $0.2-0.3\%$ at $l_{max}=28$, depending on the specific component separation method used to extract the CMB signal.

The directionality of this parity asymmetry was probed by Zhao (2014), where the ratio $R(l_{max})$ and its variants were computed in different sky directions to obtain a map of the even-odd power asymmetry with a chosen $l_{max}$. Curiously, the minimum of the odd parity excess statistic, $R(l_{max})$, was found to occur in the direction of the CMB dipole.

Here we analyse the even and odd multipoles separately in a wider multipole range, to explore any preferred directions associated with these point parity (a)symmetry modes.

2 Power tensor, Power entropy, and Alignment Statistics

The Power tensor is a robust diagnostic to test isotropy of CMB data (Ralston & Jain, 2004; Samal et al., 2008, 2009). The CMB temperature is conventionally expanded in terms of spherical harmonics $Y_{lm}(\hat{n})$:

[TABLE]

Here $a_{lm}$ are the spherical harmonic coefficients, $\Delta T(\hat{n})$ denotes the CMB temperature anisotropies after subtracting the monopole and dipole, and $\hat{n}$ is the position vector on the dome of the sky.

In Dirac notation, the coefficients of the spherical harmonic expansion are

[TABLE]

where $|l,m\rangle$ represent eigenstates of the angular momentum operators $J^{2}$ and $J_{z}$. Under a small rotation, the $a_{lm}$’s change to

[TABLE]

where the infinitesimal change is given by $|\delta a(l)\rangle\ =-i{\bf J}\cdot{\bf\Theta}|a(l)\rangle$. Here $J_{i}$ ($i=1\cdots 3$) are the angular momentum matrices in spin$-l$ representation, and $\Theta_{i}$ are the angles of rotation. To find the axes along which the maximum change is achieved, compute the Hessian, which is

[TABLE]

The eigenvectors of $A_{ij}$ define the frame to which maximal change is developed under rotations. The corresponding statistic $A_{ij}$ that we call Power tensor, is defined as

[TABLE]

Under the assumption of statistical isotropy, different spherical harmonic coefficients are uncorrelated i.e., $\langle a_{lm}a^{*}_{l^{\prime}m^{\prime}}\rangle=C_{l}\delta_{ll^{\prime}}\delta_{mm^{\prime}}$, and hence $\langle A_{ij}\rangle=(C_{l}/3)\delta_{ij}$. Thus, in an ensemble realizations of an uncorrelated, statistically isotropic CMB sky, the eigenvalues of the Power tensor are randomly distributed about the mean value of $C_{l}/3$. The Power tensor eigenvectors are also distributed uniformly over the sky.

Thus, Power tensor maps the complicated pattern of each multipole on the sky onto an ellipsoid whose axes lengths are given by its eigenvalues, and the three ellipsoid axes by its eigenvectors. Hence, Power tensor can be used to characterise axiality, planarity, as well as consistency with isotropy of each multipole by comparing the ratio of its eigenvalues (ie., shape of the ellipsoid), with the corresponding eigenvector denoting the direction of isotropy breakdown.

In any given realization, the eigenvalues of the Power tensor will not be equal. Let the eigenvalues and eigenvectors corresponding to a multipole be denoted $\Lambda_{\alpha}$ and $e^{i}_{\alpha}$, where ‘$\alpha$’ denotes the three eigen-indices and ‘$i$’ denotes the components of each eigenvector ${\bf e}_{\alpha}$. We also define the principal eigenvector (PEV) as the eigenvector associated with the largest eigenvalue. Each PEV is then taken as the anisotropy axis corresponding to a multipole $l$.

The significance of anisotropy/axiality represented by a PEV can be quantified using Power entropy, defined as

[TABLE]

where $\lambda_{\alpha}=\Lambda_{\alpha}/\sum_{\beta}\Lambda_{\beta}$ are the normalized eigenvalues of the Power tensor. In the limit that a multipole is highly anisotropic, one normalized eigenvalue will tend to being ‘$1$’. Correspondingly, the Power entropy $S\rightarrow 0$. If statistical isotropy holds, then each normalized eigenvalue is equal to $1/3$, and $S\rightarrow\ln(3)\approx 1.0986$, which is the maximum possible value.

The PEV’s make it possible to compare the orientations of different multipoles, which a priori contain information, that is independent of the power. A typical statistic is the dot-product-squared of PEV’s from two distinct multipoles $l$ and $l^{\prime}$. Squaring the dot-product removes the arbitrary sign convention of eigenvectors.

To quantify correlations in a set of PEV’s from a range $l_{min}\leq l\leq l_{max}$, we use the Alignment tensor $X$, which is defined as

[TABLE]

where $\tilde{\bf e}_{l}$ is the principal eigenvector of a multipole $l$. Let $\zeta_{\alpha}$ and ${\bf f}_{\alpha}$ denote the normalized eigenvalues and eigenvectors of this Alignment tensor. The eigenvalues are normalized to remove the trivial effect of the $l$-range. One then computes the Alignment entropy, $S_{X}$, which is a rotationally invariant summary of the ratios of $\zeta_{\alpha}$, that is given by

[TABLE]

When the PEV’s over the range are uncorrelated, $X_{ij}\sim\delta_{ij}$ and all $\zeta_{\alpha}$ are equal. In the extreme opposite case when the PEV’s over the set of multipoles are all parallel to a single eigenvector, then all but one $\zeta_{\alpha}\rightarrow 0$. That leads to the maximal range of Alignment entropy as $0\leq S_{X}\leq\ln(3)$. The lower limit $S_{X}\rightarrow 0$ represents the maximum possible correlation. The upper limit $S_{X}\rightarrow\ln(3)$ represents the completely uncorrelated hypothesis of the standard Big Bang. We define the collective alignment vector of a set of multipoles as the principal eigenvector of the corresponding Alignment tensor ($\tilde{\bf f}_{\alpha}$). It’s significance is assessed using Alignment entropy. The reader may refer to Ralston & Jain (2004); Samal et al. (2008) for more details about the Power tensor method, as well as it’s relation to axes inferred from other statistics viz. the angular momentum dispersion maximization (de Oliveira-Costa et al., 2004) and Maxwell’s multipole vectors (Schwarz et al., 2004).

3 Description of procedure and data sets

3.1 Analysis procedure

The Power tensor and Alignment tensor allow us to probe any underlying anisotropy axis associated with CMB anisotropies from a desired multipole range or a set of multipoles.

Under point inversion, $a_{lm}\rightarrow(-1)^{l}a_{lm}$, and so the even(odd) multipoles are symmetric(antisymmetric) under such operation. In the present work, we apply the Alignment tensor statistic to even and odd multipoles separately. Thus we can explore any common preferred axes underlying these modes separately.

We first compute the principal eigenvector (PEV) corresponding to each multipole in a chosen multipole range $l=[l_{min},l_{max}]$. The PEVs are separated between even and odd multipoles to construct the Alignment tensor for each parity set separately. The PEV of the Alignment tensor will provide the common anisotropy axis corresponding to each set of parity even/odd multipoles under study. The significance of anisotropy represented by this axis is measured using Alignment entropy. This is done by computing the lower tail probability deduced from simulations in comparison to the observed entropy value from data. We also study alignments in cumulative multipole bins, by varying the upper and lower end of the $l-$range being considered.

3.2 Real and mock data used

For this study we use the full sky Commander CMB map, derived from PLANCK 2015 data, that is made publicly available111http://irsa.ipac.caltech.edu/data/Planck/release_2/all-sky-maps/matrix_cmb.html. It is a maximum likelihood estimate of the CMB map, along with various astrophysical components such as galactic synchrotron, thermal dust, their spectral indices, etc., that uses multi-frequency CMB observations and external observations/templates for various galactic emission types (Eriksen et al., 2004, 2008; Planck Collaboration IX, 2016; Planck Collaboration X, 2016).

The Commander map is available at a resolution of HEALPix222http://healpix.jpl.nasa.gov/ $N_{side}=2048$. However, we downgrade the map to a lower resolution of $N_{side}=256$, and smooth it to have a Gaussian beam $FWHM=1^{\circ}$ (degrees). Since we are interested in large angular scales, this is sufficient for our purposes.

We also prepare the mock data accordingly. The PLANCK collaboration has also provided sets of CMB realizations that have the appropriate instrument effects such as beam smoothing, as well as noise realizations for public use333http://crd.lbl.gov/cmb-data. These are referred to as Full Focal Plane (FFP) simulations. We use the FFP8 and FFP8.1 simulation sets for our purpose. The set FFP8 was an initial release that complements the PLANCK 2015 full mission data release. However due to a slight mismatch in the theoretical power spectrum of CMB used to generate these realizations, with the angular power spectrum consistent with final PLANCK 2015 cosmological parameters, the CMB realizations were updated with a new set denoted as FFP8.1 that match PLANCK 2015 cosmology (Planck Collaboration XII, 2016). Hence we use simulated CMB skies from the set FFP8.1, but will use the FFP8 realizations for noise.

The FFP simulations of CMB and noise that are provided, correspond to a specific frequency channel, and are not readily usable. These simulation sets do not constitute individual component separated maps corresponding to various cleaning algorithms used by PLANCK such as Commander, SMICA, etc., to obtain clean CMB maps from the raw satellite data (Planck Collaboration IX, 2016). Thus, to obtain a set of realistic CMB maps, we process this ensemble of multi-channel maps as follows.

We downgrade all the CMB and noise realizations to a common resolution of HEALPix $N_{side}=256$, and smooth to have a uniform beam resolution of $FWHM=1^{\circ}$ (degrees) Gaussian beam. We apply the HEALPix facilities anafast, alteralm and synfast in that order to bring them to the afore mentioned common HEALPix resolution and beam smoothing. We used the circularized beam transfer functions corresponding to each PLANCK frequency channel, that are provided with the second public release of PLANCK data. We then compute the noise rms corresponding to each channel using these smoothed/downgraded realizations. These noise rms maps are used to combine the smoothed/downgraded individual frequency specific CMB and noise realizations through inverse noise variance weighting. Thus we are considering only the diagonal part of the full covariance matrix that results from beam smoothing. However since we are interested in studying large angular scales, the coadded CMB and effective noise maps thus obtained would sufficiently represent the observed sky.

A set of 1000 CMB and noise Monte Carlo realizations are provided with appropriate instrument and noise characteristics through PLANCK public release 2. Correspondingly we generate 1000 co-added CMB maps with noise from the FFP realizations following this procedure.

4 Results

We are interested in any preferred directional correlations associated with even versus odd multipoles corresponding to large angular scales of the CMB sky. We use the multipole range $l=[2,61]$ for this study. Before proceeding we discuss the anomalous alignment of quadrupole and octopole modes of the CMB seen in WMAP as well as PLANCK data, that have received considerable attention (see Bennett et al. (2013); Planck Collaboration XXIII (2014) for the assessment of the WMAP and PLANCK collaborations).

4.1 Quadrupole-Octopole alignment

The alignment of the quadrupole ($l=2$) and octopole ($l=3$) anisotropy axes as seen in the PLANCK full mission Commander map deserves comment. The directions inferred from the principal eigenvector (PEV) corresponding to $l=2$ and $3$ multipoles are listed in Table 1. Since eigenvectors of the Power tensor are headless vectors, we report the direction of these axes from only one of the hemispheres. We find that these two modes are well aligned with a separation of only $\approx 6.2^{\circ}$ (degrees). This corresponds to a random chance occurrence probability of $1-\cos(6.2^{\circ})\approx 0.0058$, which is close to a $3\sigma$ significance. Together with the CMB dipole, the quadrupole and octopole modes point towards the Virgo cluster (Ralston & Jain, 2004). These axes are shown in subsequent plots as some of the reference anisotropy directions seen in the CMB sky. Note that the alignment of CMB temperature quadrupole and octopole modes was found to improve by appying any additional corrections such as residual galactic bias correction (Aluri et al., 2011) or kinetic quadrupole correction, frequency independent (Schwarz et al., 2004) or frequency dependent (Notari & Quartin, 2015). The PLANCK 2015 foreground cleaned maps have the frequency independent kinetic Doppler boost contribution subtracted (Planck Collaboration IX, 2016). Here we used the PLANCK’s Commander 2015 CMB map as provided.

4.2 Parity alignments

Using the PEVs computed for each ‘$l$’ from the multipole range of our interest, we construct the Alignment tensor defined in Eq. [7] for even and odd multipoles separately.

First we present results for the case of varying $l_{max}$, meaning, we fix $l_{min}=2$ and vary $l_{max}=[7,61]$. So, the smallest range considered is $l=2\cdots 7$, and the Alignment tensor is computed separately for even and odd multipoles using $l=2,4,6$ and $l=3,5,7$ PEVs respectively. Then we keep increasing the multipole range up to $l_{max}=61$ by two multipoles each time (so that there are an equal number of even and odd multipoles for computing the Alignment tensor), and obtain the common anisotropy axis for the set of even/odd multipoles in the current range every time. The results are shown in Fig. [2].

There seems to be an apparent clustering of even multipoles, denoted by $+$’s, broadly oriented along the CMB kinetic dipole ($l=1$) direction. By progressively adding more multipoles to the Alignment tensor, the derived PEV moves closer to the CMB dipole direction. On the other hand, the common alignment axis of odd multipole PEVs, plotted in the same figure using $\bullet$ point types, steadily drifts from being close to the southern galactic pole towards the galactic plane.

We assess the significance of these collective alignment axes of even/odd multipoles using the Alignment entropy ($S_{X}$) defined in Eq. [8]. The value of the Alignment entropy obtained from the data is compared with the same quantity computed from simulations. The $p-$value plot for the observed value of $S_{X}$ as a function of $l_{max}$ is shown in Fig. [2]. We find that the apparent clustering indicated by the common alignment axes of even multipoles (black curve) is not significant, as the $p-$value curve is always within $2\sigma$ in the multipole range considered. However, it could be an indication of a remnant anisotropy (or a leakage) that is resulting in the apparent clustering of these axes towards CMB dipole.

In the same plot, Fig. [2], we also show the significances of odd multipole alignment axes as a function of $l_{max}$ (red curve). We find that these axes are highly directional, despite the change in their orientation steadily with the addition of more multipoles. The significance fluctuates about the $2\sigma$ confidence level up to $l_{max}=27$, and becomes insignificant thereafter. So, by adding more multipoles, the directionality of common alignment axis of odd multipoles seen at low$-l$ is weakened.

For reference, we also plot other interesting anisotropy directions seen in the CMB data with different point types in black. The quadrupole and octopole axes listed in Table 1 of the present analysis are denoted by up and inverted triangles respectively. The CMB dipole direction, and the low$-l$ hemispherical power asymmetry axis - that is obtained from the analysis of PLANCK 2015 data using the BipoSH framework (Planck Collaboration XVI, 2016), are highlighted using a black circle and a cross respectively. A set of interesting anisotropy axes corresponding to a mirror parity (a)symmetry are also found in the CMB data (Planck Collaboration XVI, 2016). However, only the mirror asymmetry axis is found to be anomalous. The maximum mirror symmetry axis is labeled $max(S_{+})$, and the maximum mirror asymmetry axis is labeled as $max(S_{-})$. These two axes are highlighted using a black diamond and a square respectively in Fig. [2].

It is interesting to note that the even/odd multipoles’ common axes span two broad regions of the sky in an apparently non-random/non-overlapping manner. One can readily see that the common alignment axes of even multipole PEVs found here and the (insignificant) even mirror parity direction - $max(S_{+})$, are broadly aligned with the CMB kinetic dipole direction. The region spanned by the odd multipole common alignment axes contain the odd mirror parity axis - $max(S_{-})$, and the odd parity low$-l$ dipole modulation axis.

Aluri & Jain (2012) found that the significance of the even-odd multipole power asymmetry in CMB angular power spectrum significantly decreases when the first few multipoles are omitted. We now test for low multipole contributions to the $\sim 2\sigma$ significance seen for the directionality of odd multipole alignment axes. We repeat the calculations, while choosing different $l_{min}$ values i.e., $l_{min}=4,6$ and $8$. The results are shown in Fig. [3] in the left column. We find that the distribution of common alignment axes still persists for different low$-l$ cuts i.e., using different $l_{min}$, but varying the other end of the multipole window upto $l_{max}=61$.

However, similar to what was observed by Aluri & Jain (2012), we find that the significance of odd multipole PEV alignments quickly disappears when $l_{min}$ of the multipole window is varied. The $p-$value plots corresponding to choosing different $l_{min}$ are shown in the right column of Fig. [3]. The even-multipole alignments remain insignificant in this case as well.

To study the alignment preferences of high$-l$ in the multipole range under consideration, we fix $l_{max}$ and vary $l_{min}$. In Fig. [5], we show the collective alignment axes obtained by varying $l_{min}$ in the range $l=[2,56]$, with fixed $l_{max}=61$. The significance of these axes as a function of $l_{min}$ are plotted in Fig. [5]. This study suggests a possibility of two distinct populations for $l\lesssim 30$ compared to $30\lesssim l\leq 61$ when contrasted with varying $l_{max}$ case. We find a $\sim 2\sigma$ significance upto $l_{max}\sim 30$ in the varying $l_{max}$ case. However in the varying $l_{min}$ case, the significance keeps building up upto $l_{min}\sim 30$ which indicates two distinct populations of anisotropy axes.

We observe the alignment axis of even multipole PEVs drifting towards the galactic plane as more and more low$-l$ are discarded. In comparison, the odd multipole PEVs’ alignment axis now seem to have settled at the galactic plane. The significance of the common alignment axis becomes acute for $l_{min}\sim 28$. A residual foreground bias may explain the clustering of these axes in the galactic plane, and also the corresponding anomalous significance. We pursue this aspect later in the paper.

Now we probe the observed clustering of common alignment axes of even multipole PEVs further. The absolute scalar product of the common axes obtained from the smallest and largest subset of multipole bins of even/odd ‘$l$’ PEVs from the whole multipole range $l=[2,61]$ is computed. This product denoted by $\cos(\alpha)$ is taken as representative of these axes being closer or scattered away from each other. The frequency plots of $\cos(\alpha)$ corresponding to even and odd multipoles, as obtained from simulations, are shown in Fig. [6]. The two cases of varying $l_{max}$ and $l_{min}$ while fixing the other end of the multipole window are shown in that figure, in the left and right panels respectively. The observed value of the inner product of the same axes from the data are denoted by vertical dashed lines in respective colours. From the histogram plot, we see that the clustering of even multipole common axes is not statistically significant in both cases of varying $l_{max}$ and $l_{min}$. In contrast with this, the scalar product of odd multipoles’ common axes from the smallest and largest subsets is statistically significant in comparison to simulations.

The simulations suggest that the collective alignment axes, computed using Alignment tensor, from the smallest and largest multipole bin windows, tend to be closer to each other. This could be because the small multipole bin window is a subset of the larger multipole window, and thus correlated, leading to this preference. Upon extending the multipole window range ($l_{max}$), we observe that the distributions tend towards being uniform, as expected.

We tested the stability of alignment axes by applying galactic masks with different sky fractions, and inpainting the masked CMB maps using iSAP software444http://www.cosmostat.org/software/isap/ (Starck, Rassat & Fadili, 2013). The publicly available PLANCK HFI masks were used which exclude $1\%,3\%,10\%$, $20\%$ and $30\%$ of the sky fraction555http://irsa.ipac.caltech.edu/data/Planck/release_2/ancillary-data/. We find that the odd multipole alignment axes are stable up to an exclusion of $10\%$ of the sky in the galactic plane. However, the even multipole common axes are found to be sensitive to galactic cuts. They progressively move towards or away from the galactic plane in the varying $l_{max}$ and $l_{min}$ cases respectively, while remaining broadly clustered. Applying a galactic mask with $80\%$ or less sky fraction is found to destroy the collective orientation of these axes. This analysis is presented in Appendix A.

We then tested the effect of including more multipoles by extending the multipole range to $l_{max}=71,81,91$ and $101$. Any significant alignments seen in studying the multipole window $l=[2,61]$ vanish. This is not unexpected, as it could be a simple consequence of diluting the signal.

Finally, we analysed clean CMB maps obtained using other cleaning procedures and data sets. We find a similar behaviour for the even/odd multipole common axes in WMAP provided nine year Internal Linear Combination666https://lambda.gsfc.nasa.gov/product/map/current/ (ILC) map (Bennett et al., 2013), and the Local-generalized Morphological Component Analysis (LGMCA) map that was produced using both the WMAP and PLANCK full mission observations777http://www.cosmostat.org/product/lgmca_cmb/ (Bobin, Sureau & Starck, 2016).

We also checked collective alignment axes in multipole blocks of $\Delta l=6$ from the same range $l=[2,61]$, with three even/odd multipoles in each block. The alignment axes thus inferred for even/odd multipoles accordingly span the same region, from lowest multipole bin to the highest, as seen in varying $l_{max}$ and $l_{min}$ cases. However the cumulative statistics are better suited for our purpose i.e., to probe the widest possible correlations across (even/odd) multipoles.

4.3 Dissecting cumulative statistics

The cumulative statistics do not give much information on which regions of the data dominate the analysis. The Alignment entropy is also just a single-number summary that cannot completely identify the source of this anomaly. To glean more information about the observed alignments, we look inside the cumulative statistics in this section, while also introducing an independent statistic for testing isotropy.

To make a more informative statistic from PEVs $|\tilde{e}_{l}\rangle$, we first observe that normalized eigenvectors are equivalent to rank-1 projection operators $\Pi_{l}=|\tilde{e}_{l}\rangle\langle\tilde{e}_{l}|$. We can then define a Hilbert-Schmidt inner product (HSIP) (Reed & Simon, 1972) as

[TABLE]

For a set of ‘$n$’ unit vectors, there will be a total of ‘$n(n-1)/2$’ such independent inner products possible. The distribution of these independent HSIPs treated as a random variable, $B_{ll^{\prime}}\rightarrow x$ (for all $l$, and $l^{\prime}<l$), has an analytic form given by $f(x)=1/(2\sqrt{x})$ for $0\leq x\leq 1$ (see Appendix B for details). Correspondingly, its cumulative distribution function is given by $F(x)=\sqrt{x}$. We refer to the analytic isotropic null distribution function as aPDF, and the corresponding cumulative distribution function as aCDF. Analogously, we refer to the empirical counterparts as ePDF and eCDF, respectively. The aPDF in this form is normalized to have unit area under the curve.

Before proceeding further we first check that $f(x)=1/(2\sqrt{x})$ is the true PDF of Hilbert-Schmidt inner products of isotropically distributed unit vectors. We generate 1000 sets of $n=30$ units vectors. All possible HSIPs among these unit vectors are computed for each set of $30$ normalized vectors which will be a total of $30\times 29/2=435$. Then the mean empirical distribution function is built by taking the average of individual ePDF histograms of 1000 sets of $30$ isotropic unit vectors to compare with the analytic distribution function. The mean and analytic PDFs are shown in Fig. [7]. The $435$ independent HSIPs for each set of $30$ isotropic unit vectors are sorted into $50$ bins to compare the aPDF and ePDF. We find excellent agreement between the two distribution functions.

Now we evaluate the ePDF and eCDF of HSIPs from the data (PLANCK 2015 Commander map) and compare them with their analytic forms. We illustrate the distributions for three representative multipole ranges $l=[2,25]$, $[2,61]$ and $[26,61]$. There are a total of $12$, $30$ and $18$ even or odd multipole PEVs in these three sets. Thus $12\times 11/2=66$, $30\times 29/2=435$ and $18\times 17/2=153$ independent HSIPs are possible, respectively, in each set of multipoles among even or odd multipole PEVs. Recall that, in the cumulative statistics, we chose the multipole range such that there are equal number of even/odd multipoles available in the $l-$range being considered. These are then sorted into $20$ bins to build the ePDF and eCDF. The results are shown in Fig. [8] and [9], for the three multipole ranges mentioned above. In describing these plots below, we only highlight a visual discrepancy. Later, we use Anderson-Darling (AD) test statistic to find whether the data conforms with the isotropic null distribution function or not, and also quantify it’s significance using simulations.

The eCDF plots highlight the peculiarity of odd multipole PEV alignments rather more dramatically than ePDF plots. One notices that there is a mild deficit at low HSIP bin values, and a mild excess at intermediate HSIP bin values in the empirical PDF of odd multipole PEV alignments for the range $l=[2,25]$ in Fig. [8]. The discrepancy with the isotropic hypothesis is more striking in the empirical cumulative distribution function of odd multipole PEV HSIPs for the same range compared to the analytic distribution in Fig. [9]. With larger $l_{max}=61$, the discrepancy nearly vanishes. The diagonal dashed line is the reference curve about which the data statistic coming from the null distribution is expected to fluctuate. The empirical CDF of even multipole PEV HSIPs essentially criss-crosses this reference curve in Fig. [9], in agreement with our findings from previous sections. However, as noted above, the odd multipole alignments deviate significantly. Our earlier observation on the presence of two populations of anisotropy axes is also corroborated by the eCDF curves for $l=[2,25]$ and $l=[26,61]$ that are non-overlapping in multipole range.

The Anderson-Darling ($AD$) test (Anderson & Darling, 1954; Bohm & Zech, 2010) quantifies the agreement of the data with the isotropic null distribution. The Anderson-Darling statistic is defined as

[TABLE]

where ‘$N$’ is the number of sample points, and $F(x_{i})$ is the analytic cumulative distribution function evaluated for the data sample point $x_{i}$. For our specific case of HSIPs, $F(x_{i})=\sqrt{x_{i}}$, and for a set of ‘$n$’ even/odd multipole PEVs, there are $N=n(n-1)/2$ number of independent inner products possible. Similar to the case of varying $l_{max}$ discussed in the previous section, the $AD$ statistic is obtained as a function of $l_{max}$ from the multipole range $l=[2,61]$. At each $l_{max}$, the $AD$ statistic is computed from the even/odd multipole PEV sub sets of the current $l-$range separately. Likewise, we also show the results for varying $l_{min}$ case.

The $AD$ statistic values as a function of $l_{max}$ are shown in the left panel of Fig. [10], and as a function of $l_{min}$ in the right panel in the same figure. The expected value of the $AD$ statistic is denoted by a (blue) dashed line. It is computed from 1000 ILC-like noisy CMB maps obtained from FFP simulations described in Sec. [3.2]. The mean $AD$ statistic from simulations is evaluated in both cases for even and odd multipoles separately. Since the two curves are indistinguishable, as expected, only one of them is shown to avoid redundancy. From Fig. [10], one can readily see that the $AD$ statistic for $l_{max}=9,19,23$ and $25$ acquires very high values, hinting at the origin of the $~{}2\sigma$ level significance seen for the common alignment axes of odd multipole PEVs on large angular scales. From Fig. [5], we see that many of the collective alignment axes in the case of varying $l_{min}$ settled in the galactic plane. Correspondingly, in the right-hand panel of Fig. [10], we see that the distribution of the HSIPs quantified by the $AD$ statistic is very high compared to its expectation in the same multipole range, in the varying $l_{min}$ case.

The $p-$values of the $AD$ statistic for the PLANCK 2015 Commander map derived HSIPs as a function of $l_{max}$ are shown in the left-hand panel of Fig. [11]. The significances of the $AD$ statistic for even and odd multipole PEV HSIPs are computed separately, and are shown in black and red solid lines with square and circle point types respectively.

The Anderson-Darling statistic gives independent confirmation that the odd multipole PEV alignments are anomalous on large angular scales. Significance exceeding $2\sigma$ confidence level is found for $l_{max}=9,19,23$ and $25$ which are found to have high values for $AD$ statistic from the left plot of Fig. [10]. The even multipole PEV HSIPs show no significant signal of differing from the isotropic null distribution in this analysis, consistent with the finding from preceding section. Thus there are some anomalous alignments among odd multipole anisotropy axes on large angular scales represented by their principal eigenvectors that are resulting in the high significance of our test statistic. Owing to the highly deviant $AD$ statistic in the varying $l_{min}$ case, the $AD$ statistic is found to be anomalous for the same range of multipoles. The $p-$value plot for the same is shown in right panel of Fig. [11], which follows a trend similar to the significances found in Fig. [5].

5 Conclusions

We have compared alignment statistics of parity even and odd multipoles with several independent methods. We used the clean CMB signal estimate from PLANCK 2015 data obtained using the Commander algorithm. Analysis was restricted to the first sixty multipoles i.e., $l=[2,61]$. Power tensor and Alignment tensor statistics were used to probe the alignments of even and odd parity multipoles, separately.

We studied the data in several ways. The collective alignment axes of even and odd multipoles show different behaviors. The anisotropy axes of even-parity multipoles from large angular scales are broadly clustered near the direction of the CMB dipole. The anisotropy axes of odd multipoles are much less concentrated, but are significantly directional as quantified by Alignment entropy.

We constructed cumulative statistical measures that fixed the lower limit $l_{min}=2$, while varying the upper limit to reach $l_{max}=61$. The Alignment entropy, $S_{X}$, of even-parity multipoles was as expected from an uncorrelated isotropic distribution. The odd–parity multipole $S_{X}$ was unusually small on large angular scales with significance exceeding $2\sigma$ magnitude. As $l_{max}$ was increased above $l_{max}\sim 27$ the significance disappeared, apparently by dilution in the larger set. This significance nevertheless disappears by ignoring the first few multipoles. A similar effect was seen in studying even-odd multipole power asymmetry, using the WMAP seven year temperature power spectrum (Aluri & Jain, 2012). To understand the alignment preferences of small angular scales in the range being studied, we fixed the upper limit at $l_{max}=61$ while varying the lower limit $l_{min}$. A regime of multipoles with small $S_{X}$ at $2\sigma$ or more significance for odd-parity multipoles was observed, with lowest $p-$value for $S_{X}$ occurring at $l_{min}\gtrsim 26$. The two different effects from varying $l_{min}$ and $l_{max}$ analysis in a single data set pose a puzzle. The resolution may involves two different populations separated by a middle range of $l\sim 27$, with each population diluting a distinctive signal of the other when populations are mixed. The observation that the axes of the $l>27$ set settled at the galactic plane may be an indication of a residual galactic bias in this subset.

These results are further tested against potential residual contamination in the full sky map by excising different fractions of the sky, and then inpainting the masked region. The odd multipoles’ common axes are stable against galactic cuts up to excluding (and then inpainting) $10\%$ of the sky, whereas the even multipoles are found to be sensitive to galactic cuts.

An independent statistic was used to dissect the cumulative statistical studies. The Hilbert-Schmidt inner products (HSIP) are rotationally invariant statistics with an analytic isotropic null distribution. The distribution of the data compared to the HSIP null was computed using Anderson-Darling (AD) test statistic. For the odd multipole PEVs, the AD statistic for the data HSIPs shows a significance similar to that found using the Alignment entropy method. The AD method pinpoints $l_{max}=9,19,23$ and $25$ as containing unusual alignments that are rendering the AD statistic anomalous at a significance of $2\sigma$ or more.

Interestingly, we find that the even mirror parity axis from the PLANCK 2015 results, and the even multipoles’ common axes from large angular scales computed here, broadly point in the CMB dipole direction. Likewise, the odd mirror parity axis from the PLANCK 2015 analysis, and the odd parity low$-l$ hemispherical power asymmetry axis fall in the region spanned by the odd multipole alignment axes. From these observations, we speculate that these anomalous axes may have a common origin in their peculiar parity (a)symmetry properties.

We plan to investigate these speculations more in a later work.

Acknowledgements

We acknowledge the use of freely available HEALPix888http://healpix.jpl.nasa.gov/ (Gorski et al., 2005) package and iSAP software999http://www.cosmostat.org/software/isap/ in this work. Part of the results presented here are based on observations obtained with PLANCK101010http://www.esa.int/Planck, an ESA science mission with instruments and contributions directly funded by ESA Member States, NASA, and Canada. We also acknowledge the use of WMAP data made available from Legacy Archive for Microwave Background Data Analysis111111https://lambda.gsfc.nasa.gov/product/map/dr5/ (LAMBDA) site that is a part of NASA’s High Energy Astrophysics Science Archive Research Center (HEASARC). This research used resources of the National Energy Research Scientific Computing (NERSC) Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.

PKA is funded by the post-doctoral fellowship program of the Claude Leon Foundation, South Africa at UCT. This work is based on the research supported by the South African Research Chairs Initiative of the Department of Science and Technology and the National Research Foundation of South Africa as well as the Competitive Programme for Rated Researchers (Grant Number 91552) (AW). Any opinion, finding and conclusion or recommendation expressed in this material is that of the authors and the National Research Foundation (NRF) of South Africa does not accept any liability in this regard.

PKA also thanks Pankaj Jain for helpful exchanges on an earlier version of the paper. AW would like to thank David Spergel for helpful discussions on this work.

We thank the anonymous referee for a careful reading and helpful comments on our paper.

Appendix A Stability of alignment axes

Here we probe the stability of the even/odd multipole alignment axes using different foreground exclusion masks. We used PLANCK 2015 HFI masks with varying sky fractions, that are provided along with the second public release of PLANCK data121212http://irsa.ipac.caltech.edu/data/Planck/release_2/ancillary-data/. The respective sky fractions of the masks used are 1%, 3%, 10%, 20% and 30%. The excluded regions corresponding to these masks are shown in Fig. [12].

We used these masks at their native resolution of HEALPix $N_{side}=2048$ on the PLANCK 2015 Commander CMB temperature map which is also made available at the same resolution. The masked CMB map is then inpainted using the freely available iSAP software131313http://www.cosmostat.org/software/isap/ (see Starck, Rassat & Fadili (2013)). We used the default settings of the mrs_alm_inpainting facility of iSAP to inpaint the CMB sky.

Following the same procedure as described in the main analysis, the inpainted CMB map is then downgraded to $N_{side}=256$ and simultaneously smoothed to have a beam beam resolution of $FWHM=1^{\circ}$ (degree) Gaussian beam.

The common alignment axes of even and odd multipole PEVs from masking and inpainting 3%, 10% and 20% of the CMB sky are shown in Fig. [13]. Here we performed a qualitative analysis only. By visual inspection we see that the odd multipole alignment axes are broadly stable up to 10% of the sky being masked and inpainted. However the even multipole PEV alignment axes steadily drift towards galactic plane in the varying $l_{max}$, and move towards the poles in the case of varying $l_{min}$. Applying galactic cuts with 20% or more masking fraction (followed by inpainting the masked sky) is found to destroy the alignment patterns seen otherwise.

Appendix B The Isotropic Null $HSIP$ Distribution

Let $|\tilde{e}_{l}\rangle$ be a random eigenvector from an isotropic distribution. Since eigenvectors have no magnitude and no sign, $|\tilde{e}_{l}\rangle$ is equivalent to the rank-one projector $\Pi_{l}=|\tilde{e}_{l}\rangle\langle\tilde{e}_{l}|$. Consider the distribution of $x=Tr\{\Pi_{l}^{\dagger}\Pi_{l^{\prime}}\}=\langle\tilde{e}_{l}|\tilde{e}_{l^{\prime}}\rangle^{2}$ (for all $l\neq l^{\prime}$). Choose coordinates where the first instance $|\tilde{e}_{1}\rangle$ is along the $z$ axis, so that $\langle\tilde{e}_{l}|\tilde{e}_{1}\rangle=\cos\theta_{l}$. In an isotropic ensemble the distribution of $\cos\theta_{l}$ is constant over the range $-1\leq\cos\theta\leq 1$ as shown by the solid angle measure $d\Omega_{l}=d\cos\theta_{l}d\phi_{l}$. Averaging over all cases we can drop the index $l$. For each $x=\cos^{2}\theta$ there are two signs of $\cos\theta$. The distribution of $x$ over the range $0\leq x\leq 1$ is then

[TABLE]

The same result comes from $f(x)=(2\pi/4\pi)\int_{-1}^{1}d\cos\theta\,\delta(\cos^{2}\theta-x)$, accounting for two solutions of the delta function.