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We review the recently found large-scale anomalies in the maps of temperature anisotropies in the cosmic microwave background. These include alignments of the largest modes of CMB anisotropy with each other and with geometry and direction of motion of the solar ssystem, and the unusually low power at these largest scales. We discuss these findings in relation to expectation from standard inflationary cosmology, their statistical significance, the tools to study them, and the various attempts to explain them.

The Copernican principle states that the Earth does not occupy a special place in the universe and that observations made from Earth can be taken to be broadly characteristic of what would be seen from any other point in the universe at the same epoch. The microwave sky is isotropic, apart from a Doppler dipole and a microwave foreground from the Milky Way. Together with the Copernican principle and some technical assumptions, an oft-inferred consequence is the so-called cosmological principle. It states that the distributions of matter and light in the Universe are homogeneous and isotropic at any epoch and thus also defines what we mean by cosmic time.

This set of assumptions is a crucial, implicit ingredient in obtaining most important results in quantitative cosmology, for example, it allows us to treat cosmic microwave background (CMB) temperature fluctuations in different directions in the sky as multiple probes of a single statistical ensemble, leading to the precision determinations of cosmological parameters that we have today.

Although we have some observational evidence that homogeneity and isotropy are reasonably good approximations to reality, neither of these are actual logical consequences of the Copernican principle, for example, the geometry of space could be homogeneous but anisotropic—like the surface of a sharp mountain ridge, with a gentle path ahead but the ground dropping steeply away to the sides. Indeed, three-dimensional space admits not just the three well known homogeneous isotropic geometries (Euclidean, spherical and hyperbolic—

Similarly, although the Earth might not occupy a privileged place in the universe, it is not necessarily true that all points of observation are equivalent, for example, the topology of space may not be simply connected—we could live in a three dimensional generalization of a torus so that if you travel far enough in certain directions you come back to where you started. While such three-spaces generically admit locally homogeneous and isotropic geometries, certain directions or points might be singled out when nonlocal measurements are considered, for example the length of the shortest closed nontrivial geodesic through a point depends on the location of that point within the fundamental domain. Similarly, the inhomogeneity and anisotropy of eigenmodes of differential operators on such spaces are likely to translate into statistically inhomogeneous and anisotropic large scale structure, in the manner of Chladni figures on vibrating plates.

The existence of nontrivial cosmic topology and of anisotropic geometry are questions that can only be answered observationally. In this regard, it is worth noting that our record at predicting the gross properties of the universe on large scales from first principles has been rather poor. According to the standard concordance model of cosmology, over

The stakes are set even higher with the recent discovery of dark energy that makes the universe undergo accelerated expansion. It is known that dark energy can affect the largest scales of the universe, for example, the clustering scale of dark energy may be about the horizon size today. Similarly, inflationary models can induce observable effects on the largest scales via either explicit or spontaneous violations of statistical isotropy. It is reasonable to suggest that statistical isotropy and homogeneity should be substantiated observationally, not just assumed. More generally, testing the cosmological principle should be one of the key goals of modern observational cosmology.

With the advent of high signal-to-noise maps of the cosmic microwave background anisotropies and with the conduct of nearly-full-sky deep galaxy surveys, statistical isotropy

In this brief paper, we describe the large-scale anomalies in the CMB data, some of which were first reported on by the Cosmic Background Explorer (COBE) Differential Microwave Radiometer (DMR) collaboration in the mid 1990s. In particular, we report on alignments of the largest modes of CMB anisotropy with each other, and with geometry and direction of motion of the Solar System, as well as on unusually low angular correlations at the largest angular scales. We discuss these findings and, as this is not meant to be a comprehensive review and we emphasize results based on our own work in the area, we refer the reader to literature for all developments in the field. This paper extends an earlier review on the subject by Huterer [

The paper is organized as follows. In Section

A fixed angular scale on the sky probes the physics of the universe at a range of physical distances corresponding to the range of observable redshifts. This is illustrated in Figure

The comoving length within the context of the concordance model of an arc seen at a fixed angle and the comoving Hubble length as functions of redshift. Linear perturbation theory is expected to work well outside the shaded region, in which the large scale structure (LSS) forms.

What do we expect for the large angular scales of the CMB? A crucial ingredient of cosmology’s concordance model is cosmological inflation—a period of accelerating cosmic expansion in the early universe. If we assume that inflationary expansion persisted for sufficiently many e-folds, then we expect to live in a homogeneous and isotropic universe within a domain larger than our Hubble volume. This homogeneity and isotropy will not be exact but should characterize both the background and the statistical distributions of matter and metric fluctuations around that background. These fluctuations are made visible as anisotropies of the CMB temperature and polarization, which are expected to inherit the underlying statistical isotropy. The temperature

Statistical isotropy implies that the expectation values of all

If inflation is driven by a single dynamically relevant degree of freedom with appropriate properties (minimal coupling, Minkowski vacuum in UV limit, etc.), then we can reduce the quantization of matter and space-time fluctuations during inflation to the problem of quantizing free scalar fields. For free fields the only nontrivial object is the two-point correlation (the propagator), and all higher correlation functions either vanish or are just some trivial combination of the two-point function. This property is mapped onto the temperature field of the CMB. A classical random field with these properties is a Gaussian with mean

Another generic feature of inflation is the almost scale invariance of the power spectrum of fluctuations. This can be understood easily, as the Hubble scale is approximately constant during inflation as the wavelengths of observable modes are redshifted beyond the horizon. Given that fluctuations of modes on horizon exit are related to the Hubble parameter,

At the level of the angular power spectrum, exact scale invariance implies the Sachs-Wolfe “plateau” (i.e., constancy of

As we can measure only one sky, it is important to find the best estimators of

With these same assumptions, the variance of the two-point correlation function is easily shown to be

Putting the results of this section together allows us to come up with a generic prediction of inflationary cosmology for

Mean and (cosmic) variance of the angular two-point correlation function as expected from cosmological inflation (arbitrary normalization). Only statistical isotropy, Gaussianity and scale invariance are assumed. Tensors, spectral tilt, reionization and the integrated Sachs-Wolfe effect are neglected for the purpose of this plot. Comparison to the prediction from the best-fit

Multipole vectors of our sky, based on WMAP five-year full-sky ILC map and with galactic plane coinciding with the plane of the page. The temperature pattern at each multipole

In brief, the upshot of the previous section is that the twin assumptions of statistical isotropy and Gaussianity are the starting point of

Let us assume that the various methods that have been developed to get rid of the Galactic foreground in single frequency band maps of the microwave sky are reliable (though we argue below that this might not be the case). Our review of alignments will be based on the internal linear combination (ILC) map produced by the WMAP team, which is based on a minimal variance combination of the WMAP frequency bands. The weights for the five frequency band maps are adjusted in 12 regions of the sky, one region lying outside the Milky Way and 11 regions along the Galactic plane.

To study the orientation and alignment of CMB multipoles, Copi et al. [

These two forms for representing

An efficient algorithm to compute the multipole vectors for low-

The relation between multipole vectors and the usual harmonic basis is very much the same as that between Cartesian and spherical coordinates of standard geometry: both are complete bases, but specific problems are much more easily addressed in one basis than the other. In particular, we and others have found that multipole vectors are particularly well suited for tests of planarity and alignment of the CMB anisotropy pattern. Moreover, a number of interesting theoretical results have been found; for example, Dennis and Phys [

Tegmark et al. [

In the work of Copi et al. [

the four area vectors of the quadrupole and octopole are mutually close (i.e., the quadrupole and octopole planes are aligned) at the

the quadrupole and octopole planes are orthogonal to the ecliptic at the

the normals to these four planes are aligned with the direction of the cosmological dipole (and with the equinoxes) at a level inconsistent with Gaussian random, statistically isotropic skies at

the ecliptic threads between a hot and a cold spot of the combined quadrupole and octopole map, following a node line across about

Quadrupole and octopole (

These numbers refer to the WMAP ILC map from three years of data; other maps give similar results. Moreover, correction for the kinematic quadrupole—slight modification of the quadrupole due to our motion through the CMB rest frame—must be made and increases significance of the alignments. See [

While not all of these alignments are statistically independent, their combined statistical significance is certainly greater than their individual significances; for example, given their mutual alignments, the conditional probability of the four normals lying so close to the ecliptic is less than 2%; the combined probability of the four normals being both so aligned with each other and so close to the ecliptic is less than

Particularly puzzling are the alignments with solar system features. CMB anisotropy should clearly not be correlated with our local habitat. While the observed correlations seem to hint that there is contamination by a foreground or perhaps by the scanning strategy of the telescope, closer inspection reveals that there is no obvious way to explain the observed correlations. Moreover, if their explanation is that they are a foreground, then that will likely exacerbate other anomalies that we will discuss in Section

Our studies (see [

Finally, it is important to make sure that the results are unbiased by unfairly chosen statistics. We have studied this issue extensively in [

To define statistics we first compute the three dot-products between the quadrupole area vector and the three octopole area vectors:

Alternatively, generalizing the definition in [

To test alignments of multipole planes with physical directions, we find the plane whose normal,

The study of alignments in the low

This is strongly suggestive of an unknown systematic in the data reduction; however, careful scrutiny has revealed no such systematic (except the mentioned modification of the radiometer gain model, that leads to a reduction of ecliptic alignment); see Sections

The usual CMB analysis solely involves the spherical harmonic decomposition and the two-point angular power spectrum. There are many reasons for this. Firstly, when working with a statistically isotropic universe the angular power spectrum contains all of the physical information. Secondly, the standard theory predicts the

The two-point angular correlation function provides another means of analyzing CMB observations and should not be ignored even if, in principle, it contains the same information as the angular power spectrum. Thus, even in the case of full-sky observations and/or statistical isotropy there are benefits in looking at the data in different ways. The situation is similar to a function in one dimension where it is widely appreciated that features easily found in the real space analysis can be very difficult to find in the Fourier transform, and vice versa. Furthermore, the two-point angular correlation function highlights behavior at large-angles (small

Care should be taken when discussing statistical quantities of the CMB and their estimators. It rarely is in the literature. Here we follow the notation of Copi et al. [

Spergel et al. [

We have revisited the angular two-point function in the 3-yr WMAP data in [

All of the cut-sky map curves are very similar to each other, and they are also very similar to the Legendre transform of the pseudo-

The most striking feature of the cut-sky (and pseudo-

Two-point angular correlation function,

In order to be more quantitative about these observations we adopt the

Applying this statistic we have found that the two-point function computed from the various cut-sky maps shows an even stronger lack of power, for WMAP 5 year maps significant at the 0.037%–0.025% level depending on the map used; see Figure (

There are actually two interesting questions one can ask.

Is the correlation function measured on the cut-sky compatible with cut-sky expectation from the Gaussian random, isotropic underlying model?

Is the reconstruction of the full-sky correlation function from partial information compatible with the expectation from the Gaussian random, isotropic underlying model?

Our results refer to the first question above. The second question, while also extremely interesting, is more difficult to be robustly resolved because the reconstruction uses assumptions about statistical isotropy (see the next subsection).

The little large-angle correlation that does appear in the full-sky maps (e.g., the solid, black line in Figure

The two-point angular correlation function from the WMAP 5-year results. Plotted are

The two-point angular correlation function,

Various approaches have been taken to incorporate the standard model in the analysis, for example; Hajian [

Another approach advocated by Efstathiou et al. [

Whether or not reconstructing the full-sky is a “more optimal” approach than direct calculation of the cut-sky

The striking feature of the two-point angular correlation function as seen in Figure

We also note that the vanishing of power is much more apparent in real space (as in

In [

It is for this reason that theoretical efforts to explain “low power on large scales” must focus on explaining the low

Finally, one might ask if the observed lack of correlation and the alignment of quadrupole and octopole are correlated. This issue was studied by Rakić & Schwarz [

Understanding the origin of CMB anomalies is clearly important. Both the observed alignments of the low-

astrophysical foregrounds,

artifacts of faulty data analysis,

instrumental systematics,

theoretical/cosmological.

In this section, we review these four classes of explanations, giving examples from each. First, however, we discuss two generic ways to break statistical isotropy and affect the intrinsic (true) CMB signal—additive and multiplicative modulations—and illustrate in general terms why it has been so difficult to

Why is it difficult to explaining the observed CMB anomalies? There are three basic reasons:

Most explanations work by

Unaccounting for sources of CMB fluctuations in the foreground, even if possessing/causing aligned low-

The alignments of the quadrupole and octopole are with respect to the ecliptic plane and near the dipole direction. It is generally difficult to have these directions naturally be picked out by any class of explanations (though there are exceptions to this—see the instrumental example below).

Gordon et al. [

In contrast to the additive models, the multiplicative mechanisms, where the intrinsic temperature is multiplied by a spatially varying modulation, are phenomenologically more promising. As a proof of principle, a toy-model modulation [

A realization of the multiplicative model where the quadrupole (left column) and octopole (right column) exhibit an alignment similar to WMAP. (a): intrinsic (unmodulated) sky from a Gaussian random isotropic realization. (b) (single column): the quadrupolar modulation with

One fairly obvious possibility is that there is a pernicious foreground that contaminates the primordial CMB and leads to the observed anomalies. Such foregrounds are, of course, additive mechanisms, in the sense of the preceding section, and so suffer from the shortcomings described therein. Moreover, most such foregrounds are Galactic, while the observed alignments are with respect to the ecliptic plane. One would expect that Galactic foregrounds should lead to Galactic and not ecliptic foregrounds. This simple expectation was confirmed in [

Moreover, in [

A number of authors have attempted to explain the observed quadrupole-octopole correlations in terms of a

It is also interesting to ask if any known or unknown Solar system physics could lead to the observed alignments. Dikarev et al. [

Finally, it has often been suggested to some of us in private communications that the anomalies may not reflect an unknown foreground that has been neglected, but rather the “missubtraction” of a known foreground. However, it has never quite been clear to us how this leads to the observed alignments or lack of large-angle correlations, and we are unaware of any literature that realizes this suggestion successfully.

Most of the results discussed so far have been obtained using reconstructed full-sky maps of the WMAP observations [

A different kind of explanation of missing large-scale power, or missing large-angle correlations, has been taken by Efstathiou et al. [

Are instrumental artifacts the cause of the observed alignments (and/or the low large-scale power)? One possible scenario would go as follows. WMAP avoids making observations near the Sun, therefore covering regions away from the ecliptic more than those near the ecliptic. While the corresponding variations in the noise per pixel are well known (e.g., as the number of observations per pixel,

Another possibility is that an imperfect instrument couples with dominant signals from the sky to create anomalies. Let us review an example given in [

To summarize, even though the ecliptic alignments (and the north-south power asymmetry) hint at a systematic effect due to some kind of coupling of an observational strategy and the instrument, to date no plausible proposal of this sort has been put forth.

The most exciting possibility is that the observed anomalies have primordial origin, and potentially inform us about the conditions in the early universe. One expects that in this case the alignments with the dipole, or with the solar system, would be statistical flukes.

The breaking of statistical isotropy implies that the usual relation

There are many possibilities for how the absence of statistical isotropy might arise; for example, in a nontrivial spatial topology, the fundamental domain would not be rotationally invariant, and so the spherical harmonics (times an appropriate radial function) would not be a basis of independent eigenmodes of the fluctuations. This would certainly lead to a correlation among

Alternately, in the early universe, asymmetry in the stress-energy tensor of dark energy [

A commonly used mechanism to explain such anomalies is inflationary models that contain implicit breaking of isotropy [

A very reasonable approach is to describe breaking of the isotropy with a phenomenological model, measure the parameters of the model, and then try to draw inferences about the underlying physical mechanism. For example, a convenient approach is to describe the breaking of isotropy via the direction-dependent power spectrum of dark matter perturbations [

As with the other attempts to explain the anomalies, we conclude that, while there have been some interesting and even promising suggestions, no cosmological explanation to date has been compelling.

While future WMAP data is not expected to change any of the observed results, our understanding and analysis techniques are likely to improve. The most interesting test will come from the Planck satellite, whose temperature maps, obtained with a completely different instrument and observational technique than WMAP, could shed significant new light on the alignments. Moreover, polarization information could be extremely useful in distinguishing between different models and classes of explanations in general; for example, Dvorkin et al. [

In their seven year data release the WMAP team explicitly discusses several CMB anomalies [

an accidental downward fluctuation of the SW sufficient for the ISW of local structure to dominate and cause an alignment,

an accidental cancellation in angular correlation between the SW and ISW temperature patterns.

Neither the WMAP team nor Francis and Peacock estimate the likelihood of these two newly created puzzles.

Regarding the second major issue—the lack of angular correlation—the WMAP team refers to a recent work by Efstathiou et al. [

These arguments from the WMAP team offer neither new nor convincing explanations of the observed anomalies discussed in this paper. At best they replace one set of anomalies by another.

The CMB is widely regarded as offering strong substantiating evidence for the concordance model of cosmology. Indeed the agreement between theory and data is remarkable—the patterns in the two-point correlation functions (TT, TE, and EE) of Doppler peaks and troughs are reproduced in detail by fitting with only six (or so) cosmological parameters. This agreement should not be taken lightly; it shows our precise understanding of the causal physics on the last scattering surface. Even so, the cosmological model we arrive at is baroque, requiring the introduction at different scales and epochs of three sources of energy density that are only detected gravitationally—dark matter, dark energy and the inflaton. This alone should encourage us to continuously challenge the model and probe the observations particularly on scales larger than the horizon at the time of last scattering.

At the very least, probes of the large-angle (low-

If indeed the observed

While the further WMAP data is not expected to change any of the observed results, our understanding and analysis techniques are likely to improve. Much work remains to study the large-scale correlations using improved foreground treatment, accounting even for the subtle systematics and in particular studying the time-ordered data from the spacecraft. The Planck experiment will be of great importance, as it will provide maps of the largest scales obtained using a very different experimental approach than WMAP—measuring the absolute temperature rather than temperature differences. Polarization maps, when available at high enough signal-to-noise at large scales (which may not be soon), will be a fantastic independent test of the alignments, as each explanation for the alignments, in principle, also predicts the statistics of the polarization pattern on the sky.

D. Huterer is supported by DOE OJI grant under Contract DE-FG02-95ER40899 and NSF under contract AST-0807564. D. Huterer and C. J. Copi are supported by NASA under Contract NNX09AC89G; D. J. Schwarz is supported by Deutsche Forschungsgemeinschaft (DFG); G. D. Starkman is supported by a grant from the US Department of Energy; both G. D. Starkman and C. J. Copi are supported by NASA under Cooperative Agreement NNX07AG89G.