Large-Angle Anomalies in the Microwave Background
aa r X i v : . [ a s t r o - ph . C O ] J un LARGE-ANGLE ANOMALIES IN THE MICROWAVE BACKGROUND
E.F. BUNN
Physics Department, University of Richmond28 Westhampton Way, Richmond, VA 23173, USA
Several claims have been made of anomalies in the large-angle properties of the cosmic mi-crowave background anisotropy as measured by WMAP. In most cases, the statistical signifi-cance of these anomalies is hard or even impossible to assess, due to the fact that the statisticsused to quantify the anomalies were chosen a posteriori. On the other hand, the possibilityof detecting new physics on the largest observable scales is so exciting that, in my opinion,it is worthwhile to examine the claims carefully. I will focus on three particular claims: thelack of large-angle power, the north-south power asymmetry, and multipole alignments. In allcases, the problem of a posteriori statistics can best be solved by finding a new data set thatprobes similar physical scales to the large-angle CMB. This is a difficult task, but there aresome possible routes to achieving it.
Our understanding of cosmology has advanced extremely rapidly in the past decade, due inlarge part to observations of cosmic microwave background (CMB) anisotropy, particularly thedata from the Wilkinson Microwave Anisotropy Probe (WMAP).1 , , , , , , , , ,
11 andhemispheric asymmetries12 , ,
14. The WMAP collaboration has conducted a thorough reviewand analysis of the claimed anomalies.15The significance of and explanations for these puzzles are uncertain, largely because of theproblem of a posteriori statistics. The typical sequence of events in the discovery of a CMBanomaly is as follows: some unusual feature is noticed in the data, and afterwards (a posteriori)a statistic is devised to quantify the unusualness of this feature. The p -values from such astatistic cannot be taken at face value: in any moderately large data set, it is always possible tofind something that looks odd, and a statistic engineered to capture that oddness will have anartificially low probability. Chance fluctuations can therefore incorrectly seem to be in need ofexplanation.Figure 1 illustrates this with a simple toy example. A Gaussian random map was made igure 1: A toy example of the dangers of a posteriori statistics. The left panel shows a Gaussian, statisticallyisotropic simulation of a CMB map, smoothed to show only large-scale anisotropy. As the right-panel shows,the one-point probability distribution of the anisotropy has an unusually high skewness. This skewness can be“explained” by noting the presence of a pair of almost perfectly antipodal extreme hot spots. that by chance has noticeable positive skewness. Someone looking for an explanation for thisskewness might be tempted to examine the most extreme hot spots in the map and would findthat they are almost perfectly antipodal. The probability that a map’s two most extreme hotspots are as far apart as in this map is less than 1%. One might be tempted to speculate on thepossible explanations for this unlikely pair of hot spots, but of course there is none.One can (and from a formal statistical point of view, arguably one must) dismiss the entiresubject of CMB anomalies because of this problem, but I believe that a more nuanced view iscalled for. Scientists often by necessity use non-rigorous (or even “invalid”) statistical methods,especially in preliminary analyses. As long as we maintain a skeptical stance and seek furthertests that can be done of any hypotheses that result from such an analysis, these methodscan yield fruitful insights. Considering the importance of finding ways to test the largest-scaleproperties of the Universe, I suggest that is neither necessary nor wise to dismiss the subjectout of hand.In this paper, I will not discuss all of the claims of anomalies but rather focus on thethree that in various ways seem to me most interesting: the large-scale power deficit, evidencefor hemispheric power modulation, and alignment of low-order multipoles. I will also discusspossible future directions for testing hypotheses arising from these anomalies. Ever since the first all-sky CMB maps were made by the COBE satellite,16 questions have beenraised about the low amplitude of fluctuations on the largest angular scales. In the observedangular power spectrum, the quadrupole (the largest-scale data point) is lower than expected,although given the large cosmic variance in the quadrupole as well as the need to mask part ofthe sky, this discrepancy is not extremely statistically significant. The lack of large-scale powerappears more striking when viewed in real space rather than the spherical harmonic space ofthe power spectrum. As Figure 2 illustrates, the two-point correlation function is very close tozero for all angles θ & ◦ , unlike typical simulations.To quantify this behavior, the statistic S / is defined to be17 S / = Z / − [ C ( θ )] d cos θ. (1)The value of this statistic is low compared to simulations at a confidence level of approximately
20 40 60 80 100 120 140 160 180 θ (degrees) -400-300-200-1000100200300400500600700 C ( θ ) ( µ K ) QVWILC (kp0)ILC (full)from (MLE) C l LCDM
Figure 2: From Copi et al.7 two-point correlation function for the three-year WMAP data. The blue band showsthe expected range from simulations. the chief way CMB anisotropy was quantified. The qualitative difference between the observedWMAP correlation function and theoretical predictions is therefore intriguing.If we tentatively assume that the lack of large-scale power does require an explanation, thenit is natural to ask what form that explanation might take. We can rule out one broad class ofexplanations, namely those that involve a statistically independent additive contaminant to thedata.7 ,
18 The reason is that such a contaminant always biases the expected amount of large-scalepower up, rendering the low observed value less likely, not more. It is clear that a contaminantalways adds power in an rms sense: the quadrupole (or any other multipole) is simply thequadrature sum of the contributions from the true CMB anisotropy and the contaminant. ButI am making the stronger statement that the entire probability distribution shifts in such a waythat a low value of the power becomes more improbable (e.g., Figure 3). This statement istrue whether the lack of large-scale power is quantified by the observed quadrupole7 or by thestatistic S / .18 It is independent of any assumptions about the statistics of the contaminant,as long as it is independent of the intrinsic CMB anisotropy.For example, this result rules out an undiagnosed foreground as an explanation for the powerdeficit. While it is possible that a foreground could cancel the intrinsic large-scale power, such achance cancellation is always less likely than the power coming out low without the contaminant.In fact, if a foreground contaminant is invoked for some other reason (e.g., to explain one of theother anomalies), it will exacerbate the large-scale power deficit problem.In addition to foreground contaminants, some more exotic models also fall into the categoryruled out by this model, such as ellipsoidal models19, some models with large-scale magneticfields20, etc. igure 3: From Bunn & Bourdon18. The solid curve shows the cumulative probability distribution for the statistic S / . The dashed curves show the distributions for various anisotropic (ellipsoidal) models. At any given valueof S / , the probability is always largest for the standard model. In the standard cosmological paradigm, CMB anisotropy is generated by a statistically isotropicrandom process, meaning that all directions should be, on average, identical. However, thereappears to be a large-scale modulation in power in the WMAP sky maps: more fluctuationpower is seen in one hemisphere than in the opposite hemisphere. Figure 4a illustrates thispower asymmetry.Figure 4b shows an important test of this anomaly performed by Hansen et al.14. TheWMAP data are filtered into non-overlapping ranges of multipoles: l = 2 − , − , . . . .In each case, the direction is found that maximizes the power asymmetry (i.e., maximizing theratio of power in a hemisphere centered on that direction to power in the opposite hemisphere).In the standard model, these directions should all be independent random variables, but theyare clearly closely correlated with each other. Even if the initial detection of a power asymmetryis contaminated by the problem of a posteriori statistics, this close correlation provides a largelyindependent test that is relatively free of this contamination.I believe that this test provides strong evidence that the power asymmetry is truly presentin the data, but this does not mean that it is cosmological. As indicated in the Figure, thepower-maximizing direction is quite close to the south ecliptic pole. If this alignment is not acoincidence, then the hemisphere asymmetry has a local cause, perhaps related to the WMAPscan strategy. The alignment of low-order multipoles, especially the quadrupole and octopole ( l = 2 ,
3) mayhave received the most attention of all of the claimed anomalies. For each multipole, one candefine a plane in which the fluctuations preferentially lie, using an angular-momentum statistic8or multipole vectors10. These directions are expected to be independent of each other but aresurprisingly closely aligned. In addition, the directions perpendicular to these planes are close toboth the CMB dipole and the ecliptic plane (see Figure 5). Depending on which of these surprisesone chooses to consider and how one chooses to quantify them, it is easy to get p -values of 10 − or less9. (Of course, as with the hemisphere asymmetry, if there is a cosmological explanationfor the alignments, then the alignment with the ecliptic must be a mere coincidence.)Yet again the problem of a posteriori statistics rears its head. The lowest p -values arise from Figure 4: The left panel (from Eriksen et al.12) illustrates the hemisphere asymmetry seen in WMAP. The colorof eac large disk indicates the ratio of power in a hemisphere centered on the disk to power in the oppositehemisphere, considering only multipoles l = 2-63. The right panel (from Hansen et al.14) shows the directionsyielding maximum power asymmetry for different ranges of multipoles. considering alignment of things that have been seen to be aligned. It is difficult to know howto correct this for the various alignments that could have been seen but weren’t. Reasonablepeople can (and do) differ over how much weight to give to the various multipole alignments. Inmy opinion, it is impossible to be confident that the observed alignments are significant, but itis reasonable to use them to generate hypotheses for future examination and then look for waysto test these hypotheses with new data sets that are independent of the large-angle CMB. If we tentatively assume that a given anomaly is “real” (i.e., not merely a statistical fluctuationamplified by an a posteriori statistic), then it is natural to ask what explanations might bepossible. Possibilities include systematic errors, foreground contaminants (although not in thecase of the power deficit), or more exotic explanations involving new physics. Examples from thelatter category are theories that define a preferred direction in space, either through spontaneousisotropy breaking21 or by the presence of a vector field during inflation22.Because the alternative theories generally have additional free parameters (and usually in-clude the standard model as a limiting case), they typically can provide better fits to the data.A model selection criterion is required to decide whether the improved goodness of fit is worththe “cost” of a more complex theory. Perhaps the most natural such criterion is the Bayesianevidence ratio23 , , , , , ,
29, which is essentially the factor by which the posterior prob-ability ratio of two theories is increased, in comparison to the prior ratio, by the acquisitionof the new data. The evidence ratio automatically disfavors complicated theories (i.e., thosewith large parameter spaces) unless the improvement in fit is correspondingly large; in otherwords, it automatically incorporates a form of Occam’s razor. Recent work26 ,
29 has attemptedto quantify the Bayesian evidence ratios for certain classes of theories. Figure 6 illustrates anexample, in which we quantify the degree to which the quadrupole-octopole alignment improvesthe likelihood of spontaneous isotropy breaking and preferred direction models. Although theevidence ratios exceed 1, indicating that the more complicated models go up in probability asa result of the multipole alignment, the improvement is extremely modest. In general, one doesnot pay much attention to Bayesian evidence ratios unless they are far larger than these values23.Because of the uncertainties surrounding the interpretation of the statistics of the variousCMB anomalies, they should be regarded chiefly as potentially useful guides in formulatinghypotheses for further testing. The essential next step, therefore, is to find new data setsthat can be used to test any such hypotheses. To be specific, we need to find data sets that igure 5: From Schwarz et al.9 The quadrupole plus octupole of the WMAP data. Several directions that can becomputed from these multipoles are indicated, along with the orientations of the ecliptic and dipole. probe comparable physical scales to the large-angle CMB (i.e., gigaparsec scales in comovingcoordinates) but that are independent of the CMB anisotropy modes, which have already beenmeasured to the cosmic-variance limit. Finding such data sets is nontrivial, of course, but itmay not be impossible.The first natural place to look is to CMB polarization. For any given anomaly, one canimagine devising (a priori) statistical tests to be performed on polarization maps to look forthe anomaly’s presence. For example, one can compute the two-point correlation function of amap of E-type CMB polarization, and see if it shows the same lack of large-angle power as thetemperature anisotropy. Because there are correlations between temperature and E polarization,this is not, strictly speaking, an independent test, but in practice it is nearly so: as Figure 7shows, the predicted probability distribution for an S / statistic computed from a polarizationmap is essentially independent of the value of S / for temperature. When CMB polarizationdata are good enough to allow reliable estimation of the correlation function, we can compute S / . If it is anomalously low, then we have found independent evidence that this puzzle requiresan explanation.Unfortunately, this test is likely to be of less value than it might initially appear: the bulk ofthe large-angle power in CMB polarization data comes from photons that last scattered at lowredshift (after reionization), and hence probes far smaller length scales than the correspondingtemperature data. Thus even if there is a cosmological explanation for the lack of large-scaletemperature correlations, we would probably not expect to find confirmation of it in polarization.The same conclusion would likely apply to tests of other anomalies. A polarization map doesin principle contain information on large length scales that is independent of the temperaturedata, but it is not obvious (at least to me) that this information can be separated from thereionization signal in a way that would allow a clear test of the anomalies.There are other possibilities for independent probes of perturbations on gigaparsec scales.Since the CMB anisotropy primarily probes the surface of last scattering, methods that samplethe interior of our horizon volume will generically provide independent data sets. One methodthat might prove promising in the future is the Kamionkowski-Loeb effect:30 by measuring thepolarization of the Sunyaev-Zel’dovich-scattered photons coming from a galaxy cluster, one caninfer the CMB temperature quadrupole measured at that cluster’s location and look-back time.A sample of such cluster measurements can be used to reconstruct modes on length scalescorresponding to the CMB modes at l ∼ ,
32 This is a challenging task, but the generationof telescopes currently being developed, such as SPTPol and ACTPol, are capable of achievingit. - log Σ max L Figure 6: An illustration of the use of Bayesianevidence to decide whether anomalies in the datawarrant adoption of a more complicated model.The quantity plotted is the Bayesian evidence ratiocomparing various anisotropic models (two basedon the idea of spontaneous isotropy breaking andone involving a preferred direction during inflation)with the standard model. The statistic used incomputing the evidence ratio is based on the mul-tipole vector method of quantifying quadrupole-octopole alignment. For further details, see Zheng& Bunn.29 Figure 7: The cumulative probability distribu-tion of the statistic S / for E-mode polarization,as predicted by the standard model. The solidcurve shows the overall probability distribution, de-rived from simulations based on a standard ΛCDMmodel. The dashed curve is the probability distri-bution conditioned on extremely low values (firstpercentile) of the temperature S / statistic. Theclose correspondence of the curves shows that a lowvalue of S / for polarization may be regarded asindependent, to a good approximation, of the mea-surement of a low S / in temperature. The subject of large-angle CMB anomalies remains controversial, largely because of the diffi-culty in interpreting a posteriori statistics. While reasonable people can and do conclude thatthe correct attitude is to dismiss the subject entirely, I believe that a more nuanced view isappropriate, in which we view some anomalies as providing hints of possible new directions toexplore. With this attitude, it is of course essential to seek rigorous tests of any hypothesesgenerated. Such tests may be difficult but not impossible to find.
Acknowledgments
This work was supported by US National Science Foundation Awards 0507395 and 0908319. Ithank the Laboratoire Astroparticule-Cosmologie of the Universit´e Paris VII for their hospitalityduring some of the time this work was prepared.
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