## A Bayesian Look at Cosmic Anomalies

Posted in Cosmic Anomalies with tags , , , on March 3, 2019 by telescoper

I’ve posted a few times on this blog about Cosmic Anomalies, by which I mean apparent departures from the predictions of the standard cosmological model. From time to time I also talk about this subject at seminars and conferences.

There’s an interesting new paper on this topic on the arXiv now by Shaikh et al., with the following abstract:

You can click on the image to make it larger. You can also find the PDF version of the full paper here.

I find this Bayesian analysis of two of the apparent anomalies (low amplitude in the power spectrum at large angular scales and hemispherical power asymmetry) may be different manifestations of the same underlying phenomenon, which would make them easier to account for without invoking new physics. Rather than being two independent statistical flukes these measurements might both be the result of one, which would be more likely to occur in the standard model. This analysis however suggests that this might not be the case after all, and these are two different things after all. This presupposes, however, that the model chosen to describe the asymmetries is appropriate. Anyway, this paper is well worth a read if you’re into Bayesian model testing (which you should be)…

This also gives me the excuse to post the following poll, which has been running for several years (even longer than Brexit):

## Stokes, Lonsdale and DCU

Posted in Cosmic Anomalies, Maynooth, Talks and Reviews, The Universe and Stuff with tags , , , on November 2, 2018 by telescoper

On Wednesday I took a trip from Maynooth into Dublin to give a talk at the Centre for Astrophysics and Relativity at Dublin City University (DCU). I’ve stolen the above picture, which someone took near the start of the talk, from Twitter.

My talk was very general, as it was not a specialist cosmology audience, and was similar to the talks I was giving a few years ago about the Axle of Elvis Axis of Evil. If anyone is interested in the slides, here they are.

Confusingly, Dublin City University (DCU) consists of the same combination of quarks as University College Dublin (UCD), but I managed to find my way to the correct campus via Drumcondra Railway Station (which is next to historic Croke Park). Anyway, there was quite a big audience and not all of them fell asleep (even though I did go on too long) so by that measure at least the talk was moderately successful. Thanks to everyone there for their hospitality during the afternoon!

Incidentally, my talk was in the Lonsdale Building which is right next to the Stoke Building. Both are named in honour of famous Irish-born scientists. physicist George Stokes (who was born in Skreen, in County Sligo, but spent most of his adult life in Cambridge) and crystallographer Kathleen Lonsdale (who was born in Newbridge, County Kildare, but moved to England when she was only five).

## The Supervoid and the Cold Spot

Posted in Astrohype, Cosmic Anomalies, The Universe and Stuff with tags , , , , , on April 21, 2015 by telescoper

While I was away at the SEPnet meeting yesterday a story broke in the press broke about the discovery of a large underdensity in the distribution of galaxies. The discovery is described in a paper by Szapudi et al. in the journal Monthly Notices of the Royal Astronomical Society. The claim is that this structure in the galaxy distribution can account for the apresence of a mysterious cold spot in the cosmic microwave background, shown here (circled) in the map generated by Planck:

The abstract of the latest paper is here:

We use the WISE-2MASS infrared galaxy catalogue matched with Pan-STARRS1 (PS1) galaxies to search for a supervoid in the direction of the cosmic microwave background (CMB) cold spot (CS). Our imaging catalogue has median redshift z ≃ 0.14, and we obtain photometric redshifts from PS1 optical colours to create a tomographic map of the galaxy distribution. The radial profile centred on the CS shows a large low-density region, extending over tens of degrees. Motivated by previous CMB results, we test for underdensities within two angular radii, 5°, and 15°. The counts in photometric redshift bins show significantly low densities at high detection significance, ≳5σ and ≳6σ, respectively, for the two fiducial radii. The line-of-sight position of the deepest region of the void is z ≃ 0.15–0.25. Our data, combined with an earlier measurement by Granett, Szapudi & Neyrinck, are consistent with a large Rvoid = (220 ± 50) h−1 Mpc supervoid with δm ≃ −0.14 ± 0.04 centred at z = 0.22 ± 0.03. Such a supervoid, constituting at least a ≃3.3σ fluctuation in a Gaussian distribution of the Λ cold dark matter model, is a plausible cause for the CS.

The result is not entirely new: it has been discussed at various conferences over the past year or so (e.g this one) but this is the first refereed paper showing details of the discovery.

This gives me the excuse to post this wonderful cartoon, the context of which is described here. Was that really in 1992? That was twenty years ago!

Anyway, I just wanted to make a few points about this because some of the press coverage has been rather misleading. I’ve therefore filed this one in the category Astrophype.

First, the “supervoid” structure that has been discovered is not a “void”, which would be a region completely empty of galaxies. As the paper makes clear it is less dramatic than that: it’s basically an underdensity of around 14% in the density of galaxies. It is (perhaps) the largest underdensity yet found on such a large scale – though that depends very much on how you define a void – but it is not in itself inconsistent with the standard cosmological framework. Such large underdensities are expected to be rare, but rare things do occur if you survey a large enough volume of the universe. Large overdensities also arise as statistical fluctuations in large volumes.

Second, and probably most importantly, although this “supervoid” is in the direction of the CMB Cold Spot it cannot on its own explain the Cold Spot; the claim in the abstract that it provides a plausible explanation of the cold spot is simply incorrect. A void can affect the measured temperature of the CMB through the Integrated Sachs-Wolfe effect: photons travelling through such a structure are redshifted as they travel through the underdense region, so the CMB looks cooler in the direction of the void. However, even optimistic calculations of the magnitude of the effect suggest that this particular “void” can only account for about 10% of the signal associated with the Cold Spot. This is a reasonably significant contribution but it does not account for the signal on its own.

This is not to say however that it is irrelevant. It could well be that the supervoid actually sits in front of a region of the CMB sky that was already cold, as a result of a primordial fluctuation rather than a line-of-sight effect. Such an effect could well arise by chance, at least with some probability. If the original perturbation were a “3σ” temperature fluctuation then the additional effect of the supervoid would turn it into a 3.3σ effect. Since this pushes the event further out into the tail of the probability distribution it makes a reasonably uncommon feature look  less probable. Because the tail of a Gaussian distribution drops off very quickly this has quite a large effect on the probability. For example, a fluctuation of 3.3σ or greater has a probability of 0.00048 whereas one of 3.0σ has a probability of 0.00135, about a factor of 2.8 larger. That’s an effect, but not a large one.

In summary, I think the discovery of this large underdensity is indeed interesting but it is not a plausible explanation for the CMB Cold Spot. Not, that is, unless there’s some new physical process involved in the propagation of light that we don’t yet understand.

Now that would be interesting…

## Why the Universe is (probably) not rotating

Posted in Cosmic Anomalies, The Universe and Stuff with tags , , , , , on October 1, 2013 by telescoper

Just a quick post to point you towards a nice blog post by Jason McEwen entitled Is the Universe rotating? It’s a general rule that if  an article has a question for a title then the answer to that question is probably “no”, and “probably no” is indeed the answer in this case.

The item relates to a paper by McEwen et al whose abstract is given here:

We perform a definitive analysis of Bianchi VII_h cosmologies with WMAP observations of the cosmic microwave background (CMB) temperature anisotropies. Bayesian analysis techniques are developed to study anisotropic cosmologies using full-sky and partial-sky, masked CMB temperature data. We apply these techniques to analyse the full-sky internal linear combination (ILC) map and a partial-sky, masked W-band map of WMAP 9-year observations. In addition to the physically motivated Bianchi VII_h model, we examine phenomenological models considered in previous studies, in which the Bianchi VII_h parameters are decoupled from the standard cosmological parameters. In the two phenomenological models considered, Bayes factors of 1.7 and 1.1 units of log-evidence favouring a Bianchi component are found in full-sky ILC data. The corresponding best-fit Bianchi maps recovered are similar for both phenomenological models and are very close to those found in previous studies using earlier WMAP data releases. However, no evidence for a phenomenological Bianchi component is found in the partial-sky W-band data. In the physical Bianchi VII_h model we find no evidence for a Bianchi component: WMAP data thus do not favour Bianchi VII_h cosmologies over the standard Lambda Cold Dark Matter (LCDM) cosmology. It is not possible to discount Bianchi VII_h cosmologies in favour of LCDM completely, but we are able to constrain the vorticity of physical Bianchi VII_h cosmologies at $(\omega/H)_0 < 8.6 \times 10^{-10}$ with 95% confidence.

For non-experts the Bianchi cosmologies are based on exact solutions of Einstein’s equations for general relativity which obey the condition that they are spatially homogeneous but not necessarily isotropic. If you find that concept hard to understand, imagine a universe which looks the same everywhere but which is pervaded by a uniform magnetic field: that would be homogeneous (because every place is identical) but anisotropic (because there is a preferred direction – along the magnetic field lines). Another example of would be s a universe which is, for reasons known only to itself, rotating; the preferred direction here is the axis of rotation. The complete classification of all Bianchi space-times is discussed here. I also mentioned them and showed some pictures on this blog here.

As Jason’s post explains, observations of the cosmic microwave background by the Wilkinson Microwave Anisotropy Probe (WMAP) suggest  that there is something a little bit fishy about it: it seems to be have an anomalous large-scale asymmetry not expected in the standard cosmology. These suggestions seem to be confirmed by Planck, though the type of analysis done for WMAP has not yet been performed for Planck. The paper mentioned above investigates whether the WMAP asymmetry could be accounted for by one particular Bianchi cosmology, i.e. Bianchi VII_h. This is quite a complicated model which has negative spatial curvature, rotation (vorticity) and shear; formally speaking, it is the most general Bianchi model of any type that includes the standard Friedmann cosmology as a special case.

The question whether such a complicated model actually provides a better fit to the data than the much simpler standard model is one naturally answered by Bayesian techniques that trade off the increased complexity of a more sophisticated model  against the improvement in goodness-of-fit achieved by having more free parameters.  Using this approach McEwen et al. showed that, in simple  terms, while a slight improvement in fit is indeed gained by adding a Bianchi VII_h component to the model,  the penalty paid in terms of increased complexity means that the alternative model is not significantly more probable than the simple one. Ockham’s Razor strikes again! Although this argument does not definitively exclude the possibility that the Universe is rotating, it does put limits on how much rotation there can be. It also excludes one possible explanation of the  peculiar pattern  of the temperature fluctuations seen by WMAP.

So what does cause the anomalous behaviour of the cosmic microwave background?

I have no idea.

## Planck, Pointillism and the Axle of Elvis

Posted in Art, Biographical, Cosmic Anomalies, Open Access, The Universe and Stuff with tags , , , , , , on March 21, 2013 by telescoper

The reason I was out of the office yesterday was that I was in Cambridge, doing a PhD oral in the Cavendish Laboratory so the first thing to say is congratulations Dr Johnston! It was one of those viva voce examinations that turned out to be less of an examination than an interesting chat about physics. In fact the internal examiner, Prof. Steve Gull, seemed to spend more time asking me questions rather than the candidate!

Afterwards I met up with Anthony Lasenby, the candidate’s supervisor. Not surprisingly the main topic of our brief discussion was today’s impending announcement of results from Planck. Anthony is one of the folks who have been involved with Planck for about twenty years, since it began as a twinkle in the eye of COBRAS/SAMBA. I was looking forward to getting in bright and early this morning to watch the live streaming of the Planck press conference from Paris.

Unfortunately however, I could feel a bit of a lurgy coming on as I travelled to Cambridge yesterday. It got decidedly worse on the way home – it must have been the Cambridge air – and I even ended up passing out on the train from Victoria to Brighton. Fortunately, Brighton was the terminus so someone woke me up when we got there and I got home, coughing and spluttering. I suspect many cosmologists didn’t sleep well last night because of excitement about the Planck results, but in my case it was something else that kept me awake. Anyway, I didn’t make it in this morning so had to follow the announcements via Twitter. Fortunately there’s a lot of press coverage too; see the ESA site and a nice piece by the BBC’s redoubtable Jonathan Amos.

Anyway, without further ado, here’s Planck’s map of the cosmic microwave background:

It’s rather beautiful, in a pointillist kind of way, I think…

It will take me a while in my weakened state to complete a detailed study of the results – and I’m sure to return to them many times in the future, but I will make a couple of points now.

The first is that the papers and data products are all immediately available online. The papers will all appear on the arXiv. Open Access sceptics please take note!

The second is that the most interesting result (as far as I’m concerned) is that at least some of the cosmic anomalies I’ve blogged about in the past, such as the Axle of Elvis Axis of Evil and the famous colder-than-it-should-be cold spot, are still present in the Planck data:

The other results excite me less because, at a quick reading, they all seem to be consistent with the standard cosmological model. Of course, the north-south asymmetry is a small effect on could turn out to be a foreground (e.g. zodiacal emission) or an artefact of the scanning strategy. But if it isn’t a systematic it could be very important. I suspect there’ll be a rush of papers about this before long!

I’m sure to p0st much more about the Planck results in due course, but I think I’ll leave it there for now. Please feel free to post comments and reactions through the box below.

## “Cosmic Anomalies” Talk, Copenhagen, August 2011

Posted in Art, Books, Talks and Reviews, Cosmic Anomalies, The Universe and Stuff with tags , , on August 31, 2011 by telescoper

I think I’m getting the hang of this slideshare malarky so I thought I’d try it out by posting the slides I used for my (short) talk at the workshop in Copenhagen I told you about two or three weeks ago. I’m not sure how useful they will be to anyone, as I suppose it will be quite hard to reconstruct the talk using only the small amount of information I bother to put on the slides..

If you’re wondering about the presence of various apparently random works of art then what can I say? I like paintings!

## DEUS

Posted in Cosmic Anomalies, The Universe and Stuff with tags , , , on August 12, 2011 by telescoper

Well, I’m back home from Copenhagen after a very interesting and stimulating workshop called “DEUS” (subtitled “Current and Future Challenges of the Dark and Early Universes”). I just thought I’d post a brief message to thank the organizers for inviting me and for arranging such an interesting and varied programme, and especially for giving so many young researchers the chance to give talks (as well as some old farts like me).

Although I’d originally planned to talk about something else, I evenually decided to do a variation on the theme of cosmic anomalies, a topic I’ve blogged about at various times over the past couple of years. In a nutshell this was a quick overview of various features of the observed universe that seem to suggest departures from the standard “Lambda-CDM”  (or LCDM, for short) cosmological model, including the famous WMAP Cold Spot, the Axis of Evil, and various other statistical hints of anomalous behaviour in present-day observations.

To add a bit of audience participation I gave those attending my talk the chance to vote on what they thought about these – I was genuinely interested to see what this particular audience felt about whether the standard model is threatened or not.  I asked specifically about these in order to exclude other niggling worries people might have about LCDM from other astrophysical arguments, such as galaxy formation. Anyway, I thought it might be fun to repeat the poll here, so feel free to add your vote here:

As for the results of the vote during my presentation, I was somewhat surprised to see a roughly equal division between A and B, but there were even a few in C. I had assumed the vast majority would vote “A”….

Anyway, thanks again to the organizers of a fun meeting. That’s three trips to Copenhagen in as many months. I guess it will be a while before I go back again. 😦

## Hints of Bubbles in the Background?

Posted in Astrohype, Cosmic Anomalies, The Universe and Stuff with tags , , , on August 4, 2011 by telescoper

Looking around for a hot cosmological topic for a brief diversionary post, I came across a news item on the BBC website entitled ‘Multiverse theory suggested by microwave background‘. I’ll refer you to the item itself for a general description of the study and to the actual paper (by Feeney et al.), which has been accepted for publication in Physical Review D, for technical details.

I will, however, flagrantly steal Auntie Beeb’s nice picture which shows the location on the sky of a number of allegedly anomalous features; they being the coloured blobs that look like Smarties in the bottom right. The greyed out bits of the map are areas of the sky masked out to avoid contamination from our own Galaxy or various other foreground sources.

One possible explanation of the Smarties from Outer Space is furnished by a variant of the theory known as chaotic inflation in which the universe comprises a collection of mini-universes  which nucleate and expand rather like bubbles in a glass of champagne. Assuming this “multiverse” picture is correct – a very big “if”, in my opinion –  it is just possible that two bubbles might collide just after nucleation leaving a sort of dent in space that we see in the microwave background.

It’s a speculative idea, of course, but there’s nothing wrong with such things. Everything starts off with speculation, really. I’ve actually read the paper, and I think it’s an excellent piece of work.  I can’t resist commenting, however, that there’s a considerable gap between the conclusions of the study and the title of the BBC article, either the present Multiverse  theory suggested by microwave background’ or the original one Study hints at bubble universes’.

My point is that the authors  concede that they do not find any statistically significant evidence for the bubble collision interpretation, i.e. this is essentially  a null result. I’m not sure how “study fails to find evidence for..” turned into “study hints at…”.

Nonetheless, it’s an interesting paper and there’s certainly a possibility that better, cleaner and less noisy data  may find evidence where WMAP couldn’t. Yet another reason to look forward to future data from Planck!

## Doubts about the Evidence for Penrose’s Cyclic Universe

Posted in Bad Statistics, Cosmic Anomalies, The Universe and Stuff with tags , , , , , , on November 28, 2010 by telescoper

A strange paper by Gurzadyan and Penrose hit the Arxiv a week or so ago. It seems to have generated quite a lot of reaction in the blogosphere and has now made it onto the BBC News, so I think it merits a comment.

The authors claim to have found evidence that supports Roger Penrose‘s conformal cyclic cosmology in the form of a series of (concentric) rings of unexpectedly low variance in the pattern of fluctuations in the cosmic microwave background seen by the Wilkinson Microwave Anisotropy Probe (WMAP). There’s no doubt that a real discovery of such signals in the WMAP data would point towards something radically different from the standard Big Bang cosmology.

I haven’t tried to reproduce Gurzadyan & Penrose’s result in detail, as I haven’t had time to look at it, and I’m not going to rule it out without doing a careful analysis myself. However, what I will say here is that I think you should take the statistical part of their analysis with a huge pinch of salt.

Here’s why.

The authors report a hugely significant detection of their effect (they quote a “6-σ” result; in other words, the expected feature is expected to arise in the standard cosmological model with a probability of less than 10-7. The type of signal can be seen in their Figure 2, which I reproduce here:

Sorry they’re hard to read, but these show the variance measured on concentric rings (y-axis) of varying radius (x-axis) as seen in the WMAP W (94 Ghz) and V (54 Ghz) frequency channels (top two panels) compared with what is seen in a simulation with purely Gaussian fluctuations generated within the framework of the standard cosmological model (lower panel). The contrast looks superficially impressive, but there’s much less to it than meets the eye.

For a start, the separate WMAP W and V channels are not the same as the cosmic microwave background. There is a great deal of galactic foreground that has to be cleaned out of these maps before the pristine primordial radiation can be isolated. The fact similar patterns can be found in the BOOMERANG data by no means rules out a foreground contribution as a common explanation of anomalous variance. The authors have excluded the region at low galactic latitude (|b|<20°) in order to avoid the most heavily contaminated parts of the sky, but this is by no means guaranteed to eliminate foreground contributions entirely. Here is the all-sky WMAP W-band map for example:

Moreover, these maps also contain considerable systematic effects arising from the scanning strategy of the WMAP satellite. The most obvious of these is that the signal-to-noise varies across the sky, but there are others, such as the finite size of the beam of the WMAP telescope.

Neither galactic foregrounds nor correlated noise are present in the Gaussian simulation shown in the lower panel, and the authors do not say what kind of beam smoothing is used either. The comparison of WMAP single-channel data with simple Gaussian simulations is consequently deeply flawed and the significance level quoted for the result is certainly meaningless.

Having not looked looked at this in detail myself I’m not going to say that the authors’ conclusions are necessarily false, but I would be very surprised if an effect this large was real given the strenuous efforts so many people have made to probe the detailed statistics of the WMAP data; see, e.g., various items in my blog category on cosmic anomalies. Cosmologists have been wrong before, of course, but then so have even eminent physicists like Roger Penrose…

Another point that I’m not sure about at all is even if the rings of low variance are real – which I doubt – do they really provide evidence of a cyclic universe? It doesn’t seem obvious to me that the model Penrose advocates would actually produce a CMB sky that had such properties anyway.

Above all, I stress that this paper has not been subjected to proper peer review. If I were the referee I’d demand a much higher level of rigour in the analysis before I would allow it to be published in a scientific journal. Until the analysis is done satisfactorily, I suggest that serious students of cosmology shouldn’t get too excited by this result.

It occurs to me that other cosmologists out there might have looked at this result in more detail than I have had time to. If so, please feel free to add your comments in the box…

IMPORTANT UPDATE: 7th December. Two papers have now appeared on the arXiv (here and here) which refute the Gurzadyan-Penrose claim. Apparently, the data behave as Gurzadyan and Penrose claim, but so do proper simulations. In otherwords, it’s the bottom panel of the figure that’s wrong.

ANOTHER UPDATE: 8th December. Gurzadyan and Penrose have responded with a two-page paper which makes so little sense I had better not comment at all.

## Cosmology on its beam-ends?

Posted in Cosmic Anomalies, The Universe and Stuff with tags , , , , on June 14, 2010 by telescoper

Interesting press release today from the Royal Astronomical Society about a paper (preprint version here) which casts doubt on whether the Wilkinson Microwave Anisotropy Probe supports the standard cosmological model to the extent that is generally claimed. Apologies if this is a bit more technical than my usual posts (but I like occasionally to pretend that it’s a science blog).

The abstract of the paper (by Sawangwit & Shanks) reads

Using the published WMAP 5-year data, we first show how sensitive the WMAP power spectra are to the form of the WMAP beam. It is well known that the beam profile derived from observations of Jupiter is non-Gaussian and indeed extends, in the W band for example, well beyond its 12.’6 FWHM core out to more than 1 degree in radius. This means that even though the core width corresponds to wavenumber l ~ 1800, the form of the beam still significantly affects the WMAP results even at l~200 which is the scale of the first acoustic peak. The difference between the beam convolved Cl; and the final Cl is ~ 70% at the scale of the first peak, rising to ~ 400% at the scale of the second.  New estimates of the Q, V and W-band beam profiles are then presented, based on a stacking analysis of the WMAP5 radio source catalogue and temperature maps. The radio sources show a significantly (3-4σ) broader beam profile on scales of 10′-30′ than that found by the WMAP team whose beam analysis is based on measurements of Jupiter. Beyond these scales the beam profiles from the radio sources are too noisy to give useful information. Furthermore, we find tentative evidence for a non-linear relation between WMAP and ATCA/IRAM 95 GHz source fluxes. We discuss whether the wide beam profiles could be caused either by radio source extension or clustering and find that neither explanation is likely. We also argue against the possibility that Eddington bias is affecting our results. The reasons for the difference between the radio source and the Jupiter beam profiles are therefore still unclear. If the radio source profiles were then used to define the WMAP beam, there could be a significant change in the amplitude and position of even the first acoustic peak. It is therefore important to identify the reasons for the differences between these two beam profile estimates.

The press release puts it somewhat more dramatically

New research by astronomers in the Physics Department at Durham University suggests that the conventional wisdom about the content of the Universe may be wrong. Graduate student Utane Sawangwit and Professor Tom Shanks looked at observations from the Wilkinson Microwave Anisotropy Probe (WMAP) satellite to study the remnant heat from the Big Bang. The two scientists find evidence that the errors in its data may be much larger than previously thought, which in turn makes the standard model of the Universe open to question. The team publish their results in a letter to the journal Monthly Notices of the Royal Astronomical Society.

I dare say the WMAP team will respond in due course, but this paper spurred me to mention some work on this topic that was done by my friend (and former student) Lung-Yih Chiang. During his last visit to Cardiff we discussed this at great length and got very excited at one point when we thought we had discovered an error along the lines that the present paper claims. However, looking more carefully into it we decided that this wasn’t the case and we abandoned our plans to publish a paper on it.

Let me show you a few slides from a presentation that Lung-Yih gave to me a while ago. For a start here is the famous power-spectrum of the temperature fluctuations of the cosmic microwave background which plays an essential role in determining the parameters of the standard cosmology:

The position of the so-called “acoustic peak” plays an important role in determining the overall curvature of space-time on cosmological scales and the higher-order peaks pin down other parameters. However, it must be remembered that WMAP doesn’t just observe the cosmic microwave background. The signal it receives is heavily polluted by contamination from within our Galaxy and there is also significant instrumental noise.  To deal with this problem, the WMAP team exploit the five different frequency channels with which the probe is equipped, as shown in the picture below.

The CMB, being described by a black-body spectrum, has a sky temperature that doesn’t vary with frequency. Foreground emission, on the other hand, has an effective temperature that varies with frequency in way that is fairly well understood. The five available channels can therefore be used to model and subtract the foreground contribution to the overall signal. However, the different channels have different angular resolution (because they correspond to different wavelengths of radiation). Here are some sample patches of sky illustrating this

At each frequency the sky is blurred out by the “beam” of the WMAP optical system; the blurring is worse at low frequencies than at high frequencies. In order to do the foreground subtraction, the WMAP team therefore smooth all the frequency maps to have the same resolution, i.e. so the net effect of optical resolution and artificial smoothing produces the same overall blurring (actually 1 degree).  This requires accurate knowledge of the precise form of the beam response of the experiment to do it accurately. A rough example (for illustration only) is given in the caption above.

Now, here are the power spectra of the maps in each frequency channel

Note this is Cl not l(l+1)Cl as in the first plot of the spectrum. Now you see how much foreground there is in the data: the curves would lie on top of each other if the signal were pure CMB, i.e. if it did not vary with frequency. The equation at the bottom basically just says that the overall spectrum is a smoothed version of the CMB plus the foregrounds  plus noise. Note, crucially,  that the smoothing suppresses the interesting high-l wiggles.

I haven’t got space-time enough to go into how the foreground subtraction is carried out, but once it is done it is necessary to “unblur” the maps in order to see the structure at small angular scales, i.e. at large spherical harmonic numbers l. The initial process of convolving the sky pattern with a filter corresponds to multiplying the power-spectrum with a “window function” that decreases sharply at high l, so to deconvolve the spectrum one essentially has to divide by this window function to reinstate the power removed at high harmonics.

This is where it all gets very tricky. The smoothing applied is very close to the scale of the acoustic peaks so you have to do it very carefully to avoid introducing artificial structure in Cl or obliterating structure that you want to see. Moreover, a small error in the beam gets blown up in the deconvolution so one can go badly wrong in recovering the final spectrum. In other words, you need to know the beam very well to have any chance of getting close to the right answer!

The next picture gives a rough model for how much the “recovered” spectrum depends on the error produced by making even a small error in the beam profile which, for illustration only, is assumed to be Gaussian. It also shows how sensitive the shape of the deconvolved spectrum is to small errors in the beam.

Incidentally, the ratty blue line shows the spectrum obtained from a small patch of the sky rather than the whole sky. We were interested to see how much the spectrum varied across the sky so broke it up into square patches about the same size as those analysed by the Boomerang experiment. This turns out to be a pretty good way of getting the acoustic peak position but, as you can see, you lose information at low l (i.e. on scales larger than the patch).

The WMAP beam isn’t actually Gaussian – it differs quite markedly in its tails, which means that there’s even more cross-talk between different harmonic modes than in this example – but I hope you get the basic point. As Sawangwit & Shanks say, you need to know the beam very well to get the right fluctuation spectrum out. Move the acoustic peak around only slightly and all bets are off about the cosmological parameters and, perhaps, the evidence for dark energy and dark matter. Lung-Yih looked at the way the WMAP had done it and concluded that if their published beam shape was right then they had done a good job and there’s nothing substantially wrong with the results shown in the first graph.

Sawangwit & Shanks suggest the beam isn’t right so the recovered angular spectrum is suspect. I’ll need to look a bit more at the evidence they consider before commenting on that, although if anyone else has worked through it I’d be happy to hear from them through the comments box!