Archive for Cold Spot

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:

Planck_CMB_large

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:

_66524456_66524455

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.

The Laws of Extremely Improbable Things

Posted in Bad Statistics, The Universe and Stuff with tags , , , , , , , , on June 9, 2011 by telescoper

After a couple of boozy nights in Copenhagen during the workshop which has just finished, I thought I’d take things easy this evening and make use of the free internet connection in my hotel to post a short item about something I talked about at the workshop here.

Actually I’ve been meaning to mention a nice bit of statistical theory called Extreme Value Theory on here for some time, because not so many people seem to be aware of it, but somehow I never got around to writing about it. People generally assume that statistical analysis of data revolves around “typical” quantities, such as averages or root-mean-square fluctuations (i.e. “standard” deviations). Sometimes, however, it’s not the typical points that are interesting, but those that appear to be drawn from the extreme tails of a probability distribution. This is particularly the case in planning for floods and other natural disasters, but this field also finds a number of interesting applications in astrophysics and cosmology. What should be the mass of the most massive cluster in my galaxy survey? How bright the brightest galaxy? How hot the hottest hotspot in the distribution of temperature fluctuations on the cosmic microwave background sky? And how cold the coldest? Sometimes just one anomalous event can be enormously useful in testing a theory.

I’m not going to go into the theory in any great depth here. Instead I’ll just give you a simple idea of how things work. First imagine you have a set of n observations labelled X_i. Assume that these are independent and identically distributed with a distribution function F(x), i.e.

\Pr(X_i\leq x)=F(x)

Now suppose you locate the largest value in the sample, X_{\rm max}. What is the distribution of this value? The answer is not F(x), but it is quite easy to work out because the probability that the largest value is less than or equal to, say, z is just the probability that each one is less than or equal to that value, i.e.

F_{\rm max}(z) = \Pr \left(X_{\rm max}\leq z\right)= \Pr \left(X_1\leq z, X_2\leq z\ldots, X_n\leq z\right)

Because the variables are independent and identically distributed, this means that

F_{\rm max} (z) = \left[ F(z) \right]^n

The probability density function associated with this is then just

f_{\rm max}(z) = n f(z) \left[ F(z) \right]^{n-1}

In a situation in which F(x) is known and in which the other assumptions apply, then this simple result offers the best way to proceed in analysing extreme values.

The mathematical interest in extreme values however derives from a paper in 1928 by Fisher \& Tippett which paved the way towards a general theory of extreme value distributions. I don’t want to go too much into details about that, but I will give a flavour by mentioning a historically important, perhaps surprising, and in any case rather illuminating example.

It turns out that for any distribution F(x) of exponential type, which means that

\lim_{x\rightarrow\infty} \frac{1-F(x)}{f(x)} = 0

then there is a stable asymptotic distribution of extreme values, as n \rightarrow \infty which is independent of the underlying distribution, F(x), and which has the form

G(z) = \exp \left(-\exp \left( -\frac{(z-a_n)}{b_n} \right)\right)

where a_n and b_n are location and scale parameters; this is called the Gumbel distribution. It’s not often you come across functions of the form e^{-e^{-y}}!

This result, and others, has established a robust and powerful framework for modelling extreme events. One of course has to be particularly careful if the variables involved are not independent (e.g. part of correlated sequences) or if there are not identically distributed (e.g. if the distribution is changing with time). One also has to be aware of the possibility that an extreme data point may simply be some sort of glitch (e.g. a cosmic ray hit on a pixel, to give an astronomical example). It should also be mentioned that the asymptotic theory is what it says on the tin – asymptotic. Some distributions of exponential type converge extremely slowly to the asymptotic form. A notable example is the Gaussian, which converges at the pathetically slow rate of \sqrt{\ln(n)}! This is why I advocate using the exact distribution resulting from a fully specified model whenever this is possible.

The pitfalls are dangerous and have no doubt led to numerous misapplications of this theory, but, done properly, it’s an approach that has enormous potential.

I’ve been interested in this branch of statistical theory for a long time, since I was introduced to it while I was a graduate student by a classic paper written by my supervisor. In fact I myself contributed to the classic old literature on this topic myself, with a paper on extreme temperature fluctuations in the cosmic microwave background way back in 1988..

Of course there weren’t any CMB maps back in 1988, and if I had thought more about it at the time I should have realised that since this was all done using Gaussian statistics, there was a 50% chance that the most interesting feature would actually be a negative rather than positive fluctuation. It turns out that twenty-odd years on, people are actually discussing an anomalous cold spot in the data from WMAP, proving that Murphy’s law applies to extreme events…

The Cold Spot

Posted in Cosmic Anomalies, The Universe and Stuff with tags , , , , on August 16, 2009 by telescoper

Musing yesterday about the rapidly approaching restart of the academic year reminded me that I really ought to get on and finish the bunch of papers sitting on my desk and on various computers. I’ve also got a book to finish before October so I’d better get cracking with that too.

More importantly, however, it reminded me to congratulate my PhD student Rockhee Sung who has just had her first paper published (in the journal Classical and Quantum Gravity). The paper is available online here and it’s free to download for a month even if you don’t have a personal or institutional subscription to the journal.

The idea of this paper came a while ago but it has taken us a long time to get everything in place to start writing it up. In the meantime other papers have been written on the subject, but Rockhee and I have done this our own way – or rather she has, as she put most of the hard work into actually doing the calculations.

About four years ago, during the course of careful statistical analysis of data from the Wilkinson Microwave Anisotropy Probe (WMAP), a group based in Santander (Spain) published a paper drawing attention to the existence of an anomalous “Cold Spot” in the data. This phenomenon has now acquired its own Wikipedia entry (here), so I won’t repeat all the details except to say that it is about 5° across and that it is colder than one would expect if the temperature fluctuations are Gaussian, as is predicted in the simplest models of the early Universe involving cosmological inflation. The spot is to the bottom right, and is marked with an arrow on the picture below.

It’s worth digressing a little here to explain that a fluctuating field of course contains both hot spots and cold spots. Because there CMB temperature fluctuations comprise a wide range of wavelengths there are also spots on different scales. Assessing the statistical significance of a single isolated feature like the cold spot is not particularly easy. Based on the brute force method of simulating skies according to the Gaussian hypothesis and then repeating the approach that led to the original discovery, the result is that around 1% of Gaussian CMB skies have a cold spot as cold as that observed in the real data. Before the non-Bayesians among you get too excited, I’ll remind you that this means that the probability of a Cold spot given the standard model is about 1%, i.e. P(Cold Spot | Standard Model)=0.01. This is NOT the same as saying that the probability of the standard model being correct is 0.01…

A probability of 1% is an in-between kind of level: not too small to be decisive, and not too large to be instantly dismissed as just being a chance fluctuation. My personal opinion is that the Cold Spot is an interesting feature that deserves to be investigated further, but is not something that in itself should cause anyone to doubt the standard model. I include it among the list of cosmological anomalies that I’ve blogged about before (for example, here, here and here). I find them interesting but don’t lose sleep worrying that the standard model is about to fall to pieces. Not yet, anyway.

Not all theorists are as level-headed as me, however, and within weeks of the discovery of the cold spot suggestions were already being put forward as to how it could be “explained” theoretically. Some of these are described in the Wikipedia entry, so I won’t rehash the list. However, one suggestion not included there was the idea that the anomalous cold spot might be there because the Universe were not isotropic, i.e. if the Cosmological Principle were violated.

Way back when I was a lad doing my own PhD, my supervisor John Barrow had been interested in globally anisotropic (but nevertheless homogeneous) cosmologies. These are models in which any observer sees different things in different directions, but the pattern seen by observers in different places is always the same. I never worked on these at the time – they seemed a bit too esoteric even for me – but I remembered bits and pieces about them from conversations.

A complete classification of all the space-times  possessing this property was completed over a hundred years ago (before General Relativity was invented) by the Italian mathematician Luigi Bianchi, and cosmological models based on them are called the Bianchi models.

This isn’t the place to go into detail about the Bianchi models: the classification is based on the mathematical properties of Lie groups, which would take me ages to explain. However, it is worth pointing out that only five Bianchi types actually contain the cosmologically principled Friedmann-Lemaître-Robertson-Walker universe as a special case: I, V, VII0 ,VIIh and IX. If you really want to know what the classes are you’ll have to look them up! Since we know our Universe is very close to being homogeneous and isotropic, it seems reasonable to look at those models capable of describing small departures from that case so the above list provides a useful subset of the models to explore.

Rockhee’s PhD project was to explore  the patterns of cosmic microwave background  fluctuations that can arise in that set of Bianchi cosmologies, not just in the temperature (which had been done before) but also in polarization (which hadn’t). I’ve already posted some of the temperature patterns Rockhee computed here.

The reason for extending wanting to extend this work to include polarization was the following. The microwave background radiation is partly linearly polarized because of the way radiation is scattered by electrons. If an electron is immersed in a radiation bath which is isotropic there is no net polarization, but if the radiation field is anisotrpic – in particular if it varies on an angular scale of 90º (i.e. a quadrupole) – then the scattered radiation will be partly polarized. In the standard cosmology the variations in the radiation field are random fluctuations so each electron “sees” a different quadupole. The net polarization field is therefore produced incoherently, by adding stochastic contributions. In  a  Bianchi model the situation is different. Each electron in this case sees the same quadupole. The polarization pattern produced is therefore coherent. Not only do anisotropic universes produce characteristic radiation patterns, they also produce a corresponding pattern in polarization.

So what does this all have to do with the Cold Spot? Well, in anisotropic spaces that are also curved, it is possible for light rays to get focussed in such a way that the entire pattern of flucuations present at least-scattering winds up concentrated in a small patch of the sky as seen by a late-time observer. for this to happen the space has to be negatively curved. Only two of the Bianchi types can do this, as there are only two that are both near-FLRW and negatively curved: V and VIIh. Both of these models could, in principle, therefore produce a cold spot by geometrical, rather than stochastic means. In the little figure below, taken from our paper, you can see examples of Bianchi VIIh (top) and Bianchi V (bottom) showing the temperature (left) and polarization (right) in each case. We’ve oriented the model to put the cold spot in approximately the right location as the observed one.

 

cold

 

The point is that there is a pretty heavy price to be paid for producing the cold spot in this way: an enormous, coherent signal in the polarized radiation field.

As often happens in such situations, somebody else had the idea to investigate these models and we were scooped to a large extent by Andrew Pontzen and Anthony Challinor from Cambridge, who recently published a paper showing that the polarization produced in these models is already excluded by experimental upper limits. They concentrated on the Bianchi VIIh case, as this appears to have a more general structure than V and it was the model first advocated as an explanation of the cold spot. In this model the combined effect of vorticity and shear introduces a swirly pattern into the radiation field that you can see clearly in the top two panels of the figure as well as focussing it into a small patch. Bianchi V doesn’t produce the same kind of pattern either in temperature or polarization: it looks more like a simple quadrupole squeezed into a small part of the sky. A particularly interesting aspect of this is that the Bianchi VIIh case clearly has a definite “handedness” while the Bianchi V one doesn’t.

The moral of all this is that the polarization of the cosmic microwave background provides key additional information that could prove decisive in eliminating (or perhaps even confirming) models of the Universe more exotic than the standard one. That’s one of the areas in which  we expect Planck to produce the goods!

In the meantime Rockhee and I will be submitting a couple of much larger papers in due course, one containing a wider discussion of the possible pattern morphologies that can be produced in these models, and another about their detailed statistical properties.

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