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A Little Bit of Chaos

Posted in The Universe and Stuff with tags astronomy, Carl Heiles, Chaos, Laplace, Michel Henon, phase space, Physics, Sir Isaac Newton, Sir William Hamilton on November 21, 2009 by telescoper

The era of modern physics could be said to have begun in 1687 with the publication by Sir Isaac Newton of his great Philosophiae Naturalis Principia Mathematica, (Principia for short). In this magnificent volume, Newton presented a mathematical theory of all known forms of motion and, for the first time, gave clear definitions of the concepts of force and momentum. Within this general framework he derived a new theory of Universal Gravitation and used it to explain the properties of planetary orbits previously discovered but unexplained by Johannes Kepler. The classical laws of motion and his famous “inverse square law” of gravity have been superseded by more complete theories when dealing with very high speeds or very strong gravity, but they nevertheless continue supply a very accurate description of our everyday physical world.

Newton’s laws have a rigidly deterministic structure. What I mean by this is that, given precise information about the state of a system at some time then one can use Newtonian mechanics to calculate the precise state of the system at any later time. The orbits of the planets, the positions of stars in the sky, and the occurrence of eclipses can all be predicted to very high accuracy using this theory.

At this point it is useful to mention that most physicists do not use Newton’s laws in the form presented in the Principia, but in a more elegant language named after Sir William Rowan Hamilton. The point about Newton’s laws of motion is that they are expressed mathematically as differential equations: they are expressed in terms of rates of changes of things. For instance, the force on a body gives the rate of change of the momentum of the body. Generally speaking, differential equations are very nasty things to solve which is a shame because most a great deal of theoretical physics involves them. Hamilton realised that it was possible to express Newton’s laws in a way that did not involve clumsy mathematics of this type. His formalism was equivalent, in the sense that one could obtain the basic differential equations from it, but easier to use in general situations. The key concept he introduced – now called the Hamiltonian – is a single mathematical function that depends on both the positions q and momenta p of the particles in a system, say H(q,p). This function is constructed from the different forms of energy (kinetic and potential) in the system, and how they depend on the p’s and q’s, but the details of how this works out don’t matter. Suffice to say that knowing the Hamiltonian for a system is tantamount to a full classical description of its behaviour.

Hamilton was a very interesting character. He was born in Dublin in 1805 and showed an astonishing early flair for languages, speaking 13 of them by the time he was 13. He graduated from Trinity College aged 22, at which point he was clearly a whiz-kid at mathematics as well as languages. He was immediately made professor of astronomy at Dublin and Astronomer Royal for Ireland. However, he turned out to be hopeless at the practicalities of observational work. Despite employing three of his sisters to help him in the observatory he never produced much of astronomical interest. Mathematics and alcohol seem to have been the two real loves of his life.

It is a fascinating historical fact that the development of probability theory during the late 17^th and early 18^th century coincided almost exactly with the rise of Newtonian Mechanics. It may seem strange in retrospect that there was no great philosophical conflict between these two great intellectual achievements since they have mutually incompatible views of prediction. Probability applies in unpredictable situations; Newtonian Mechanics says that everything is predictable. The resolution of this conundrum may owe a great deal to Laplace, who contributed greatly to both fields. Laplace, more than any other individual, was responsible to elevated the deterministic world-view of Newton to a scientific principle in its own right. To quote:

We ought then to regard the present state of the Universe as the effect of its preceding state and as the cause of its succeeding state.

According to Laplace’s view, knowledge of the initial conditions pertaining at the instant of creation would be sufficient in order to predict everything that subsequently happened. For him, a probabilistic treatment of phenomena did not conflict with classical theory, but was simply a convenient approach to be taken when the equations of motion were too difficult to be solved exactly. The required probabilities could be derived from the underlying theory, perhaps using some kind of symmetry argument.

The s-called “randomizing” devices used in all traditional gambling games – roulette wheels, dice, coins, bingo machines, and so on – are in fact well described by Newtonian mechanics. We call them “random” because the motions involved are just too complicated to make accurate prediction possible. Nevertheless it is clear that they are just straightforward mechanical devices which are essentially deterministic. On the other hand, we like to think the weather is predictable, at least in principle, but with much less evidence that it is so!

But it is not only systems with large numbers of interacting particles (like the Earth’s atmosphere) that pose problems for predictability. Some deceptively simple systems display extremely erratic behaviour. The theory of these systems is less than fifty years old or so, and it goes under the general title of nonlinear dynamics. One of the most important landmarks in this field was a study by two astronomers, Michel Hénon and Carl Heiles in 1964. They were interested in what would happens if you take a system with a known analytical solutions and modify it.

In the language of Hamiltonians, let us assume that H₀ describes a system whose evolution we know exactly and H₁ is some perturbation to it. The Hamiltonian of the modified system is thus

$H(q_i,p_i)=H_0(q_i, p_i) + H_1 (q_i, p_i)$

What Hénon and Heiles did was to study a system whose unmodified form is very familiar to physicists: the simple harmonic oscillator. This is a system which, when displaced from its equilibrium, experiences a restoring force proportional to the displacement. The Hamiltonian description for a single simple harmonic oscillator system involves a function that is quadratic in both p and q:

$H=\frac{1}{2} \left( q_1^2+p_1^2\right)$

The solution of this system is well known: the general form is a sinusoidal motion and it is used in the description of all kinds of wave phenomena, swinging pendulums and so on.

The case Henon and Heiles looked at had two degrees of freedom, so that the Hamiltonian depends on q₁, q₂, p₁ and p_2:

$H=\frac{1}{2} \left( q_1^2+p_1^2 + q_2^2+p_2^2\right)$

However, in this example, the two degrees of freedom are independent, meaning that there is uncoupled motion in the two directions. The amplitude of the oscillations is governed by the total energy of the system, which is a constant of the motion. Other than this, the type of behaviour displayed by this system is very rich, as exemplified by the various Lissajous figures shown in the diagram below. Note that all these figures are produced by the same type of dynamical system of equations: the different shapes are consequences of different initial conditions and different coefficients (which I set to unity in the form above).

If the oscillations in each direction have the same frequency then one can get an orbit which is a line or an ellipse. If the frequencies differ then the orbits can be much more complicated, but still pretty. Note that in all these cases the orbit is just a line, i.e. a one-dimensional part of the two-dimensional space drawn on the paper.

More generally, one can think of this system as a point moving in a four-dimensional phase space defined by the coordinates q₁, q₂, p₁ and p₂; taking slices through this space reveals qualitatively similar types of orbit for, say, p₂ and q₂ as for p₁ and p₂. The motion of the system is confined to a lower-dimensional part of the phase space rather than filling up all the available phase space. In this particular case, because each degree of freedom moves in only one of its two available dimensions, the system as a whole moves in a two-dimensional part of the four-dimensional space.

This all applies to the original, unperturbed system. Hénon and Heiles took this simple model and modified by adding a term to the Hamiltonian that was cubic rather than quadratic and which coupled the two degrees of freedom together. For those of you interested in the details their Hamiltonian was of the form

$H=\frac{1}{2} \left( q_1^2+p_1^2 + q_2^2+p_2^2\right) +q_1^2q_2+ \frac{1}{3}q_2^3$

The first set of terms in the brackets is the unmodified form, describing a simple harmonic oscillator; the other two terms are new. The result of this simple alteration is really quite surprising. They found that, for low energies, the system continued to behave like two uncoupled oscillators; the orbits were smooth and well-behaved. This is not surprising because the cubic modifications are smaller than the original quadratic terms if the amplitude is small. For higher energies the motion becomes a bit more complicated, but the phase space behaviour is still characterized by continuous lines, as shown in the left hand part of the following figure.

However, at higher values of the energy (right), the cubic terms become more important, and something very striking happens. A two-dimensional slice through the phase space no longer shows the continuous curves that typify the original system, but a seemingly disorganized scattering of dots. It is not possible to discern any pattern in the phase space structure of this system: it appear to be random.

Nowadays we describe the transition from these two types of behaviour as being accompanied by the onset of chaos. It is important to note that this system is entirely deterministic, but it generates a phase space pattern that is quite different from what one would naively expect from the behaviour usually associated with classical Hamiltonian systems. To understand how this comes about it is perhaps helpful to think about predictability in classical systems. It is true that precise knowledge of the state of a system allows one to predict its state at some future time. For a single particle this means that precise knowledge of its position and momentum, and knowledge of the relevant H, will allow one to calculate the position and momentum at all future times.

But think a moment about what this means. What do we mean by precise knowledge of the particle’s position? How precise? How many decimal places? If one has to give the position exactly then that could require an infinite amount of information. Clearly we never have that much information. Everything we know about the physical world has to be coarse-grained to some extent, even if it is only limited by measurement error. Strict determinism in the form advocated by Laplace is clearly a fantasy. Determinism is not the same as predictability.

In “simple” Hamiltonian systems what happens is that two neighbouring phase-space paths separate from each other in a very controlled way as the system evolves. In fact the separation between paths usually grows proportionally to time. The coarse-graining with which the input conditions are specified thus leads to a similar level of coarse-graining in the output state. Effectively the system is predictable, since the uncertainty in the output is not much larger than in the input.

In the chaotic system things are very different. What happens here is that the non-linear interactions represented in the Hamiltonian play havoc with the initial coarse-graining. Phase-space orbits that start out close to each other separate extremely violently (typically exponentially) and in a way that varies from one part of the phase space to another. What happens then is that particle paths become hopelessly scrambled and the mapping between initial and final states becomes too complex to handle. What comes out the end is practically impossible to predict.

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The Academic Journal Racket

Posted in Open Access, Science Politics with tags Academic Journals, arXiv, astronomy, Institute of Physics, Physics, Publishing on November 18, 2009 by telescoper

I’ve had this potential rant simmering away at the back of my mind for a while now, since our last staff meeting to be precise. In common, I suspect, with many other physics and astronomy departments, here at Cardiff we’re bracing ourselves for an extended period of budget cuts to help pay for our government’s charitable donations of taxpayer’s money to the banking sector.

English universities are currently making preparations for a minimum 10% reduction in core funding, and many are already making significant numbers of redundancies. We don’t know what’s going to happen to us here in Wales yet, but I suspect it will be very bad indeed.

Anyway, one of the items of expenditure that has been identified as a source of savings as we try to tighten our collective belts is the cost of academic journals. I nearly choked when the Head of School revealed how much we spend per annum on some of the journal subscriptions for physics and astronomy. In fact, I think university and departmental libraries are being taken to the cleaners by the academic publishing industry and it’s time to make a stand.

Let me single out one example. Like many learned societies, the Institute of Physics (the professional organisation for British physicists) basically operates like a charity. It does, however, have an independent publishing company that is run as a profit-making enterprise. And how.

In 2009 we paid almost £30K (yes, THIRTY THOUSAND POUNDS) for a year’s subscription to the IOP Physics package, a bundled collection of mainstream physics journals. This does not include Classical and Quantum Gravity or the Astrophysical Journal (both of which I have published in occasionally) which require additional payments running into thousands of pounds.

The IOP is not the only learned society to play this game. The Royal Astronomical Society also has a journal universally known as MNRAS (Monthly Notices of the Royal Astronomical Society) which earns it a considerable amount of revenue from its annual subscription of over £4K per department. Indeed, I don’t think it is inaccurate to say that without the income from MNRAS the RAS itself would face financial oblivion. I dare say MNRAS also earns a tidy sum for its publisher Wiley…

If you’re not already shocked by the cost of these subscriptions, let me outline the way academic journal business works, at least in the fields of physics and astronomy. I hope then you’ll agree that we’re being taken to the cleaners.

First, there is the content. This consists of scientific papers submitted to the journal by researchers, usually (though not exclusively) university employees. If the paper is accepted for publication the author receives no fee whatsoever and in some cases even has to pay “page charges” for the privilege of seeing the paper in print. In return for no fee, the author also has to sign over the copyright for the manuscript to the publisher. This is entirely different from the commercial magazine market, where contributors are usually paid a fee for writing a piece, or book publishing, where authors get a royalty on sales (and sometimes an advance).

Next there is the editorial process. The purpose of an academic journal – if there is one – is to ensure that only high quality papers are published. To this end it engages a Board of Editors to oversee this aspect of its work. The Editors are again usually academics and, with a few exceptions, they undertake the work on an unpaid basis. When a paper arrives at the journal which lies within the area of expertise of a particular editor, he or she identifies one or more suitable referees drawn from the academic community to provide advice on whether to publish it. The referees are expected to read the paper and provide comments as well as detailed suggestions for changes. The fee for referees? You guess it. Zilch. Nada.

The final part of the business plan is to sell the content (supplied for free), suitably edited (for free) and refereed (for free) back to the universities paying the wages of the people who so generously donated their labour. Not just sell, of course, but sell at a grossly inflated price.

Just to summarise, then: academics write the papers, do the refereeing and provide the editorial oversight for free and we then buy back the product of our labours at an astronomical price. Why do we participate in this ridiculous system? Am I the only one who detects the whiff of rip-off? Isn’t it obvious that we (I mean academics in universities) are spending a huge amout of time and money achieving nothing apart from lining the pockets of these exploitative publishers?

And if it wasn’t bad enough, there’s also the matter of inflation. There used to be a myth that advances in technology should lead to cheaper publishing.Nowadays authors submit their manuscripts electronically, they are sent electronically to referees and they are typset automatically if and when accepted. Most academics now access journals online rather than through paper copies; in fact some publications are only published electronically these days. All this may well lead to cheaper publishing but it doesn’t lead to cheaper subscriptions. The forecast inflation rate for physics journals over this year is about 8.5%, way above the Retail Price Index, which is currently negative.

Where is all the money going? Right into the pockets of the journal publishers. Times are tough enough in the university sector without us giving tens of thousands of pounds per year, plus free editoral advice and the rest, to these rapacious companies. Enough is enough.

It seems to me that it would be a very easy matter to get rid of academic journals entirely (at least from the areas of physics and astronomy that I work in). For a start, we have an excellent free repository (the arXiv) where virtually every new research paper is submitted. There is simply no reason why we should have to pay for journal subscriptions when papers are publically available there. In the old days, the journal industry had to exist in order for far flung corners of the world to have access to the latest research. Now everyone with an internet connection can get it all. Journals are redundant.

The one thing the arXiv does not do is provide editorial control, which some people argue is why we have to carry on being fleeced in the way I have described. If there is no quality imprint from an established journal how else would researchers know which papers to read? There is a lot of dross out there.

For one thing, not all referees put much effort into their work so there’s a lot of dross in refereed journals anyway. And, frustratingly, many referees sit on papers for months on end before sending in a report that’s only a couple of sentences. Far better, I would say, to put the paper on the arXiv and let others comment on it, either in private with the authors or perhaps each arXiv entry should have a comments facility, like a blog, so that the paper could be discussed interactively. The internet is pushing us in a direction in which the research literature should be discussed much more openly than it is at present, and in which it evolves much more as a result of criticisms and debate.

Finally, the yardstick by which research output is now being measured – or at least one of the metrics – is not so much a count of the number of refereed papers, but the number of citations the papers have attracted. Papers begin to attract citations – through the arXiv – long before they appear in a refereed journal and good papers get cited regardless of where they are eventually published.

If you look at citation statistics for refereed journals you will find it very instructive. A sizeable fraction of papers published in the professional literature receive no citations at all in their lifetime. So we end up paying over the odds for papers that nobody even bothers to read. Madness.

It could be possible for the arXiv (or some future version of it) to have its own editorial system, with referees asked to vet papers voluntarily. I’d be much happier giving my time in this way for a non-profit making system than I am knowing that I’m aiding and abetting racketeers. However, I think I probably prefer the more libertarian solution. Put it all on the net with minimal editorial control and the good stuff will float to the top regardless of how much crud there is.

Anyway, to get back to the starting point of this post, we have decided to cancel a large chunk of our journal subscriptions, including the IOP Physics package which is costing us an amount close to the annual salary of a lecturer. As more and more departments decide not to participate in this racket, no doubt the publishers will respond by hiking the price for the remaining customers. But it seems to me that this lunacy will eventually have to come to an end.

And if the UK university sector has to choose over the next few years between sacking hundreds of academic staff and ditching its voluntary subsidy to the publishing industry, I know what I would pick…

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Ergodic Means…

Posted in The Universe and Stuff with tags anthropic principle, Cosmology, ergodic hypothesis, multiverse, Physics, probability, statistical physics on October 19, 2009 by telescoper

The topic of this post is something I’ve been wondering about for quite a while. This afternoon I had half an hour spare after a quick lunch so I thought I’d look it up and see what I could find.

The word ergodic is one you will come across very frequently in the literature of statistical physics, and in cosmology it also appears in discussions of the analysis of the large-scale structure of the Universe. I’ve long been puzzled as to where it comes from and what it actually means. Turning to the excellent Oxford English Dictionary Online, I found the answer to the first of these questions. Well, sort of. Under etymology we have

ad. G. ergoden (L. Boltzmann 1887, in Jrnl. f. d. reine und angewandte Math. C. 208), f. Gr.

I say “sort of” because it does attribute the origin of the word to Ludwig Boltzmann, but the greek roots (εργον and οδοσ) appear to suggest it means “workway” or something like that. I don’t think I follow an ergodic path on my way to work so it remains a little mysterious.

The actual definitions of ergodic given by the OED are

Of a trajectory in a confined portion of space: having the property that in the limit all points of the space will be included in the trajectory with equal frequency. Of a stochastic process: having the property that the probability of any state can be estimated from a single sufficiently extensive realization, independently of initial conditions; statistically stationary.

As I had expected, it has two meanings which are related, but which apply in different contexts. The first is to do with paths or orbits, although in physics this is usually taken to meantrajectories in phase space (including both positions and velocities) rather than just three-dimensional position space. However, I don’t think the OED has got it right in saying that the system visits all positions with equal frequency. I think an ergodic path is one that must visit all positions within a given volume of phase space rather than being confined to a lower-dimensional piece of that space. For example, the path of a planet under the inverse-square law of gravity around the Sun is confined to a one-dimensional ellipse. If the force law is modified by external perturbations then the path need not be as regular as this, in extreme cases wandering around in such a way that it never joins back on itself but eventually visits all accessible locations. As far as my understanding goes, however, it doesn’t have to visit them all with equal frequency. The ergodic property of orbits is intimately associated with the presence of chaotic dynamical behaviour.

The other definition relates to stochastic processes, i.e processes involving some sort of random component. These could either consist of a discrete collection of random variables {X₁…X_n} (which may or may not be correlated with each other) or a continuously fluctuating function of some parameter such as time t, i.e. X(t) or spatial position (or perhaps both).

Stochastic processes are quite complicated measure-valued mathematical entities because they are specified by probability distributions. What the ergodic hypothesis means in the second sense is that measurements extracted from a single realization of such a process have a definition relationship to analagous quantities defined by the probability distribution.

I always think of a stochastic process being like a kind of algorithm (whose workings we don’t know). Put it on a computer, press “go” and it spits out a sequence of numbers. The ergodic hypothesis means that by examining a sufficiently long run of the output we could learn something about the properties of the algorithm.

An alternative way of thinking about this for those of you of a frequentist disposition is that the probability average is taken over some sort of statistical ensemble of possible realizations produced by the algorithm, and this must match the appropriate long-term average taken over one realization.

This is actually quite a deep concept and it can apply (or not) in various degrees. A simple example is to do with properties of the mean value. Given a single run of the program over some long time T we can compute the sample average

$\bar{X}_T\equiv \frac{1}{T} \int_0^Tx(t) dt$

the probability average is defined differently over the probability distribution, which we can call p(x)

$\langle X \rangle \equiv \int x p(x) dx$

If these two are equal for sufficiently long runs, i.e. as T goes to infinity, then the process is said to be ergodic in the mean. A process could, however, be ergodic in the mean but not ergodic with respect to some other property of the distribution, such as the variance. Strict ergodicity would require that the entire frequency distribution defined from a long run should match the probability distribution to some accuracy.

Now we have a problem with the OED again. According to the defining quotation given above, ergodic can be taken to mean statistically stationary. Actually that’s not true. ..

In the one-parameter case, “statistically stationary” means that the probability distribution controlling the process is independent of time, i.e. that p(x,t)=p(x,t+Δt) . It’s fairly straightforward to see that the ergodic property requires that a process X(t) be stationary, but the converse is not the case. Not every stationary process is necessarily ergodic. Ned Wright gives an example here. For a higher-dimensional process, such as a spatially-fluctuating random field the analogous property is statistical homogeneity, rather than stationarity, but otherwise everything carries over.

Ergodic theorems are very tricky to prove in general, but there are well-known results that rigorously establish the ergodic properties of Gaussian processes (which is another reason why theorists like myself like them so much). However, it should be mentioned that even if the ergodic assumption applies its usefulness depends critically on the rate of convergence. In the time-dependent example I gave above, it’s no good if the averaging period required is much longer than the age of the Universe; in that case even ergodicity makes it difficult to make inferences from your sample. Likewise the ergodic hypothesis doesn’t help you analyse your galaxy redshift survey if the averaging scale needed is larger than the depth of the sample.

Moreover, it seems to me that many physicists resort to ergodicity when there isn’t any compelling mathematical grounds reason to think that it is true. In some versions of the multiverse scenario, it is hypothesized that the fundamental constants of nature describing our low-energy turn out “randomly” to take on different values in different domains owing to some sort of spontaneous symmetry breaking perhaps associated a phase transition generating cosmic inflation. We happen to live in a patch within this structure where the constants are such as to make human life possible. There’s no need to assert that the laws of physics have been designed to make us possible if this is the case, as most of the multiverse doesn’t have the fine tuning that appears to be required to allow our existence.

As an application of the Weak Anthropic Principle, I have no objection to this argument. However, behind this idea lies the assertion that all possible vacuum configurations (and all related physical constants) do arise ergodically. I’ve never seen anything resembling a proof that this is the case. Moreover, there are many examples of physical phase transitions for which the ergodic hypothesis is known not to apply. If there is a rigorous proof that this works out, I’d love to hear about it. In the meantime, I remain sceptical.

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Alarm Bells at STFC

Posted in Science Politics with tags astronomy, Physics, STFC on September 30, 2009 by telescoper

The financial catastrophe engulfing the Science and Technology Facilities Council (STFC) has suddenly reared its (very ugly) head again.

Here is a statement posted yesterday on their webpage.

STFC Council policy on grants

STFC Council examined progress of its current science and technology prioritisation exercise at a strategy session on 21 and 22 September. Without prejudging the outcome of the prioritisation, Council agreed that prudent financial management required a re-examination of upcoming grants.

Council therefore agreed that new grants will be issued only to October 2010 in the first instance. This temporary policy is in place pending the outcome of the prioritisation exercise, expected in the New Year.

According to the e-astronomer the STFC has written to all Vice-chancellors and Principals of UK universities to tell them about this move. I gather the intention is that this measure will be temporary, but it looks deeply ominous to me. Those of us whose rolling grant requests for 5 years from April 2010 are currently being assessed face the possibility of receiving grants for only 6 months of funding. On the other hand, I’m told that what is more likely is that our grant won’t be announced until January or February, after the hitlist prioritisation exercise has been completed in the New Year. Hardest hit will be the particle physicists whose rolling grants start on 1st October 2009 (tomorrow), which will have only a year’s funding on them…

It seems that STFC has finally realised the scale of its budgetary problems and payback time is looming. I honestly think we could be doomed…

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Index Rerum

Posted in Biographical, Science Politics with tags astronomy, Astrophysics, bibliometric, citations, Cosmology, Ed Witten, g-index, h-index, Physics, research impact on September 29, 2009 by telescoper

Following on from yesterday’s post about the forthcoming Research Excellence Framework that plans to use citations as a measure of research quality, I thought I would have a little rant on the subject of bibliometrics.

Recently one particular measure of scientific productivity has established itself as the norm for assessing job applications, grant proposals and for other related tasks. This is called the h-index, named after the physicist Jorge Hirsch, who introduced it in a paper in 2005. This is quite a simple index to define and to calculate (given an appropriately accurate bibliographic database). The definition is that an individual has an h-index of h if that individual has published h papers with at least h citations. If the author has published N papers in total then the other N-h must have no more than h citations. This is a bit like the Eddington number. A citation, as if you didn’t know, is basically an occurrence of that paper in the reference list of another paper.

To calculate it is easy. You just go to the appropriate database – such as the NASA ADS system – search for all papers with a given author and request the results to be returned sorted by decreasing citation count. You scan down the list until the number of citations falls below the position in the ordered list.

Incidentally, one of the issues here is whether to count only refereed journal publications or all articles (including books and conference proceedings). The argument in favour of the former is that the latter are often of lower quality. I think that is in illogical argument because good papers will get cited wherever they are published. Related to this is the fact that some people would like to count “high-impact” journals only, but if you’ve chosen citations as your measure of quality the choice of journal is irrelevant. Indeed a paper that is highly cited despite being in a lesser journal should if anything be given a higher weight than one with the same number of citations published in, e.g., Nature. Of course it’s just a matter of time before the hideously overpriced academic journals run by the publishing mafia go out of business anyway so before long this question will simply vanish.

The h-index has some advantages over more obvious measures, such as the average number of citations, as it is not skewed by one or two publications with enormous numbers of hits. It also, at least to some extent, represents both quantity and quality in a single number. For whatever reasons in recent times h has undoubtedly become common currency (at least in physics and astronomy) as being a quick and easy measure of a person’s scientific oomph.

Incidentally, it has been claimed that this index can be fitted well by a formula h ~ sqrt(T)/2 where T is the total number of citations. This works in my case. If it works for everyone, doesn’t it mean that h is actually of no more use than T in assessing research productivity?

Typical values of h vary enormously from field to field – even within each discipline – and vary a lot between observational and theoretical researchers. In extragalactic astronomy, for example, you might expect a good established observer to have an h-index around 40 or more whereas some other branches of astronomy have much lower citation rates. The top dogs in the field of cosmology are all theorists, though. People like Carlos Frenk, George Efstathiou, and Martin Rees all have very high h-indices. At the extreme end of the scale, string theorist Ed Witten is in the citation stratosphere with an h-index well over a hundred.

I was tempted to put up examples of individuals’ h-numbers but decided instead just to illustrate things with my own. That way the only person to get embarrased is me. My own index value is modest – to say the least – at a meagre 27 (according to ADS). Does that mean Ed Witten is four times the scientist I am? Of course not. He’s much better than that. So how exactly should one use h as an actual metric, for allocating funds or prioritising job applications, and what are the likely pitfalls? I don’t know the answer to the first one, but I have some suggestions for other metrics that avoid some of its shortcomings.

One of these addresses an obvious deficiency of h. Suppose we have an individual who writes one brilliant paper that gets 100 citations and another who is one author amongst 100 on another paper that has the same impact. In terms of total citations, both papers register the same value, but there’s no question in my mind that the first case deserves more credit. One remedy is to normalise the citations of each paper by the number of authors, essentially sharing citations equally between all those that contributed to the paper. This is quite easy to do on ADS also, and in my case it gives a value of 19. Trying the same thing on various other astronomers, astrophysicists and cosmologists reveals that the h index of an observer is likely to reduce by a factor of 3-4 when calculated in this way – whereas theorists (who generally work in smaller groups) suffer less. I imagine Ed Witten’s index doesn’t change much when calculated on a normalized basis, although I haven’t calculated it myself.

Observers complain that this normalized measure is unfair to them, but I’ve yet to hear a reasoned argument as to why this is so. I don’t see why 100 people should get the same credit for a single piece of work: it seems like obvious overcounting to me.

Another possibility – if you want to measure leadership too – is to calculate the h index using only those papers on which the individual concerned is the first author. This is a bit more of a fiddle to do but mine comes out as 20 when done in this way. This is considerably higher than most of my professorial colleagues even though my raw h value is smaller. Using first author papers only is also probably a good way of identifying lurkers: people who add themselves to any paper they can get their hands on but never take the lead. Mentioning no names of course. I propose using the ratio of unnormalized to normalized h-indices as an appropriate lurker detector…

Finally in this list of bibliometrica is the so-called g-index. This is defined in a slightly more complicated way than h: given a set of articles ranked in decreasing order of citation numbers, g is defined to be the largest number such that the top g articles altogether received at least g² citations. This is a bit like h but takes extra account of the average citations of the top papers. My own g-index is about 47. Obviously I like this one because my number looks bigger, but I’m pretty confident others go up even more than mine!

Of course you can play with these things to your heart’s content, combining ideas from each definition: the normalized g-factor, for example. The message is, though, that although h definitely contains some information, any attempt to condense such complicated information into a single number is never going to be entirely successful.

Comments, particularly with suggestions of alternative metrics are welcome via the box. Even from lurkers.

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Cosmic Haiku

Posted in Poetry, The Universe and Stuff with tags astronomy, Cosmology, Haiku, Physics on September 6, 2009 by telescoper

I haven’t had much time to post today and will probably be too busy next week for anything too substantial, so I thought I’d resort to a bit of audience participation. How about a few Haiku on themes connected to astronomy, cosmology or physics?

Don’t be worried about making the style of your contributions too authentic, just make sure they are 17 syllables in total, and split into three lines of 5, 7 and 5 syllables respectively.

Here’s a few of my own to give you an idea!

Quantum Gravity:
The troublesome double-act
Of Little and Large

Gravity’s waves are
Traceless; which does not mean they
Can never be found

The Big Bang wasn’t
So big, at least not when you
Think in decibels.

Cosmological
Constant and Dark Energy
Are vacuous names

Microwave Background
Photons remember a time
When they were hotter

Isotropic and
Homogeneous metric?
Robertson-Walker

Galaxies evolve
In a complicated way
We don’t understand

Acceleration:
Type Ia Supernovae
Gave us the first clue

Cosmic Inflation
Could have stretched the Universe
And made it flatter

Astrophysicist
Is what I’m told is my Job
Title. Whatever.

Contributions welcome via the comments box. The best one gets a chance to win Bully’s star prize.

12 Comments »

The Inductive Detective

Posted in Bad Statistics, Literature, The Universe and Stuff with tags A Study in Scarlet, astronomy, David Hume, Karl Popper, Physics, probability, Science, Sherlock Holmes on September 4, 2009 by telescoper

I was watching an old episode of Sherlock Holmes last night – from the classic Granada TV series featuring Jeremy Brett’s brilliant (and splendidly camp) portrayal of the eponymous detective. One of the things that fascinates me about these and other detective stories is how often they use the word “deduction” to describe the logical methods involved in solving a crime.

As a matter of fact, what Holmes generally uses is not really deduction at all, but inference (a process which is predominantly inductive).

In deductive reasoning, one tries to tease out the logical consequences of a premise; the resulting conclusions are, generally speaking, more specific than the premise. “If these are the general rules, what are the consequences for this particular situation?” is the kind of question one can answer using deduction.

The kind of reasoning of reasoning Holmes employs, however, is essentially opposite to this. The question being answered is of the form: “From a particular set of observations, what can we infer about the more general circumstances that relating to them?”. The following example from a Study in Scarlet is exactly of this type:

From a drop of water a logician could infer the possibility of an Atlantic or a Niagara without having seen or heard of one or the other.

The word “possibility” makes it clear that no certainty is attached to the actual existence of either the Atlantic or Niagara, but the implication is that observations of (and perhaps experiments on) a single water drop could allow one to infer sufficient of the general properties of water in order to use them to deduce the possible existence of other phenomena. The fundamental process is inductive rather than deductive, although deductions do play a role once general rules have been established.

In the example quoted there is an inductive step between the water drop and the general physical and chemical properties of water and then a deductive step that shows that these laws could describe the Atlantic Ocean. Deduction involves going from theoretical axioms to observations whereas induction is the reverse process.

I’m probably labouring this distinction, but the main point of doing so is that a great deal of science is fundamentally inferential and, as a consequence, it entails dealing with inferences (or guesses or conjectures) that are inherently uncertain as to their application to real facts. Dealing with these uncertain aspects requires a more general kind of logic than the simple Boolean form employed in deductive reasoning. This side of the scientific method is sadly neglected in most approaches to science education.

In physics, the attitude is usually to establish the rules (“the laws of physics”) as axioms (though perhaps giving some experimental justification). Students are then taught to solve problems which generally involve working out particular consequences of these laws. This is all deductive. I’ve got nothing against this as it is what a great deal of theoretical research in physics is actually like, it forms an essential part of the training of an physicist.

However, one of the aims of physics – especially fundamental physics – is to try to establish what the laws of nature actually are from observations of particular outcomes. It would be simplistic to say that this was entirely inductive in character. Sometimes deduction plays an important role in scientific discoveries. For example, Albert Einstein deduced his Special Theory of Relativity from a postulate that the speed of light was constant for all observers in uniform relative motion. However, the motivation for this entire chain of reasoning arose from previous studies of eletromagnetism which involved a complicated interplay between experiment and theory that eventually led to Maxwell’s equations. Deduction and induction are both involved at some level in a kind of dialectical relationship.

The synthesis of the two approaches requires an evaluation of the evidence the data provides concerning the different theories. This evidence is rarely conclusive, so a wider range of logical possibilities than “true” or “false” needs to be accommodated. Fortunately, there is a quantitative and logically rigorous way of doing this. It is called Bayesian probability. In this way of reasoning, the probability (a number between 0 and 1 attached to a hypothesis, model, or anything that can be described as a logical proposition of some sort) represents the extent to which a given set of data supports the given hypothesis. The calculus of probabilities only reduces to Boolean algebra when the probabilities of all hypothesese involved are either unity (certainly true) or zero (certainly false). In between “true” and “false” there are varying degrees of “uncertain” represented by a number between 0 and 1, i.e. the probability.

Overlooking the importance of inductive reasoning has led to numerous pathological developments that have hindered the growth of science. One example is the widespread and remarkably naive devotion that many scientists have towards the philosophy of the anti-inductivist Karl Popper; his doctrine of falsifiability has led to an unhealthy neglect of an essential fact of probabilistic reasoning, namely that data can make theories more probable. More generally, the rise of the empiricist philosophical tradition that stems from David Hume (another anti-inductivist) spawned the frequentist conception of probability, with its regrettable legacy of confusion and irrationality.

My own field of cosmology provides the largest-scale illustration of this process in action. Theorists make postulates about the contents of the Universe and the laws that describe it and try to calculate what measurable consequences their ideas might have. Observers make measurements as best they can, but these are inevitably restricted in number and accuracy by technical considerations. Over the years, theoretical cosmologists deductively explored the possible ways Einstein’s General Theory of Relativity could be applied to the cosmos at large. Eventually a family of theoretical models was constructed, each of which could, in principle, describe a universe with the same basic properties as ours. But determining which, if any, of these models applied to the real thing required more detailed data. For example, observations of the properties of individual galaxies led to the inferred presence of cosmologically important quantities of dark matter. Inference also played a key role in establishing the existence of dark energy as a major part of the overall energy budget of the Universe. The result is now that we have now arrived at a standard model of cosmology which accounts pretty well for most relevant data.

Nothing is certain, of course, and this model may well turn out to be flawed in important ways. All the best detective stories have twists in which the favoured theory turns out to be wrong. But although the puzzle isn’t exactly solved, we’ve got good reasons for thinking we’re nearer to at least some of the answers than we were 20 years ago.

I think Sherlock Holmes would have approved.

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Flame Academy

Posted in Biographical, The Universe and Stuff with tags Athens, chemistry, Fires, Flames, Great Fire of London, Los Angeles, Physics, Zel'dovich on September 2, 2009 by telescoper

I heard on the radio this morning from that nice Mr Cowan that today is the anniversary of the start of the Great Fire of London which burned for four days in 1666. That provides for a bit of delayed synchronicity with yesterday’s post about the dreadful fires in the outskirts of Los Angeles and a similar conflagration in Athens (which now thankfully appears to be under control).

Fires are of course terrifying phenomena, and it must be among most people’s nightmares to be caught in one. The cambridge physicist Steve Gull experienced this at first hand when his boat exploded and caught fire recently. I’ll take this opportunity to wish him a speedy recovery from his injuries.

But frightening as such happenings are, a flame (the visible, light emitting part of a fire) can also be a very beautiful and fascinating spectacle. Flames are stable long-lived phenomena involving combustion in which a “fuel”, often some kind of hydrocarbon, reacts with an oxidizing element which, in the case of natural wildfires at any rate, is usually oxygen. However, along the way, many intermediate radicals are generated and the self-sustaining nature of the flame is maintained by intricate reaction kinetics.

The shape and colour of a flame is determined not just by its temperature but also, in a complicated way, by diffusion, convection and gravity. In a diffusion flame, the fuel and the oxidizing agent diffuse into each other and the rate of diffusion consequently limits the rate at which the flame spreads. Usually combustion takes place only at the edge of the flame: the interior contains unburnt fuel. A candle flame is usually relatively quiescent because the flow of material in it is predominantly laminar. However, at higher speeds you can find turbulent flames, like in the picture below!

Sometimes convection carries some of the combustion products away from the source of the flame. In a candle flame, for example, incomplete combustion forms soot particles which are convected upwards and then incandesce inside the flame giving it a yellow colour. Gravity limits the motion of heavier products away from the source. In a microgravity environment, flames look very different!

All this stuff about flames also gives me the opportunity to mention the great Russian physicist Yakov Borisovich Zel’dovich. To us cosmologists he is best known for his work on the large-scale structure of the Universe, but he only started to work on that subject relatively late in his career during the 1960s. He in fact began his career as a physical chemist and arguably his greatest contribution to science was that he developed the first completely physically based theory of flame propagation (together with Frank-Kamenetskii). No doubt he used insights gained from this work, together with his studies of detonation and shock waves, in the Soviet nuclear bomb programme in which he was a central figure.

But one thing even Zel’dovich couldn’t explain is why fires are such fascinating things to look at. I remember years ago having a fire in my back garden to get rid of garden rubbish. The more it burned the more things I wanted to throw on it, to see how well they would burn rather than to get rid of them. I ended up spending hours finding things to burn, building up a huge inferno, before finally retiring indoors, blackened with soot.

I let the fire die down, but it smouldered for three days.

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A Unified Quantum Theory of the Sexual Interaction

Posted in The Universe and Stuff with tags Particle Physics, Physics, Quantum Mechanics, Sexual Orientation on May 20, 2009 by telescoper

Recent changes to the criteria for allocating research funding require particle physicists and astronomers to justify the wider social, cultural and economic impact of their science. In view of the directive to engage in work more directly relevant to the person in the street, I’ve decided to share with you my latest results, which involve the application of ideas from theoretical physics in the wider field of human activity. That is, if you’re one of those people who likes to have sex in a field.

In the simplest theories of the sexual interaction, the eigenstates of the Hamiltonian describing all allowed forms of two-body coupling are identified with the conventional gender states, “Male” and “Female” denoted |M> and |F> in the Dirac bra-ket notation; note that the bra is superfluous in this context so, as usual, we dispense with it at the outset. Interactions between |M> and |F> states are assumed to be attractive while those between |M> and |M> or |F> and |F> are supposed either to be repulsive or, in some theories, entirely forbidden.

Observational evidence, however, strongly suggests that two-body interactions involving either F-F or M-M coupling, though suppressed in many situations, are by no means ruled out in the manner one would expect from the simplest theory outlined above. Furthermore, experiments indicate that the relevant channel for M-M interactions appears to have a comparable cross-section to that of the standard M-F variety, so a similar form of tunneling is presumably involved. This suggests that a more complete theory could be obtained by a relatively simple modification of the version presented above.

Inspired by the recent Nobel prize awarded for the theory of quark mixing, we are now able to present a new, unified theory of the sexual interaction. In our theory the “correct” eigenstates for sexual behaviour are not the conventional |M> and |F> gender states but linear combinations of the form

|M>=cosθ|S> + sinθ|G>

|F>=-sinθ|G>+cosθ|S>

where θ is the Cabibbo mixing angle or, more appropriately in this context, the sexual orientation (measured in degrees). Extension to three states is in principle possible (but a bit complicated) and we will not discuss this issue further.

In this theory each |M> or |F> state is regarded as a linear combination of heterosexual (straight, S) and homosexual (gay, G) states represented by a rotation of the basis by an angle θ, exactly the same mechanism that accounts for the charge-changing weak interactions between quarks.

For a purely heterosexual state, this angle is zero, in which case we recover the simple theory outlined above. At θ=90° only the G component manifests itself; in this state only classically forbidden interactions are permitted. The general state is however, one with a value of the orientation angle somewhere between these two limits and this permits all forms of interaction, at least with some probability.

Note added in proof: the |G> states do not appear in standard QFT but are motivated by some versions of string theory, expecially those involving G-strings.

One immediate consequence of this theory is that a “pure” gender state should be generally regarded as a quantum superposition of “straight” and “gay” states. This differs from a classical theory in that the true state can not be known with certainty; only the relative frequency of straight and gay behaviour (over a large number of interactions) can be predicted, perhaps explaining the large number of married men to be found on gaydar. The state at any given time is thus entirely determined by a sum over histories up to that moment, taking into account the appropriate action. In the Copenhagen interpretation, collapse one way or another occurs only when a measurement is made (or when enough Carlsberg is drunk).

If there is a difference in energy of the basis states a pure |M> state can oscillate between |S> and |G> according to a time-dependent phase factor arising when the two states interfere with each other:

|M(t)>=cosθ|S>exp(-iE₁t) + sinθ|G>exp(-iE₂t);

(obviously we are using natural units here, so that it all looks cleverer than it actually is). This equation is the origin of the expressions “it’s just a phase he’s going through” and “he swings both ways”. In physics parlance this means that the eigenstates of the sexual interaction do not coincide with the conventional gender types, indicating that sexual behaviour is not necessarily time-invariant for a given body.

Whether single-body phenomena (i.e. self-interactions) can provide insights into this theory depends, as can be seen from the equation, on the energies of the relevant states (as is also the case in neutrino oscillations). If they are equal then there is no oscillation. However, a detailed discussion of the role of degeneracy is beyond the scope of this analysis.

Self- interactions involving a solitary phase are generally difficult to observe, although examples have been documented that involve short-lived but highly-excited states accompanied by various forms of stimulated emission. Unfortunately, however, the resulting fluxes are not often well measured. This form of interaction also appears to be the current preoccupation of string theorists.

More definitive evidence for the theory might emerge from situations involving some form of entanglement, such as in the examples of M-M and F-F coupling mentioned above. Non-local interactions of a sexual type are possible in principle, but causality and simultaneity issues exist and most researchers consequently prefer to focus on local interactions, which are generally supposed to be more satisfactory from the point-of-view of reproducibility.

Although the theory is qualitatively successful we need more experimental data to pin down the parameters needed for a robust fit. It is not known, for example, whether the rates of M-M and F-F coupling are similar or, indeed, whether the peak intensity of these interactions, when resonance is reached, is similar to those of the standard M-F form. It is generally accepted, however, that the rate of decay from peak intensity is rather slower for processes involving |F> states than for|M> which is not so easy to model in this theory, although with a bit of renormalization we can probably explain anything.

Answers to these questions can perhaps be gleaned from observations of many-body processes (i.e. those with N≥3), especially if they involve a multiplicity of hardon states (i.e. collective excitations). Only these permit a full exploration of all possible degrees of freedom, although higher-order Feynman diagrams are needed to depict them and they require more complicated group theoretical techniques. Examples like the one shown above – representing a threesome – are not well understood, but undoubtedly contribute significantly to the bi-spectrum.

One might also speculate that in these and other highly excited states, the sexual interaction may be described by something more like the electroweak theory in which all forms of interaction occur in a much more symmetric fashion and at much higher rates than at lower energies. That sounds like some kind of party…

It is worth remarking that there may be finer structure than this model takes into account. For example, the |G> state is generally associated with singlet configurations like those shown on the right. However, G-G coupling is traditionally described in terms of “top” |t> and “bottom” |b> states, with b-t coupling the preferred mode, leading to the possibility of doublets or even triplets. It may be even prove necessary to introduce a further mixing angle φ of the form

|G>=cosφ |t> + sinφ |b>

so that the general state of |G> is “versatile”. However, whether G-G interactions can be adequately described even in this extended theory is a matter for debate until the intensity of t-t and b-b coupling is more accurately measured.

Finally, we should like to point out the difference between our model and that of the usual quark sextet, in which interacting states are described in terms of three pairs: the bottom (b) and top (t) which we have mentioned already; the strange (s) and charmed (c); and the up (u) and down (d). While it is clear that |b> and |t> do exhibit strong interactions and it appears plausible that |s> and |c> might do likewise, the sexual interaction clearly breaks the isospin symmetry between the |u> and the |d> in both M-M and M-F cases. The “up” state is definitely preferred in all forms of coupling and, indeed, the “down” has only ever been known to engage in weak interactions.

We have recently submitted an application to the Science and Technology Facilities Council for a modest sum (£754 million) to build a large-scale UK facility in order to carry out hands-on experimental tests of some aspects of the theory. We hope we can rely on the support of the physics community in agreeing to close down their labs and quit their jobs in order to release the funding needed to support it.

Follow @telescoper

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How Loud was the Big Bang?

Posted in The Universe and Stuff with tags Big Bang, Cosmic Microwave Background, Cosmology, decibels, Physics, sound waves, WMAP on April 26, 2009 by telescoper

The other day I was giving a talk about cosmology at Cardiff University’s Open Day for prospective students. I was talking, as I usually do on such occasions, about the cosmic microwave background, what we have learnt from it so far and what we hope to find out from it from future experiments, assuming they’re not all cancelled.

Quite a few members of staff listened to the talk too and, afterwards, some of them expressed surprise at what I’d been saying, so I thought it would be fun to try to explain it on here in case anyone else finds it interesting.

As you probably know the Big Bang theory involves the assumption that the entire Universe – not only the matter and energy but also space-time itself – had its origins in a single event a finite time in the past and it has been expanding ever since. The earliest mathematical models of what we now call the Big Bang were derived independently by Alexander Friedman and George Lemaître in the 1920s. The term “Big Bang” was later coined by Fred Hoyle as a derogatory description of an idea he couldn’t stomach, but the phrase caught on. Strictly speaking, though, the Big Bang was a misnomer.

Friedman and Lemaître had made mathematical models of universes that obeyed the Cosmological Principle, i.e. in which the matter was distributed in a completely uniform manner throughout space. Sound consists of oscillating fluctuations in the pressure and density of the medium through which it travels. These are longitudinal “acoustic” waves that involve successive compressions and rarefactions of matter, in other words departures from the purely homogeneous state required by the Cosmological Principle. The Friedman-Lemaitre models contained no sound waves so they did not really describe a Big Bang at all, let alone how loud it was.

However, as I have blogged about before, newer versions of the Big Bang theory do contain a mechanism for generating sound waves in the early Universe and, even more importantly, these waves have now been detected and their properties measured.

The above image shows the variations in temperature of the cosmic microwave background as charted by the Wilkinson Microwave Anisotropy Probe about five years ago. The average temperature of the sky is about 2.73 K but there are variations across the sky that have an rms value of about 0.08 milliKelvin. This corresponds to a fractional variation of a few parts in a hundred thousand relative to the mean temperature. It doesn’t sound like much, but this is evidence for the existence of primordial acoustic waves and therefore of a Big Bang with a genuine “Bang” to it.

A full description of what causes these temperature fluctuations would be very complicated but, roughly speaking, the variation in temperature you corresponds directly to variations in density and pressure arising from sound waves.

So how loud was it?

The waves we are dealing with have wavelengths up to about 200,000 light years and the human ear can only actually hear sound waves with wavelengths up to about 17 metres. In any case the Universe was far too hot and dense for there to have been anyone around listening to the cacophony at the time. In some sense, therefore, it wouldn’t have been loud at all because our ears can’t have heard anything.

Setting aside these rather pedantic objections – I’m never one to allow dull realism to get in the way of a good story- we can get a reasonable value for the loudness in terms of the familiar language of decibels. This defines the level of sound (L) logarithmically in terms of the rms pressure level of the sound wave P_rms relative to some reference pressure level P_ref

L=20 log₁₀[P_rms/P_ref]

(the 20 appears because of the fact that the energy carried goes as the square of the amplitude of the wave; in terms of energy there would be a factor 10).

There is no absolute scale for loudness because this expression involves the specification of the reference pressure. We have to set this level by analogy with everyday experience. For sound waves in air this is taken to be about 20 microPascals, or about 2×10^-10 times the ambient atmospheric air pressure which is about 100,000 Pa. This reference is chosen because the limit of audibility for most people corresponds to pressure variations of this order and these consequently have L=0 dB. It seems reasonable to set the reference pressure of the early Universe to be about the same fraction of the ambient pressure then, i.e.

P_ref~2×10^-10 P_amb

The physics of how primordial variations in pressure translate into observed fluctuations in the CMB temperature is quite complicated, and the actual sound of the Big Bang contains a mixture of wavelengths with slightly different amplitudes so it all gets a bit messy if you want to do it exactly, but it’s quite easy to get a rough estimate. We simply take the rms pressure variation to be the same fraction of ambient pressure as the averaged temperature variation are compared to the average CMB temperature, i.e.

P_rms~ a few ×10^-5P_amb

If we do this, scaling both pressures in logarithm in the equation in proportion to the ambient pressure, the ambient pressure cancels out in the ratio, which turns out to be a few times 10^-5.

AudiogramsSpeechBanana

With our definition of the decibel level we find that waves corresponding to variations of one part in a hundred thousand of the reference level give roughly L=100dB while part in ten thousand gives about L=120dB. The sound of the Big Bang therefore peaks at levels just over 110 dB. As you can see in the Figure above, this is close to the threshold of pain, but it’s perhaps not as loud as you might have guessed in response to the initial question. Many rock concerts are actually louder than the Big Bang, at least near the speakers!

A useful yardstick is the amplitude at which the fluctuations in pressure are comparable to the mean pressure. This would give a factor of about 10¹⁰ in the logarithm and is pretty much the limit that sound waves can propagate without distortion. These would have L≈190 dB. It is estimated that the 1883 Krakatoa eruption produced a sound level of about 180 dB at a range of 100 miles. By comparison the Big Bang was little more than a whimper.

PS. If you would like to read more about the actual sound of the Big Bang, have a look at John Cramer’s webpages. You can also download simulations of the actual sound. If you listen to them you will hear that it’s more of a “Roar” than a “Bang” because the sound waves don’t actually originate at a single well-defined event but are excited incoherently all over the Universe.

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In the Dark