Archive for anthropic principle

Insignificance

Posted in The Universe and Stuff with tags , , , , , , , on January 4, 2011 by telescoper

I’m told that there was a partial eclipse of the Sun visible from the UK this morning, although it was so cloudy here in Cardiff that I wouldn’t have seen anything even if I had bothered to get up in time to observe it. For more details of the event and pictures from people who managed to see it, see here. There’s also a nice article on the BBC website. The BBC are coordinating three days of programmes alongside a host of other events called Stargazing Live presumably timed to coincide with this morning’s eclipse. It’s taking a chance to do live broadcasts about astronomy given the British weather, but I hope they are successful in generating interest especially among the young.

As a spectacle a partial solar eclipse is pretty exciting – as long as it’s not cloudy – but even a full view of one can’t really be compared with the awesome event that is a total eclipse. I’m lucky enough to have observed one and I can tell you it was truly awe-inspiring.

If you think about it, though, it’s a very strange thing that such a thing is possible at all. In a total eclipse, the Moon passes between the Earth and the Sun in such a way that it exactly covers the Solar disk. In order for this to happen the apparent angular size of the Moon (as seen from Earth) has to be almost exactly the same as that of the Sun (as seen from Earth). This involves a strange coincidence: the Moon is small (about 1740 km in radius) but very close to the Earth in astronomical terms (about 400,000 km away). The Sun, on the other hand, is both enormously large (radius 700,000 km) and enormously distant (approx. 150,000,000 km).  The ratio of radius to distance from Earth of these objects is almost identical at the point of a a total eclipse, so the apparent disk of the Moon almost exactly fits over that of the Sun. Why is this so?

The simple answer is that it is just a coincidence. There seems no particular physical reason why the geometry of the Earth-Moon-Sun system should have turned out this way. Moreover, the system is not static. The tides raised by the Moon on the Earth lead to frictional heating and a loss of orbital energy. The Moon’s orbit  is therefore moving slowly outwards from the Earth. I’m not going to tell you exactly how quickly this happens, as it is one of the questions I set my students in the module Astrophysical Concepts I’ll be starting in a few weeks, but eventually the Earth-Moon distance will be too large for total eclipses of the Sun by the Moon to be possible on Earth, although partial and annular eclipses may still be possible.

It seems therefore that we just happen to be living at the right place at the right time to see total eclipses. Perhaps there are other inhabited moonless planets whose inhabitants will never see one. Future inhabitants of Earth will have to content themselves with watching eclipse clips on Youtube.

Things may be more complicated than this though. I’ve heard it argued that the existence of a moon reasonably close to the Earth may have helped the evolution of terrestrial life. The argument – as far as I understand it – is that life presumably began in the oceans, then amphibious forms evolved in tidal margins of some sort wherein conditions favoured both aquatic and land-dwelling creatures. Only then did life fully emerge from the seas and begin to live on land. If it is the case that the existence of significant tides is necessary for life to complete the transition from oceans to solid ground, then maybe the Moon played a key role in the evolution of dinosaurs, mammals, and even ourselves.

I’m not sure I’m convinced of this argument because, although the Moon is the dominant source of the Earth’s tides, it is not overwhelmingly so. The effect of the Sun is also considerable, only a factor of three smaller than the Moon. So maybe the Sun could have done the job on its own. I don’t know.

That’s not really the point of this post, however. What I wanted to comment on is that astronomers basically don’t question the interpretation of the occurence of total eclipses as simply a coincidence. Eclipses just are. There are no doubt many other planets where they aren’t. We’re special in that we live somewhere where something apparently unlikely happens. But this isn’t important because eclipses aren’t really all that significant in cosmic terms, other than that the law of physics allow them.

On the other hand astronomers (and many other people) do make a big deal of the fact that life exists in the Universe. Given what  we know about fundamental physics and biology – which admittedly isn’t very much – this also seems unlikely. Perhaps there are many other worlds without life, so the Earth is special once again. Others argue that the existence of life is so unlikely that special provision must have been made to make it possible.

Before I find myself falling into the black hole marked “Anthropic Principle” let me just say that I don’t see the existence of life (including human life) as being of any greater significance than that of a total eclipse. Both phenomena are (subjectively) interesting to humans, both are contingent on particular circumstances, and both will no doubt cease to occur at some point in perhaps not-too-distant the future. Neither tells us much about the true nature of the Universe.

Let’s face it. We’re just not significant.


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Ergodic Means…

Posted in The Universe and Stuff with tags , , , , , , 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…Xn} (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.

Multiversalism

Posted in The Universe and Stuff with tags , , on June 17, 2009 by telescoper

The word “cosmology” is derived from the Greek κόσμος (“cosmos”) which means, roughly speaking, “the world as considered as an orderly system”. The other side of the coin to “cosmos” is Χάος (“chaos”). In one world-view the Universe comprised two competing aspects: the orderly part that was governed by laws and which could (at least in principle) be predicted, and the “random” part which was disordered and unpredictable. To make progress in scientific cosmology we do need to assume that the Universe obeys laws. We also assume that these laws apply everywhere and for all time or, if they vary, then they vary in accordance with another law.  This is the cosmos that makes cosmology possible.  However, with the rise of quantum theory, and its applications to the theory of subatomic particles and their interactions, the field of cosmology has gradually ceded some of its territory to chaos.

In the early twentieth century, the first mathematical world models were constructed based on Einstein’s general theory of relativity. This is a classical theory, meaning that it describes a system that evolves smoothly with time. It is also entirely deterministic. Given sufficient information to specify the state of the Universe at a particular epoch, it is possible to calculate with certainty what its state will be at some point in the future. In a sense the entire evolutionary history described by these models is not a succession of events laid out in time, but an entity in itself. Every point along the space-time path of a particle is connected to past and future in an unbreakable chain. If ever the word cosmos applied to anything, this is it.

But as the field of relativistic cosmology matured it was realised that these simple classical models could not be regarded as complete, and consequently that the Universe was unlikely to be as predictable as was first thought. The Big Bang model gradually emerged as the favoured cosmological theory during the middle of the last century, between the 1940s and the 1960s. It was not until the 1960s, with the work of Hawking and Penrose, that it was realised that expanding world models based on general relativity inevitably involve a break-down of known physics at their very beginning. The so-called singularity theorems demonstrate that in any plausible version of the Big Bang model, all physical parameters describing the Universe (such as its density, pressure and temperature) all become infinite at the instant of the Big Bang. The existence of this “singularity” means that we do not know what laws if any apply at that instant. The Big Bang contains the seeds of its own destruction as a complete theory of the Universe. Although we might be able to explain how the Universe subsequently evolves, we have no idea how to describe the instant of its birth. This is a major embarrassment. Lacking any knowledge of the laws we don’t even have any rational basis to assign probabilities. We are marooned with a theory that lets in water.

The second important development was the rise of quantum theory and its incorporation into the description of the matter and energy contained within the Universe. Quantum mechanics (and its development into quantum field theory) entails elements of unpredictability. Although we do not know how to interpret this feature of the theory, it seems that any cosmological theory based on quantum theory must include things that can’t be predicted with certainty.

As particle physicists built ever more complete descriptions of the microscopic world using quantum field theory, they also realised that the approaches they had been using for other interactions just wouldn’t work for gravity. Mathematically speaking, general relativity and quantum field theory just don’t fit together. It might have been hoped that quantum gravity theory would help us plug the gap at the very beginning of the Universe, but that has not happened yet because there isn’t such a theory. What we can say about the origin of the Universe is correspondingly extremely limited and mostly speculative, but some of these speculations have had a powerful impact on the subject.

One thing that has changed radically since the early twentieth century is the possibility that our Universe may actually be part of a much larger “collection” of Universes. The potential for semantic confusion here is enormous. The Universe is, by definition, everything that exists. Obviously, therefore, there can only be one Universe. The name given to a Universe that consists of bits and pieces like this is the multiverse.

 There are various ways a multiverse can be realised. In the “Many Worlds” interpretation of quantum mechanics there is supposed to be a plurality of versions of our Universe, but their ontological status is far from clear (at least to me). Do we really have to accept that each of the many worlds is “out there”, or can we get away with using them as inventions to help our calculations?

 On the other hand, some plausible models based on quantum field theory do admit the possibility that our observable Universe is part of collection of mini-universes, each of which “really” exists. It’s hard to explain precisely what I mean by that, but I hope you get my drift. These mini-universes form a classical ensemble in different domains of a single-space time, which is not what happens in quantum multiverses.

According to the Big Bang model, the Universe (or at least the part of it we know about) began about fourteen billion years ago. We do not know whether the Universe is finite or infinite, but we do know that if it has only existed for a finite time we can only observe a finite part of it. We can’t possibly see light from further away than fourteen billion light years because any light signal travelling further than this distance would have to have set out before the Universe began. Roughly speaking, this defines our “horizon”: the maximum distance we are in principle able to see. But the fact that we can’t observe anything beyond our horizon does not mean that such remote things do not exist at all. Our observable “patch” of the Universe might be a tiny part of a colossal structure that extends much further than we can ever hope to see. And this structure might be not at all homogeneous: distant parts of the Universe might be very different from ours, even if our local piece is well described by the Cosmological Principle.

Some astronomers regard this idea as pure metaphysics, but it is motivated by plausible physical theories. The key idea was provided by the theory of cosmic inflation, which I have blogged about already. In the simplest versions of inflation the Universe expands by an enormous factor, perhaps 1060, in a tiny fraction of a second. This may seem ridiculous, but the energy available to drive this expansion is inconceivably large. Given this phenomenal energy reservoir, it is straightforward to show that such a boost is not at all unreasonable. With inflation, our entire observable Universe could thus have grown from a truly microscopic pre-inflationary region. It is sobering to think that everything galaxy, star, and planet we can see might from a seed that was smaller than an atom. But the point I am trying to make is that the idea of inflation opens up ones mind to the idea that the Universe as a whole may be a landscape of unimaginably immense proportions within which our little world may be little more than a pebble. If this is the case then we might plausibly imagine that this landscape varies haphazardly from place to place, producing what may amount to an ensemble of mini-universes. I say “may” because there is yet no theory that tells us precisely what determines the properties of each hill and valley or the relative probabilities of the different types of terrain.

Many theorists believe that such an ensemble is required if we are to understand how to deal probabilistically with the fundamentally uncertain aspects of modern cosmology. I don’t think this is the case. It is, at least in principle, perfectly possible to apply probabilistic arguments to unique events like the Big Bang using Bayesian inference. If there is an ensemble, of course, then we can discuss proportions within it, and relate these to probabilities too. Bayesians can use frequencies if they are available but do not require them. It is one of the greatest fallacies in science that probabilities need to be interpreted as frequencies.

At the crux of many related arguments is the question of why the Universe appears to be so well suited to our existence within it. This fine-tuning appears surprising based on what (little) we know about the origin of the Universe and the many other ways it might apparently have turned out. Does this suggest that it was designed to be so or do we just happen to live in a bit of the multiverse nice enough for us to have evolved and survived in?  

Views on this issue are often boiled down into a choice between a theistic argument and some form of anthropic selection.  A while ago I gave a talk at a meeting in Cambridge called God or Multiverse? that was an attempt to construct a dialogue between theologians and cosmologists. I found it interesting, but it didn’t alter my view that science and religion don’t really overlap very much at all on this, in the sense that if you believe in God it doesn’t mean you have to reject the multiverse, or vice-versa. If God can create a Universe, he could create a multiverse to0. As it happens, I’m agnostic about both.

So having, I hope, opened up your mind to the possibility that the Universe may be amenable to a frequentist interpretation, I should confess that I think one can actually get along quite nicely without it.  In any case, you will probably have worked out that I don’t really like the multiverse. One reason I don’t like it is that it accepts that some things have no fundamental explanation. We just happen to live in a domain where that’s the way things are. Of course, the Universe may turn out to be like that -  there definitely will be some point at which our puny monkey brains  can’t learn anything more – but if we accept that then we certainly won’t find out if there is really a better answer, i.e. an explanation that isn’t accompanied by an infinite amount of untestable metaphysical baggage. My other objection is that I think it’s cheating to introduce an infinite thing to provide an explanation of fine tuning. Infinity is bad.

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