## The Power Spectrum and the Cosmic Web

Posted in Bad Statistics, The Universe and Stuff with tags , , , , , , on June 24, 2014 by telescoper

One of the things that makes this conference different from most cosmology meetings is that it is focussing on the large-scale structure of the Universe in itself as a topic rather a source of statistical information about, e.g. cosmological parameters. This means that we’ve been hearing about a set of statistical methods that is somewhat different from those usually used in the field (which are primarily based on second-order quantities).

One of the challenges cosmologists face is how to quantify the patterns we see in galaxy redshift surveys. In the relatively recent past the small size of the available data sets meant that only relatively crude descriptors could be used; anything sophisticated would be rendered useless by noise. For that reason, statistical analysis of galaxy clustering tended to be limited to the measurement of autocorrelation functions, usually constructed in Fourier space in the form of power spectra; you can find a nice review here.

Because it is so robust and contains a great deal of important information, the power spectrum has become ubiquitous in cosmology. But I think it’s important to realise its limitations.

Take a look at these two N-body computer simulations of large-scale structure:

The one on the left is a proper simulation of the “cosmic web” which is at least qualitatively realistic, in that in contains filaments, clusters and voids pretty much like what is observed in galaxy surveys.

To make the picture on the right I first  took the Fourier transform of the original  simulation. This approach follows the best advice I ever got from my thesis supervisor: “if you can’t think of anything else to do, try Fourier-transforming everything.”

Anyway each Fourier mode is complex and can therefore be characterized by an amplitude and a phase (the modulus and argument of the complex quantity). What I did next was to randomly reshuffle all the phases while leaving the amplitudes alone. I then performed the inverse Fourier transform to construct the image shown on the right.

What this procedure does is to produce a new image which has exactly the same power spectrum as the first. You might be surprised by how little the pattern on the right resembles that on the left, given that they share this property; the distribution on the right is much fuzzier. In fact, the sharply delineated features  are produced by mode-mode correlations and are therefore not well described by the power spectrum, which involves only the amplitude of each separate mode. In effect, the power spectrum is insensitive to the part of the Fourier description of the pattern that is responsible for delineating the cosmic web.

If you’re confused by this, consider the Fourier transforms of (a) white noise and (b) a Dirac delta-function. Both produce flat power-spectra, but they look very different in real space because in (b) all the Fourier modes are correlated in such away that they are in phase at the one location where the pattern is not zero; everywhere else they interfere destructively. In (a) the phases are distributed randomly.

The moral of this is that there is much more to the pattern of galaxy clustering than meets the power spectrum…

## Illustris, Cosmology, and Simulation…

Posted in The Universe and Stuff with tags , , , , , , on May 8, 2014 by telescoper

There’s been quite a lot of news coverage over the last day or two emanating from a paper just out in the journal Nature by Vogelsberger et al. which describes a set of cosmological simulations called Illustris; see for example here and here.

The excitement revolves around the fact that Illustris represents a bit of a landmark, in that it’s the first hydrodynamical simulation with sufficient dynamical range that it is able to fully resolve the formation and evolution of  individual galaxies within the cosmic web of large-scale structure.

The simulations obviously represent a tremendous piece or work; they were run on supercomputers in France, Germany, and the USA; the largest of them was run on no less than 8,192 computer cores and took 19 million CPU hours. A single state-of-the-art desktop computer would require more than 2000 years to perform this calculation!

There’s even a video to accompany it (shame about the music):

The use of the word “simulation” always makes me smile. Being a crossword nut I spend far too much time looking in dictionaries but one often finds quite amusing things there. This is how the Oxford English Dictionary defines SIMULATION:

1.

a. The action or practice of simulating, with intent to deceive; false pretence, deceitful profession.

b. Tendency to assume a form resembling that of something else; unconscious imitation.

2. A false assumption or display, a surface resemblance or imitation, of something.

3. The technique of imitating the behaviour of some situation or process (whether economic, military, mechanical, etc.) by means of a suitably analogous situation or apparatus, esp. for the purpose of study or personnel training.

So it’s only the third entry that gives the meaning intended to be conveyed by the usage in the context of cosmological simulations. This is worth bearing in mind if you prefer old-fashioned analytical theory and want to wind up a simulationist! In football, of course, you can even get sent off for simulation…

Reproducing a reasonable likeness of something in a computer is not the same as understanding it, but that is not to say that these simulations aren’t incredibly useful and powerful, not just for making lovely pictures and videos but for helping to plan large scale survey programmes that can go and map cosmological structures on the same scale. Simulations of this scale are needed to help design observational and data analysis strategies for, e.g., the  forthcoming Euclid mission.

## One Hundred Years of Zel’dovich

Posted in The Universe and Stuff with tags , , , , on March 12, 2014 by telescoper

Lovely weather today, but it’s also been an extremely busy day with meetings and teachings. I did realize yesterday however that I had forgotten to mark a very important centenary at the weekend. If I hadn’t been such a slacker that I took last Saturday off work I would probably have been reminded…

The great Russian physicist Yakov Borisovich Zel’dovich (left) was born on March 8th 1914, so had he lived he would have been 100 years old last Saturday. To us cosmologists Zel’dovich  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 life in research 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 also 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, and which no doubt led to the chestful of medals he’s wearing in the photograph.

My own connection with Zel’dovich is primarily through his scientific descendants, principally his former student Sergei Shandarin, who has a faculty position at the University of Kansas. For example, I visited Kansas back in 1992 and worked on a project with Sergei and Adrian Melott which led to a paper published in 1993, the abstract of which makes it clear the debt it owed to the work of Ze’dovich.

The accuracy of various analytic approximations for following the evolution of cosmological density fluctuations into the nonlinear regime is investigated. The Zel’dovich approximation is found to be consistently the best approximation scheme. It is extremely accurate for power spectra characterized by n = -1 or less; when the approximation is ‘enhanced’ by truncating highly nonlinear Fourier modes the approximation is excellent even for n = +1. The performance of linear theory is less spectrum-dependent, but this approximation is less accurate than the Zel’dovich one for all cases because of the failure to treat dynamics. The lognormal approximation generally provides a very poor fit to the spatial pattern.

The Zel’dovich Approximation referred to in this abstract is based on an extremely simple idea but which, as we showed in the above paper, turns out to be extremely accurate at reproducing the morphology of the “cosmic web” of large-scale structure.

Zel’dovich passed away in 1987. I was a graduate student at that time and had never had the opportunity to meet him. If I had done so I’m sure I would have found him fascinating and intimidating in equal measure, as I admired his work enormously as did everyone I knew in the field of cosmology.  Anyway, a couple of years after his death a review paper written by himself and Sergei Shandarin was published, along with the note:

The Russian version of this review was finished in the summer of 1987. By the tragic death of Ya. B.Zeldovich on December 2, 1987, about four-fifths of the paper had been translated into English. Professor Zeldovich would have been 75 years old on March 8, 1989 and was vivid and creative until his last day. The theory of the structure of the universe was one of his favorite subjects, to which he made many note-worthy contributions over the last 20 years.

As one does if one is vain I looked down the reference list to see if any of my papers were cited. I’d only published one paper before Zel’dovich died so my hopes weren’t high. As it happens, though, my very first paper (Coles 1986) was there in the list. That’s still the proudest moment of my life!

Anyway, this post gives me the opportunity to advertise that there is a special meeting called The Zel’dovich Universe coming up this summer in Tallinn, Estonia. It looks a really interesting conference and I really hope I can find the time to fit it into my schedule. I’ve never been to Estonia…

## The Cosmic Web at Sussex

Posted in Books, Talks and Reviews, The Universe and Stuff with tags , on December 10, 2013 by telescoper

Yesterday I had the honour of giving an evening lecture for staff and students at the School of Mathematical and Physical Sciences at the University of Sussex. The event was preceded by a bit of impromptu twilight stargazing with the new telescope our students have just purchased:

You can just about see Venus in the second picture, just to the left of the street light.

Anyway, after briefly pretending to be a proper astronomer it was down to my regular business as a cosmologist and my talk entitled The Cosmic Web. Here is the abstract:

The lecture will focus on the large-scale structure of the Universe and the ideas that physicists are weaving together to explain how it came to be the way it is. Over the last few decades, astronomers have revealed that our cosmos is not only vast in scale – at least 14 billion light years in radius – but also exceedingly complex, with galaxies and clusters of galaxies linked together in immense chains and sheets, surrounding giant voids of (apparently) empty space. Cosmologists have developed theoretical explanations for its origin that involve such exotic concepts as ‘dark matter’ and ‘cosmic inflation’, producing a cosmic web of ideas that is, in some ways, as rich and fascinating as the Universe itself.

And for those of you interested, here are the slides I used for your perusal:

It was quite a large (and  very mixed) audience; it’s always difficult to pitch a talk at the right level in those circumstances so that it’s not too boring for the people who know something already but not too challenging for those who don’t know anything at all. A couple of people walked out about five minutes into the talk, which doesn’t exactly inspire a speaker with confidence, but overall it seemed to go down quite well.

Most of all, thank you to the organizers for the very nice reward of a bottle of wine!

## Power versus Pattern

Posted in Bad Statistics, The Universe and Stuff with tags , , , , , on June 15, 2012 by telescoper

One of the challenges we cosmologists face is how to quantify the patterns we see in galaxy redshift surveys. In the relatively recent past the small size of the available data sets meant that only relatively crude descriptors could be used; anything sophisticated would be rendered useless by noise. For that reason, statistical analysis of galaxy clustering tended to be limited to the measurement of autocorrelation functions, usually constructed in Fourier space in the form of power spectra; you can find a nice review here.

Because it is so robust and contains a great deal of important information, the power spectrum has become ubiquitous in cosmology. But I think it’s important to realise its limitations.

Take a look at these two N-body computer simulations of large-scale structure:

The one on the left is a proper simulation of the “cosmic web” which is at least qualitatively realistic, in that in contains filaments, clusters and voids pretty much like what is observed in galaxy surveys.

To make the picture on the right I first  took the Fourier transform of the original  simulation. This approach follows the best advice I ever got from my thesis supervisor: “if you can’t think of anything else to do, try Fourier-transforming everything.”

Anyway each Fourier mode is complex and can therefore be characterized by an amplitude and a phase (the modulus and argument of the complex quantity). What I did next was to randomly reshuffle all the phases while leaving the amplitudes alone. I then performed the inverse Fourier transform to construct the image shown on the right.

What this procedure does is to produce a new image which has exactly the same power spectrum as the first. You might be surprised by how little the pattern on the right resembles that on the left, given that they share this property; the distribution on the right is much fuzzier. In fact, the sharply delineated features  are produced by mode-mode correlations and are therefore not well described by the power spectrum, which involves only the amplitude of each separate mode.

If you’re confused by this, consider the Fourier transforms of (a) white noise and (b) a Dirac delta-function. Both produce flat power-spectra, but they look very different in real space because in (b) all the Fourier modes are correlated in such away that they are in phase at the one location where the pattern is not zero; everywhere else they interfere destructively. In (a) the phases are distributed randomly.

The moral of this is that there is much more to the pattern of galaxy clustering than meets the power spectrum…

## Merseyside Astronomy Day

Posted in Books, Talks and Reviews, The Universe and Stuff with tags , , , , on May 11, 2012 by telescoper

I’m just about to head by train off up to Merseyside (which, for those of you unfamiliar with the facts of British geography, is in the Midlands). The reason for this trip is that I’m due to give a talk tomorrow morning (Saturday 12th May) at Merseyside Astronomy Day, the 7th such event. It promises to be a MAD occasion.

My lecture, entitled The Cosmic Web, is an updated version of a talk I’ve given a number of times now; it will focus on the large scale structure of the Universe and the ideas that physicists are weaving together to explain how it came to be the way it is. Over the last few decades astronomers have revealed that our cosmos is not only vast in scale – at least 14 billion light years in radius – but also exceedingly complex, with galaxies and clusters of galaxies linked together in immense chains and sheets, surrounding giant voids of empty space. Cosmologists have developed theoretical explanations for its origin that involve such exotic concepts as ‘dark matter’ and ‘cosmic inflation’, producing a cosmic web of ideas that is in some ways as rich and fascinating as the Universe itself.

Anyway, I’m travelling to Liverpool this afternoon so I can meet the organizers for dinner this evening and stay overnight because there won’t be time to get there by train from Cardiff tomorrow morning. It’s not all that far from Cardiff to Liverpool as the crow flies, but unfortunately I’m not going by crow by train. I am nevertheless looking forward to seeing the venue, Spaceport, which I’ve never seen before.

If perchance any readers of this blog are planning to attend MAD VII please feel free to say hello. No doubt you will also tell me off for referring to Liverpool as the Midlands…

## Astronomy in Darkness

Posted in The Universe and Stuff with tags , , , , , , , , on January 14, 2012 by telescoper

Yesterday, being the second Friday of the month, was the day for the Ordinary Meeting of the Royal Astronomical Society (followed by dinner at the Athenaeum for members of the RAS Club). Living and working in Cardiff it’s difficult for me to get the specialist RAS Meetings earlier in the day, but if I get myself sufficiently organized I can usually get to Burlington House in time for the 4pm start of the Ordinary Meeting, which is open to the public.

The distressing news we learnt on Thursday about the events of Wednesday night cast a shadow over the proceedings. Given that I was going to dinner afterwards, for which a jacket and tie are obligatory, I went through my collection of (rarely worn) ties, and decided that a black one would be appropriate. When I arrived at Burlington House I was just in time to hear a warm tribute paid by a clearly upset Professor Roger Davies, President of the RAS and Oxford colleague of the late Steve Rawlings. There then followed a minute’s silence in his memory.

The principal reaction to this news amongst the astronomers present was one of disbelief and/or incomprehension. Some  friends and colleagues of Steve clearly knew much more about what had happened than has so far appeared in the press, but I don’t think it’s appropriate for me to make these public at this stage. We will know the facts soon enough. A colleague also pointed out to me that Steve had spent most of his recent working life as a central figure in the project to build the Square Kilometre Array, which will be the world’s largest radio telescope. He has died just a matter of days before the announcement will be made of where the SKA will actually be built. It’s sobering to think that one can spend so many years working on a project, only for something wholly unforeseen to prevent one seeing it through to completion.

Anyway, the meeting included an interesting talk by Tom Kitching of the University of Edinburgh who talked about recent results from the Canada-France-Hawaii Telescope Lensing Survey (CHFTLenS). The same project was the subject of a press release because the results were presented earlier in the week at the American Astronomical Society meeting in Austin, Texas. I haven’t got time to go into the technicalities of this study – which exploits the phenomenon of weak gravitational lensing to reconstruct the distribution of unseen (dark) matter in the Universe through its gravitational effect on light from background sources – but Tom Kitching actually contributed a guest post to this blog some time ago which will give you some background.

In the talk he presented one of the first dark matter maps obtained from this survey, in which the bright colours represent regions of high dark matter density

Getting maps like this is no easy process, so this is mightily impressive work, but what struck me is that it doesn’t look very filamentary. In other words, the dark matter appears to reside predominantly in isolated blobs with not much hint of the complicated network of filaments we call the Cosmic Web. That’s a very subjective judgement, of course, and it will be necessary to study the properties of maps like this in considerable detail in order to see whether they really match the predictions of cosmological theory.

After the meeting, and a glass of wine in Burlington House, I toddled off to the Athenaeum for an extremely nice dinner. It being the Parish meeting of the RAS Club, afterwards we went through a number of items of Club business, including the election of four new members.

Life  goes on, as does astronomy, even in darkness.

## SDSS-III and the Cosmic Web

Posted in The Universe and Stuff with tags , , , , , on January 12, 2011 by telescoper

It’s typical, isn’t it? You wait weeks for an interesting astronomical result to blog about and then two come along together…

Another international conference I’m not at is the 217th Meeting of the American Astronomical Society in the fine city of Seattle, which yesterday saw the release of some wonderful things produced by SDSS-III, the third incarnation of the Sloan Digital Sky Survey. There’s a nice article about it in the Guardian, followed by the usual bizarre selection of comments from the public.

I particularly liked the following picture of the cosmic web of galaxies, clusters and filaments that pervades the Universe on scales of hundreds of millions of lightyears, although it looks to me like a poor quality imitation of a Jackson Pollock action painting:

The above image contains about 500 million galaxies, which represents an enormous advance in the quest to map the local structure of the Universe in as much detail as possible. It will also improve still further the precision with which cosmologists can analyse the statistical properties of the pattern of galaxy clustering.

The above represents only a part (about one third) of the overall survey; the following graphic shows how much of the sky has been mapped. It also represents only the imaging data, not the spectroscopic information and other information which is needed to analyse the galaxy distribution in full detail.

There’s also a short video zooming out from one galaxy to the whole Shebang.

The universe is a big place.

## Colour in Fourier Space

Posted in The Universe and Stuff with tags , , , , , on February 9, 2010 by telescoper

As I threatened promised after Anton’s interesting essay on the perception of colour, a couple of days ago, I thought I’d write a quick item about something vaguely relevant that relates to some of my own research. In fact, this ended up as a little paper in Nature written by myself and Lung-Yih Chiang, a former student of mine who’s now based in his homeland of Taiwan.

This is going to be a bit more technical than my usual stuff, but it also relates to a post I did some time ago concerning the cosmic microwave background and to the general idea of the cosmic web, which has also featured in a previous item. You may find it useful to read these contributions first if you’re not au fait with cosmological jargon.

Or you may want to ignore it altogether and come back when I’ve found another look-alike

The large-scale structure of the Universe – the vast chains of galaxies that spread out over hundreds of millions of light-years and interconnect in a complex network (called the cosmic web) – is thought to have its origin in small fluctuations generated in the early universe by quantum mechnical effects during a bout of cosmic inflation.

These fluctuations in the density of an otherwise homogeneous universe are usually expressed in dimensionless form via the density contrast, defined as$\delta({\bf x})=(\rho({\bf x})-\bar{\rho})/\bar{\rho},$ where $\bar{\rho}$ is the mean density. Because it’s what physicists always do when they can’t think of anything better, we take the Fourier transform of this and write it as $\tilde{\delta}$, which is a complex function of the wavevector ${\bf k}$, and can therefore be written

$\tilde{\delta}({\bf k})=A({\bf k}) \exp [i\Phi({\bf k})],$

where $A$ is the amplitude and $\Phi$ is the phase belonging to the wavevector ${\bf k}$; the phase is an angle between zero and $2\pi$ radians.

This is a particularly useful thing to do because the simplest versions of inflation predict that the phases of each of the Fourier modes should be randomly distributed. Each is independent of the others and is essentially a random angle designating any point on the unit circle. What this really means is that there is no information content in their distribution, so that the harmonic components are in a state of maximum statistical disorder or entropy. This property also guarantees that fluctuations from place to place have a Gaussian distribution, because the density contrast at any point is formed from a superposition of a large number of independent plane-wave modes  to which the central limit theorem applies.

However, this just describes the initial configuration of the density contrast as laid down very early in the Big Bang. As the Universe expands, gravity acts on these fluctuations and alters their properties. Regions with above-average initial density ($\delta >0$) attract material from their surroundings and get denser still. They then attract more material, and get denser. This is an unstable process that eventually ends up producing enormous concentrations of matter ($\delta>>1$) in some locations and huge empty voids everywhere else.

This process of gravitational instability has been studied extensively in a variety of astrophysical settings. There are basically two regimes: the linear regime covering the early stages when $\delta << 1$ and the non-linear regime when large contrasts begin to form. The early stage is pretty well understood; the latter isn’t. Although many approximate analytical methods have been invented which capture certain aspects of the non-linear behaviour, general speaking we have to  run N-body simulations that calculate everything numerically by brute force to get anywhere.

The difference between linear and non-linear regimes is directly reflected in the Fourier-space behaviour. In the linear regime, each Fourier mode evolves independently of the others so the initial statistical form is preserved. In the non-linear regime, however, modes couple together and the initial Gaussian distribution begins to distort.

About a decade ago, Lung-Yih and I started to think about whether one might start to understand the non-linear regime a bit better by looking at the phases of the Fourier modes, an aspect of the behaviour that had been largely neglected until then. Our point was that mode-coupling effects must surely generate phase correlations that were absent in the initial random-phase configuration.

In order to explore the phase distribution we hit upon the idea of representing the phase of each Fourier mode using a  colour model. Anton’s essay discussed the  RGB (red-green-blue) parametrization of colour is used on computer screens as well as the CMY (Cyan-Magenta-Yellow) system preferred for high-quality printing.

However, there are other systems that use parameters different to those representing basic tones in these schemes. In particular, there are colour models that involve a parameter called the hue, which represents the position of a particular colour on the colour wheel shown left. In terms of the usual RGB framework you can see that red has a hue of zero, green is 120 degrees, and blue is 240. The complementary colours cyan, magenta and yellow lie 180 degrees opposite their RGB counterparts.

This representation is handy because it can be employed in a scheme that uses colour to represent Fourier phase information. Our idea was simple. The phases of the initial conditions should be random, so in this representation the Fourier transform should just look like a random jumble of colours with equal amounts of, say, red green and blue. As non-linear mode coupling takes hold of the distribution, however, a pattern should emerge in the phases in a manner which is characteristic of gravitational instability.

I won’t go too much further into the details here, but I will show a picture that proves that it works!

What you see here are four columns. The leftmost shows (from top to bottom) the evolution of a two-dimensional simulation of gravitational clustering. You can see the structure develops hierarchically, with an increasing characteristic scale of structure as time goes on.

The second column shows a time sequence of (part of) the Fourier transform of the distribution seen in the first; for the aficianados I should say that this is only one quadrant of the transform and that the rest is omitted for reasons of symmetry. Amplitude information is omitted here and the phase at each position is represented by an appropriate hue. To represent on this screen, however, we had to convert back to the RGB system.

The pattern is hard to see on this low resolution plot but two facts are noticeable. One is that a definite texture emerges, a bit like Harris Tweed, which gets stronger as the clustering develops. The other is that the relative amount of red green and blue does not change down the column.

The reason for the second property is that although clustering develops and the distribution of density fluctuations becomes non-Gaussian, the distribution of phases remains uniform in the sense that binning the phases of the entire Fourier transform would give a flat histogram. This is a consequence of the fact that the statistical properties of the fluctuations remain invariant under spatial translations even when they are non-linear.

Although the one-point distribuition of phases stays uniform even into the strongly non-linear regime, they phases do start to learn about each other, i.e. phase correlations emerge. Columns 3 and 4 illustrate this in the simplest possible way; instead of plotting the phases of each wavemode we plot the differences between the phases of neighbouring modes in the x  and y directions respectively.

If the phases are random then the phase differences are also random. In the initial state, therefore, columns 3 and 4 look just like column 2. However, as time goes on you should be able to see the emergence of a preferred colour in both columns, showing that the distribution of phase differences is no longer random.

The hard work is to describe what’s going on mathematically. I’ll spare you the details of that! But I hope I’ve at least made the point that this is a useful way of demonstrating that phase correlations exist and of visualizing some of their properties.

It’s also – I think – quite a lot of fun!

P.S. If you’re interested in the original paper, you will find it in Nature, Vol. 406 (27 July 2000), pp. 376-8.

## The Cosmic Web

Posted in The Universe and Stuff with tags , , , , , on November 23, 2009 by telescoper

When I was writing my recent  (typically verbose) post about chaos  on a rainy saturday afternoon, I cut out a bit about astronomy because I thought it was too long even by my standards of prolixity. However, walking home this evening I realised I could actually use it in a new post inspired by a nice email I got after my Herschel lecture in Bath. More of that in a minute, but first the couple of paras I edited from the chaos item…

Astronomy provides a nice example that illustrates how easy it is to make things too complicated to solve. Suppose we have two massive bodies orbiting in otherwise empty space. They could be the Earth and Moon, for example, or a binary star system. Each of the bodies exerts a gravitational force on the other that causes it to move. Newton himself showed that the orbit followed by each of the bodies is an ellipse, and that both bodies orbit around their common centre of mass. The Earth is much more massive than the Moon, so the centre of mass of the Earth-Moon system is rather close to the centre of the Earth. Although the Moon appears to do all the moving, the Earth orbits too. If the two bodies have equal masses, they each orbit the mid-point of the line connecting them, like two dancers doing a waltz.

Now let us add one more body to the dance. It doesn’t seem like too drastic a complication to do this, but the result is a mathematical disaster. In fact there is no known mathematical solution for the gravitational three-body problem, apart from a few special cases where some simplifying symmetry helps us out. The same applies to the N-body problem for any N bigger than 2. We cannot solve the equations for systems of gravitating particles except by using numerical techniques and very big computers. We can do this very well these days, however, because computer power is cheap.

Computational cosmologists can “solve” the N-body problem for billions of particles, by starting with an input list of positions and velocities of all the particles. From this list the forces on each of them due to all the other particles can be calculated. Each particle is then moved a little according to Newton’s laws, thus advancing the system by one time-step. Then the forces are all calculated again and the system inches forward in time. At the end of the calculation, the solution obtained is simply a list of the positions and velocities of each of the particles. If you would like to know what would have happened with a slightly different set of initial conditions you need to run the entire calculation again. There is no elegant formula that can be applied for any input: each laborious calculation is specific to its initial conditions.

Now back to the Herschel lecture I gave, called The Cosmic Web, the name given to the frothy texture of the large-scale structure of the Universe revealed by galaxy surveys such as the 2dFGRS:

One of the points I tried to get across in the lecture was that we can explain the pattern – quite accurately – in the framework of the Big Bang cosmology by a process known as gravitational instability. Small initial irregularities in the density of the Universe tend to get amplified as time goes on. Regions just a bit denser than average tend to pull in material from their surroundings faster, getting denser and denser until they collapse in on themselves, thus forming bound objects.

This  Jeans instability  is the dominant mechanism behind star formation in molecular clouds, and it leads to the rapid collapse of blobby extended structures  to tightly bound clumps. On larger scales relevant to cosmological structure formation we have to take account of the fact that the universe is expanding. This means that gravity has to fight against the expansion in order to form structures, which slows it down. In the case of a static gas cloud the instability grows exponentially with time, whereas in an expanding background it is a slow power-law.

This actually helps us in cosmology because the process of structure formation is not so fast that it destroys all memory of the initial conditions, which is what happens when stars form. When we look at the large-scale structure of the galaxy distribution we are therefore seeing something which contains a memory of where it came from. I’ve blogged before about what started the whole thing off here.

Here’s a (very low-budget) animation of the formation of structure in the expanding universe as computed by an N-body code. The only subtlety in this is that it is in comoving coordinates, which expand with the universe: the box should really be getting bigger but is continually rescaled with the expansion to keep it the same size on the screen.

You can see that filaments form in profusion but these merge and disrupt in such a way that the characteristic size of the pattern evolves with time. This is called hierarchical clustering.

One of the questions I got by email after the talk was basically that if the same gravitational instability produced stars and large-scale structure, why wasn’t the whole universe just made of enormous star-like structures rather than all these strange filaments and things?

Part of the explanation is that the filaments are relatively transient things. The dominant picture is one in which the filaments and clusters
become incorporated in larger-scale structures but really dense concentrations, such as the spiral galaxies, which do
indeed look a bit like big solar systems, are relatively slow to form.

When a non-expanding cloud of gas collapses to form a star there is also some transient filamentary structure  but the processes involved go so rapidly that it is all swept away quickly. Out there in the expanding universe we can still see the cobwebs.