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.
In the Dark 