## Recycled Bach

Posted in Music with tags , , , , , , on March 31, 2017 by telescoper

I had the office to myself this morning so I was listening to Essential Classics presented by Rob Cowan on BBC Radio 3 earlier on. During the course of the programme he pointed out that Johann Sebastian Bach was not averse to a bit of recycling and gave the following example. I’m sure that everyone has heard of Bach’s Mass in B Minor (BWV232), which is widely regarded as one of the greatest works ever composed in the entire history of music.

However, although this work is often depicted as a kind of culmination of Bach’s career as a composer and it wasn’t completed until 1749 (the year before Bach’s death), many sections were in fact recycled from much earlier compositions.

For example, give a listen to this. It is the Aria Ach, bleibe doch, mein liebstes Leben from the Cantata Lobet Gott in seinen Reichen (BWV11), often called the Ascension Oratorio, which was first performed in 1735. Apart from the fact that it sets a different text in a different language – the B Minor Mass is a setting of the complete `Ordinary’ of the Latin mass – and there are one or two musical differences here and there, this is instantly recognizable as an earlier incarnation of the sublime Agnus Dei from the B Minor Mass..

Oh, and if you’ve got half an hour to spare you could watch this video of a sparkly and sprightly performance of the entire cantana.

p.s. It’s Bach’s birthday today: he was born on March 31 1685.

## The Zel’dovich Lens

Posted in The Universe and Stuff with tags , , , , on June 30, 2014 by telescoper

Back to the grind after an enjoyable week in Estonia I find myself with little time to blog, so here’s a cute graphic by way of  a postscript to the IAU Symposium on The Zel’dovich Universe. I’ve heard many times about this way of visualizing the Zel’dovich Approximation (published in Zeldovich, Ya.B. 1970, A&A, 5, 84) but this is by far the best graphical realization I have seen. Here’s the first page of the original paper:

In a nutshell, this daringly simple approximation considers the evolution of particles in an expanding Universe from an early near-uniform state into the non-linear regime as a sort of ballistic, or kinematic, process. Imagine the matter particles are initial placed on a uniform grid, where they are labelled by Lagrangian coordinates $\vec{q}$. Their (Eulerian) positions at some later time $t$ are taken to be

$\vec{r}(\vec(q),t) = a(t) \vec{x}(\vec{q},t) = a(t) \left[ \vec{q} + b(t) \vec{s}(\vec{q},t) \right].$

Here the $\vec{x}$ coordinates are comoving, i.e. scaled with the expansion of the Universe using the scale factor $a(t)$. The displacement $\vec{s}(\vec{q},t)$ between initial and final positions in comoving coordinates is taken to have the form

$\vec{s}(\vec{q},t)= \vec{\nabla} \Phi_0 (\vec{q})$

where $\Phi_0$ is a kind of velocity potential (which is also in linear Newtonian theory proportional to the gravitational potential).If we’ve got the theory right then the gravitational potential field defined over the initial positions is a Gaussian random field. The function $b(t)$ is the growing mode of density perturbations in the linear theory of gravitational instability.

This all means that the particles just get a small initial kick from the uniform Lagrangian grid and their subsequent motion carries on in the same direction. The approximation predicts the formation of caustics  in the final density field when particles from two or more different initial locations arrive at the same final location, a condition known as shell-crossing. The caustics are identified with the walls and filaments we find in large-scale structure.

Despite its simplicity this approximation is known to perform extremely well at reproducing the morphology of the cosmic web, although it breaks down after shell-crossing has occurred. In reality, bound structures are formed whereas the Zel’dovich approximation simply predicts that particles sail straight through the caustic which consequently evaporates.

Anyway the mapping described above can also be given an interpretation in terms of optics. Imagine a uniform illumination field (the initial particle distribution) incident upon a non-uniform surface (e.g. the surface of the water in a swimming pool). Time evolution is represented by greater depths within the pool.  The light pattern observed on the bottom of the pool (the final distribution) displays caustics with a very similar morphology to the Cosmic Web, except in two dimensions, obviously.

Here is a very short  but very nice video by Johan Hidding showing how this works:

In this context, the Zel’dovich approximation corresponds to the limit of geometrical optics. More accurate approximations can presumably be developed using analogies with physical optics, but this programme has only just begun.