## R.I.P. Derek Mahon (1941-2020)

Posted in Covid-19, Poetry with tags , , , , on October 2, 2020 by telescoper

The poet Derek Mahon has died, so it seems apt to pay tribute by posting some examples of his poetry.

This poem, Everything is going to be all right, was read on the main news on RTÉ television when the national lockdown was announced back in March, sounding a note of optimism to a worried nation. I’m not sure everything is going to be all right, but it’s an excellent poem:

How should I not be glad to contemplate
the clouds clearing beyond the dormer window
and a high tide reflected on the ceiling?
There will be dying, there will be dying,
but there is no need to go into that.
The poems flow from the hand unbidden
and the hidden source is the watchful heart.
The sun rises in spite of everything
and the far cities are beautiful and bright.
I lie here in a riot of sunlight
watching the day break and the clouds flying.
Everything is going to be all right.

Sadly he didn’t live to see the end of the pandemic. Over the years I have posted a few poems by Derek Mahon. Here are two more. This one is called The Thunder Shower

a rumor, a grumble of white rain
growing in volume, rustling over the ground,
drenching the gravel in a wash of sound.
Drops tap like timpani or shine
like quavers on a line.

It rings on exposed tin,
a suite for water, wind and bin,
plinky Poulenc or strongly groaning Brahms’
rain-strings, a whole string section that describes
the very shapes of thought in warm
self-referential vibes

the whispering roar is a recital.
Jostling rain-crowds, clamorous and vital,
struggle in runnels through the afternoon.
The rhythm becomes a regular beat;
steam rises, body heat—

and now there’s city noise,
bits of recorded pop and rock,
the drums, the strident electronic shock,
a vast polyphony, the dense refrain
of wailing siren, truck and train
and incoherent cries.

All human life is there
in the unconfined, continuous crash
whose slow, diffused implosions gather up
car radios and alarms, the honk and beep,
and tiny voices in a crèche
piercing the muggy air.

the rackety global-franchise rush,
oil wars and water wars, the diatonic
crescendo of a cascading world economy
are audible in the hectic thrash

The voice of Baal explodes,
raging and rumbling round the clouds,
frantic to crush the self-sufficient spaces
and re-impose his failed hegemony
in Canaan before moving on
to other simpler places.

At length the twining chords
run thin, a watery sun shines out,
the deluge slowly ceases, the guttural chant
subsides; a thrush sings, and discordant thirds
diminish like an exhausted concert
on the subdominant.

The angry downpour swarms
growling to far-flung fields and farms.
The drains are still alive with trickling water,
a few last drops drip from a broken gutter;
but the storm that created so much fuss
has lost interest in us.

And this one, about the noble self-sacrifice of Captain Lawrence Oates,  is called Antarctica

‘I am just going outside and may be some time.’
The others nod, pretending not to know.
At the heart of the ridiculous, the sublime.
He leaves them reading and begins to climb,
Goading his ghost into the howling snow;
He is just going outside and may be some time.
The tent recedes beneath its crust of rime
And frostbite is replaced by vertigo:
At the heart of the ridiculous, the sublime.
Need we consider it some sort of crime,
This numb self-sacrifice of the weakest? No,
He is just going outside and may be some time
In fact, for ever. Solitary enzyme,
Though the night yield no glimmer there will glow,
At the heart of the ridiculous, the sublime.

Rest in Peace Derek Mahon (1941-2020)

## What the Power Spectrum misses

Posted in The Universe and Stuff with tags , , , , , , , on August 2, 2017 by telescoper

Just taking a short break from work I chatted over coffee to one of the students here at the Niels Bohr Institute about various things to do with the analysis of signals in the Fourier domain (as you do). That discussion reminded me of this rather old post (from 2009) which I thought might be worth a second airing (after a bit of editing). The discussion is all based on past cosmological data (from WMAP) rather than the most recent (from Planck), but that doesn’t change anything qualitatively. So here you are.

The picture above shows the all-sky map of fluctuations in the temperature of the cosmic microwave background across the sky as revealed by the Wilkinson Microwave Anisotropy Probe, known to its friends as WMAP.

I spent many long hours fiddling with the data coming from the WMAP experiment, partly because I’ve never quite got over the fact that such wonderful data actually exists. When I started my doctorate in 1985 the whole field of CMB analysis was so much pie in the sky, as no experiments had yet been performed with the sensitivity to reveal the structures we now see. This is because they are very faint and easily buried in noise. The fluctuations in temperature from pixel to pixel across the sky are of order one part in a hundred thousand of the mean temperature (i.e. about 30 microKelvin on a background temperature of about 3 Kelvin). That’s smoother than the surface of a billiard ball. That’s why it took such a long time to make the map shown above, and why it is such a triumphant piece of science.

I blogged a while ago about the idea that the structure we see in this map was produced by sound waves reverberating around the early Universe. The techniques cosmologists use to analyse this sound are similar to those used in branches of acoustics except that we only see things in projection on the celestial sphere which requires a bit of special consideration.

One of the things that sticks in my brain from my undergraduate years is being told that `if you don’t know what you’re doing as a physicist you should start by making a Fourier transform of everything. This approach breaks down the phenomenon being studied into a set of  plane waves with different wavelengths corresponding to analysing the different tones present in a complicated sound.

It’s often very good advice to do such a decomposition for one-dimensional time series or fluctuation fields in three-dimensional Cartesian space, even you do know what you’re doing, but it doesn’t work with a sphere because plane waves don’t fit properly on a curved surface. Fortunately, however, there is a tried-and-tested alternative involving spherical harmonics rather than plane waves.

Spherical harmonics are quite complicated beasts mathematically but they have pretty similar properties to Fourier harmonics in many respects. In particular they are represented as complex numbers having real and imaginary parts or, equivalently, an amplitude and a phase (usually called the argument by mathematicians),

$Z=X+iY = R \exp(i\phi)$

This latter representation is the most useful one for CMB fluctuations because the simplest versions of inflationary theory predict that the phases φ of each of the spherical harmonic modes should be randomly distributed. What this really means is that there is no information content in their distribution so that the harmonic modes are in a state of maximum statistical disorder or entropy. This property also guarantees that the distribution of fluctuations over the sky should have a Gaussian distribution.

If you accept that the fluctuations are Gaussian then only the amplitudes of the spherical harmonic coefficients are useful. Indeed, their statistical properties can be specified entirely by the variance of these amplitudes as a function of mode frequency. This pre-eminently important function is called the power-spectrum of the fluctuations, and it is shown here for the WMAP data:

Although the units on the axes are a bit strange it doesn”t require too much imagination to interpret this in terms of a sound spectrum. There is a characteristic tone (at the position of the big peak) plus a couple of overtones (the bumps at higher frequencies). However these features are not sharp so the overall sound is not at all musical.

If the Gaussian assumption is correct then the power-spectrum contains all the useful statistical information to be gleaned from the CMB sky, which is why so much emphasis has been placed on extracting it accurately from the data.

Conversely, though, the power spectrum is completely insensitive to any information in the distribution of spherical harmonic phases. If something beyond the standard model made the Universe non-Gaussian it would affect the phases of the harmonic modes in a way that would make them non-random.

However,I will now show you how important phase information could actually be, if only we could find a good way of exploiting it. Let’s start with a map of the Earth, with the colour representing height of the surface above mean sea level:

You can see the major mountain ranges (Andes, Himalayas) quite clearly as red in this picture and note how high Antarctica is…that’s one of the reasons so much astronomy is done there.

Now, using the same colour scale we have the WMAP data again (in Galactic coordinates).

The virture of this representation of the map is that it shows how smooth the microwave sky is compared to the surface of the Earth. Note also that you can see a bit of crud in the plane of the Milky Way that serves as a reminder of the difficulty of cleaning the foregrounds out.

Clearly these two maps have completely different power spectra. The Earth is dominated by large features made from long-wavelength modes whereas the CMB sky has relatively more small-scale fuzz.

Now I’m going to play with these maps in the following rather peculiar way. First, I make a spherical harmonic transform of each of them. This gives me two sets of complex numbers, one for the Earth and one for WMAP. Following the usual fashion, I think of these as two sets of amplitudes and two sets of phases. Note that the spherical harmonic transformation preserves all the information in the sky maps, it’s just a different representation.

Now what I do is swap the amplitudes and phases for the two maps. First, I take the amplitudes of WMAP and put them with the phases for the Earth. That gives me the spherical harmonic representation of a new data set which I can reveal by doing an inverse spherical transform:

This map has exactly the same amplitudes for each mode as the WMAP data and therefore possesses an identical power spectrum to that shown above. Clearly, though, this particular CMB sky is not compatible with the standard cosmological model! Notice that all the strongly localised features such as coastlines appear by virtue of information contained in the phases but absent from the power-spectrum.

To understand this think how sharp features appear in a Fourier transform. A sharp spike at a specific location actually produces a broad spectrum of Fourier modes with different frequencies. These modes have to add in coherently at the location of the spike and cancel out everywhere else, so their phases are strongly correlated. A sea of white noise also has a flat power spectrum but has random phases. The key difference between these two configurations is not revealed by their spectra but by their phases.

Fortunately there is nothing quite as wacky as a picture of the Earth in the real data, but it makes the point that there are more things in Heaven and Earth than can be described in terms of the power spectrum!

Finally, perhaps in your mind’s eye you might consider what it might look lie to do the reverse experiment: recombine the phases of WMAP with the amplitudes of the Earth.

If the WMAP data are actually Gaussian, then this map is a sort of random-phase realisation of the Earth’s power spectrum. Alternatively you can see that it is the result of running a kind of weird low-pass filter over the WMAP fluctuations. The only striking things it reveals are (i) a big blue hole associated with foreground contamination, (ii) a suspicious excess of red in the galactic plane owing to the same problem, and (iiI) a strong North-South asymmetry arising from the presence of Antarctica.

There’s no great scientific result here, just a proof that spherical harmonic phases are potentially interesting because of the information they contain about strongly localised features

PS. These pictures were made by a former PhD student of mine, Patrick Dineen, who has since quit astrophysics  to work in the financial sector for Winton Capital, which has over the years recruited a number of astronomy and cosmology graduates and also sponsors a Royal Astronomical Society prize. That shows that the skills and knowledge obtained in the seemingly obscure field of cosmological data analysis have applications elsewhere!

## Antarctica

Posted in Literature with tags , , , on February 11, 2012 by telescoper

‘I am just going outside and may be some time.’
The others nod, pretending not to know.
At the heart of the ridiculous, the sublime.
He leaves them reading and begins to climb,
Goading his ghost into the howling snow;
He is just going outside and may be some time.
The tent recedes beneath its crust of rime
And frostbite is replaced by vertigo:
At the heart of the ridiculous, the sublime.
Need we consider it some sort of crime,
This numb self-sacrifice of the weakest? No,
He is just going outside and may be some time
In fact, for ever. Solitary enzyme,
Though the night yield no glimmer there will glow,
At the heart of the ridiculous, the sublime.

by Derek Mahon (b. 1941).

## Terra Nova

Posted in Art, History, The Universe and Stuff with tags , , , , , , , on February 3, 2012 by telescoper

We’re currently enduring a spell of cold weather here in Cardiff, although I think it might be rather milder here then elsewhere in the UK. My garden thermometer showed a mere -5 C when I looked at it at 7.15 this morning. The other day we had a meeting of half-a-dozen people in one of our large teaching rooms and it was absolutely freezing. I don’t know what was wrong with the heating. Yesterday I actually did a lecture in the same room, but with 80-odd “warm bodies” (or “students” as they are sometimes known) in there, it was bearable.

The cold here of course is nothing compared with that endured by Captain Scott‘s ill-fated expedition to the South Pole, but I mention it here for a number of reasons. First, the centenary of the death of Scott and his companions is coming up next month; the tragedy unfolded in March 1912. There’s actually a very special concert coming up next week, featuring Vaughan Williams’ wonderful music written for the classic film Scott of the Antarctic (which, incidentally, you can actually watch in full on Youtube). I’m definitely going along, and will probably review the performance next week, but quite a number of my colleagues are also going, for reasons which will become obvious..

The concert is special because of the very strong connections between the Scott Expedition and the City of Cardiff. Much of the financial support needed to fund the trek to the South Pole was raised from Cardiff businessmen and Scott’s ship, the Terra Nova, actually set sail from Cardiff (in June 1910) on its journey, first to New Zealand and thence to Antarctica.

Incidentally, an article in this morning’s Western Mail relates to a historic painting of the departure of the Terra Nova which is about to be auctioned:

Cardiff Bay has certainly changed a great deal since 1910, but quite a lot is recognizable, especially the Pierhead Building, which can be seen to the right. The actual docks, the locations of which are revealed by the lines of masts of tall ships, are now mainly filled in. But there is at least one other reminder of this occasion to be found at Cardiff Bay, a large waterfront bar itself called Terra Nova

There’s also a deep connection with the South Pole, and the Antarctic generally, for many members of the Astronomy Instrumentation Group here in the School of Physics & Astronomy at Cardiff University, quite a few of whom have actually been to the South Pole in connection with various experiments, including Quad,  Boomerang and BLAST, because of the unique observing conditions there.

## Power isn’t Everything

Posted in The Universe and Stuff with tags , , , , , , , on January 6, 2009 by telescoper

The picture above shows the latest available all-sky map of fluctuations in the temperature of the cosmic microwave background across the sky as revealed by the Wilkinson Microwave Anisotropy Probe, known to its friends as WMAP.

I’ve spent many long hours fiddling with the data coming from the WMAP experiment, partly because I’ve never quite got over the fact that such wonderful data actually exists. When I started my doctorate in 1985 the whole field of CMB analysis was so much pie in the sky, as no experiments had yet been performed with the sensitivity to reveal the structures we now see. This is because they are very faint and easily buried in noise. The fluctuations in temperature from pixel to pixel across the sky are of order one part in a hundred thousand of the mean temperature (i.e. about 30 microKelvin on a background temperature of about 3 Kelvin). That’s smoother than the surface of a billiard ball. That’s why it took such a long time to make the map shown above, and why it is such a triumphant piece of science.

I blogged a few days ago about the idea that the structure we see in this map was produced by sound waves reverberating around the early Universe. The techniques cosmologists use to analyse this sound are similar to those used in branches of acoustics except that we only see things in projection on the celestial sphere which requires a bit of special consideration.

One of the things that sticks in my brain from my undergraduate years is being told that if a physicist doesn’t know what they are doing they should start by making a Fourier transform. This breaks down the phenomenon being studied into a set of independent plane waves with different wavelengths corresponding to the different tones present in a complicated sound.

It’s often very good advice to do such a decomposition for one-dimensional time series or fluctuation fields in three-dimensional Cartesian space, even you do know what you’re doing, but it doesn’t work with a sphere because plane waves don’t fit properly on a curved surface. Fortunately, however, there is a tried-and-tested alternative involving spherical harmonics rather than plane waves.

Spherical harmonics are quite complicated beasts mathematically but they have pretty similar properties to Fourier harmonics in many respects. In particular they are represented as complex numbers having real and imaginary parts or, equivalently, an amplitude and a phase (usually called an argument by mathematicians). The latter representation is the most useful one for CMB fluctuations because the simplest versions of inflation predict that the phases of each of the spherical harmonic modes should be randomly distributed. What this really means is that there is no information content in their distribution so that the harmonic modes are in a state of maximum statistical disorder or entropy. This property also guarantees that the distribution of fluctuations over the sky should have a Gaussian distribution.

If you accept that the fluctuations are Gaussian then only the amplitudes of the spherical harmonic coefficients are useful. Indeed, their statistical properties can be specified entirely by the variance of these amplitudes as a function of mode frequency. This pre-eminently important function is called the power-spectrum of the fluctuations, and it is shown here for the WMAP data:

Although the units on the axes are a bit strange it doesn”t require too much imagination to interpret this in terms of a sound spectrum. There is a characteristic tone (at the position of the peak) plus a couple of overtones. However these features are not sharp so the overall sound is not at all musical.

If the Gaussian assumption is correct then the power-spectrum contains all the useful statistical information to be gleaned from the CMB sky, which is why so much emphasis has been placed on extracting it accurately from the data.

Conversely, though, the power spectrum is completely insenstive to any information in the distribution of spherical harmonic phases. If something beyond the standard model made the Universe non-Gaussian it would affect the phases of the harmonic modes in a way that would make them non-random.

So far, so good. It sounds like it should be a straightforward job to work out whether the WMAP phases are random or not. Unfortunately, though, this task is heavily complicated by the presence of noise and systematics which can be quite easily cleaned from the spectrum but not from more sophisticated descriptors. All we can say so far is that the data seem to be consistent with a Gaussian distribution.

However, I thought I’d end with a bit of fun and show you how important phase information could actually be, if only we could find a good way of exploiting it. Let’s start with a map of the Earth, with the colour representing height of the surface above mean sea level:

You can see the major mountain ranges (Andes, Himalayas) quite clearly as red in this picture and note how high Antarctica is…that’s one of the reasons so much astronomy is done there.

Now, using the same colour scale we have the WMAP data again (in Galactic coordinates).

The virture of this map is that it shows how smooth the microwave sky is compared to the surface of the Earth. Note also that you can see a bit of crud in the plane of the Milky Way that serves as a reminder of the difficulty of cleaning the foregrounds out.

Clearly these two maps have completely different power spectra. The Earth is dominated by large features made from long-wavelength modes whereas the CMB sky has relatively more small-scale fuzz.

Now I’m going to play with these maps in the following rather peculiar way. First, I make a spherical harmonic transform of each of them. This gives me two sets of complex numbers, one for the Earth and one for WMAP. Following the usual fashion, I think of these as two sets of amplitudes and two sets of phases. Note that the spherical harmonic transformation preserves all the information in the sky maps, it’s just a different representation.

Now what I do is swap the amplitudes and phases for the two maps. First, I take the amplitudes of WMAP and put them with the phases for the Earth. That gives me the spherical harmonic representation of a new data set which I can reveal by doing an inverse spherical transform:

This map has exactly the same amplitudes for each mode as the WMAP data and therefore possesses an identical power spectrum to that shown above. Clearly, though, this particular CMB sky is not compatible with the standard cosmological model! Notice that all the strongly localised features such as coastlines appear by virtue of information contained in the phases but absent from the power-spectrum.

To understand this think how sharp features appear in a Fourier transform. A sharp spike at a specific location actually produces a broad spectrum of Fourier modes with different frequencies. These modes have to add in coherently at the location of the spike and cancel out everywhere else, so their phases are strongly correlated. A sea of white noise also has a flat power spectrum but has random phases. The key difference between these two configurations is not revealed by their spectra but by their phases.

Fortunately there is nothing quite as wacky as a picture of the Earth in the real data, but it makes the point that there are more things in Heaven and Earth than can be described in terms of the power spectrum!

Finally, perhaps in your mind’s eye you might consider what it might look lie to do the reverse experiment: recombine the phases of WMAP with the amplitudes of the Earth.

If the WMAP data are actually Gaussian, then this map is a sort of random-phase realisation of the Earth’s power spectrum. Alternatively you can see that it is the result of running a kind of weird low-pass filter over the WMAP fluctuations. The only striking things it reveals are (i) a big blue hole associated with foreground contamination, (ii) a suspicious excess of red in the galactic plane owing to the same problem, and (iiI) a strong North-South asymmetry arising from the presence of Antarctica.

There’s no great scientific result here, just a proof that spherical harmonics can be fun.

PS. These pictures were made by a former PhD student of mine, Patrick Dineen, who has since quit astronomy to work in high finance. I hope he is weathering the global financial storm!