Power isn’t Everything
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!