In reports of studies of the Cosmic Microwave Background, such as this from WMAP, we read the phrase "angular (power) spectrum", and can view charts (graphs) like this.

But what is an angular power spectrum?

How easy it is to make one?

What sort of input data do you need?

This thread is devoted to answering these questions.

It is not, repeat, NOT, about the CMB per se; rather, it is about the data you need to make one of these, the mathematical tools (operations, analyses, etc) you'd use, and (maybe) some things to be careful of.

Let's start with a simple question: what's the difference between an angular power spectrum and a Fourier transform?

OK, so what's a Fourier transform (in simple terms)?

(And while we're at it, what's the relationship between an angular power spectrum and spherical harmonics, again, in simple terms?)

Nereid. I'm going to add a question until I have time when school gets out in three weeks. At what size will the cooling ejecta cloud from a type 2 supernova explosion going off in a previously excavated spatial volume (from an earlier SN)..most closely mimic the acoustical spherical harmonics seen in the WMAP data of the CMB? This assumes a slight dipole due to parity effects in the explosion that does not necessarily align itself with the inferred motion of the observing station....10 AU's? a parsec? 10 parsecs, 100? 1000?

pete

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Input date: a measure of intensity of microwave radiation at angular positions all over the sky. If we identify each point on the celestial sphere with two angles -- (RA, Dec) -- then we can write:

Intensity(i) = intensity at the i'th position

where i = 1 ... N.

So, you have an array of hundreds and hundreds, or thousands and thousands, of measurements.

Now, consider EVERY possible pair of measurements -- if there are N measurements, you'll have N(N-1)/2 unique pairs. For every pair of measurements, compute some quantity which depends on the intensity of the two positions, something like:

Q(i, j) = I(i)*I(j)

and also compute the angular distance between these two points:

angdist(i, j) = distance between point i and j on sky

Now, if the intensity of the radiation were to vary in a random manner over the entire sky, then the quantity Q(i, j) would also vary randomly. There would be no difference between the value for points right next to each other, and points far apart. On the other hand, if there is structure in the radiation -- say, blobs which are typically 3 degrees in diameter -- then we'll notice something: the quantity Q(i, j) will be large when the two points are less than about 3 degrees apart (because both points may fall into the same bright blob), but vary randomly when the two points are more than 3 degrees apart.


So, make a plot of the quantity Q(i, j) as a function of distance between the two points. That graph will represent the two-point correlation function of the radiation in a picture.

If you want to represent this relationship with an equation, rather than with a picture, you can try to find a mathematical formula which describes the quantity Q(i, j) as a function of the angular distance between the points. One choice for describing this quantity Q(i, j) are the spherical harmonic functions: they are combinations of sines and cosines of two angles. If there is no preferred direction for the radiation, then all the information on "variation of radiation as a function of angular distance" can be expressed in terms of the coefficients of a subset of the spherical harmonic functions.

The "angular power spectrum" is a plot of (a simple function of) these coefficients of the spherical harmonic functions which best describe the angular correlations of the radiation.

So, to recap briefly,

- compute a quantity which expresses how well the CMB at one
location in the sky predicts the CMB at another location,
as a function of the angle between the two locations

- use the spherical harmonic functions to approximate the very
complicated, detailed data values from millions of pairs of
CMB measurements

- calculate a quantity using the coefficients of this model fit

- plot that quantity

The final plot contains what is commonly called "the angular power spectrum of the CMB."

Very nice explanation, StupendousMan.

Trinitree, it seems like you're trying to raise vague issues without presenting any kind of serious case. If you think that type II supernovae could somehow be responsible for observed fluctuations in the CMB, it would seem that the onus would fall on you to actually show how that would work, quantitatively (and such a claim would probably be best presented in the AtM section).

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No kidding! That was a lot more clear than trying to figure out the angular portion of an orthogonal set of solutions to Laplace's equation represented in a system of spherical coordinates! (Equation pictured below).

Attached Images

eqn.png (3.0 KB, 37 views)

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Thanks a lot StupenduousMan, very clear explanation.

Can you clear up a few more things? I was looking at this page and there are a couple of things I'm not quite sure about. It says that angular power is really multipole moment. I'm familiar with dipoles and other multipoles from electrostatics and I wonder if this means that the CMB spectrum can be thought of as a superposition of multipoles (i.e for a quadrupole two points with high intensity and two points with low intensity at the ends of a + sign) with the angular scale spectrum representing the strength of the given multipole? Is this the right interpretation or am I completely off?

The other question is that I can't help noticing that there are only a few data points on the figure. Why can't (or don't) they calculate angular power for arbitrary separations?

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Yes, that's the idea.

There's a tradeoff: you have many pairs of points. You can choose to arrange them in lots and lots of bins, with small steps in angular distance -- in that case there will be only a few points in each bin; or you can make only a few, wide bins -- in which case there will be many pairs in each bin. The former gives you many points, each with large error bars; the latter gives you a few points, with very small error bars.

The people who made the graph you mention chose a relatively few, wide bins.

In my OP, I forgot to include another aspect: just how easy is it to produce an angular power spectrum of some (well-defined) input data?

Taking the WMAP data (loosely) as input, we have pixels of size ~1 square degree*, so there are ~40k pixels for the whole sky**.

Assuming no preferred direction(s), what part of the process of generating a (1D) angular power spectrum from ~40k inputs would consume the most computing resources?

From StupendousMan's description (assuming you understood it!), and assuming you have some familiarity with writing computer code, how hard do you think it would be to write pseudo-code for the number crunching part of a program that would produce an angular power spectrum?

If you didn't understand StupendousMan's description, what parts were hardest to grasp? Where did you 'lose the plot'?

FWIW, here are some historical papers, tracing (part of) the ancestry of the WMAP CMB angular power spectrum (the links are to arXiv abstracts):

Angular Power Spectrum of the Microwave Background Anisotropy seen by the COBE Differential Microwave Radiometer; Wright, Smoot, Bennett, Lubin (1994)

2-Point Correlations in the COBE DMR 4-Year Anisotropy Maps; Hinshaw, Banday, Bennett, Gorski, Kogut, Lineweaver, Smoot, Wright (1996)

Angular Power Spectra of the COBE DIRBE Maps; Wright (1997)

First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Angular Power Spectrum; Hinshaw et al. (2003) (>10 "al"s!)

Three-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Temperature Analysis; Hinshaw et al (2006) (> 20 "al"s!)

*OK, the resolution is better than that, more like ~0.6^o, but this is only illustrative.
**Why?

I think I understood it, but as near as I can tell, it's in error

StupendousMan seems to be describing an attempt at a cross-correlation function--which the angular power spectrum is not.

What Eckelston thought were data points, are not, they are the computed spectrum coefficients, fit by a curve.

Much like we can take the spectrum of an electronic signal by integrating it with Fourier functions, we can integrate the CMB with spherical harmonics to produce the angular power spectrum.

I am describing the mathematical operations one performs in order to produce the angular power spectrum -- and one (common) step involves computing a correlation function. You can see them in any number of places:

Chapter 9 of Ryden's "Introduction to Cosmology"

Or the abstract to the article "Fast Cosmic Microwave Background Analyses via Correlation Functions" by Szapudi et al., ApJ 548, L115 (2001), which begins as follows:

We propose and implement a fast, universally applicable method for extracting the angular power spectrum 𝒞ℓ from cosmic microwave background temperature maps by first estimating the correlation function ξ(&thetas.

or in many other places. After computing the two-point correlation function, one fits a model to the many, many data.

Thanks for the clarification StupendousMan, sorry about that. I am not involved in the CMB research or study, but I have been involved with some signal processing, electronic and geophysical. The geophysics (shape of the earth) relies upon spherical harmonics.

I think I can answer some of the questions about spherical harmonics. And I just noticed that Eckelston's link is to the same chart that Nereid linked to in the OP, her third link.

I found this, Max Tegmark's webpage about it, it seems to be an informal and high level approach.

Wow. Lots of good links in this thread. Getting a better feel for this creature, the power spectrum of the CMB. Not to forget the bolded point below....

It is important to remember that although certain projects (such as trying to determine the shape of the inflaton potential by measuring the spectral index n, the tensor spectral index, and the tensor-to-scalar ratio) require previously untested assumptions about untested high-energy physics, there are many other parameters that can be measured in a robust fashion by assuming little else than that we understand gravity and the behavior of hydrogen and photons at a few thousand degrees (Hu & White 1996). For instance, the spacing between the power spectrum peaks provides a fairly clean probe of the angle-distance relationship (which fixes a certain combination of Omega and Lambda). Whatever the true power spectrum turns out to look like, it is likely to help us clean up among the profusion of cosmological models that are currently on the market.

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hay
I have a 3D map of particles and I try to create a power spectrum of it. This is then something like a density power spectrum. From your very detailed explanations above I suppose I have to do the following:
1. Create a density profile, by defining pixels (3D cubes) and counting the particles within each box. The number of particles would be I(i).
2. Calculating Q(i,j)=I(i)*I(j) for each pair of 3D cubes.
3. plot Q(i,j) against the distance in pc (Mpc).
is this correct? I ask, because my plot looks strange and other plots which I use for comparisons look much different. Could it be, that the analogy between angular power spectrum and matter power spectrum is not as easy as I suggested above???
Thanks for helping
florian

You need to provide many, many details for us to be able to help you. What are you trying to do? Why are you trying to do it? What is your goal?

sorry for the incomplete question. I am trying to write a tool which evaluates a special type of data. The data is always a map of particles with a special distribution. I implemented already a tool to calculate the correlation function, but I was told to add a matter power spectra. This is the first time that I work with power spectra, so I just searched in the web and found your page. I have a plot which I try to reproduce, the plot is attached.
So my goal is to reproduce this plot.
You can see that the x-axis is in pc^{-1} what contradicts my guess that the x-axis is the distance. The y-axis is in pc^3 what puzzles me even more....
thanks for helping
florian

Attached Images

pspect2.png (7.2 KB, 3 views)

First, this seems a bit peculiar. If you were told to add a matter power spectrum, then I am guessing that you are a student of some sort working with an astronomer. Why not ask the astronomer instead of us? Is she scary?

Second, if you don't know why the plot has the units that it has, I think you may need some more background than I can give in a posting on this website. Let me suggest that you go find a copy of Peebles' book "Principles of Physical Cosmology", or "Large Scale Structure in the Universe." Any large university library should have them. Read the sections of this book which talk about the power spectrum as a tool to investigate cosmology.

Good luck. I think you'll find reading that material will help you greatly.

I can't ask him, because I am in Germany at the moment and he is in Australia. Furthermore I would like to do it without his help.
Thanks for the books you mentioned. I will try to gte them as soon as possible, but I am not at any university at the moment and probably I will not be able to get them within the next 2 month. Is there any source of information at the web?
thanks
and best regards
florian

supplement: I found both books in the google book database. "Large Scale Structure in the Universe" explaines the power spectra, but I have problems to understand it. It would be very helpful if anybody could help me to interpret the equations.
http://books.google.de/books?id=O_BP...erse#PPA378,M1
thanks and best regards
florian

In its simplist pedagogical form, is it correct to say that the power spectrum reveals the specifics to the density variations (anisotropy) at the time of recombination? Of course, the related anisotropy of the temperature, polarization, etc. come as a result of this density anisotropy, as well as, angular size observed.

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The CMB angular power spectrum certainly contains that signal ... but there may - or should - be others too (depends on the details, of the angular resolution, for example): the ISW (integrated Sachs-Wolfe effect), and the SZE (Sunyev-Zel'dovich effect), to name just two.