Angular Power Spectrum of CMB
Chalky
other respects, I am still struggling to understand the angular power
spectrum data for the CMB at even the most elementary level. For
example, in the relevant graph at http://en.wikipedia.org/wiki/Cosmic_microwave_background_radiation,
what, precisely, is being measured in the vertical axis?
Is this the total power measured at a given angular scale, or the
deviation from some reference power? Is this measured over the entire
spectrum, or over a narrow reference band of frequencies.
C
Stupendous_Man
> Although comments from other contributors have been very helpful in
> other respects, I am still struggling to understand the angular power
> spectrum data for the CMB at even the most elementary level. For
> what, precisely, is being measured in the vertical axis?
>
> Is this the total power measured at a given angular scale, or the
> deviation from some reference power? Is this measured over the entire
> spectrum, or over a narrow reference band of frequencies.
>
> C
Let me suggest that you read this thread on the BAUT forum:
http://www.bautforum.com/questions-answers/59109-angular-power-spectrum-what-how-make-one.html
There is a link in the thread to Max Tegmark's CMB pages, which
might help you. You could also look at Wayne Hu's CMB tutorial --
just search for it.
Good luck.
Kent Paul Dolan
> Although comments from other contributors have
> been very helpful in other respects, I am still
> struggling to understand the angular power
> spectrum data for the CMB at even the most
> elementary level. For example, in the relevant
> graph at
> http://en.wikipedia.org/wiki/Cosmic_microwave_background_radiation,
> what, precisely, is being measured in the vertical
> axis?
> Is this the total power measured at a given
> angular scale, or the deviation from some
> reference power?
No, it's a _difference_ of power between points at a
give separation, averaged across the data space.
> Is this measured over the entire spectrum, or over
> a narrow reference band of frequencies.
"Power" would strongly suggest "across the whole
blackbody spectrum", but in practice, limited to
whatever the sensors could capture.
The graph is pretty clearly labeled as being in
micro-Kelvins squared (squared, presumably both to
give a pretty graph and to remove numerical sign
issues).
http://en.wikipedia.org/wiki/Image:WMAP_TT_power_spectrum.png
It is also a power spectrum, and angular, so the
reasonable interpretation is that it is the
difference between radiation intensities at points
in receiver aiming coordinates (directional
stereradians?) separated by the horizontal scale's
indicated angular separation, which at some angular
separations is on average small, at others, on
average large, those separations corresponding to
real, if tiny, differences in the photon intensity
emitted out of different volumes of the infant
universe we see when "looking" in different
directions.
If I recall correctly, these are "acoustic" because
they correspond to compressions/rarefactions
(accoustic waves, that is) in the average density of
the universe at the point it became transparent to
photons.
That is the 10^-4 to 10^-5 inhomogeneity spoken of
in the article.
[Probably something very like a 2D Fourier analysis
of the whole dataset produced the displayed peaks,
though Monte Carlo sampling across it would have
worked just as well.]
This seems to be the salient original document, and
section 7, where the angular power spectrum is
discussed, is fairly lucid.
http://www.arxiv.org/PS_cache/astro-ph/pdf/0603/0603451v2.pdf
xanthian.
eb...@lfa221051.richmond.edu
Kent Paul Dolan <xant...@well.com> wrote:
>Chalky wrote:
>
> > Is this the total power measured at a given
> > angular scale, or the deviation from some
> > reference power?
>
>No, it's a _difference_ of power between points at a
>give separation, averaged across the data space.
No, it's not.
The power spectrum at any given wavenumber l is, more or less, the
mean of the absolute squares of the Fourier transform of the temperature
map at that wavenumber. (To be slightly more precise, it's based
on a spherical harmonic transform, not a Fourier transform, but that makes
a big difference only at very low l.)
Others have suggested some good places to go to read more about what
this means. In a bit of shameless self-promotion, I'll mention
one more: a set of lectures I gave at a summer school a while back,
available at
http://arxiv.org/abs/astro-ph/9607088
A lot of the content in there is similar to things on Wayne Hu's
web pages, which someone mentioned earlier. But I suspect that,
on the particular topic of what a power spectrum is and how you measure it,
my writing might be easier to follow than Wayne's.
These lectures are more than 10 years old. Of course, the state of
the art has advanced a lot since then! But if the goal is an
understanding of the fundamentals (e.g., what an angular power
spectrum is), I suspect this is still useful.
>The graph is pretty clearly labeled as being in
>micro-Kelvins squared (squared, presumably both to
>give a pretty graph and to remove numerical sign
>issues).
These reasons aren't correct. The power spectrum is given in
units of temperature squared because power spectra are by definition
quadratic quantities.
>[Probably something very like a 2D Fourier analysis
>of the whole dataset produced the displayed peaks,
That's right. That's part of the definition of a power spectrum.
>though Monte Carlo sampling across it would have
>worked just as well.]
That's not true.
>This seems to be the salient original document, and
>section 7, where the angular power spectrum is
>discussed, is fairly lucid.
>
>http://www.arxiv.org/PS_cache/astro-ph/pdf/0603/0603451v2.pdf
That's certainly the definitive description of the WMAP angular power
spectrum, although I'd be surprised if people who didn't already know
quite a bit about the "standard lore" in the field could get a full
understanding from this article.
-Ted
--
[E-mail me at na...@domain.edu, as opposed to na...@machine.domain.edu.]