Re: Question about inflation theory

Greg Hennessy wrote:
On 2006-04-11, Thomas Smid <thomas.smid@xxxxxxxxx> wrote:
Greg Hennessy wrote:
You are correct that *IF* you fit a power spectrum to the amplitude of
the temperature fluctuations you would get a biased answer. But
because that way gets a biased answer WMAP calculates a "cross power"
spectrum from two maps, rather than using an "auto power" from one
map. The auto power is subject to a bias from the noise that the cross
power is not subject to.

Whether you cross- or auto-correlate the actual signal doesn't make any
difference at all. Cross-correlation is only applied here to eliminate
receiver noise.

Excuse me, but you started this thread by claiming the WMAP power
spectra were wrong because WMAP ignored a bias in the differential
measurements. Craig Markwardt and I have patiently worked to correct
your misstatements of the statistics, especially where you disagreed
that the mean signal from subtracting two equal signals is zero. If
after this painstaking work you want to finally agree that the WMAP
power spectra isn't affected by the bias you claimed was there it
seems rather strange to simply claim that the differences between the
cross and auto correlated signal doesn't make any difference at all.

I thought I had made this clear already: the bias is not affected by
the cross-correlation process as it is in the signal proper and thus
correlated in all receivers. The corresponding peak near 0.3 deg
appears therefore here as well as in the auto-correlated power
spectrum.


Actually, the latest WMAP releases very much confirm my point: from
Fig.3 in
http://map.gsfc.nasa.gov/m_mm/pub_papers/temperature/wmap_3yr_temp.pdf
, it is obvious that the 'map' of the spatial fluctuations is actually
not constant in time.

That does not logically follow. The data processing for the three year
map is different from the one year map, so there will be differences
in the difference between the one year and three year map which isn't
related to the cosmic signal at all. Even if the two data sets were
run through the same software, which they haven't been, you expect a
difference map to be of order of the noise in the noisier map.

The order of the 'noise' level (i.e. the difference between the two
maps) is of the order of the amplitude of the second peak in the
spectrum, so it is not noise that can be neglected here. In fact it
suggests that the two are essentially the same. Why don't they publish
a power spectrum of the difference maps? It might reveal some
interesting features.


The difference between the three-year and one
year 'maps' shows residuals of +-20 microKelvin, which is exactly the
magnitude of the second peak near 0.3 deg in the power spectrum

No, the peak in the power spectrum doesn't even have the same units as
the residuals in the difference map.

What do you mean 'not the same units'? The figure caption for the
difference map of indicates clearly a fluctuation of +-20microK. The
amplitude of the second peak (near l=500) in
http://map.gsfc.nasa.gov/m_ig/060918/grandspectrum640.jpg is 50 microK
(in most of the power spectra the square of the temperatures are given
instead, i.e. in this case 2500 microK^2).


If you want to take the difference map, and then use the WMAP
algorithms to calculate a power spectra, and if *that* power spectra
had a peak at about the second acoustic peak, *THEN* you might have a
point to worry about.

On the contrary, I would have to worry about it if the difference
contains *no* peak.


considering the circumstance that they 'smoothed' the map with a 1 deg
radius (which should decrease the amplitude of the peak by about 1/2).

How much the amplitude of the peak is reduces would be a strong
function of which radiometer, since the resolution varies from about
.8 deg to .2 deg as you go from K to W band.

The lowest resolutions are not actually used for the power spectra,
only the Q,V and W bands (which all have a half-width of less than 0.3
deg).



So this shows that this peak is actually not a stationary feature but
merely a statistical fluctuation both in space and time.

You have not shown the 2nd acoustic peak to be a statistical
fluctuation.

I have shown it and the data show it (also the polarization data, as
already mentioned earlier in this thread (post #37 by date)).
Anyway, there is nothing that would show that it is *not* due to a
statistical fluctuation.



By the way I have now also worked out that the CMB fluctuation
corresponding to this peak near 0.3 deg is what you would expect from a
purely statistical fluctuation of the signal:

Your calculation is for a single observation, WMAP observed the same
spot on the sky thousands of times over the course of the 3 year
period that data has been released. That increases the signal to noise
by about a factor of 30.

As mentioned above, the integration time for each data point is merely
0.1 sec. Averaging over the data points shouldn't improve their signal
to noise ratio:
going back to the 'coin tossing' analogy used above. if you toss a coin
which has values of 2 and 4 N times and determined the actual average
of the score, then you get a result corresponding to a random
distribution with a standard deviation from the mean value (3) of
1/sqrt(N). Repeating you series of N tosses several times enables you
to determine the standard deviation more accurately but doesn't change
its value.

Anyway, my calculation is still somewhat uncertain as I don't know for
sure the frequency bandwidth of the WMAP receivers. If it is
substantially different from 10 GHz, then the number of photons sampled
(and hence the statistical error) would obviously change accordingly.

Thomas

.

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