Re: Question about inflation theory

Greg Hennessy wrote:
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.

No. First of all each receiver has an independent noise response as
compared to the other receivers. Secondly the technique used to
calculate the power spectrum is a cross power (not cross correlation)
between data channels with uncorrelated noise. This removes the noise
bias from the estimate of the power spectrum.

It removes the noise bias only if it is statistically uncorrelated from
receiver to receiver i.e. for the internal receiver noise (that's the
assumption made in
http://lambda.gsfc.nasa.gov/product/map/pub_papers/firstyear/powspec/wmap_powspec.pdf
). It does not remove noise-like behaviour in the actual signal as this
is correlated in all receivers. If it would be uncorrelated, then,
according to your own admission, the auto- and cross-power spectra
would be different (which they are not according to Fig.6 in that
paper).


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.

The difference between the one year and three year maps is not noise,
indeed it is immediatly obvious, especially in the K and Ka band
images, that the differences majorly in the galactic plane.

In the V and W-bands (which are the most important ones for the angular
power spectra) there is no galactic emission at all visible in Fig. 3
in
http://map.gsfc.nasa.gov/m_mm/pub_papers/temperature/wmap_3yr_temp.pdf
.. The overall fluctuation of +-20 microK in the difference map is also
evident from Fig.6 in this paper.
As I said already, I am pretty sure that a power spectrum for the
(unsmoothed) difference map would prove my point that the second peak
is not a static feature.

If you think that averaging data points doesn't improve signal to
noise you need a remedial course in data analysis. If you don't think
that increased exposure time improves signal to noise doesn't increase
signal to noise when you aren't limited by systematics you need a
remedial course in data analysis.

You are still missing the point here: due to the differential system
there is no proper signal but only one that has noise characteristics:
if telescope 1 observes a temperature T1=T0+dT1 and telescope 2
T2=T0+dT2, then the difference signal is T1-T2=dT1-dT2 . If dT1 and dT2
vary randomly in space and time, then the mean of the absolute value of
dT1-dT1 i.e. <|dT1-dT2|> = sqrt(2)*sigma where sigma is the standard
deviation of dT1 and dT2 individually (assuming both are equal). Now
<|dT1-dT2|>=sqrt(2)*sigma is itself only an expected mean value which
would only be realized if one data point would be covered an infinite
number of times. Otherwise this quantity itself would have an
statistical error (e.g. 10% for 100 measurements 1% for 10000). The
latter corresponds to the error bars given in the power spectra. The
error bars just merely indicate the statistical uncertainty of the bias
sqrt(2)*sigma, i.e. smaller error bars (as in the latest analysis)
actually confirm the bias rather than invalidate it.

Anyway, my calculation is still somewhat uncertain as I don't know for
sure the frequency bandwidth of the WMAP receivers.

In about 30 seconds of looking I can find from the WMAP exploratory
suppliment that the bandwidth of the K, Ka, Q, V, and W receivers are
4, 5, 8, 13 and 19 GHz.

Yes, well done, thanks. I might get back to you in case I am stuck
again with finding a particular reference.

Thomas

.

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