Bearing these caveats in mind, it is certainly of interest to begin this process of quantitative comparison of CMB data with theoretical curves. Figure 21 shows
a set of recent data points, many of them discussed above, put
on a common scale (which may effectively be treated as
), and compared with an analytical representation of the first
Doppler peak in a CDM model. The work required to convert the
data to this common framework is substantial, and is discussed in
Hancock
et al.
(1997) [54], from where this figure was taken. The analytical version of
the power spectrum is parameterised by its location in height and
left/right position, and enables one to construct a likelihood
surface for the parameters
and
, where
is the height of the peak, and is related to a combination of
and
, as discussed above. The dotted and dashed extreme curves in
Figure
21
indicate the best fit curves corresponding to varying the
Saskatoon calibration by
. The central fit yields a
confidence interval of
with a maximum likelihood point of
after marginalisation over the value of
. Incorporating nucleosynthesis information as well, as sketched
above (specifically the Copi
et al.
[43] bounds of
are assumed), a
confidence interval for
of
is obtained. This range ignores the Saskatoon calibration
uncertainty. Generally, in the range of parameters of current
interest, increasing
lowers the height of the peak. Thus taking the Saskatoon
calibration to be lower than nominal, for example by the 14%
figure quoted as the one-sigma error, enables us to raise the
allowed range for
. By this means, an upper limit closer to
is obtained.
The best angular resolution offered by MAP is 12 arcmin, in
its highest frequency channel at 90 GHz, and the median
resolution of its channels is more like 30 arcmin. This means
that it may have difficulty in pining down the full shape of the
first and certainly secondary Doppler peaks in the power
spectrum. On the other hand, the angular resolution of the Planck
Surveyor extends down to 5 arcmin, with a median (across the six
channels most useful for CMB work) of about 10 arcmin. This means
that it will be able to determine the power spectrum to good
accuracy, all the way into the secondary peaks, and that
consequently very good accuracy in determining cosmological
parameters will be possible. Figure
19, taken from the Planck Surveyor Phase A study document, shows
the accuracy to which
,
and
can be recovered, given coverage of 1/3 of the sky with
sensitivity
in
per pixel. The horizontal scale represents the resolution of the
satellite. From this we can see that the good angular resolution
of the Planck Surveyor should mean a joint determination of
and
to
accuracy is possible in principle. Figure
22
show the likelihood contours for two experiments with different
resolutions.
These figures do not, however, take into account any reduction in sensitivity as a result of the need to separate Galactic foregrounds from the CMB. Nevertheless, simulations using a maximum entropy separation algorithm (Hobson, Jones, Lasenby & Bouchet, in press) suggest that for the Planck Surveyor the reduction in the final sensitivity to the CMB is very small indeed, and that the accuracy of the cosmological parameters estimates indicated in Figure 19 may be attainable.
One additional problem is that of degeneracy. It is possible to formulate two models with similar power spectra, but different underlying physics. For example, standard CDM and a model with a non zero cosmological component and a gravity wave component can have almost identical power spectra (to within the accuracy of the MAP satellite). To break the degeneracy more accuracy is required (like the Planck Surveyor) or information about the polarisation of the CMB photons can be used. This extra information on polarisation is very good at discriminating between theories but requires very sensitive polarimeters.
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The Cosmic Microwave Background
Aled W. Jones and Anthony N. Lasenby http://www.livingreviews.org/lrr-1998-11 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |