Acknowledgements6 Analysis of CMB data6.2 Wiener filtering

6.3 The CMB power spectrum versus experimental points

It will have become apparent in the preceding sections that the CMB data are approaching the point where meaningful comparison between theory and prediction, as regards the shape and normalisation of the power spectrum, can be made. This is particularly the case with the new availability of the recent CAT and Saskatoon results, where the combination of scales they provide is exactly right to begin tracing out the shape of the first Doppler peak. (If this exists, and if tex2html_wrap_inline1999 .) Before embarking on this exercise, some proper cautions ought to be given. First, the current CMB data is not only noisy, with in some cases uncertain calibration, but will still have present within it residual contamination, either from the Galaxy, or from discrete radio sources, or both. Experimenters make their best efforts to remove these effects, or to choose observing strategies that minimise them, but the process of getting really `clean' CMB results, free of these effects to some guaranteed level of accuracy, is still only in its infancy. Secondly, in any comparison of theory and data where parameters are to be estimated, the results for the parameters are only as good as the underlying theoretical models and assumptions that went into them. If CDM turns out not to be a viable theory for example, then the bounds on tex2html_wrap_inline1565 derived below will have to be recomputed for whatever theory replaces it. Many of the ingredients which go into the form of the power spectrum are not totally theory-specific (this includes the physics of recombination, which involves only well-understood atomic physics), so that one can hope that at least some of the results found will not change too radically.

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

  

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Figure 21: Analytic fit to power spectrum versus experimental points. (From Hancock et al. [54Jump To The Next Citation Point In The Article], 1997.)

a set of recent data points, many of them discussed above, put on a common scale (which may effectively be treated as tex2html_wrap_inline2003), 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 tex2html_wrap_inline1565 and tex2html_wrap_inline2007, where tex2html_wrap_inline2007 is the height of the peak, and is related to a combination of tex2html_wrap_inline1641 and tex2html_wrap_inline1639, as discussed above. The dotted and dashed extreme curves in Figure  21 indicate the best fit curves corresponding to varying the Saskatoon calibration by tex2html_wrap_inline2015 . The central fit yields a tex2html_wrap_inline2017 confidence interval of

equation581

with a maximum likelihood point of tex2html_wrap_inline2019 after marginalisation over the value of tex2html_wrap_inline2007 . Incorporating nucleosynthesis information as well, as sketched above (specifically the Copi et al. [43] bounds of tex2html_wrap_inline2023 are assumed), a tex2html_wrap_inline2017 confidence interval for tex2html_wrap_inline1639 of

equation586

is obtained. This range ignores the Saskatoon calibration uncertainty. Generally, in the range of parameters of current interest, increasing tex2html_wrap_inline1639 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 tex2html_wrap_inline1639 . By this means, an upper limit closer to tex2html_wrap_inline2033 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 tex2html_wrap_inline1565, tex2html_wrap_inline1639 and tex2html_wrap_inline1641 can be recovered, given coverage of 1/3 of the sky with sensitivity tex2html_wrap_inline2041 in tex2html_wrap_inline2043 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 tex2html_wrap_inline1565 and tex2html_wrap_inline1639 to tex2html_wrap_inline2049 accuracy is possible in principle. Figure  22 show the likelihood contours for two experiments with different resolutions.

  

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Figure 22: The contours show 50, 5, 2 and 0.1 percentile likelihood contours for pairs of parameters determined from fits to the CMB power spectrum. The figures to the left show results for an experiment with resolution tex2html_wrap_inline2051 . Those to the right for a higher resolution experiment with tex2html_wrap_inline1563 plotted on the same scale (central column) and with expanded scales (rightmost column). This figure is taken from Bersanelli et al. 1996 [37].

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.



Acknowledgements6 Analysis of CMB data6.2 Wiener filtering

image The Cosmic Microwave Background
Aled W. Jones and Anthony N. Lasenby
http://www.livingreviews.org/lrr-1998-11
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