Primordial fluctuations of the CMB contain a great deal of information about physical processes occurring in the early universe. Since some of these processes depend upon the value of H0, then obviously there is a hope that by getting accurate enough information about the CMB, we may be able to pin down H0 quite independently of either optical determinations, or the physical methods like SZ and gravitational lensing. Figure 7 shows the power spectrum for primordial fluctuations of the CMB predicted from inflationary theory and the cold dark matter (CDM) model. The quantities plotted are l(l + 1)cl, versus l where cl is defined via
and the Ylm are standard spherical harmonics. Increasing l corresponds to decreasing angular scale , with a rough relationship between the two of 2/l radians. We see that on large angular scales ( 2°) we expect the CMB power spectrum to reflect the initially near scale-invariant spectrum coming out of inflation; on intermediate angular scales we expect to see a series of peaks, and on smaller angular scales ( 10 arcmin) we expect to see a sharp decline in amplitude. In terms of the diameter of corresponding proto-objects imprinted in the CMB, then a rich cluster of galaxies corresponds to a scale of about 8 arcmin, while the angular scale corresponding to the largest scale of clustering we know about in the universe today corresponds to 1/2 to 1 degree. The first large peak in the power spectrum, at l's near 200, and therefore angular scales near 1°, is known as the `Doppler,' or `Sakharov,' or `acoustic' peak.
Figure 7. Power spectrum for standard CDM. Parameters assumed are = 1, n = 1, H0 = 50 km s-1 Mpc-1 and a baryon fraction of b = 0.04.
Now we come to a very interesting point. The position of this Doppler peak, and of the smaller secondary peaks, is determined by the value of the total , and varies as lpeak -1/2 (This behavior is determined by the linear size of the causal horizon at recombination, and the usual formula for angular diameter distance). This means that if we were able to determine the position (in a left/right sense) of this peak, and we were confident in the underlying model assumptions, then we could read off the value of the total density of the universe (In the case where the cosmological constant was non-zero we would effectively be reading off the combination matter + ). This would be a determination of free of all the usual problems encountered in local determinations using velocity fields etc. Similar remarks apply to the Hubble constant. The height of the Doppler peak is controlled by a combination of H0 and the density of the universe in baryons, b. We have a constraint on the combination b H20 from nucleosynthesis, and thus using this constraint and the peak height we can determine H0 within a band compatible with both nucleosynthesis and the CMB. Alternatively, if we have the power spectrum available to good accuracy covering the secondary peaks as well, then it is possible to read off the values of tot, b and H0 independently, without having to bring in the nucleosynthesis information. The overall point here, is that the power spectrum of the CMB contains a wealth of physical information, and that once we have it to good accuracy, and have become confident that an underlying model, such as inflation and CDM, is correct then we can use the spectrum to obtain the values of parameters in the model, potentially to high accuracy.
Experimentally, the CMB data is approaching the point where meaningful comparison between theory and prediction, as regards the shape and normalization of the power spectrum, can be made. This is particularly the case with the new availability of the recent CAT (Scott et al. (1996)) and Saskatoon (Netterfield et al., submitted) 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 tot = 1). 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 minimize 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 + inflation turns out not to be a viable theory for example, then the bounds on H0 and 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.
This said, it is certainly of interest to begin this process of quantitative comparison of CMB data with theoretical curves. Fig. 8 shows a set of recent data points, some of them discussed above, put on a common scale (which may effectively be treated as sqrt[l(l + 1)cl)], 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., submitted, from where this figure was taken. The analytical version of the power spectrum is parameterized by its location in height and left/right position, and enables one to construct a likelihood surface for the parameters and Apeak, where Apeak is the height of the peak, and is related to a combination of b and H0, as discussed above. The dotted and dashed extreme curves in Fig. 8 correspond to the fact that the calibration of the Saskatoon experiment is not well established (Netterfield et al., submitted), and indicate the best fits obtained by varying the amplitude of the Saskatoon points by ± 14%. The central fit yields a 68% confidence interval of
with a maximum likelihood point of = 0.7 after marginalization over the value of Apeak. Incorporating nucleosynthesis information as well, as sketched above (specifically we assume the Copi et al. (1995) bounds of 0.009 bh2 0.02), we obtain a 68% confidence interval for H0 of
This range ignores the Saskatoon calibration uncertainty. Generally, in the range of parameters of current interest, increasing H0 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 H0. By this means, an upper limit closer to 70 km s-1 Mpc-1 is obtained.
Figure 8. Analytic fit to power spectrum versus experimental points (Taken from Hancock et al., submitted).
These first results are only schematic, but considering this is a totally new method, give a very encouraging agreement with the range of H0 picked out by other methods, though with a slightly lower mean that most optical determinations, in agreement with the Ryle SZ results so far.
What of the future? Although the Cosmic Anisotropy Telescope (CAT) in Cambridge has already provided maps of CMB anisotropy on scales ~ 0°.4, these are relatively poor as images due to the limited number of baseline lengths and pixels available. In fact, the CAT is a prototype for a considerably more advanced instrument, the Very Small Array (VSA). The objectives of the VSA are to obtain detailed maps of the CMB with a sensitivity approaching 5µK and covering a range of angular scales from 10' to 2°. The good accuracy available over a scale range that is well-matched to the positions of the first and secondary Doppler peaks in the power spectrum, should enable measurements of and H0 to be made to an accuracy of better than 10%. The instrument is currently under construction at Cambridge and Jodrell Bank, and it is hoped it will be operational in Tenerife by the middle of the year 1999.
Two new satellite experiments to study the CMB have recently been selected as future missions. These are MAP, or Microwave Anisotropy Probe, which has been selected by NASA as a Midex mission, for launch probably in 2001, and COBRAS/SAMBA, which has been selected by ESA as an M3 mission, and will be launched hopefully soon after 2004.
A crucial feature of a satellite experiment is the potential all-sky coverage that it affords, and the ability to map features on large angular scales ( 10°). Neither of these facilities are possible from the ground, due to problems with the atmosphere. On the other hand a satellite experiment has more problems in attaining resolution at the smaller angular scales, because of the limited dish size possible within the confines of the launcher. In this respect high frequency capability is an advantage. The best angular resolution offered by MAP is 18 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 pinning down the full shape of the first and certainly secondary Doppler peaks in the power spectrum. On the other hand, the angular resolution of COBRAS/SAMBA extends down to 4 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. In fact if COBRAS/SAMBA can measure one third of the CMB sky to an accuracy (after foreground subtraction) of 2 x 10-6 in T/T per pixel, then a joint determination of and H0 to ~ 1% accuracy is possible in principle. There seems little chance of being able to do this by any other means, and in conjunction with the other powerful capabilities of the satellite, balloon and ground-based experiments to come, represents a tremendously exciting prospect for the future.
We thank all the other colleagues at Cambridge and Jodrell Bank involved in the Ryle, CAT and VSA work and in particular thank Keith Grainge for permission to use material from his thesis. We also thank Stephen Hancock, Graca Rocha and Carlos Gutierrez for permission to quote from joint results before publication.