6. Caveats and summary
Our low h measurements are in disagreement with current measurements of the Hubble constant. Possible explanations for this discrepancy could be unidentified systematics in the CMB data or the local h measurements. Galactic foregrounds could be a problem for the CMB while the distance ladder may need some readjusting for the local h measurements (e.g., Feast & Catchpole 1997). The best way to address these problems is with more and better data. This is being done rapidly. New detectors and better designed observations are improving the quality of both the CMB data and the more direct h measurements. SZ and supernovae observations are also increasing the redshift of the h measurements.
A simple answer to the discrepancy between direct h measurements and our results is that the Universe is not well-described by the models considered here. It is possible that one or more of our basic assumptions is wrong, or we could simply be looking at too restricted a region of parameter space. The shape of the primordial power spectrum may be more complicated than the family of models we are using. Inflation may be wrong and structure may not have formed from adiabatic curvature fluctuations. Topological defects may be the origin of structure.
We emphasize that we have only considered a particular type of cosmological scenario, although arguably the simplest; the results we have presented here are valid under the assumption of inflation-based, Gaussian adiabatic initial conditions in a critical density universe () with no cosmological constant. We have not considered any early reionization scenarios or gravitational wave contribution. We have also not included any hot dark matter.
Our assumption can be considered very restrictive since plausible values for in the range can change the power spectrum significantly. For example, the position of the Doppler peak, is roughly proportional to . Thus low , by pushing higher may permit higher h values to accomodate the high of the CMB data. We are in the process of checking this assumption; we consider open models in Lineweaver & Barbosa (1998).
Reionization models can affect the power spectra significantly by lowering the Doppler peak but this can be compensated by n values larger than 1. For example, de Bernardis et al. (1997) have looked at reionization models and find a best-fit early reionization at and .
In paper I we took a brief look at flat- models. The supernovae results of Perlmutter et al. (1997) can constrain better than the CMB data. The combination of the CMB, BBN and supernovae constraints in flat- models yields .
If gravitational waves or any other effect plays an important role in CMB anisotropy formation, we expect that the inclusion of this effect in the family of models tested, will improve the resulting fits. However the inclusion of gravitational waves seems to make the fits slightly worse without changing the location of the minimum (Liddle et al. 1996b). Bond and Jaffe (1997) analyzed the combined DMR (Bennett et al. 1996), South Pole (Gunderson et al. 1995) and Saskatoon (Netterfield et al. 1997) data using signal-to-noise eigenmodes. They looked at the parameters h, n and in a variety of models. The inclusion of tensor modes for models seems to produce small shifts in the likelihood surfaces.
There may be extra-relativistic degrees of freedom (hot dark matter (HDM) or mixed dark matter (MDM)). de Bernardis et al. (1997) found that current CMB anisotrophy measurements cannot distinguish between CDM and MDM models. We agree with this assessment and add that HDM and CDM models also cannot usefully be distinguished with current CMB data.
In addition to the h, , n and Q considered here, regions of a larger dimensional parameter space deserve further investigation including , , , early reionization parameters such as , tensor mode parameters and T, iso-curvature or adiabatic initial conditions and topological defect models with their additional parameters such as the coherence length. The fact that we obtain acceptable values in our small 4-D parameter space lends some support to the idea that we may be close to the right model. However establishing error bars on broad-band power estimates is a relatively new science. If the Universe is not well-described by these models then as the data improve, work like this will show poor fits and other regions of parameter space may be preferred.
CMB measurements have become sensitive enough to constrain cosmological parameters in a restricted class of models. The results we have presented here are valid under the assumption of Gaussian adiabatic initial conditions in a critical density () universe with no cosmological constant. We have explored the 4-dimensional parameter space of h, , n and Q. Our CMB-derived constraints on h, , n and Q exclude large regions of parameter space. We obtain a low value for Hubble's constant: . The CMB data constrain only weakly: . For the slope and normalization of the power spectrum we obtain and K. The error bars on each parameter are for the case where the other 3 parameters have been marginalized. When we condition on we obtain the normalization K.
CMB constraints are independent of other cosmological tests of these parameters and are thus particularly important. The fact that reasonable values are obtained means that the current CMB data are consistent with inflationary-based CDM models with a low Hubble constant. In the context of the models considered, the CMB results are consistent with four other independent cosmological measurements but are in disagreement with more direct measurements of h.
© European Southern Observatory (ESO) 1998
Online publication: December 16, 1997