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Astron. Astrophys. 329, 799-808 (1998)

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3. Results for h and [FORMULA]

The permitted regions of the 4-D parameter space are presented in a series of 2-D projections which contain likelihood contours from a combination of the most recent CMB measurements. There are four groups of figures corresponding to the four planes [FORMULA], [FORMULA], [FORMULA] and [FORMULA] ; Figs. 1-3, 5, 6-7 and 8-9 respectively. The thick `X ' in each figure marks the minimum. Areas within the [FORMULA] contours have been shaded.

The best-fit values and confidence intervals displayed in the figures are summarized in Table 1 which thus contains the main results of this work. The values of h, [FORMULA], n and Q at the minimum of the 4-D [FORMULA] are given with error bars from the projection onto 1-D of the [FORMULA] surface. In Figs. 1 through 4 the region preferred by Big Bang nucleosynthesis (BBN) is shaded light grey ([FORMULA], see Sect. 4.1).


Table 1. parameter results

In Fig. 1 we have conditioned on [FORMULA] and [FORMULA] K. The results are [FORMULA], [FORMULA]. At [FORMULA], [FORMULA] and [FORMULA].

[FIGURE] Fig. 1. Likelihood contours in the [FORMULA] plane from recent CMB measurements. We have conditioned on [FORMULA] and [FORMULA] K. The `X ' marks the minimum. The contours are at levels [FORMULA] where [FORMULA] (see Sect. 2.2). When projected onto either of the axes these regions give the approximate size of the 1, 2, 3 and [FORMULA] confidence intervals respectively. The area within the [FORMULA] contour has been shaded dark grey. The result, [FORMULA], [FORMULA], is given in Table 1. Big Bang nucleosynthesis predictions favor the light grey band defined by [FORMULA] (see Sect. 4.1).

The contours and notation of Fig. 2 are the same as in Fig. 1 except that we have let the normalization Q be a free parameter. That is, for each value of h and [FORMULA], Q takes on the value (within the discretely sampled range) which minimizes [FORMULA]. The minimum and the [FORMULA] errors on h and [FORMULA] are the same as in Fig. 1; h stays low. The 2, 3, and [FORMULA] contours are noticeably larger than in Fig. 1. Within the [FORMULA] contours, the higher values of [FORMULA] permitted correspond to [FORMULA] K.

[FIGURE] Fig. 2. Contours and notation as in previous figure except here the normalization Q has become a free parameter, i.e., for a given h and [FORMULA], Q takes on the value that minimizes the value of [FORMULA] at that point. The minimum and the [FORMULA] errors on h and [FORMULA] are the same as in Fig. 1 however the 2, 3, and [FORMULA] contours are noticeably larger. The higher values of [FORMULA] permitted here correspond to [FORMULA] K.

Fig. 3 displays the main result for h of this paper. The result is more general than the results of Figs. 1 and 2since no restrictions on n and Q are used. Examining Figs. 1, 2 and 3 sequentially, the dark grey [FORMULA] contours can be seen to get larger as we first condition on and then marginalize over n and Q. With both n and Q marginalized we obtain [FORMULA] and [FORMULA] where the error bars are approximately [FORMULA]. At the minimum, [FORMULA] and [FORMULA] K. The [FORMULA] value at this minimum is 16. The number of degrees of freedom is 23 (= 27 data points - 1 nuisance parameter - 3 marginalized parameters). The probability of finding a [FORMULA] value this low or lower is [FORMULA]. Thus the fit obtained is "good".

[FIGURE] Fig. 3. Same as previous figure except here both Q and n are free parameters. The preferred value of h stays low: [FORMULA] but at [FORMULA], h can assume all values tested, i.e., the projection of the [FORMULA] region onto the h axis covers the entire axis. We also find [FORMULA]. At the minimum [FORMULA] and [FORMULA] K. In contrast to the previous figure the higher values of [FORMULA] permitted correspond to [FORMULA] K and [FORMULA]. On the right, the high h, [FORMULA] region within the [FORMULA] contour has [FORMULA] and [FORMULA] K.

The region of the [FORMULA] plane acceptable to both the BBN and CMB have low values of h. Large values of h, especially in the BBN region (lower right of plot) are not favored by the CMB data. However at [FORMULA], h is unconstrained since the projection of the [FORMULA] contours onto the h axis spans the entire axis.

Since n is a free parameter the amplitude, but not the location, of the Doppler peak can vary substantially. This explains the shape of the 68% region allowed by the data: the possible range of [FORMULA] is enlarged, but not the confidence interval of the Hubble constant, which seems to be predominantly determined by the position of the Doppler peak.

The disjoint [FORMULA] contour region on the right is characterized by the parameter values [FORMULA], [FORMULA], [FORMULA] and [FORMULA] K. The minimum [FORMULA] of this region is at [FORMULA].

3.1. Robustness of results to data analysis choices

Since there is a [FORMULA] inconsistency between the MSAM and the Saskatoon data and since there may be unknown systematic errors, we have performed some checks to see how dependent our results are on the various ways of analyzing the data.

The figures and Table 1 results include the MSAM data points, but we have also performed [FORMULA] calculations without the MSAM data points. When the MSAM points are not included the h and [FORMULA] minimum and contours do not change significantly. For example the results from Fig. 3 without MSAM are [FORMULA] and [FORMULA].

[FORMULA] Saskatoon calibration treatment
We have treated the Saskatoon calibration as a nuisance parameter from a Gaussian distribution around the nominal Saskatoon calibration with a dispersion of [FORMULA]. The values of u at the minimum [FORMULA] in this technique are [FORMULA]. Netterfield et al. (1997) have compared the Saskatoon results to the MSAM first flight results in the north polar region observed by both experiments. They find a best-fit calibration of [FORMULA] which is equivalent to the 0.82 discussed above. Thus, there is some evidence for a lower nominal Saskatoon calibration.

However, preliminary results based on a relative calibration between Jupiter and Cas A at 32 GHz imply that a [FORMULA] Saskatoon calibration is appropriate (Leitch et al. 1997). Using this calibration changes the results slightly. For example, the analog of Fig. 3 yields tighter contours around the unchanged h minimum: [FORMULA] and [FORMULA] with [FORMULA] error bars larger than the range probed. The preference for [FORMULA] 0.15-0.20 is increased (independent of h) and the avoidance of the high h, low [FORMULA], BBN region is increased. The [FORMULA] value of the minimum increases from [FORMULA] to [FORMULA] thus the fit is still good; [FORMULA] probability of having a lower [FORMULA].

We have also let u be a free parameter from a uniform distribution, i.e., a free-floating Saskatoon calibration. The analog of Fig. 3 yields [FORMULA], [FORMULA]. At the minimum [FORMULA], [FORMULA] and the probability of obtaining a lower [FORMULA] is [FORMULA]. We have also obtained results assuming three different Saskatoon calibrations; the nominal value, 14% higher and 14% lower. The minimum [FORMULA] stays at [FORMULA] in all three cases.

Thus several plausible choices of data selection and data analysis producing [FORMULA] variations in the amplitude of the Doppler peak, do not strongly affect the low h results from the CMB. The many measurements on the Doppler slope in the interval [FORMULA] contribute strongly to determining the position of the peak and thus to the preference for low h (see Figs. 3 and 4 of paper I). [FORMULA] appears to be a fairly robust CMB result for the critical density CDM models tested here.

3.2. [FORMULA] results

Our CMB constraints on [FORMULA] are weaker than our constraints on h ; the [FORMULA] contours in Fig. 3 are elongated vertically and yield [FORMULA]. Comparing Figs. 1 and 2 with 3, one sees that it is the marginalization over n which opens the [FORMULA] region (where [FORMULA]). White et al. (1996) highlight the merits of high baryonic fraction [FORMULA] models. We confirm that the CMB [FORMULA] [FORMULA] region is centered near this range but the valley of minima is very shallow. In the context of our models, non-CMB data can still constrain [FORMULA] slightly better than the CMB. See our discussion of Fig. 4 in Sect. 4.5 where we report [FORMULA].

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© European Southern Observatory (ESO) 1998

Online publication: December 16, 1997