Astron. Astrophys. 329, 799-808 (1998) 3. Results for h andThe 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 , , and ; Figs. 1-3, 5, 6-7 and 8-9 respectively. The thick `X ' in each figure marks the minimum. Areas within the 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, , n and Q at the minimum of the 4-D are given with error bars from the projection onto 1-D of the surface. In Figs. 1 through 4 the region preferred by Big Bang nucleosynthesis (BBN) is shaded light grey (, see Sect. 4.1). Table 1. parameter results In Fig. 1 we have conditioned on and K. The results are , . At , and .
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 , Q takes on the value (within the discretely sampled range) which minimizes . The minimum and the errors on h and are the same as in Fig. 1; h stays low. The 2, 3, and contours are noticeably larger than in Fig. 1. Within the contours, the higher values of permitted correspond to 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 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 and where the error bars are approximately . At the minimum, and K. The 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 value this low or lower is . Thus the fit obtained is "good".
The region of the 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 , h is unconstrained since the projection of the 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 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 contour region on the right is characterized by the parameter values , , and K. The minimum of this region is at . 3.1. Robustness of results to data analysis choicesSince there is a 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. Without MSAM Saskatoon calibration treatment However, preliminary results based on a relative calibration between Jupiter and Cas A at 32 GHz imply that a 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: and with error bars larger than the range probed. The preference for 0.15-0.20 is increased (independent of h) and the avoidance of the high h, low , BBN region is increased. The value of the minimum increases from to thus the fit is still good; probability of having a lower . 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 , . At the minimum , and the probability of obtaining a lower is . We have also obtained results assuming three different Saskatoon calibrations; the nominal value, 14% higher and 14% lower. The minimum stays at in all three cases. Thus several plausible choices of data selection and data analysis producing 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 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). appears to be a fairly robust CMB result for the critical density CDM models tested here. 3.2. resultsOur CMB constraints on are weaker than our constraints on h ; the contours in Fig. 3 are elongated vertically and yield . Comparing Figs. 1 and 2 with 3, one sees that it is the marginalization over n which opens the region (where ). White et al. (1996) highlight the merits of high baryonic fraction models. We confirm that the CMB 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 slightly better than the CMB. See our discussion of Fig. 4 in Sect. 4.5 where we report . © European Southern Observatory (ESO) 1998 Online publication: December 16, 1997 |