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

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2. Method

2.1. Data

The data used are described in paper I, however we have updated some data points and now include several more measurements:
[FORMULA] updated Tenerife point (Hancock et al. 1997): [FORMULA] at [FORMULA],
[FORMULA] added BAM point (Tucker et al. 1997): [FORMULA] at [FORMULA],
[FORMULA] updated the two Python points (Platt et al. 1997): [FORMULA] at [FORMULA] and [FORMULA] at [FORMULA],
[FORMULA] added MSAM single- and double-difference points (Cheng et al. 1996) [FORMULA] at [FORMULA] [FORMULA] at [FORMULA].

With the exception of the Saskatoon points, the calibration uncertainties of the experiments were added in quadrature to the error bars on the points. The MSAM values are the weighted averages of the first and second MSAM flights. The error bars assigned encompass the [FORMULA] limits from both flights. In paper I, we did not include the MSAM points because of possible correlations with the Saskatoon results. However the MSAM results are substantially lower than the Saskatoon results in this crucial high- [FORMULA] region of the power spectrum and it is not clear that avoiding such correlations is more important than the additional information provided by the MSAM data. The figures presented in this paper include the MSAM points however we have also performed [FORMULA] calculations without MSAM. We discuss these results in Sect. 3.1.

2.2. Calculation

A two-dimensional version of our [FORMULA] calculation is described in paper I. In this work we generalize to 5 dimensions and use a likelihood approach to determine the parameter ranges. We treat the correlated calibration uncertainty of the 5 Saskatoon points as a nuisance parameter "u ". For each point in 5-D space we obtain a value for [FORMULA]. We assume u comes from a Gaussion distribution about its nominal value with a dispersion of [FORMULA]. This Gaussian assumption amounts to adding [FORMULA] to the [FORMULA] calculation described in paper I. For example, in paper I, our notation Sk-14, Sk0 and Sk+14 corresponds to [FORMULA] respectively (however we did not assume a Gaussian distribution and so did not add the extra factor to the [FORMULA] values).

For each point in the 4-D space of interesting parameters, u takes on the value which minimizes the [FORMULA] (Avni 1976, Wright 1994). At the minimum in 4-D, [FORMULA], the parameter values are the best-fit parameters. To obtain error bars on these values, we determine the 4-D surfaces which satisfy [FORMULA] where [FORMULA]. Under the assumption that the errors on the data points are Gaussian (cf. de Bernardis et al. 1997), the [FORMULA] ellipsoid can be projected onto any of the axes to get the [FORMULA] confidence interval for the parameter of that axis. If one is not interested in 1-D intervals but rather in the confidence regions for 2 parameters simultaneously, then one would use [FORMULA] (see Press et al. 1989 for details). To make the figures, we project the 4-D surfaces onto the two dimensions of our choice and obtain contours which we project onto either axis.

We have normalized the power spectra using the conversion [FORMULA], where [FORMULA] is the power spectrum output of the Boltzmann code.

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© European Southern Observatory (ESO) 1998

Online publication: December 16, 1997