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Abstract:

We use a frequentist statistical approach to set confidence intervals on the values of cosmological parameters using the MAXIMA-1 and COBE measurements of the angular power spectrum of the cosmic microwave background. We define a $\Delta \chi^{2}$ statistic, simulate the measurements of MAXIMA-1 and COBE, determine the probability distribution of the statistic, and use it and the data to set confidence intervals on several cosmological parameters. We compare the frequentist confidence intervals to Bayesian credible regions. The frequentist and Bayesian approaches give best estimates for the parameters that agree within 15%, and confidence interval-widths that agree within 30%. The results also suggest that a frequentist analysis gives slightly broader confidence intervals than a Bayesian analysis. The frequentist analysis gives values of \Omega=0.89{+0.26\atop -0.19}, \Omega_{\rm B}h^2=0.026{+0.020\atop -0.011} and n=1.02{+0.31\atop -0.10}, and the Bayesian analysis gives values of \Omega=0.98{+0.14\atop -0.19}, \Omega_{\rm B}h^2=0.0.029{+0.015\atop-0.010}, and $n=1.18{+0.10\atop -0.23}$, all at the 95% confidence level.

Abstract:

We discuss the cosmological implications of the new constraints on the power spectrum of the Cosmic Microwave Background Anisotropy derived from a new high resolution analysis of the MAXIMA-1 measurement (Lee et al. 2001). The power spectrum shows excess power at $\ell \sim 860$ over the average level of power at $411 \le\ell \le 785.$ This excess is statistically significant on the 95% confidence level. Such a feature is consistent with the presence of a third acoustic peak, which is a generic prediction of inflation-based models. The height and the position of the excess power match the predictions of a family of inflationary models with cosmological parameters that are fixed to fit the CMB data previously provided by BOOMERANG-LDB and MAXIMA-1 experiments (e.g., Jaffe et al.2001). Our results, therefore, lend support for inflationary models and more generally for the dominance of coherent perturbations in the structure formation of the Universe. At the same time, they seem to disfavor a large variety of the non-standard (but still inflation-based) models that have been proposed to improve the quality of fits to the CMB data and consistency with other cosmological observables. Within standard inflationary models, our results combined with the COBE-DMR data give best fit values and 95% confidence limits for the baryon density, $\Omega_b h^2\simeq 0.033{\pm 0.013}$, and the total density, $\Omega=0.9{+0.18\atop -0.16}$. The primordial spectrum slope ($n_s$) and the optical depth to the last scattering surface ($\tau_c$) are found to be degenerate and to obey the relation $n_s \simeq 0.46 \tau_c + (0.99 \pm 0.14)$, for $\tau_c \le 0.5$ (all 95% c.l.).