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

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1. Introduction

The standard picture of structure formation relies on the gravitational amplification of initially small perturbations in the matter distribution. The origin of these fluctuations is unclear, but a popular assumption is that these fluctuations originate in the very early universe during an inflationary epoch. The most straightforward incarnation of this inflationary scenario predicts that the fluctuations are adiabatic, Gaussian, Harrison-Zeldovich ([FORMULA]) and that the Universe is spatially flat (Kolb & Turner 1990). To avoid violating primordial nucleosynthesis constraints, the Universe should be dominated by non-baryonic matter. The cold dark matter (CDM) model has been the preferred model in the inflationary scenario (Peebles 1982, Liddle and Lyth, 1993).

The statistical properties of CMB fluctuations provide an ideal tool for testing CDM models. CMB data offer valuable information not only on the scenario of the origin of cosmic structures, but also on the early physics of the Universe and the cosmological parameters that characterize the Universe. Using the CMB to determine these parameters is the beginning of a new era in cosmology. This truly cosmological method probes scales much larger and epochs much earlier ([FORMULA]) than more traditional techniques which rely on supernovae, galaxies, galaxy clusters and other low-redshift objects. The CMB probes the entire observable universe.

Acoustic oscillations of the baryon-photon fluid at recombination produce peaks and valleys in the CMB power spectrum at sub-degree angular scales. Measurements of these model-dependent peaks and valleys have the potential to determine many important cosmological parameters to the few percent level (Jungman et al. 1996, Zaldarriaga et al. 1997). Within the next decade, increasingly accurate sub-degree scale CMB observations from the ground, from balloons and particularly from two new satellites (MAP: Wright et al. 1996, Planck Surveyor: Bersanelli et al. 1996) will tell us the ultimate fate of the Universe ([FORMULA]), what the Universe is made of ([FORMULA], [FORMULA]) and the age and size of the Universe ([FORMULA])) with unprecedented precision.

In preparation for the increasingly fruitful harvests of data, it is important to determine what the combined CMB data can already tell us about the cosmological parameters. In Lineweaver et al. (1997), (henceforth "paper I"), we compared the most recent CMB data to predictions of COBE normalized flat universes with Harrison-Zel'dovich ([FORMULA]) power spectra. We used predominantly goodness-of-fit statistics to locate the regions of parameter space preferred by the CMB data. We explored the [FORMULA] plane and the [FORMULA] plane.

In the present paper we focus on the range for h favored by the CMB in the context of CDM critical density universes and we broaden the scope of our exploration to the 4-dimensional parameter space: h, [FORMULA], n and Q. Our motivation for choosing this 4-D subspace of the higher dimensional parameter space is that (i) it is the largest dimensional subspace that we can explore at a reasonable resolution with the means available and (ii) it is centered on the simplest CDM model: [FORMULA], [FORMULA], [FORMULA]. This model is arguably the simplest scenario for the formation of large-scale structure. One of our goals is to see what is required of such a model if it is to explain the current set of large-scale structure data, and what could eventually force us to accept the fine-tuning demanded by the inclusion of another cosmological parameter, such as the cosmological constant.

Hubble's constant [FORMULA] is possibly the most important parameter in cosmology, giving the expansion rate, age and size of the Universe. Recent, direct, low-redshift measurements fall in the range [45-90] but may be subject to unidentified systematic errors. Thus it is important to have different methods which may not be subject to the same systematics. For example, CMB determinations of h are distance-ladder-independent. Current CMB data are not of high enough quality to draw definitive model-independent conclusions, however in the restricted class of models considered here, the CMB data are already able to provide interesting constraints.

The quantity [FORMULA] is important because we would like to know what the Universe is made of and how much normal baryonic matter exists in it. The combination [FORMULA] is relatively well-constrained by the observations and theory of primordial nucleosynthesis, but the uncertainty on the Hubble constant means that the value of [FORMULA] is rather poorly constrained. The question of just how many baryons there are in the Universe has received close attention recently due to estimates of the baryon fraction in galaxy clusters and attempts to constrain [FORMULA] by measuring the deuterium in high-redshift quasar absorption systems.

The parameter n is the primordial power spectrum slope that remains equal to its primordial value at the largest scales (low [FORMULA]). It is important because it's measurement is a glimpse at the primordial universe. Although generic inflation predicts [FORMULA], a larger set of plausible inflationary models is consistent with [FORMULA]. Model power spectra and particularly the amplitude of the first peak depend strongly on n. Thus, an important limitation of paper I was the restriction to [FORMULA]. By adding n as a free parameter we obtain observational limits on n and quantify the reduced constraining ability of the CMB observations when n is marginalized.

The power spectrum quadrupole normalization Q is important because it normalizes all models. Here we treat Q as a free parameter.

We examine how the contraints on any one of these parameters change as we condition on and marginalize over the other parameters. We obtain [FORMULA] minimization values and likelihood intervals for h, [FORMULA], n and Q. As in paper I, we take advantage of the recently available fast Boltzmann code (Seljak & Zaldarriga 1996) to make a detailed exploration of parameter space.

All the results reported here were obtained under a restrictive set of assumptions. We assume inflation-based CDM models of structure formation with Gaussian adiabatic initial conditions in critical density universes ([FORMULA]) with no cosmological constant ([FORMULA]). We have ignored the possibility of early reionization and any gravity wave contribution to the spectra. We do not test topological defect models. We use no hot dark matter. We have used the helium fraction [FORMULA] and a mean CMB temperture [FORMULA] K. Although we have not yet looked carefully at how dependent our results are on these assumptions we make some informed estimates in Sect. 6.1where we also discuss previous work using similar data sets and similar methods to look at different families of models.

In Sect. 2we describe the data analysis. In Sect. 3we present our results for h and [FORMULA] and discuss their dependence on some plausible variations in the data analysis. We discuss non-CMB constraints and compare them with our results in Sect. 4. In Sect. 5we present our results for n and Q. In Sect. 6we add some caveats and summarize.

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© European Southern Observatory (ESO) 1998

Online publication: December 16, 1997
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