Toward a standard cosmological model?

2.2 Toward a standard cosmological model?

Within the context of a big bang creation scenario in which an inflationary phase is followed by expansion within the confines of General Relativity, there are at least 11 parameters that define a cosmological model in the post inflation era [36 ]. These 11+ parameters affect what happens in the first few minutes during which nucleosynthesis of the lightest elements occurs (BBN), how the geometry of space-time develops, and how structure forms through gravitational enhancement of primordial density inhomogeneities which come out of the inflationary phase. The density inhomogeneities will leave an imprint on the microwave background radiation that survives from the era of recombination, some 300,000 years after the big bang. Subsequently, they grow through gravitational instability to give rise to the structure we see in the distribution of visible objects in the Universe today. Hence, by bringing together observations that relate to BBN, space-time evolution, microwave background anisotropies, and large-scale structure, it is possible to define inter-related regions of the 11 parameter phase space that are consistent with all the information. Tegmark, Zalderriaga, and Hamilton [137

] did this just after the first release of new CMB data from BOOMERANG [42

] and MAXIMA [67

]. The parameter space used was

where $τ$ is the reionisation optical depth, $As$ , $ns$ , $r$ , $nt$ are the primodial amplitudes and tilts of the scalar and tensor inhomogeneities, $b$ is a bias factor relating rms galaxy fluctuations to the underlying rms matter fluctuations, $Ωk$ and $ΩΛ$ are the contributions to the overall density from curvature and the cosmological constant, $ωd$ and $ωb$ are the physical densities of dark matter (both hot and cold together) and baryonic matter, and finally $fν$ is the fraction of dark matter in the form of hot dark matter. The primary observational data that they used were all available CMB data [54 ] and the recently released IRAS Point Source Catalogue Redshift Survey (PSCz) [113 ] from which they derived the large scale structure power spectrum. Simultaneous fits were then done to these data allowing all 11 parameters to vary [136 ].

Table 1: Allowable ranges of values of density parameters within the standard cosmological model derived from the first release CMB data of BOOMERANG and MAXIMA (left-hand column, [137

]) with corresponding values (where quoted) from the newest data sets (right-hand column, [143

]). $h$ is the Hubble parameter and values of $0.74 ± 0.08$ and $0.72 ± 0.08$ were used in [137 ] and [143

], respectively.

$0.49 ≤ Ω ≤ 0.74 Λ$	$0.49 ≤ Ω ≤ 0.76 Λ$
$0.20 ≤ Ωm ≤ 0.50$
$0.11 ≤ h2Ωd ≤ 0.17$	$0.09 ≤ h2 Ωd ≤ 0.17$
$0.00 ≤ h2Ωhdm ≤ 0.12$
$0.10 ≤ h2Ωcdm ≤ 0.32$
$0.020 ≤ h2Ωb ≤ 0.037$	$0.01 ≤ h2 Ωb ≤ 0.03$

Table 1 shows the acceptable range of values for the key parameters that came out of those fits. In the table the matter density, $Ωm = Ωb + Ωcdm + Ωhdm$ , includes both baryonic matter and dark matter, and moreover the dark matter can be classed either as “hot” or “cold” depending on whether it was relativistic or not in the early Universe. The total dark matter density is $Ωd = Ωhdm + Ωcdm$ , and $h2Ωd = ωd$ . Of particular note for this review were that the cold dark matter density is non-zero and that the baryonic density has a range that just accomodates the constraints from BBN [27 ] at its lowest end, but with significantly better fits for higher values. The hot dark matter density can only be a minor component.

Very recently there have been significant new CMB data released from BOOMERANG [96 ], MAXIMA [64 ], DASI [77 ], and CBI [101 ]. These data have given better definition to the second and third peaks in the CMB power spectrum. Wang, Tegmark and Zaldarriaga [143 ] subsequently repeated the above analysis using a combination of these and previously available CMB data.

Figure 1: Allowable parameter spaces for $Ω Λ$ , $Ωk$ , $ωb$ , $ωd$ , $fν$ , $τ$ and $r$ . The figure is taken from [143

] and the dashed lines mark the 95% confidence limits.

The two left-hand plots in the top row in Figure 1 are the most relevant for the dark matter. We see that the dark matter density is again non-zero, with a similar range of values as before, and that the fraction of dark matter as “hot dark matter” is less than 35%, assuming no constraints on the hubble parameter, $h$ . The allowable fraction of hot dark matter drops to only 20% if the preferred hubble parameter value is imposed. The right-hand column in Table 1 lists the quoted 95% confidence limits for comparison with the earlier analysis. The most striking difference is in the baryon density. While previously the allowable range of $Ωb$ was only just compatible with the upper limit derived from BBN [27 ], it now comfortably embraces it. This is illustrated in the left-hand panel in Figure 2 , which shows the combined constraints on the baryonic matter and dark matter densities. The white central region is the allowed parameter space when all constraints are applied, except for BBN of course. Relaxing the constraints by not using the PSCz data enlarges the allowed region to include the cyan coloured area. If, in addition, no assumptions are made about the value of the Hubble constant, then the green area also becomes allowed. If all constraints are accepted then Figure 2 implies there is between 4.5 and 9 times as much dark matter in the Universe as there is baryonic matter.

Figure 2: The left-hand panel from [143

] shows the joint constraints on the baryonic matter and dark matter densities, together with the allowed band of baryonic density from BBN models. The right-hand panel from [41

] shows the joint constraints on $Ω λ$ and $Ω m$ which result from combined use of CMB data and high-redshift supernova data.

Constraints on cosmological models can also be derived from the observations of high red-shift Type 1a supernovae [59 ]. When combined with data from the CMB anisotropies, these limits give reasonable agreement with those cited earlier in Table 1. A recent result from de Bernardis et al. [41 ] is shown in the right-hand panel in Figure 2 . This time what is shown are the joint constraints on $Ωλ$ and $Ωm$ . The solid curves are the 1 to 3 $σ$ combined likelihood contours and these can be compared with the values in the table. A somewhat weaker constraint on the Hubble constant was used.

Hence, from the above, there does indeed seem to be a cosmological model that can simultaneously satisfy all the observational evidence used. The ranges of values for the key parameters relevant to dark matter searches have been summarized in Table 1. Rotation curves of galaxies can also be explained with this type of cosmological model. Numerous N-body simulations have been performed to verify whether structure formation occurs properly in a number of different types of models. Gawiser and Silk [53 ] reviewed the situation with regard to large-scale structure. Simulations of gravitational collapse on the scale of galaxies have resulted in universal rotation curves that match reasonably well those observed in a wide range of galaxies [93 , 94 ].

From Table 1 the main features of the emerging standard cosmology from the point of view of dark matter are:

A $Λ$ CDM scenario where the dominant energy density is from $ΩΛ$ .
The matter density $Ωm$ greatly exceeds that which can be present in a baryonic form, and hence there is a significant dark matter component.
The dark matter density $Ωd$ is mostly comprised of cold dark matter.
N-body simulations suggest that galaxies are comprised of similar mixes of baryonic matter and dark matter as the universe as a whole.

The origin of $Ω Λ$ remains a topic of current debate, with a great deal of interest in quintessence [29 , 9 ].