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8. NONSTANDARD CBR: S neq 1, Nnu neq 3

The CBR temperature fluctuation anisotropy spectrum is sensitive to the early-Universe radiation density (rhoR) as well as to the overall expansion rate. The early Universe is radiation dominated so that rho approx rhoR propto 1 + 0.135 Delta Nnu (see Eq. 6 and recall that rho propto H2). The late Universe is matter dominated (MD) (omegaM ident OmegaM h2) and the crossover from RD to MD, important for the growth of fluctuations and for the age/size of the Universe at recombination, occurs for a redshift

Equation 8 (8)

If the matter content is kept fixed while the radiation content is increased, corresponding to a faster than standard expansion rate, matter-radiation equality is delayed, modifying the growth of fluctuations prior to recombination and, also, the Universe is younger at recombination and has a smaller sound horizon, shifting the angular location of the acoustic peaks. The degeneracy between the radiation density (Delta Nnu or S) and omegaM is broken by the requirement that OmegaM + OmegaLambda = 1 and the HST Key Project determination of the Hubble parameter (see [Barger et al. (2003a)] for details and further references). In Figure 10 are shown the 1sigma and 2sigma contours in the Delta Nnu - eta plane from the CBR (WMAP) data; note the very different Delta Nnu scales and ranges in Figures 9 and 10. As is the case for BBN (see Section 7), the CBR favors a slightly slower than standard expansion. However, while the "best" fit value for the expansion rate factor is at S < 1 (Delta Nnu < 0), the CBR likelihood distribution of Delta Nnu values is very shallow and the WMAP data are fully consistent with S = 1 (Delta Nnu = 0).

Figure 10

Figure 10. The 1sigma and 2sigma contours in the eta - Delta Nnu plane from the CBR (WMAP) data. The best fit point (eta10 = 6.3, Delta Nnu = -0.25) is indicated by the cross.

Comparing Figures 9 and 10, it is clear that for this variant of the standard cosmology there is excellent overlap between the eta - Delta Nnu confidence contours from BBN and those from the CBR (see Barger et al. 2003a). This variant of SBBN (S neq 1) is consistent with the CBR. In Figure 11 (from Barger et al. 2003a) are shown the confidence contours in the eta - Delta Nnu plane for a joint BBN - CBR fit. Again, while the best fit value for Delta Nnu is negative (driven largely by the adopted value for YP), Delta Nnu = 0 (S = 1) is quite acceptable.

Figure 11

Figure 11. As for Figure 10, but for the joint BBN - CBR fit. The best fit point (eta10 = 6.0, Delta Nnu = -0.75) is indicated by the cross.

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