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It is convenient to express the mean densities rhoi of various quantities in the Universe in terms of their fractions relative to the critical density: Omegai ident rhoi / rhocrit. The theory of cosmological inflation strongly suggests that the the total density should be very close to the critical one: Omegatot appeq 1, and this is supported by the available data on the cosmic microwave background radiation (CMB) (Bond 2003). The fluctuations observed in the CMB at a level ~ 10-5 in amplitude exhibit a peak at a partial wave ell ~ 200, as would be produced by acoustic oscillations in a flat Universe with Omegatot appeq 1. At lower partial waves, ell << 200, the CMB fluctuations are believed to be dominated by the Sachs-Wolfe effect due to the gravitational potential, and more acoustic oscillations are expected at larger ell > 200, whose relative heights depend on the baryon density Omegab. At even larger ell gtapprox 1000, these oscillations should be progressively damped away.

Fig. 1 compares measurements of CMB fluctuations made before WMAP (Bond 2003) with the WMAP data themselves (Bennett et al. 2003; Hinshaw et al. 2003), that were released shortly after this Meeting. The position of the first acoustic peak indeed corresponds to a flat Universe with Omegatot appeq 1: in particular, now WMAP finds Omegatot = 1.02 ± 0.02 (Spergel et al. 2003), and two more acoustic peaks are established with high significance, providing a new determination of Omegab h2 = 0.0224 ± 0.0009, where h ~ 0.7 is the present Hubble expansion rate H, measured in units of 100 km/s/Mpc. The likelihood functions for various cosmological parameters are shown in Fig. 2. Remarkably, there is excellent consistency between the estimate of the present-day Hubble constant H ~ 72 km/s/Mpc from WMAP (Spergel et al. 2003) with that inferred from the local distance ladder based, e.g., on Cepheid variables.

Figure 1

Figure 1. Spectrum of fluctuations in the cosmic microwave background measured by WMAP (darker points with smaller error bars), compared with previous measurements (lighter points with larger error bars, extending to greater ell) (Hinshaw et al. 2003).

Figure 2

Figure 2. The likelihood functions for various cosmological parameters obtained from the WMAP data analysis (Spergel et al. 2003). The panels show the baryon density Omegab h2, the matter density Omegam h2, the Hubble expansion rate h, the strength A, the optical depth tau, the spectral index ns and its rate of change dns / ln k, respectively.

As seen in Fig 3, the combination of CMB data with those on high-redshift Type-Ia supernovae (Perlmutter 2003; Perlmutter & Schmidt 2003) and on large-scale structure (Peacock 2003a, b) favour strongly a flat Universe with about 30 % of (mainly dark) matter and 70 % of vacuum (dark) energy. Type-Ia supernovae probe the geometry of the Universe at redshifts z ltapprox 1. They disagree with a flat Omegatot = 1 Universe that has no vacuum energy, and also with an open Omegam appeq 0.3 Universe (Perlmutter 2003; Perlmutter & Schmidt 2003). They appear to be adequate standard candles, and two observed supernovae with z > 1 argue strongly against dust or evolution effects that would be sufficient to cloud their geometrical interpretation. The supernovae indicate that the expansion of the Universe is currently accelerating, though it had been decelerating when z was > 1. There are good prospects for improving substantially the accuracy of the supernova data, by a combination of continued ground-based and subsequent space observations using the SNAP satellite project (Perlmutter 2003; Perlmutter & Schmidt 2003).

Figure 3

Figure 3. The density of matter Omegam and dark energy OmegaLambda inferred from WMAP and other CMB data (WMAPext), and from combining them with supernova and Hubble Space Telescope data (Spergel et al. 2003).

It is impressive that the baryon density inferred from WMAP data (Spergel et al. 2003) is in good agreement with the value calculated previously on the basis of Big-Bang nucleosynthesis (BBN), which depends on completely different (nuclear) physics. Fig. 4 compares the abundances of light elements calculated using the WMAP value of Omegab h2 with those inferred from astrophysical data (Cyburt et al. 2003). Depending on the astrophysical assumptions that are made in extracting the light-element abundances from astrophysical data, there is respectable overlap.

Figure 4

Figure 4. The likelihood functions for the primordial abundances of light elements inferred from astrophysical observations (lighter, yellow shaded regions) compared with those calculated using the CMB value of Omegab h2 (darker, blue shaded regions). The dashed curves are likelihood functions obtained under different astrophysical assumptions (Cyburt et al. 2003).

As we heard at this Meeting, several pillars of inflation theory have now been verified by WMAP and other CMB data (Bond 2003): the Sachs-Wolfe effect due to fluctuations in the large-scale gravitational potential were first seen by the COBE satellite, the first acoustic peak was seen in the CMB spectrum at ell ~ 210 and this has been followed by two more peaks and the intervening dips, the damping tail of the fluctuation spectrum expected at ell gtapprox 1000 has been seen, polarization has been observed, and the primary anisotropies are predominantly Gaussian. WMAP has, additionally, measured the thickness of the last scattering surface and observed the reionization of the Universe when z ~ 20 by the first generation of stars (Kogut et al. 2003). Remaining to be established are secondary anisotropies, due, e.g., to the Sunyaev-Zeldovich effect, weak lensing and inhomogeneous reionization, and tensor perturbations induced by gravity waves.

As we also heard at this meeting, the values of OmegaCDM inferred from X-ray studies of gas in rich clusters using the Chandra satellite (Rees 2003), which indicate OmegaCDM = 0.325 ± 0.34, gravitational lensing (Schneider 2003) and data on large-scale structure, e.g., from the 2dF galaxy redshift survey (Peacock 2003a, b), are very consistent with that inferred by combining CMB and supernova data. The WMAP data confirm this concordance with higher precision: OmegaCDM h2 = 0.111 ± 0.009 (Spergel et al. 2003).

The 2dF galaxy survey has examined two wedges through the Universe. Significant structures are seen at low redshifts, which die away at larger redshifts where the Universe becomes more homogeneous and isotropic. The perturbation power spectrum at these large scales matches nicely with that seen in the CMB data, whilst the structures seen at small scales would not be present in a baryon-dominated Universe, or one with a significant fraction of hot dark matter. Indeed, the 2dF data were used to infer an upper limit on the sum of the neutrino masses of 1.8 eV (Elgaroy et al. 2002), which has recently been improved using WMAP data (Spergel et al. 2003) to

Equation 1 (1.1)

as seen in Fig. 5. This impressive upper limit is substantially better than even the most stringent direct laboratory upper limit on an individual neutrino mass, as discussed in the next Section. The WMAP data also provide (Crotty et al. 2003) a new limit on the effective number of light neutrino species, beyond the three within the Standard Model:

Equation 2 (1.2)

This limit is not as stringent as that from LEP, but applies to additional light degrees of freedom that might not be produced in Z decay.

Figure 5

Figure 5. The likelihood function for the total neutrino density Omeganu h2 derived by WMAP (Spergel et al. 2003). The upper limit mnu < 0.23 eV applies if there are three degenerate neutrinos.

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