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Although it contains two ingredients - dark matter and dark energy - which have not yet been verified by laboratory experiments, the LambdaCDM model is almost universally accepted by cosmologists as the best description of present data. The basic ingredients are given by the parameters listed in Sec. 1.4, with approximate values of some of the key parameters being Omegab approx 0.04, Omegadm approx 0.26, OmegaLambda approx 0.70, and a Hubble constant h approx 0.7. The spatial geometry is very close to flat (and often assumed to be precisely flat), and the initial perturbations Gaussian, adiabatic, and nearly scale-invariant.

The most powerful single experiment is WMAP, which on its own supports all these main tenets. Values for some parameters, as given in Spergel et al. [7], are reproduced in Table 2. This model presumes a flat Universe, and so OmegaLambda is a derived quantity in this analysis, with best-fit value OmegaLambda = 0.73.

Table 2. Parameter constraints reproduced from Spergel et al. [7], both from WMAP alone and from the preferred data compilation of WMAP+CBI+ACBAR (known as WMAPext) plus 2dFGRS. The first two columns assume a power-law initial spectrum, while the third allows a running of the spectral index (in this case n is defined at a particular scale, and its value cannot be directly compared with the power-law case). Spatial flatness is assumed in the parameter fit. The parameter A is a measure of the perturbation amplitude; see Ref. [7] for details. Uncertainties are shown at one sigma, and caution is needed in extrapolating them to higher significance levels due to non-Gaussian likelihoods and assumed priors.

  WMAP alone WMAPext + 2dFGRS WMAPext + 2dFGRS
  power-law power-law running

Omegam h2 0.14 ± 0.02 0.134 ± 0.006 0.136 ± 0.009
Omegab h2 0.024 ± 0.001 0.023 ± 0.001 0.022 ± 0.001
h 0.72 ± 0.05 0.73 ± 0.03 0.71 ± 0.04
n 0.99 ± 0.04 0.97 ± 0.03 0.93+0.04-0.05
tau 0.17+0.08-0.07 0.15 ± 0.07 0.17 ± 0.06
A 0.9 ± 0.1 0.8 ± 0.1 0.84 ± 0.09
dn / d ln k - - -0.031+0.023-0.025

However, to obtain the most powerful constraints, other data sets need to be considered in addition to WMAP. A standard data compilation unites WMAP with shorter-scale CMB measurements from CBI and ACBAR, and the galaxy power spectrum from the 2dF survey. In our opinion, this combination of datasets offers the most reliable set of constraints at present. In addition, it is possible to add the Lyman-alpha forest power spectrum data, but this has proven more controversial as the interpretation of such data has not reached a secure level.

Using the extended data set without the Lyman-alpha constraints produces no surprises; as compared to WMAP alone, the best-fit values move around a little within the uncertainties, and the error bars improve somewhat, as seen in Table 2. In this table we also show the effect of allowing the spectral index to vary with scale (`running'): the running is found to be consistent with zero and there are small drifts in the values and uncertainties of the other parameters. 7

However, inclusion of the Lyman-alpha data suggests a more radical development, with the running weakly detected at around 95% confidence, the spectral index making a transition from n > 1 on large scales to n < 1 on small scales [7, 6]. The significance of this measurement is not high, and the result rather unexpected on theoretical grounds (it suggests that the power spectrum has a maximum which just happens to lie in the rather narrow range of scales that observations are able to probe, and the running is much larger than in typical inflation models giving a spectral index close to one). In our view it is premature to read much significance into this observation, though if true, it should rapidly be firmed up by new data.

The baryon density Omegab is now measured with quite high accuracy from the CMB and large-scale structure, and shows reasonable agreement with the determination from big bang nucleosynthesis; Fields and Sarkar in this volume quote the range 0.012 geq Omegab h2 geq 0.025. Given the sensitivity of the measurement, it is important to note that it has significant dependence on both the datasets and parameter sets chosen, as seen in Table 2.

While OmegaLambda is measured to be non-zero with very high confidence, there is no evidence of evolution of the dark energy density. The WMAP team find the limit w < - 0.78 at 95% confidence from a compilation of data including SNe Ia data, where they impose a prior w geq - 1, with the cosmological constant case w = - 1 giving an excellent fit to the data.

As far as inflation is concerned, the data provide good news and bad news. The good news is that WMAP supports all the main predictions of the simplest inflation models: spatial flatness and adiabatic, Gaussian, nearly scale-invariant density perturbations. But it is disappointing that there is no sign of primordial gravitational waves, with WMAP providing only a weak upper limit r < 0.53 at 95% confidence [6] (this assumes no running, and weakens significantly if running is allowed), and especially that no convincing deviations from scale-invariance have been seen. It is perfectly possible for inflation models to give n appeq 1 and r appeq 0, but in that limit, the observations give no clues as to the dynamical processes driving inflation. Tests have been made for various types of non-Gaussianity, a particular example being a parameter fNL which measures a quadratic contribution to the perturbations and is constrained to -58 < fNL < 134 at 95% confidence [41] (this looks weak, but prominent non-Gaussianity requires the product fNL DeltaR to be large, and DeltaR is of order 10-5).

Two parameters which are still uncertain are Omegam and sigma8 (see Figure 4 and Ref. [42]). The value of Omegam is beginning to be pinned down with some precision, with most observations indicating a value around 0.3, including the CMB anisotropies, the cluster number density, and gravitational lensing, though the latter two have a strong degeneracy with the amplitude of mass fluctuations sigma8. However, not all observations yet agree fully on this, for instance mass-to-light ratio measurements give Omegam approx 0.15 [43], and the fractional uncertainty remains significantly higher than one would like. Concerning sigma8, results from the cluster number density have varied quite a lot in recent years, spanning the range 0.6 to 1.0, primarily due to the uncertainties in the mass-temperature-luminosity relations used to connect the observables with theory. There is certainly scope for improving this calibration by comparison to mass measurements from strong gravitational lensing. The WMAP-alone measurements gives sigma8 = 0.9 ± 0.1. However, this is not a direct constraint; WMAP only probes larger length scales, and the constraint comes from using WMAP to estimate all the parameters of the model needed to determine sigma8. As such, their constraint depends strongly on the assumed set of cosmological parameters being sufficient.

Figure 4

Figure 4. Various constraints shown in the Omegam - sigma8 plane. [Figure provided by Sarah Bridle; see also Ref. [42].]

One parameter which is surprisingly robust is the age of the Universe. There is a useful coincidence that for a flat Universe the position of the first peak is strongly correlated with the age of the Universe. The WMAP-only result is 13.4 ± 0.3 Gyr (assuming a flat Universe). This is in good agreement with the ages of the oldest globular clusters [44] and radioactive dating [45].

7 As we were finalizing this article, an analysis of WMAP combined with the SDSS galaxy power spectrum appeared [40], giving results in good agreement with those discussed here. Back.

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