4. BRINGING OBSERVATIONS TOGETHER

Although it contains two ingredients—dark matter and dark energy—which have not yet been verified by laboratory experiments, the CDM model is almost universally accepted by cosmologists as the best description of the present data. The approximate values of some of the key parameters are _b ≈ 0.05, _c ≈ 0.25, ≈ 0.70, and a Hubble constant h ≈ 0.70. The spatial geometry is very close to flat (and usually assumed to be precisely flat), and the initial perturbations Gaussian, adiabatic, and nearly scale-invariant.

The most powerful data source is the CMB, which on its own supports all these main tenets. Values for some parameters, as given in Ade et al. [2] and Hinshaw et al. [4], are reproduced in Table 1. These particular results presume a flat Universe. The constraints are somewhat strengthened by adding additional data-sets such as BAO, as shown in the Table, though most of the constraining power resides in the CMB data. We see that the Planck and WMAP constraints are similar, though with some shifts within the uncertainties. For these six-parameter fits the parameter uncertainties are also comparable; the additional precision of Planck data versus WMAP is only really apparent when considering significantly larger parameter sets.

Table 1. Parameter constraints reproduced from Ref. 2 (Table 5) and Ref. 4 (Table 4), with some additional rounding. All columns assume the CDM cosmology with a power-law initial spectrum, no tensors, spatial flatness, and a cosmological constant as dark energy. Planck take the sum of neutrino masses fixed to 0.06eV, while WMAP set it to zero. Above the line are the six parameter combinations actually fit to the data in the Planck analysis (_MC is a measure of the sound horizon at last scattering); those below the line are derived from these. Two different data combinations including Planck are shown to highlight the extent to which additional data improve constraints. The first column is a combination of CMB data only — Planck temperature plus WMAP polarization data plus high-resolution data from ACT and SPT — while the second column adds BAO data from the SDSS, BOSS, 6dF, and WiggleZ surveys. For comparison the last column shows the final nine-year results from the WMAP satellite, combined with the same BAO data and high-resolution CMB data (which they call eCMB). Note WMAP use directly as a fit parameter rather than _MC. The perturbation amplitude ² is specified at the scale 0.05 Mpc^-1 for Planck, but 0.002 Mpc^-1 for WMAP, so the spectral index n_s needs to be taken into account in comparing them. Uncertainties are shown at 68% confidence.

	Planck+WP	Planck+WP	WMAP9+eCMB
	+highL	+highL+BAO	+BAO
b h2	0.02207 ± 0.00027	0.02214 ± 0.00024	0.02211 ± 0.00034
c h2	0.1198 ± 0.0026	0.1187 ± 0.0017	0.1162 ± 0.0020
100 MC	1.0413 ± 0.0006	1.0415 ± 0.0006	-
ns	0.958 ± 0.007	0.961 ± 0.005	0.958 ± 0.008
	0.091-0.014+0.013	0.092 ± 0.013	0.079-0.012+0.011
ln(1010 2)	3.090 ± 0.025	3.091 ± 0.025	3.212 ± 0.029
h0.673 ± 0.012	0.678± 0.008	0.688 ± 0.008
8	0.828 ± 0.012	0.826 ± 0.012	0.822-0.014+0.013
m	0.315-0.017+0.016	0.308 ± 0.010	0.293 ± 0.010
	0.685-0.016+0.017	0.692 ± 0.010	0.707 ± 0.010

If the assumption of spatial flatness is lifted, it turns out that the CMB on its own only weakly constrains the spatial curvature, due to a parameter degeneracy in the angular-diameter distance. However inclusion of other data readily removes this. For example, inclusion of BAO data, plus the assumption that the dark energy is a cosmological constant, yields a constraint on _{tot i} + of _tot = 1.0005 ± 0.0033 [2]. Results of this type are normally taken as justifying the restriction to flat cosmologies.

One parameter which is very robust is the age of the Universe, as there is a useful coincidence that for a flat Universe the position of the first peak is strongly correlated with the age. The CMB data give 13.81 ± 0.05 Gyr (assuming flatness). This is in good agreement with the ages of the oldest globular clusters and radioactive dating.

The baryon density _b is now measured with high accuracy from CMB data alone, and is consistent with the determination from BBN; Fields et al. in this volume quote the range 0.021 ≤ _b h² ≤ 0.025 (95% confidence).

While is measured to be non-zero with very high confidence, there is no evidence of evolution of the dark energy density. Mortonson et al. in this volume quote the constraint w = -1.13_-0.11^+0.13 on a constant equation of state from a compilation of CMB and BAO data, with the cosmological constant case w = -1 giving an excellent fit to the data. Allowing more complicated forms of dark energy weakens the limits.

The data provide strong support for 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 the CMB data compilation providing an upper limit r < 0.11 at 95% confidence [2] (weakening to 0.26 if running is allowed). The spectral index is clearly required to be less than one by this data, though the strength of that conclusion can weaken if additional parameters are included in the model fits.

Tests have been made for various types of non-Gaussianity, a particular example being a parameter f_NL which measures a quadratic contribution to the perturbations. Various non-gaussianity shapes are possible (see Ref. 51 for details), and current constraints on the popular `local', `equilateral', and `orthogonal' types are f_NL^local = 3 ± 6, f_NL^equil = -42 ± 75, and f_NL^ortho = -25 ± 39 (these look weak, but prominent non-Gaussianity requires the product f_NL to be large, and is of order 10^-5). Clearly none of these give any indication of primordial non-gaussianity.