arXiv:1002.3488v1 [astro-ph.CO] 18 Feb 20101. The Cosmological Parameters 11. THE COSMOLOGICAL PARAMETERSUpdated September 2009, by O. Lahav (University College London) and A.R. Liddle(University of Sussex).1.1. Parametrizing the UniverseRapid advances in observational cosmology have led to the establishment of a precisioncosmological model, with many of the key cosmological parameters determined to oneor two significant figure accuracy. Particularly prominent are measurements of cosmicmicrowave background (CMB) anisotropi es, led by the five-year results from the WilkinsonMicrowave Anisotropy Probe (WMAP) [1,2,3] . However the most accurate model o f theUniverse requires consideration of a wide range of different types of observatio n, withcomplementary probes providing consistency checks, lifting parameter degeneracies, andenabling the strongest constraints to be placed.The term ‘cosmological parameters’ is forever increasing in its scope, and nowadaysincludes the parametrization of some functions, as well as simple numbers describingproperties of the Universe. The original usage referred to the para met ers describing theglobal dynamics of the Universe, such as its expansion rat e and curvature. Also now ofgreat interest is how the matter budget of the Universe is built up from its constituent s:baryons, photons, neutrinos, dark ma tter, and dark energy. We need to describe thenature of perturbations in the Universe, through global st atistical descriptors such asthe matt er and radiation power sp ectra . There may also be parameters describing thephysical state of the Universe, such as the ionization fraction as a function of t imeduring the era since recombination. Typical comparisons of cosmological models wit hobservational data now feature between five and ten parameters.1.1.1. The global description of the Universe:Ordinarily, the Universe is taken to be a perturbed Robertson–Walker space-time withdynamics governed by Einstein’s equations. This is described in detail by Olive andPeacock in this volume. Using the density parameters Ωifor the various matter speciesand ΩΛfor the cosmological constant, the Friedmann equation can be writtenXiΩi+ ΩΛ− 1 =kR2H2, (1.1)where the sum is over all the different species of material in the Universe. This equati o napplies at any epoch, but later in thi s article we will use the symbols Ωiand ΩΛto referto the present values. A typical collection would be baryons, photons, neutrinos, anddark matter (given charge neutrality, the electron density is guaranteed to be too smallto be worth considering separately and is i ncluded with the baryons).The complete present state of the homogeneous Universe can be described by givingthe current values of all the density parameters and of the H ubble parameter h. Thesealso allow us to track the hi st ory of the Universe back in time, at least until an epochwhere interact ions allow interchanges between the densities of the different species,which is believed to have last happened at neutrino decoupling, shortly before BigBang Nucleosynthesis (BBN). To probe further back into the Universe’s history requiresassumptions a bout particle interactions, and perhaps about the nature of physical lawsthemselves.February 18, 2010 11:052 1. The Cosmological Parameters1.1.2. Neutrinos:The standard neutrino sector has three flavors. For neutrinos of mass i n the range5 × 10−4eV to 1 MeV, the density parameter in neutrinos is predicted to beΩνh2=Pmν93 eV, (1.2)where the sum is over all families with mass in that range (higher masses need a moresophisticated calcula tion). We use units with c = 1 throughout. Results on atmosphericand Solar neutrino oscillations [4] imply non-zero mass-squared differences between thethree neutrino flavors. These oscillation experiments cannot tell us the absolute neutrinomasses, but w ithin the simple assumption of a ma ss hierarchy suggest a lower limit ofΩν≈ 0.001 on the neutrino mass density parameter.For a total mass as small as 0 .1 eV, this could have a potentially observable effect onthe formation of structure, as neutrino free-streaming damps the growth of perturbations.Present cosmological observations have shown no convincing evi dence of any effects fromeither neutrino masses or a n otherwise non-standard neutrino sector, and impose quitestringent limits, which we summarize in Section 1.3.4. Accordingly, the usual assumptionis that the masses are too small to have a sig ni ficant cosmological impact at present dataaccuracy. However, we note that the inclusion of neutrino mass as a free parameter canaffect the derived values of other cosmological parameters.The cosmological effect of neutrinos can also be modified if the neutrinos have decaychannels, or if there is a l arge asymmetry in the lepton sector manifested as a differentnumber density of neutrinos versus anti-neutrinos. This latter effect would need to be oforder unity to be sig ni ficant (rather than the 10−9seen in the baryon sector), which maybe in conflict with nucleosynthesis [5].1.1.3. Inflation and perturbations:A complete model of the Universe should include a description of deviations fromhomogeneity, at least in a sta tistical way. Indeed, some of the most powerful probes ofthe paramet ers described above come from the evolution of perturbations, so their studyis naturally intertwined in the determination o f cosmological parameters.There are many different notations used to describe the perturbations, both in termsof the quantity used to describe the perturbations and the definition of the statisticalmeasure. We use the di m ensionless power spectrum ∆2as defined in Olive a nd Peacock(also denoted P in some of the literature). If the perturbatio ns obey Gaussian statistics,the power spectrum provides a complete description of their properties.From a theoretical perspective, a useful quantity to describe the perturbations is thecurvature perturbation R, which measures the spatial curva ture of a comoving slicing ofthe space-time. A case of particular interest is the Harrison–Zel’dovich sp ectrum, whichcorresponds to a constant ∆2R. More generall y, o ne can approximate the spectrum by apower-law, writing∆2R(k) = ∆2R(k∗)kk∗n−1, (1.3)February 18, 2010 11:051. The Cosmological Parameters 3where n is known as the spectral index, al way s defined so that n = 1 for the Harrison–Zel’dovich spectrum, and k∗is an arbitrarily chosen scale.