arXiv:astro-ph/0601168 v1 9 Jan 20061. The Cosmological Parameters 11. THE COSMOLOGICAL PARAMETERS 2005Written August 2003, updated September 2005, by O. Lahav ( University C ollege London)and A.R. Liddle (University of Sussex).1.1. Parametrizing the UniverseRapid advances in observational cosmology are leading to the establishment of the firstprecision cosmological model, with many of the key cosmological parameters determinedto one or two significant figure accuracy. Particularly prominent are measurements ofcosmic microwave anisotropies, led by the first results from the Wilkinson MicrowaveAnisotropy Probe (WMAP) announced in February 2003 [1]. However the most accuratemodel of the Universe requires consideration of a wide range of different types ofobservation, with complementary probes prov iding consistency checks, lifti ng parameterdegeneracies, and enabling the strongest constraints to be placed.The term ‘cosmological parameters’ is forever increasing in its scope, and nowadaysincludes the parametrization o f some functions, as well as simple numbers describingproperties of the Universe. The original usage referred to the parameters describing theglobal dynamics of the Universe, such as its expansion rate and curvature. Also now ofgreat interest is how the matter budget of the Universe is built up from its constituents:baryons, photons, neutrinos, dark matter, and dark energy. We need to describe thenature of perturbations in the Universe, through global statistical descriptions such asthe matter and radiation power spectra. There may al so b e parameters describing thephysical state of the Universe, such as the ionization fraction as a function of time duringthe era since decoupling. Typical comparisons of cosmological models with observationaldata 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. U sing the density parameters Ωifor the various matter speciesand ΩΛfor the cosmological constant, the Friedmann eq uation can be writtenXiΩi+ ΩΛ=kR2H2, (1.1)where the sum is over all the different species of matter in the Universe. This equationapplies at any epoch, but later in this 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) .The complete present state of the homog eneous Universe can be described by givingthe present values of all the density parameters and the present Hubble parameter h.These also allow us to track the history of the Universe back in time, at least untilan ep och where interactions al low interchanges between the densities of the differentspecies, which is believed to have last happened at neutrino decoupling shortly beforenucleosynthesis.January 9, 2006 09:382 1. The Cosmological ParametersTo probe further back into t he Universe’s history requires assumptions about particleinteractions, and perhaps about the nature of physical laws themselves.1.1.2. Neutrinos:The sta ndard neutrino sector has three flavors. For neutrinos of mass in the range5 × 10−4eV to 1 MeV, the density parameter in neutrinos is predicted to beΩνh2=Pmν94 eV, (1.2)where the sum is over all families with mass in that range (higher masses need amore sophisticated calculation). We use units with c = 1 throughout. Recent results onatmospheric and solar neutrino oscillations [2] imply non-zero mass-squared differencesbetween the three neutrino flavors. These oscillation experiments cannot tel l us theabsolute neutrino masses, but within the simple assumption of a mass hierarchy suggesta lower limit of Ων≈ 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 evidence of any effectsfrom either neutrino masses or an otherwise non-standard neutrino sector, and imposequite stringent limits, which we summarize in Section 1.3.4. Consequently, the standardassumption at present is t hat the masses are too small to have a significant cosmologicalimpact, but this may change i n t he near future.The cosmological effect of neutrinos can also be modified if the neutrinos have decaychannels, or if there is a large 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 significant, rather than the 10−9seen in the baryon sector, which maybe in conflict with nucleosynthesis [3].1.1.3. Inflation and perturbations:A complete model of the Universe should include a description of deviations fromhomogeneity, at least in a statistical way. Indeed, some of the most powerful probes ofthe parameters described above come from the evolution of perturbations, so their studyis naturally intertwined in the determination of cosmological parameters.There are many different notati ons used to describe the perturbations, both in termsof the quantity used to describe the perturbations and the definition of the statisticalmeasure. We use the dimensionless power spectrum ∆2as defined in Olive and Peacock(also denoted P i n some of the literature). If the perturbations obey Gaussian statistics,the power spectrum provides a complete description of their properties.From a theoretical perspective, a useful quantity to describe the perturbatio ns is thecurvature perturbation R, which measures the spatial curva ture of a comoving slicingof the space-time. A case of particular interest is the Harrison–Zel’dovich spectrum,which corresponds t o a constant spectrum ∆2R. More generally, one can approximate t hespectrum by a power-law, writing∆2R(k) = ∆2R(k∗)kk∗n−1, (1.3)January 9, 2006 09:381. The Cosmological Parameters 3where n is known as the spectral index, always defined so that n = 1 for the Harrison–Zel’dovich spectrum, and k∗is an arbitrarily chosen scale. The initial spectrum, definedat some early epoch of the Universe’s history, is usually taken to have a simple formsuch as this power-law,