2. THE PARAMETERS AND ASSUMPTIONS OF THE SMC

Let's be explicit about the standard model by giving the modern values of its basic parameters. Right now the determination of these quantities is driven by cosmic microwave background (CMB) anisotropy experiments, and in particular by results from the Planck satellite [2] (supported by many other kinds of data) on the CMB power spectra (which are discussed further in Sect. 4). Because of this, the basic parameter set is currently given in terms of the quantities that are most directly measured by CMB experiments. This means that the parameters most often discussed in relation to observational constraints are not necessarily the ones that are simplest to explain to the general public, or that are the focus of non-CMB cosmologists. These parameters are listed in Table I. The set consists of: two densities, Ω_bh² and Ω_ch² (for baryonic matter and cold dark matter separately, since they have distinct effects on the CMB power spectra), including a scaling of physical density with the dimensionless Hubble parameter, h ≡ H₀ / 100 km s⁻¹ Mpc⁻¹; a parameter θ_∗ that corresponds to the sound horizon divided by the angular diameter distance to last scattering, which quantifies sliding the CMB power spectra left and right; the amplitude A_s of the initial power spectrum of density perturbations, defined at a particular scale, and often given as a logarithm; the slope n of the initial power spectrum as a function of wavenumber; and a parameter τ describing how much the primary CMB anisotropies are scattered by the reionised medium at low redshifts.

By far the best determined of these parameters is θ_∗, with a signal-to-noise ratio (S / N) of about 2500 (from Table I, or about 2300 from the CMB alone). Then follows A_s, Ω_bh² and Ω_ch², with S / N ≃ 100, while n_s and τ only differ from their default values (of 1 and 0, respectively) at S / N ≃ 5. Other cosmological parameters that are often discussed include H₀, t₀, Ω_m, Ω_Λ, z_reion, etc., which are not independent, but can be determined from the six parameters in the context of the SMC.

Although it is often stated that there are six basic parameters, there's a seventh that is often ignored. This is the temperature of the CMB today (or equivalently the radiation density), which is constrained using data from the COBE-FIRAS instrument [5], as well as from several other experiments (see Ref. [4]). The determination is systematics dominated, with S / N ≃ 5000. It is hence more precise than other parameters, and dramatically better determined than other densities. For that reason it is usually considered to be fixed, and not a free parameter at all. However, the precision is starting to approach the cosmic-variance limit, and so if T₀ was measured with much smaller errors, we'd have to consider the fact that we can only measure parameters within our Hubble patch and not actually “background” parameter values (see Ref. [6] for discussion).

But (to be skeptical about this), we might wonder whether there are other hidden parameters. There definitely are, to some extent, but mostly any additions to the SMC are better cast as assumptions. In fact there are many of these, and it is important to be clear that the six (or seven) parameters of the SMC are only descriptive of the Universe within a specific framework. A list of these assumptions is given in Table II (and the reader can probably think of more).

All of these assumptions are testable, and they all have been investigated. Many of them are tested through putting limits on extensions to the SMC, e.g. checking whether the curvature is consistent with flat space, whether there's evidence for modified gravity, non-trivial dark energy (i.e. w ≠ −1), or non-Gaussianity, or whether there are signs of the effects of massive neutrinos or cosmic strings (e.g. see Refs. [7, 3]).

Nevertheless, this is definitely a place where we need to exercise caution. The confidence with which we know the values of the basic set of six (or seven) parameters depends on this being the full parameters space. If there are more ingredients in the actual model, then the parameters in the basic set will have larger uncertainties. For example, if we consider models that allow curvature then the constraints on w are very much weakened. Hence we need to look carefully at these tests. Right now there is no strong evidence for any additional parameter, but we fully expect that there will be more ingredients required as the data improve, e.g. that the effects of massive neutrinos or primordial gravitational waves will eventually be measured. And there may be genuine surprises of course, like multiple kinds of dark matter or dark energy, or important extra components, such as magnetic fields or isocurvature modes.

Nevertheless, there has been caution exercised, and despite attempts to find evidence for additional parameters, the basic set continues to fit very high signal-to-noise data – particularly the CMB power spectra.