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== Cosmic expansion history ==
The expansion of the universe is parameterized by a [[dimensionless]] [[scale factor (cosmology)|scale factor]] <math>a = a(t)</math> (with time <math>t</math> counted from the birth of the universe), defined relative to the present time, so <math>a_0 = a(t_0) = 1 </math>; the usual convention in cosmology is that subscript 0 denotes present-day values, so <math>t_0</math> denotes the age of the universe. The scale factor is related to the observed [[Redshift#Expansion of space|redshift]]<ref name="Dodelson"/> <math>z</math> of the light emitted at time <math>t_\mathrm{em}</math> by
The expansion rate is described by the time-dependent [[Hubble parameter]], <math>H(t)</math>, defined as
where <math>\dot a</math> is the time-derivative of the scale factor. The first [[Friedmann equations|Friedmann equation]] gives the expansion rate in terms of the matter+radiation density {{nowrap|<math>\rho</math>,}} the [[Curvature of the universe|curvature]] {{nowrap|<math>k</math>,}} and the [[cosmological constant]] {{nowrap|<math>\Lambda</math>,}}<ref name="Dodelson">{{cite book |last=Dodelson |first=Scott |title=Modern cosmology |date=2008 |publisher=[[Academic Press]] |___location=San Diego, CA |isbn=978-0-12-219141-1 |edition=4}}</ref>
where, as usual
A critical density <math>\rho_\mathrm{crit}</math> is the present-day density, which gives zero curvature <math>k</math>, assuming the cosmological constant <math>\Lambda</math> is zero, regardless of its actual value. Substituting these conditions to the Friedmann equation gives{{refn|name=constants|{{cite web|url=http://pdg.lbl.gov/2015/reviews/rpp2014-rev-astrophysical-constants.pdf |title=The Review of Particle Physics. 2. Astrophysical constants and parameters |author=K.A. Olive |collaboration=Particle Data Group |website=Particle Data Group: Berkeley Lab |date=2015 |access-date=10 January 2016 |archive-url=https://web.archive.org/web/20151203100912/http://pdg.lbl.gov/2015/reviews/rpp2014-rev-astrophysical-constants.pdf |archive-date= 3 December 2015 }}}}
<math display="block">\rho_\mathrm{crit} = \frac{3 H_0^2}{8 \pi G} = 1.878\;47(23) \times 10^{-26} \; h^2 \; \mathrm{kg{\cdot}m^{-3}},</math>
where <math> h \equiv H_0 / (100 \; \mathrm{km{\cdot}s^{-1}{\cdot}Mpc^{-1}}) </math> is the reduced Hubble constant.
If the cosmological constant were actually zero, the critical density would also mark the dividing line between eventual recollapse of the universe to a [[Big Crunch]], or unlimited expansion. For the Lambda-CDM model with a positive cosmological constant (as observed), the universe is predicted to expand forever regardless of whether the total density is slightly above or below the critical density; though other outcomes are possible in extended models where the [[dark energy]] is not constant but actually time-dependent.{{citation needed|date=February 2024}}
The present-day '''density parameter''' <math>\Omega_x</math> for various species is defined as the dimensionless ratio<ref name=Peacock-1998/>{{rp|p=74}}
where the subscript <math>x</math> is one of <math>\mathrm b</math> for [[baryon]]s, <math>\mathrm c</math> for [[cold dark matter]], <math>\mathrm{rad}</math> for [[radiation]] ([[photon]]s plus relativistic [[neutrino]]s), and <math>\Lambda</math> for [[dark energy]].{{citation needed|date=February 2024}}
Since the densities of various species scale as different powers of <math>a</math>, e.g. <math>a^{-3}</math> for matter etc.,
the [[Friedmann equation]] can be conveniently rewritten in terms of the various density parameters as
where <math>w</math> is the [[Equation of state (cosmology)|equation of state]] parameter of dark energy, and assuming negligible neutrino mass (significant neutrino mass requires a more complex equation). The various <math> \Omega </math> parameters add up to <math>1</math> by construction. In the general case this is integrated by computer to give the expansion history <math>a(t)</math> and also observable distance–redshift relations for any chosen values of the cosmological parameters, which can then be compared with observations such as [[supernovae]] and [[baryon acoustic oscillations]].{{citation needed|date=February 2024}}
In the minimal 6-parameter Lambda-CDM model, it is assumed that curvature <math>\Omega_k</math> is zero and <math> w = -1 </math>, so this simplifies to
Observations show that the radiation density is very small today, <math> \Omega_\text{rad} \sim 10^{-4} </math>; if this term is neglected
the above has an analytic solution<ref>{{cite journal|last1=Frieman|first1=Joshua A.|last2=Turner|first2=Michael S.|last3=Huterer|first3=Dragan|title=Dark Energy and the Accelerating Universe|journal=Annual Review of Astronomy and Astrophysics|year=2008|volume=46|issue=1|pages=385–432|arxiv=0803.0982|doi=10.1146/annurev.astro.46.060407.145243|bibcode=2008ARA&A..46..385F|s2cid=15117520}}</ref>
where <math> t_\Lambda \equiv 2 / (3 H_0 \sqrt{\Omega_\Lambda} ) \ ; </math>
this is fairly accurate for <math>a > 0.01</math> or <math>t > 10</math> million years.
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It follows that the transition from decelerating to accelerating expansion (the second derivative <math> \ddot{a} </math> crossing zero) occurred when
which evaluates to <math>a \sim 0.6</math> or <math>z \sim 0.66</math> for the best-fit parameters estimated from the [[Planck (spacecraft)|''Planck'' spacecraft]].{{citation needed|date=February 2024}}
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=== ''S''<sub>8</sub> tension ===
The "<math>S_8</math> tension" is a name for another question mark for the ΛCDM model.<ref name="Snowmass21"/> The <math>S_8</math> parameter in the ΛCDM model quantifies the amplitude of matter fluctuations in the late universe and is defined as
Early- (e.g. from [[Cosmic microwave background|CMB]] data collected using the Planck observatory) and late-time (e.g. measuring [[weak gravitational lensing]] events) facilitate increasingly precise values of <math>S_8</math>. However, these two categories of measurement differ by more standard deviations than their uncertainties. This discrepancy is called the <math>S_8</math> tension. The name "tension" reflects that the disagreement is not merely between two data sets: the many sets of early- and late-time measurements agree well within their own categories, but there is an unexplained difference between values obtained from different points in the evolution of the universe. Such a tension indicates that the ΛCDM model may be incomplete or in need of correction.<ref name="Snowmass21"/>
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Massimo Persic and Paolo Salucci<ref>{{Cite journal|last1=Persic|first1=M.|last2=Salucci|first2=P.|date=1992-09-01|title=The baryon content of the Universe|journal=Monthly Notices of the Royal Astronomical Society|volume=258|issue=1|pages=14P–18P|doi=10.1093/mnras/258.1.14P|arxiv=astro-ph/0502178|bibcode=1992MNRAS.258P..14P |issn=0035-8711|doi-access=free}}</ref> first estimated the baryonic density today present in ellipticals, spirals, groups and clusters of galaxies.
They performed an integration of the baryonic mass-to-light ratio over luminosity (in the following <math display="inline"> M_{\rm b}/L </math>), weighted with the luminosity function <math display="inline">\phi(L)</math> over the previously mentioned classes of astrophysical objects:
The result was:
where <math> h\
Note that this value is much lower than the prediction of standard cosmic nucleosynthesis <math> \Omega_{\rm b}\simeq0.0486 </math>, so that stars and gas in galaxies and in galaxy groups and clusters account for less than 10% of the primordially synthesized baryons. This issue is known as the problem of the "missing baryons".
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