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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
<math display="block">a(t_\text{em}) = \frac{1}{1 + z}\,.</math>
The expansion rate is described by the time-dependent [[Hubble parameter]], <math>H(t)</math>, defined as
<math display="block">H(t) \equiv \frac{\dot a}{a},</math>
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>
<math display="block">H^2 = \left(\frac{\dot{a}}{a}\right)^2 = \frac{8 \pi G}{3} \rho - \frac{kc^2}{a^2} + \frac{\Lambda c^2}{3}, </math>
where, as usual <math>c</math> is the speed of light and <math>G</math> is the [[gravitational constant]].
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}}
<math display="block">\Omega_x \equiv \frac{\rho_x(t=t_0)}{\rho_\mathrm{crit} } = \frac{8 \pi G\rho_x(t=t_0)}{3 H_0^2}</math>
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
<math display="block">H(a) \equiv \frac{\dot{a}}{a} = H_0 \sqrt{ (\Omega_{\rm c} + \Omega_{\rm b}) a^{-3} + \Omega_\mathrm{rad} a^{-4} + \Omega_k a^{-2} + \Omega_{\Lambda} a^{-3(1+w)} } ,</math>
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
<math display="block"> H(a) = H_0 \sqrt{ \Omega_{\rm m} a^{-3} + \Omega_\mathrm{rad} a^{-4} + \Omega_\Lambda } </math>
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>
<math display="block"> a(t) = (\Omega_{\rm m} / \Omega_\Lambda)^{1/3} \, \sinh^{2/3} ( t / t_\Lambda) </math>
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.
Solving for <math> a(t) = 1 </math> gives the present age of the universe <math> t_0 </math> in terms of the other parameters.{{citation needed|date=February 2024}}
It follows that the transition from decelerating to accelerating expansion (the second derivative <math> \ddot{a} </math> crossing zero) occurred when
<math display="block"> a = ( \Omega_{\rm m} / 2 \Omega_\Lambda )^{1/3} ,</math>
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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