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[[File:Aleksandr Fridman.png|thumb|236px|[[Alexander Friedmann]]]]
The '''Friedmann equations''' are a set of [[equation]]s in [[physical cosmology]] that govern the [[Metric expansion of space|expansion of space]] in [[Homogeneity (physics)|homogeneous]] and [[Isotropy|isotropic]] models of the universe within the context of [[general relativity]]. They were first derived by [[Alexander Alexandrovich Friedmann|Alexander Friedmann]] in 1922 from [[Einstein field equations|Einstein's field equations]] of [[gravitation]] for the [[Friedmann–Lemaître–Robertson–Walker metric]] and a [[perfect fluid]] with a given [[Density|mass density]] <math>\rho</math>{{mvar|[[Rho (letter)|ρ]]}} and [[pressure]] <math>{{mvar|p</math>}}.<ref name="af1922">{{cite journal |first=A |last=Friedman |author-link=Alexander Alexandrovich Friedman |title=Über die Krümmung des Raumes |journal=Z. Phys. |volume=10 |year=1922 |issue=1 |pages=377–386 |doi=10.1007/BF01332580 |bibcode = 1922ZPhy...10..377F|s2cid=125190902 |language=de}} (English translation: {{cite journal |first=A |last=Friedman |title=On the Curvature of Space |journal=General Relativity and Gravitation |volume=31 |issue=12 |year=1999 |pages= 1991–2000 |bibcode=1999GReGr..31.1991F |doi=10.1023/A:1026751225741|s2cid=122950995 }}). The original Russian manuscript of this paper is preserved in the [http://ilorentz.org/history/Friedmann_archive Ehrenfest archive].</ref> The equations for negative spatial curvature were given by Friedmann in 1924.<ref name="af1924">{{cite journal |first=A |last=Friedmann |author-link=Alexander Alexandrovich Friedman |title=Über die Möglichkeit einer Welt mit konstanter negativer Krümmung des Raumes |journal=Z. Phys. |volume=21 |year=1924 |issue=1 |pages=326–332 |doi=10.1007/BF01328280 |bibcode=1924ZPhy...21..326F|s2cid=120551579 |language=de}} (English translation: {{cite journal |first=A |last=Friedmann |title=On the Possibility of a World with Constant Negative Curvature of Space |journal=General Relativity and Gravitation |volume=31 |issue=12 |year=1999 |pages=2001–2008 |bibcode=1999GReGr..31.2001F |doi=10.1023/A:1026755309811|s2cid=123512351 }})</ref>
== Assumptions ==
{{main|Friedmann–Lemaître–Robertson–Walker metric}}
The Friedmann equations start with the simplifying assumption that the universe is spatially homogeneous and [[Isotropic manifold|isotropic]], i.e.that is, the [[cosmological principle]]; empirically, this is justified on scales larger than ~the order of 100 [[Parsec|Mpc]]. The cosmological principle implies that the metric of the universe must be of the form
:<math> ds^2 = a(t)^2 \, ds_3^2 - c^2 \, dt^2 </math>
where <{{math>ds_3^|''ds''{{su|b=3|p=2</math>}}}} is a three-dimensional metric that must be one of '''(a)''' flat space, '''(b)''' a sphere of constant positive curvature or '''(c)''' a hyperbolic space with constant negative curvature. This metric is called Friedmann–Lemaître–Robertson–Walker (FLRW) metric. The parameter <math>{{mvar|k</math>}} discussed below takes the value 0, 1, −1, or the [[Gaussian curvature]], in these three cases respectively. It is this fact that allows us to sensibly speak of a "[[Scale factor (Universe)|scale factor]]" <{{math>|''a''(''t'')</math>}}.
Einstein's equations now relate the evolution of this scale factor to the pressure and energy of the matter in the universe. From FLRW metric we compute [[Christoffel symbols]], then the [[Ricci tensor]]. With the [[stress–energy tensor]] for a perfect fluid, we substitute them into Einstein's field equations and the resulting equations are described below.
which is derived from the first together with the [[Trace (linear algebra)|trace]] of Einstein's field equations (the dimension of the two equations is time<sup>−2</sup>).
<math>{{mvar|a</math>}} is the [[scale factor (universe)|scale factor]], ''{{mvar|G''}}, {{mvar|Λ}}, and ''{{mvar|c''}} are universal constants (''{{mvar|G''}} is Newton's [[gravitational constant]], {{mvar|Λ}} is the [[cosmological constant]] (itswith dimension is length<sup>−2</sup>), and ''{{mvar|c''}} is the [[speed of light|speed of light in vacuum]]). {{mvar|ρ}} and ''{{mvar|p''}} are the volumetric mass density (and not the volumetric energy density) and the pressure, respectively. ''{{mvar|k''}} is constant throughout a particular solution, but may vary from one solution to another.
In previous equations, <math>{{mvar|a</math>}}, {{mvar|ρ}}, and ''{{mvar|p''}} are functions of time. <{{math>|{{sfrac|''k \over ''|''a^''<sup>2</mathsup>}}}} is the [[curvature|spatial curvature]] in any time-slice of the universe; it is equal to one-sixth of the spatial [[scalar curvature|Ricci curvature scalar {{mvar|R}}]] since
:<math>R = \frac{6}{c^2 a^2}(\ddot{a} a + \dot{a}^2 + kc^2)</math>
in the Friedmann model. <{{math>|''H'' \equiv≡ \frac{\dot{sfrac|''ȧ''|''a''}}}{a}</math> is the [[Hubble parameter]].
We see that in the Friedmann equations, {{math|''a''(''t'')}} does not depend on which coordinate system we chose for spatial slices. There are two commonly used choices for <math>{{mvar|a</math>}} and ''{{mvar|k''}} which describe the same physics:
*{{math|''k'' {{=}} +1, 0}} or {{math|−1}} depending on whether the [[shape of the universe]] is a closed [[3-sphere]], flat (i.e. [[Euclidean space]]) or an open 3-[[hyperboloid]], respectively.<ref>Ray A d'Inverno, ''Introducing Einstein's Relativity'', {{ISBN|0-19-859686-3}}.</ref> If {{math|''k'' {{=}} +1}}, then <math>{{mvar|a</math>}} is the [[radius of curvature]] of the universe. If {{math|''k'' {{=}} 0}}, then <math>{{mvar|a</math>}} may be fixed to any arbitrary positive number at one particular time. If {{math|''k'' {{=}} −1}}, then (loosely speaking) one can say that <{{math>|[[Imaginary unit|''i\cdot'']] · ''a</math>''}} is the radius of curvature of the universe.
*<math>{{mvar|a</math>}} is the [[Scale factor (Universe)|scale factor]] which is taken to be 1 at the present time. <math>{{mvar|k</math>}} is the current [[curvature|spatial curvature]] (when <{{math>|''a'' {{=}} 1</math> (i.e. today}}). If the [[shape of the universe]] is [[Shape of the universe#Spherical universe|hyperspherical]] and {{math|''R<mathsub>R_tt</mathsub>''}} is the radius of curvature ({{math|''R''<mathsub>R_00</mathsub>}} inat the present-day), then <{{math>|''a'' {{=}} R_t{{sfrac|''R<sub>t</R_0sub>''|''R''<sub>0</mathsub>}}}}. If <math>{{mvar|k</math>}} is positive, then the universe is hyperspherical. If <{{math>|''k</math>'' is{{=}} zero0}}, then the universe is [[Shape of the universe#Flat universe|flat]]. If <math>{{mvar|k</math>}} is negative, then the universe is [[Shape of the universe#Hyperbolic universe|hyperbolic]].
Using the first equation, the second equation can be re-expressed as
:<math>\dot{\rho} = -3 H \left(\rho + \frac{p}{c^2}\right),</math>
which eliminates <math>\Lambda</math>{{mvar|Λ}} and expresses the conservation of [[mass–energy]] :
:<math> T^{\alpha\beta}{}_{;\beta}= 0.</math>
These equations are sometimes simplified by replacing
:<math>\begin{align}
\rho &\to \rho - \frac{\Lambda c^2}{8 \pi G}</math> \\
:<math>p &\to p + \frac{\Lambda c^4}{8 \pi G}</math>
\end{align}</math>
to give:
:<math>\begin{align}
H^2 = \left(\frac{\dot{a}}{a}\right)^2 &= \frac{8 \pi G}{3}\rho - \frac{kc^2}{a^2}</math> \\
:<math>\dot{H} + H^2 = \frac{\ddot{a}}{a} &= - \frac{4\pi G}{3}\left(\rho + \frac{3p}{c^2}\right).</math>
\end{align}</math>
The simplified form of the second equation is invariant under this transformation.
Some cosmologists call the second of these two equations the '''Friedmann acceleration equation''' and reserve the term ''Friedmann equation'' for only the first equation.
== Density parameter ==<!--[[Density parameter]] links here-->
<!--[[Density parameter]] links here-->
The '''density parameter'''<!--boldface per WP:R#PLA--> <math>\Omega</math>{{mvar|Ω}} is defined as the ratio of the actual (or observed) density <math>\rho</math>{{mvar|ρ}} to the critical density {{math|''ρ''<mathsub>\rho_cc</mathsub>}} of the Friedmann universe. The relation between the actual density and the critical density determines the overall geometry of the universe; when they are equal, the geometry of the universe is flat (Euclidean). In earlier models, which did not include a [[cosmological constant]] term, critical density was initially defined as the watershed point between an expanding and a contracting Universe.
In earlier models, which did not include a [[cosmological constant]] term, critical density was initially defined as the watershed point between an expanding and a contracting Universe.
To date, the critical density is estimated to be approximately five atoms (of [[monatomic]] [[hydrogen]]) per cubic metre, whereas the average density of [[Baryons#Baryonic matter|ordinary matter]] in the Universe is believed to be 0.2–0.25 atoms per cubic metre.<ref>Rees, M., Just Six Numbers, (2000) Orion Books, London, p. 81, p. 82{{ clarify | date = September 2015 | reason =What kind of atoms? }}</ref><ref>{{cite web | publisher=[[NASA]] | title=Universe 101 | url=http://map.gsfc.nasa.gov/universe/uni_matter.html | access-date=September 9, 2015 | quote=The actual density of atoms is equivalent to roughly 1 proton per 4 cubic meters.}}</ref>
A much greater density comes from the unidentified [[dark matter]]; both ordinary and dark matter contribute in favour of contraction of the universe. However, the largest part comes from so-called [[dark energy]], which accounts for the cosmological constant term. Although the total density is equal to the critical density (exactly, up to measurement error), the dark energy does not lead to contraction of the universe but rather may accelerate its expansion. Therefore, the universe will likely expand forever.<ref name=HTUW>{{cite AV media | title=How the Universe Works 3 | volume=End of the Universe | year=2014 | publisher=[[Discovery Channel]]}}</ref>
An expression for the critical density is found by assuming {{mvar|Λ}} to be zero (as it is for all basic Friedmann universes) and setting the normalised spatial curvature, ''{{mvar|k''}}, equal to zero. When the substitutions are applied to the first of the Friedmann equations we find:
:<math>\rho_crho_\mathrm{c} = \frac{3 H^2}{8 \pi G} = 1.8788 \times 10^{-26} h^2 {\rm kg}\,{\rm m}^{-3} = 2.7754\times 10^{11} h^2 M_\odot\,{\rm Mpc}^{-3} ,</math>
:(where {{math|''h'' {{=}} ''H''<sub>o0</sub>''/(100 km/s/Mpc)}}. For {{math|''H<sub>o</sub>'' {{=}} 67.4 km/s/Mpc}}, i.e. {{math|''h'' {{=}} 0.674}}, {{math|''ρ''<sub>c</sub>'' {{=}} {{val|8.5 × 10<sup>−27</sup> 5e-27|u=kg/m<sup>3</sup>}}}}).
The density parameter (useful for comparing different cosmological models) is then defined as:
:<math>\Omega \equiv \frac{\rho}{\rho_c} = \frac{8 \pi G\rho}{3 H^2}.</math>
This term originally was used as a means to determine the [[shape of the universe|spatial geometry]] of the universe, where <{{math|''ρ''<sub>\rho_cc</mathsub>}} is the critical density for which the spatial geometry is flat (or Euclidean). Assuming a zero vacuum energy density, if <math>\Omega</math>{{mvar|Ω}} is larger than unity, the space sections of the universe are closed; the universe will eventually stop expanding, then collapse. If <math>\Omega</math>{{mvar|Ω}} is less than unity, they are open; and the universe expands forever. However, one can also subsume the spatial curvature and vacuum energy terms into a more general expression for <math>\Omega</math>{{mvar|Ω}} in which case this density parameter equals exactly unity. Then it is a matter of measuring the different components, usually designated by subscripts. According to the [[Lambda-CDM model|ΛCDM model]], there are important components of <math>\Omega</math>{{mvar|Ω}} due to [[baryon]]s, [[cold dark matter]] and [[dark energy]]. The spatial geometry of the [[universe]] has been measured by the [[Wilkinson Microwave Anisotropy Probe|WMAP]] spacecraft to be nearly flat. This means that the universe can be well approximated by a model where the spatial curvature parameter <math>k</math> is zero; however, this does not necessarily imply that the universe is infinite: it might merely be that the universe is much larger than the part we see. (Similarly, the fact that [[Earth]] is approximately flat at the scale of the [[Netherlands]] does not imply that the Earth is flat: it only implies that it is much larger than the Netherlands.)
The first Friedmann equation is often seen in terms of the present values of the density parameters, that is<ref>{{cite journal | last=Nemiroff | first=Robert J. | author-link=Robert J. Nemiroff | author2=Patla, Bijunath |arxiv = astro-ph/0703739| doi = 10.1119/1.2830536 | volume=76 | title=Adventures in Friedmann cosmology: A detailed expansion of the cosmological Friedmann equations | journal=American Journal of Physics | year=2008 | issue=3 | pages=265–276 | bibcode = 2008AmJPh..76..265N| s2cid=51782808 }}</ref>
:<math>\frac{H^2}{H_0^2} = \Omega_{0,\mathrm R} a^{-4} + \Omega_{0,\mathrm M} a^{-3} + \Omega_{0,k} a^{-2} + \Omega_{0,\Lambda}.</math>
Here <{{math|''Ω''<sub>\Omega_{0,R}</mathsub>}} is the radiation density today (i.e. when <{{math>|''a'' {{=}} 1</math>}}), {{math|''Ω''<mathsub>\Omega_{0,M}</mathsub>}} is the matter ([[dark matter|dark]] plus [[baryon]]ic) density today, <{{math|''Ω''<sub>\Omega_{0,''k}''</sub> {{=}} 1 -− \Omega_0''Ω''<sub>0</mathsub>}} is the "spatial curvature density" today, and <{{math|''Ω''<sub>\Omega_{0,\Lambda}''Λ''</mathsub>}} is the cosmological constant or vacuum density today.
== Useful solutions ==
:<math>p=w\rho c^2,</math>
where <math>{{mvar|p</math>}} is the [[pressure]], <math>\rho</math>{{mvar|ρ}} is the mass density of the fluid in the comoving frame and <math>w</math> is some constant.
In spatially flat case ({{math|''k'' {{= }} 0}}), the solution for the scale factor is
:<math> a(t)=a_0\,t^{\frac{2}{3(w+1)}} </math>
where <{{math|''a''<sub>a_00</mathsub>}} is some integration constant to be fixed by the choice of initial conditions. This family of solutions labelled by <math>{{mvar|w</math>}} is extremely important for cosmology. E.g.For <example, {{math>|''w'' {{=}} 0</math>}} describes a [[matter-dominated era|matter-dominated]] universe, where the pressure is negligible with respect to the mass density. From the generic solution one easily sees that in a matter-dominated universe the scale factor goes as
:<math>a(t)\propto t^{2/3}\frac23</math> matter-dominated
Another important example is the case of a [[radiation-dominated era|radiation-dominated]] universe, i.e.,namely when <{{math>|''w'' {{=}} {{sfrac|1/|3</math>}}}}. This leads to
:<math>a(t)\propto t^{1/2}\frac12</math> radiation -dominated
Note that this solution is not valid for domination of the cosmological constant, which corresponds to an <{{math>|''w'' {{=-1</math>}} −1}}. In this case the energy density is constant and the scale factor grows exponentially.
Solutions for other values of ''{{mvar|k''}} can be found at {{cite web|last=Tersic|first=Balsa|title=Lecture Notes on Astrophysics|url=http://nicadd.niu.edu/~bterzic/PHYS652/PHYS652_notes.pdf|access-date=20 July 2011}}.
===Mixtures===
:<math>\dot{\rho}_{f} = -3 H \left( \rho_{f} + \frac{p_{f}}{c^2} \right) \,</math>
holds separately for each such fluid ''{{mvar|f''}}. In each case,
:<math>\dot{\rho}_{f} = -3 H \left( \rho_{f} + w_{f} \rho_{f} \right) \,</math>
from which we get
:<math>{\rho}_{f} \propto a^{-3 \left(1 + w_{f}\right)} \,.</math>
For example, one can form a linear combination of such terms
:<math>\rho = A a^{-3} + B a^{-4} + C a^{0} \,</math>
where: ''{{mvar|A''}} is the density of "dust" (ordinary matter, {{math|''w'' {{= }} 0}}) when <{{math>|''a</math> '' {{= }} 1}}; ''{{mvar|B''}} is the density of radiation ({{math|''w'' {{= }} {{sfrac|1/|3}}}}) when <{{math>|''a</math> '' {{= }} 1}}; and ''{{mvar|C''}} is the density of "dark energy" ({{math|''w'' {{=}} −1}}). One then substitutes this into
:<math>\left(\frac{\dot{a}}{a}\right)^2 = \frac{8 \pi G}{3} \rho - \frac{kc^2}{a^2} \,</math>
and solves for <math>{{mvar|a</math>}} as a function of time.
===Detailed derivation===
To make the solutions more explicit, we can derive the full relationships from the first Friedman equation:
:<math>\frac{H^2}{H_0^2} = \Omega_{0,\mathrm R} a^{-4} + \Omega_{0,\mathrm M} a^{-3} + \Omega_{0,k} a^{-2} + \Omega_{0,\Lambda}</math>
with
:<math>H = \fracbegin{\dot{a}}{aalign}</math>
H &= \frac{\dot{a}}{a} \\[6px]
:<math>{H^2 } &= H_0^2 \left( \Omega_{0, \mathrm R} a^{-4} + \Omega_{0, \mathrm M} a^{-3} + \Omega_{0,k} a^{-2} + \Omega_{0,\Lambda} \right) </math>\\[6pt]▼:<math>\frac{\dot{a}}{a}H &= H_0 \sqrt{ ( \Omega_{0, \mathrm R} a^{-4} + \Omega_{0, \mathrm M} a^{-3} + \Omega_{0,k} a^{-2} + \Omega_{0,\Lambda} )} </math> \\[6pt]▼:<math>\frac{\ mathrmdot{ d}a }}{ \mathrm{d} ta} &= H_0 \sqrt{ ( \Omega_{0, \mathrm R} a^{- 24} + \Omega_{0, \mathrm M} a^{- 13} + \Omega_{0,k } a^{-2} + \Omega_{0,\Lambda }} a^2)}</math>\\[6pt]▼:<math>\frac{\mathrm{d}a = }{\mathrm{d} t } &= H_0 \sqrt{ (\Omega_{0, \mathrm R} a^{-2} + \Omega_{0, \mathrm M} a^{-1} + \Omega_{0,k} + \Omega_{0,\Lambda} a^2 )} </math> \\[6pt]▼:<math>t H_0 = \ int_mathrm{ 0d} ^{a } \frac{ &= \mathrm{d} a'}{ t H_0 \sqrt{ (\Omega_{0, \mathrm R} a '^{-2} + \Omega_{0, \mathrm M} a '^{-1} + \Omega_{0,k} + \Omega_{0,\Lambda} a '^2 )} }</math> \\[6pt]▼\end{align}</math>
Rearranging and changing to use variables <{{math >|''a' </math> '′}} and <{{math >|''t' </math>'′}} for the integration ▼▲:<math>{H^2} = H_0^2 ( \Omega_{0,R} a^{-4} + \Omega_{0,M} a^{-3} + \Omega_{0,k} a^{-2} + \Omega_{0,\Lambda}) </math>
:<math>Ht H_0 = H_0\int_{0}^{a} \frac{\mathrm{d}a'}{\sqrt{( \Omega_{0,\mathrm R} a'^{-42} + \Omega_{0,\mathrm M} a'^{-31} + \Omega_{0,k} a^{-2} + \Omega_{0,\Lambda} a'^2}}</math>
Solutions for the dependence of the scale factor with respect to time for universes dominated by each component can be found. In each we also have assumed that <{{math |''Ω''<sub> \Omega_{0, ''k }''</sub> \approx≈ 0 </math>}}, which is the same as assuming that the dominating source of energy density is <math>\approxapproximately 1 </math>. ▼▲:<math>\frac{\dot{a}}{a} = H_0 \sqrt{( \Omega_{0,R} a^{-4} + \Omega_{0,M} a^{-3} + \Omega_{0,k} a^{-2} + \Omega_{0,\Lambda})}</math>
For matter-dominated universes, where {{math|''Ω''<sub>0,M</sub> ≫ ''Ω''<sub>0,R</sub>}} and {{math|''Ω''<sub>0,''Λ''</sub>}}, as well as {{math|''Ω''<sub>0,M</sub> ≈ 1}}:
▲:<math>\frac{\mathrm{d}a }{\mathrm{d} t} = H_0 \sqrt{(\Omega_{0,R} a^{-2} + \Omega_{0,M} a^{-1} + \Omega_{0,k} + \Omega_{0,\Lambda} a^2)}</math>
:<math>\begin{align}
▲:<math>\mathrm{d}a = \mathrm{d} t H_0 \sqrt{(\Omega_{0,R} a^{-2} + \Omega_{0,M} a^{-1} + \Omega_{0,k} + \Omega_{0,\Lambda} a^2)}</math>
:<math>t H_0 &= \int_{0}^{a} \frac{\mathrm{d}a'}{\sqrt{ (\Omega_{0, \mathrm M} a'^{-1} )}} </math> \\[6px]▼t H_0 \sqrt{\Omega_{0,\mathrm M}} &= \left.\left( \frac23 a'^\frac32 \right) \,\right|^a_0 \\[6px]
:<math> \left( \ frac{3}{2}frac32 t H_0 \sqrt{\Omega_{0, \mathrm M}} \right)^ {\ frac{2}{3}}frac23 &= a(t) </math>▼\end{align}</math>
which recovers the aforementioned <{{math >|''a '' \propto∝ ''t ^''<sup>{ \frac{ sfrac|2 }{|3}}</ mathsup> }}▼▲Rearranging and changing to use variables <math>a'</math> and <math>t'</math> for the integration
For radiation-dominated universes, where {{math|''Ω''<sub>0,R</sub> ≫ ''Ω''<sub>0,M</sub>}} and {{math|''Ω''<sub>0,''Λ''</sub>}}, as well as {{math|''Ω''<sub>0,R</sub> ≈ 1}}:
▲:<math>t H_0 = \int_{0}^{a} \frac{\mathrm{d}a'}{\sqrt{(\Omega_{0,R} a'^{-2} + \Omega_{0,M} a'^{-1} + \Omega_{0,k} + \Omega_{0,\Lambda} a'^2)}}</math>
:<math>\begin{align}
▲Solutions for the dependence of the scale factor with respect to time for universes dominated by each component can be found. In each we also have assumed that <math>\Omega_{0,k} \approx 0</math>, which is the same as assuming that the dominating source of energy density is <math>\approx 1</math>.
:<math>t H_0 &= \int_{0}^{a} \frac{\mathrm{d}a'}{\sqrt{ (\Omega_{0, \mathrm R} a'^{-2} )}} </math> \\[6px]▼:<math>t H_0 \sqrt{\Omega_{0, \mathrm R}} &= \left.\frac{a'^2}{2} \,\right|^a_0 </math>\\[6px]▼:<math> \left(2 t H_0 \sqrt{\Omega_{0, \mathrm R}} \right)^ {\ frac{1}{2}}frac12 &= a(t) </math>▼\end{align}</math>
For Matter {{mvar|Λ}}-dominated universes, where <{{math|''Ω''<sub>\Omega_{0,M} >''Λ''</sub> \Omega_{≫ ''Ω''<sub>0,R}</mathsub>}} and <{{math|''Ω''<sub>\Omega_{0,\Lambda}M</mathsub>}}, as well as <{{math|''Ω''<sub>\Omega_{0,M}''Λ''</sub> \approx≈ 1}}, and where we now will change our bounds of integration from {{math|''t<sub>i</sub>''}} to {{mvar|t}} and likewise {{math|''a<sub>.i</sub>''}} to {{mvar|a}}:
:<math>\begin{align}
▲:<math>t H_0 = \int_{0}^{a} \frac{\mathrm{d}a'}{\sqrt{(\Omega_{0,M} a'^{-1})}}</math>
:<math>\left(t-t_i \right) H_0 &= \int_{a_i}^{a} \frac{\mathrm{d}a'}{\sqrt{(\Omega_{0,\Lambda} a'^2)}} </math> \\[6px]▼:<math> \left(t - t_i \right) H_0 \sqrt{\Omega_{0,\Lambda}} &= \ mathrm{bigl. \ln }(|a'| ) \,\bigr|^a_{a_i} </math> \\[6px]▼:<math>a_i \ mathrm{e}^{exp\left( (t - t_i) H_0 \sqrt{\Omega_{0,\Lambda}} }\right) &= a(t) </math>▼\end{align}</math>
The <math>\Lambda</math> {{mvar|Λ}}-dominated universe solution is of particular interest because the second derivative with respect to time is positive, non-zero; in other words implying an accelerating expansion of the universe, making <{{math |''ρ<sub> \rho_{Lambda}Λ</ mathsub> ''}} a candidate for [[dark energy]]: ▼:<math>t H_0 \sqrt{\Omega_{0,M}} = ( \frac{2}{3} a'^{\frac{3}{2}} ) |^a_0 </math>
:<math>\begin{align}
▲:<math> ( \frac{3}{2} t H_0 \sqrt{\Omega_{0,M}})^{\frac{2}{3}} = a(t) </math>
:<math>a(t) &= a_i \ mathrm{e}^{exp\left( (t - t_i) H_0 \sqrt{\Omega_{0,\Lambda}} }</math>\right) \\[6px]▼:<math>\frac{\mathrm{d}^2 a(t) }{\mathrm{d} t^2} &= a_i \left(H_0 \right)^2 \Omega_{0,\Lambda} \ mathrm{e}^{exp\left( (t - t_i) H_0 \sqrt{\Omega_{0,\Lambda}} }</math>\right)▼\end{align}</math>
Where by construction <{{math |''a<sub>i</sub> a_i'' > 0 </math>}}, our assumptions were <{{math |''Ω''<sub> \Omega_{0, \Lambda} \approx 1''Λ''</ mathsub> ≈ 1}}, and <{{math |''H''<sub> H_00</ mathsub> }} has been measured to be positive, forcing the acceleration to be greater than zero. ▼▲which recovers the aforementioned <math>a \propto t^{\frac{2}{3}}</math>
For Radiation dominated universes, where <math>\Omega_{0,R} >> \Omega_{0,M}</math> and <math>\Omega_{0,\Lambda}</math>, as well as <math>\Omega_{0,R} \approx 1</math>
▲:<math>t H_0 = \int_{0}^{a} \frac{\mathrm{d}a'}{\sqrt{(\Omega_{0,R} a'^{-2})}}</math>
▲:<math>t H_0 \sqrt{\Omega_{0,R}} = \frac{a'^2}{2} |^a_0 </math>
▲:<math> (2 t H_0 \sqrt{\Omega_{0,R}})^{\frac{1}{2}} = a(t) </math>
For <math>\Lambda</math> dominated universes, where <math>\Omega_{0,\Lambda} >> \Omega_{0,R}</math> and <math>\Omega_{0,M}</math>, as well as <math>\Omega_{0,\Lambda} \approx 1</math>, and where we now will change our bounds of integration from <math>t_i</math> to <math>t</math> and likewise <math>a_i</math> to <math>a</math>.
▲:<math>(t-t_i) H_0 = \int_{a_i}^{a} \frac{\mathrm{d}a'}{\sqrt{(\Omega_{0,\Lambda} a'^2)}}</math>
▲:<math> (t - t_i) H_0 \sqrt{\Omega_{0,\Lambda}} = \mathrm{ln}(|a'|) |^a_{a_i}</math>
▲:<math>a_i \mathrm{e}^{(t - t_i) H_0 \sqrt{\Omega_{0,\Lambda}}} = a(t)</math>
▲The <math>\Lambda</math> dominated universe solution is of particular interest because the second derivative with respect to time is positive, non-zero; in other words implying an accelerating expansion of the universe, making <math>\rho_{Lambda}</math> a candidate for [[dark energy]]:
▲:<math>a(t) = a_i \mathrm{e}^{(t - t_i) H_0 \sqrt{\Omega_{0,\Lambda}}}</math>
▲:<math>\frac{\mathrm{d}^2 a(t) }{\mathrm{d} t^2} = a_i (H_0)^2 \Omega_{0,\Lambda} \mathrm{e}^{(t - t_i) H_0 \sqrt{\Omega_{0,\Lambda}}}</math>
▲Where by construction <math>a_i > 0</math>, our assumptions were <math>\Omega_{0,\Lambda} \approx 1</math>, and <math>H_0</math> has been measured to be positive, forcing the acceleration to be greater than zero.
== Rescaled Friedmann equation ==
Set
Set <math>\tilde{a}=\frac{a}{a_0}, \;\rho_c=\frac{3H_0^2}{8\pi G},\;
:<math>\Omegatilde{a}=\frac{\rhoa}{\rho_ca_0},\; t\quad\rho_c=\frac{\tilde{t}3H_0^2}{H_08\pi G},\;quad
\Omega=\frac{\rho}{\rho_\mathrm{c}},\quad t=\frac{\tilde{t}}{H_0},\quad
\Omega_c=-\frac{kc^2}{H_0^2 a_0^2}\;</math>, where <math>a_0</math> and <math>H_0</math> are separately the [[Scale factor (Universe)|scale factor]] and the [[Hubble parameter]] today. ▼\Omega_\mathrm{c}=-\frac{kc^2}{H_0^2 a_0^2},</math>
▲\Omega_c=-\fracwhere { kc^2}{ H_0^2 a_0^2}\;</math >, where |''a''< mathsub> a_00</ mathsub> }} and {{math|''H''< mathsub> H_00</ mathsub> }} are separately the [[Scale factor (Universe)|scale factor]] and the [[Hubble parameter]] today.
Then we can have
:<math>\frac{1}{2}frac12\left( \frac{d\tilde{a}}{d\tilde{t}}\right)^2 + U_\text{eff}(\tilde{a})=\fracfrac12\Omega_\mathrm{1c}{2}\Omega_c</math>
where
:<math>U_\text{eff}(\tilde{a})=\frac{-\Omega\tilde{a}^2}{2}.</math>
where <math>U_\text{eff}(\tilde{a})=\frac{-\Omega\tilde{a}^2}{2}\;</math>. For any form of the effective potential {{math|''U''<mathsub>U_\text{eff}(\tilde{a})\;</mathsub>(''ã'')}}, there is an equation of state <{{math>|''p'' {{=}} ''p''(\rho''ρ'')</math>}} that will produce it.
== See also ==
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