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{{Short description|Equations in physical cosmology}}
{{Physical cosmology |expansion}}
[[File:Aleksandr Fridman.png|thumb|236px|[[Alexander Friedmann]]]]
The '''Friedmann equations''', also known as the '''Friedmann–Lemaître''' ('''FL''') '''equations''', are a set of [[equation]]s in [[physical cosmology]] that govern the [[
== 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]],
where
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.
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There are two independent Friedmann equations for modelling a homogeneous, isotropic universe. The first is:
<math display="block"> \frac{\dot{a}^2 + kc^2}{a^2} = \frac{8 \pi G \rho + \Lambda c^2}{3} ,</math>
which is derived from the 00 component of the [[Einstein field equations]]. The second is:
<math display="block">\frac{\ddot{a}}{a} = -\frac{4 \pi G}{3}\left(\rho+\frac{3p}{c^2}\right) + \frac{\Lambda c^2}{3}</math>
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>).
{{mvar|a}} is the [[scale factor (universe)|scale factor]], {{mvar|G}}, {{math|Λ}}, and {{mvar|c}} are universal constants ({{mvar|G}} is the [[Newtonian constant of gravitation]], {{math|Λ}} is the [[cosmological constant]] with dimension 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, {{mvar|a}}, {{mvar|ρ}}, and {{mvar|p}} are functions of time. {{math|{{sfrac|''k''|''a''<sup>2</sup>}}}} 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 display="block">R = \frac{6}{c^2 a^2}(\ddot{a} a + \dot{a}^2 + kc^2)</math>
in the Friedmann model. {{math|''H'' ≡ {{sfrac|''ȧ''|''a''}}}} 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 {{mvar|a}} 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 ([[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 {{mvar|a}} is the [[radius of curvature]] of the universe. If {{math|''k'' {{=}} 0}}, then {{mvar|a}} 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'']] · ''a''}} is the radius of curvature of the universe.
* {{mvar|a}} is the [[Scale factor (Universe)|scale factor]] which is taken to be 1 at the present time. {{mvar|k}} is the current [[curvature|spatial curvature]] (when {{math|''a'' {{=}} 1}}). If the [[shape of the universe]] is [[Shape of the universe#Spherical universe|hyperspherical]] and {{math|''R<sub>t</sub>''}} is the radius of curvature ({{math|''R''<sub>0</sub>}} at the present), then {{math|''a'' {{=}} {{sfrac|''R<sub>t</sub>''|''R''<sub>0</sub>}}}}. If {{mvar|k}} is positive, then the universe is hyperspherical. If {{math|''k'' {{=}} 0}}, then the universe is [[Shape of the universe#Flat universe|flat]]. If {{mvar|k}} 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 display="block">\dot{\rho} = -3 H \left(\rho + \frac{p}{c^2}\right),</math>
which eliminates {{math|Λ}} and expresses the conservation of [[mass–energy]]:
<math display="block"> T^{\alpha\beta}{}_{;\beta}= 0.</math>
These equations are sometimes simplified by replacing
<math display="block">\begin{align}
\end{align}</math>
to give:
<math display="block">\begin{align}
\end{align}</math>
The simplified form of the second equation is invariant under this transformation.
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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-->
The '''density parameter'''<!--boldface per WP:R#PLA-->
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>
[[File:UniverseComposition.svg|thumb|right|375px|Estimated relative distribution for components of the energy density of the universe. [[Dark energy]] dominates the total energy (74%) while [[dark matter]] (22%) constitutes most of the mass. Of the remaining baryonic matter (4%), only one tenth is compact. In February 2015, the European-led research team behind the [[Planck (spacecraft)|Planck cosmology probe]] released new data refining these values to 4.9% ordinary matter, 25.9% dark matter and 69.1% dark energy.]]
A much greater density comes from the unidentified [[dark matter]]
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 display="block">\rho_\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>
{{block indent | em = 1.5 | text = (where {{math|''h'' {{=}} ''H''<sub>0</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|8.5e-27|u=kg/m<sup>3</sup>}}}}).}}
The density parameter (useful for comparing different cosmological models) is then defined as:
This term originally was used as a means to determine the [[shape of the universe|spatial geometry]] of the universe, where
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>
Here
== Useful solutions ==
The Friedmann equations can be solved exactly in presence of a [[perfect fluid]] with equation of state
<math display="block">p=w\rho c^2,</math>
where {{mvar|p}} is the [[pressure]], {{mvar|ρ}} is the mass density of the fluid in the comoving frame and {{mvar|w}} is some constant.
In spatially flat case ({{math|''k'' {{=}} 0}}), the solution for the scale factor is
<math display="block"> a(t)=a_0\,t^{\frac{2}{3(w+1)}} </math>
where {{math|''a''<sub>0</sub>}} is some integration constant to be fixed by the choice of initial conditions. This family of solutions labelled by {{mvar|w}} is extremely important for cosmology. For example, {{math|''w'' {{=}} 0}} 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 display="block">a(t) \propto t^{2/3}</math> matter-dominated
Another important example is the case of a [[radiation-dominated era|radiation-dominated]] universe, namely when {{math|''w'' {{=}} {{sfrac|1|3}}}}. This leads to
<math display="block">a(t) \propto t^{1/2}</math> radiation-dominated
Note that this solution is not valid for domination of the cosmological constant, which corresponds to an {{math|''w'' {{=}} −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=https://www.academia.edu/5025956|access-date=24 February 2022}}
===Mixtures===
If the matter is a mixture of two or more non-interacting fluids each with such an equation of state, then
holds separately for each such fluid {{mvar|f}}. In each case,
<math display="block">\dot{\rho}_{f} = -3 H \left( \rho_{f} + w_{f} \rho_{f} \right) \,</math>
from which we get
For example, one can form a linear combination of such terms
where {{mvar|A}} is the density of "dust" (ordinary matter, {{math|''w'' {{=}} 0}}) when {{math|''a'' {{=}} 1}}; {{mvar|B}} is the density of radiation ({{math|''w'' {{=}} {{sfrac|1|3}}}}) when {{math|''a'' {{=}} 1}}; and {{mvar|C}} is the density of "dark energy" ({{math|''w'' {{=}} −1}}). One then substitutes this into
<math display="block">\left(\frac{\dot{a}}{a}\right)^2 = \frac{8 \pi G}{3} \rho - \frac{kc^2}{a^2} \,</math>
and solves for {{mvar|a}} as a function of time.
===Detailed derivation===
To make the solutions more explicit, we can derive the full relationships from the first
<math display="block">\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 display="block">\begin{align}
H &= \frac{\dot{a}}{a} \\[6px]
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) \\[6pt]
H &= H_0 \sqrt{ \Omega_{0,\mathrm R} a^{-4} + \Omega_{0,\mathrm M} a^{-3} + \Omega_{0,k} a^{-2} + \Omega_{0,\Lambda}} \\[6pt]
\frac{\dot{a}}{a} &= H_0 \sqrt{ \Omega_{0,\mathrm R} a^{-4} + \Omega_{0,\mathrm M} a^{-3} + \Omega_{0,k} a^{-2} + \Omega_{0,\Lambda}} \\[6pt]
\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} \\[6pt]
\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} \\[6pt]
\end{align}</math>
Rearranging and changing to use variables {{math|''a''′}} and {{math|''t''′}} for the integration
<math display="block">t H_0 = \int_{0}^{a} \frac{\mathrm{d}a'}{\sqrt{\Omega_{0,\mathrm R} a'^{-2} + \Omega_{0,\mathrm M} a'^{-1} + \Omega_{0,k} + \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>0,''k''</sub> ≈ 0}}, which is the same as assuming that the dominating source of energy density is approximately 1.
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 display="block">\begin{align}
t H_0 &= \int_{0}^{a} \frac{\mathrm{d}a'}{\sqrt{\Omega_{0,\mathrm M} a'^{-1}}} \\[6px]
t H_0 \sqrt{\Omega_{0,\mathrm M}} &= \left.\left( \tfrac23 {a'}^{3/2} \right) \,\right|^a_0 \\[6px]
\left( \tfrac32 t H_0 \sqrt{\Omega_{0,\mathrm M}}\right)^{2/3} &= a(t)
\end{align}</math>
which recovers the aforementioned {{math|''a'' ∝ ''t''<sup>2/3</sup>}}
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 display="block">\begin{align}
t H_0 &= \int_{0}^{a} \frac{\mathrm{d}a'}{\sqrt{\Omega_{0,\mathrm R} a'^{-2}}} \\[6px]
t H_0 \sqrt{\Omega_{0,\mathrm R}} &= \left.\frac{a'^2}{2} \,\right|^a_0 \\[6px]
\left(2 t H_0 \sqrt{\Omega_{0,\mathrm R}}\right)^{1/2} &= a(t)
\end{align}</math>
For {{mvar|Λ}}-dominated universes, where {{math|''Ω''<sub>0,''Λ''</sub> ≫ ''Ω''<sub>0,R</sub>}} and {{math|''Ω''<sub>0,M</sub>}}, as well as {{math|''Ω''<sub>0,''Λ''</sub> ≈ 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 display="block">\begin{align}
\left(t-t_i\right) H_0 &= \int_{a_i}^{a} \frac{\mathrm{d}a'}{\sqrt{(\Omega_{0,\Lambda} a'^2)}} \\[6px]
\left(t - t_i\right) H_0 \sqrt{\Omega_{0,\Lambda}} &= \bigl. \ln|a'| \,\bigr|^a_{a_i} \\[6px]
a_i \exp\left( (t - t_i) H_0 \sqrt{\Omega_{0,\Lambda}}\right) &= a(t)
\end{align}</math>
The {{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>Λ</sub>''}} a candidate for [[dark energy]]:
<math display="block">\begin{align}
a(t) &= a_i \exp\left( (t - t_i) H_0 \textstyle\sqrt{\Omega_{0,\Lambda}}\right) \\[6px]
\frac{\mathrm{d}^2 a(t)}{\mathrm{d}t^2} &= a_i {H_0}^2 \, \Omega_{0,\Lambda} \exp\left( (t - t_i) H_0 \textstyle\sqrt{\Omega_{0,\Lambda}}\right)
\end{align}</math>
Where by construction {{math|''a<sub>i</sub>'' > 0}}, our assumptions were {{math|''Ω''<sub>0,''Λ''</sub> ≈ 1}}, and {{math|''H''<sub>0</sub>}} has been measured to be positive, forcing the acceleration to be greater than zero.
== Rescaled Friedmann equation ==
Set
\rho_c = \frac{3H_0^2}{8\pi G},\quad
\Omega = \frac{\rho}{\rho_\mathrm{c}},\quad
t = \frac{\tilde{t}}{H_0},\quad
\Omega_\mathrm{k} = -\frac{kc^2}{H_0^2 a_0^2},</math>
where {{math|''a''<sub>0</sub>}} and {{math|''H''<sub>0</sub>}} are separately the [[Scale factor (Universe)|scale factor]] and the [[Hubble parameter]] today.
Then we can have
<math display="block">\frac12\left( \frac{d\tilde{a}}{d\tilde{t}}\right)^2 + U_\text{eff}(\tilde{a})=\frac12\Omega_\mathrm{k}</math>
where
<math display="block">U_\text{eff}(\tilde{a})=\frac{-\Omega\tilde{a}^2}{2}.</math>
For any form of the effective potential {{math|''U''<sub>eff</sub>(''ã'')}}, there is an equation of state {{math|''p'' {{=}} ''p''(''ρ'')}} that will produce it.
== In popular culture ==
Several students at [[Tsinghua University]] ([[Chinese Communist Party|CCP]] [[Leader of the Chinese Communist Party|leader]] [[Xi Jinping]]'s [[alma mater]]) participating in the [[2022 COVID-19 protests in China]] carried placards with Friedmann equations scrawled on them, interpreted by some as a play on the words "Free man". Others have interpreted the use of the equations as a call to “open up” China and stop its Zero Covid policy, as the Friedmann equations relate to the expansion, or “opening” of the universe.<ref>{{Cite news|url=https://www.bbc.com/news/world-asia-china-63778871|title=China's protests: Blank paper becomes the symbol of rare demonstrations|work=BBC News |date=November 28, 2022}}</ref>
== See also ==
* [[Mathematics of general relativity]]
* [[Solutions of the Einstein
* [[Warm inflation]]
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{{DEFAULTSORT:Friedmann Equations}}
[[Category:Eponymous equations of physics]]
[[Category:General relativity]]
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