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Friedmann equations

From Wikipedia, the free encyclopedia
Equations in physical cosmology
Part of a series on
Physical cosmology
Full-sky image derived from nine years' WMAP data

TheFriedmann equations, also known as theFriedmann–Lemaître (FL)equations, are a set ofequations inphysical cosmology that governcosmic expansion inhomogeneous andisotropic models of the universe within the context ofgeneral relativity. They were first derived byAlexander Friedmann in 1922 fromEinstein's field equations ofgravitation for theFriedmann–Lemaître–Robertson–Walker metric and aperfect fluid with a givenmass densityρ andpressurep.[1] The equations fornegative spatial curvature were given by Friedmann in 1924.[2]The physical models built on the Friedmann equations are called FRW or FLRW models and form theStandard Model of moderncosmology, although such a description is also associated with the further developedLambda-CDM model. The FLRW model was developed independently by the named authors in the 1920s and 1930s.

Assumptions

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Main article:Friedmann–Lemaître–Robertson–Walker metric

The Friedmann equations use three assumptions:[3]: 22.1.3 

  1. theFriedmann–Lemaître–Robertson–Walker metric,
  2. Einstein's equations forgeneral relativity, and
  3. aperfect fluid source.

The metric in turn starts with the simplifying assumption that the universe is spatially homogeneous andisotropic, that is, thecosmological principle; empirically, this is justified on scales larger than the order of 100Mpc.

The metric can be written as:[4]: 65 c2dτ2=c2dt2−R2(t)(dr2+Sk2(r)dψ2){\displaystyle c^{2}d\tau ^{2}=c^{2}dt^{2}-R^{2}(t)\left(dr^{2}+S_{k}^{2}(r)d\psi ^{2}\right)}whereS−1(r)=sinh⁡(r),S0=1,S1=sin⁡(r).{\displaystyle S_{-1}(r)=\sinh(r),S_{0}=1,S_{1}=\sin(r).}These three possibilities correspond to parameterk of(0) flat space,(+1) a sphere of constant positive curvature or(−1) a hyperbolic space with constant negative curvature.Here the radial position has been decomposed into a time-dependent scale factor,R(t){\displaystyle R(t)}, and a comoving coordinate,r{\displaystyle r}. Inserting this metric into Einstein's field equations relate the evolution of this scale factor to the pressure and energy of the matter in the universe. With thestress–energy tensor for a perfect fluid, results in the equations are described below.[4]: 73 

Equations

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General relativity
Spacetime curvature schematic

There are two independent Friedmann equations for modelling a homogeneous, isotropic universe.The first is:[3]H2≡(R˙R)2=8πGρ3−kR2+Λ3,{\displaystyle H^{2}\equiv {\left({\frac {\dot {R}}{R}}\right)}^{2}={\frac {8\pi G\rho }{3}}-{\frac {k}{R^{2}}}+{\frac {\Lambda }{3}},} and second is:R¨R=Λ3−4πG3(ρ+3p).{\displaystyle {\frac {\ddot {R}}{R}}={\frac {\Lambda }{3}}-{\frac {4\pi G}{3}}\left(\rho +3p\right).}The termFriedmann equation sometimes is used only for the first equation.[3]In these equations,H is the Hubble parameter,R(t) is thecosmological scale factor,G{\displaystyle G} is theNewtonian constant of gravitation,Λ is thecosmological constant with dimension length−2,ρ is the energy density andp is the isotropic pressure.k is constant throughout a particular solution, but may vary from one solution to another. The units set thespeed of light in vacuum to one.

In previous equations,R,ρ, andp are functions of time. If the cosmological constant,Λ, is ignored, the term−k/R2{\displaystyle -k/R^{2}} in the first Friedmann equation can be interpreted as a Newtonian total energy, so the evolution of the universe pits gravitational potential energy,8πGρ/3{\displaystyle 8\pi G\rho /3} against kinetic energy,R˙/R{\displaystyle {\dot {R}}/R}. The winner depends upon thek value in the total energy: if k is +1, gravity eventually causes the universe to contract. These conclusions will be altered if theΛ is not zero.[3]

Using the first equation, the second equation can be re-expressed as:[3]ρ˙=−3H(ρ+pc2),{\displaystyle {\dot {\rho }}=-3H\left(\rho +{\frac {p}{c^{2}}}\right),}which eliminatesΛ. Alternatively the conservation ofmass–energy:Tαβ;β=0{\displaystyle T^{\alpha \beta }{}_{;\beta }=0}leads to the same result.[3]

Spatial curvature

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The first Friedmann equation contains the discrete parameterk, the value of which determines theshape of the universe:

  • +1 is a3-sphere,[5] the universe is "closed": starting off on some paths through the universe return to the starting point - analogous to a sphere: finite but unbounded.[6]
  • 0 isflat Euclidean space[5] and infinite.[6]
  • −1 is a 3-hyperboloid[5] the universe is "open": infinite and no paths return.[6]

In the Friedmann model the choice between these different shapes is determined by a comparison between the expansion rate and the density. The expansion rate sets a critical densityρc=3H28πG,{\displaystyle \rho _{c}={\frac {3H^{2}}{8\pi G}},}whereH{\displaystyle H} is the Hubble parameter andG{\displaystyle G} is the gravitational constant. A universe at the critical density is spatially flat (k=0{\displaystyle k=0}), while higher density gives a closed universe and lower density gives an open one.[4]: 73 

Dimensionless scale factor

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A dimensionless scale factor can be defined:a(t)≡R(t)R0{\displaystyle a(t)\equiv {\frac {R(t)}{R_{0}}}}using the present day valueR0=R(now).{\displaystyle R_{0}=R({\text{now}}).}The Friedmann equations can be written in terms of this dimensionless scale factor:H2(t)=(a˙a)2=8πG3[ρ(t)+ρc−ρ0a2(t)]{\displaystyle H^{2}(t)=\left({\frac {\dot {a}}{a}}\right)^{2}={\frac {8\pi G}{3}}\left[\rho (t)+{\frac {\rho _{c}-\rho _{0}}{a^{2}(t)}}\right]}wherea˙=da/dt{\displaystyle {\dot {a}}=da/dt},ρc=3H02/8πG{\displaystyle \rho _{c}=3H_{0}^{2}/8\pi G}, andρ0=ρ(t=now){\displaystyle \rho _{0}=\rho (t={\text{now}})}.[7]: 3 

Critical density

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That value of the mass-energy density,ρ{\displaystyle \rho } that givesk=0{\displaystyle k=0} whenΛ=0{\displaystyle \Lambda =0} is called thecritical density:ρc≡3H28πG.{\displaystyle \rho _{c}\equiv {\frac {3H^{2}}{8\pi G}}.}If the universe has higher density,ρ≥ρc{\displaystyle \rho \geq \rho _{c}}, then it is called "spatially closed": in this simple approximation the universe would eventually contract. On the other hand, if has lower density,ρ≤ρc{\displaystyle \rho \leq \rho _{c}}, then it is called "spatially open" and expands forever. Therefore the geometry of the universe is directly connected to its density.[4]: 73 

Density parameter

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Thedensity parameterΩ is defined as the ratio of the actual (or observed) densityρ to the critical densityρc of the Friedmann universe:[4]: 74 Ω:=ρρc=8πGρ3H2.{\displaystyle \Omega :={\frac {\rho }{\rho _{c}}}={\frac {8\pi G\rho }{3H^{2}}}.}Both the densityρ(t){\displaystyle \rho (t)} and the Hubble parameterH(t){\displaystyle H(t)} depend upon time and thus the density parameter varies with time.[4]: 74 

The critical density is equivalent to approximately five atoms (ofmonatomichydrogen) per cubic metre, whereas the average density ofordinary matter in the Universe is believed to be 0.2–0.25 atoms per cubic metre.[8][9]

Estimated relative distribution for components of the energy density of the universe.Dark energy dominates the total energy (74%) whiledark 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 thePlanck 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 unidentifieddark matter, although both ordinary and dark matter contribute in favour of contraction of the universe. However, the largest part comes from so-calleddark energy, which accounts for the cosmological constant term. Although the total density is equal to the critical density (exactly, up to measurement error), dark energy does not lead to contraction of the universe but rather may accelerate its expansion.

An expression for the critical density is found by assumingΛ to be zero (as it is for all basic Friedmann universes) and setting the normalised spatial curvature,k, equal to zero. When the substitutions are applied to the first of the Friedmann equations given the newH0{\displaystyle H_{0}} value we find:[10]ρ=3H028πG≈1.10×10−26kgm−3≈1.88×10−26h2kgm−3≈2.78×1011h2M⊙Mpc−3{\displaystyle {\begin{aligned}\rho ={\frac {3H_{0}^{2}}{8\pi G}}&\approx 1.10\times 10^{-26}\mathrm {kg\,m^{-3}} \\&\approx 1.88\times 10^{-26}h^{2}\,{\rm {kg}}\,{\rm {m}}^{-3}\\&\approx 2.78\times 10^{11}h^{2}M_{\odot }\,{\rm {Mpc}}^{-3}\end{aligned}}}where:

Given the value of dark energy to beΩΛ=0.647{\displaystyle \Omega _{\Lambda }=0.647}This term originally was used as a means to determine thespatial geometry of the universe, whereρc is the critical density for which the spatial geometry is flat (or Euclidean). Assuming a zero vacuum energy density, ifΩ is larger than unity, the space sections of the universe are closed; the universe will eventually stop expanding, then collapse. IfΩ 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 forΩ 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 theΛCDM model, there are important components ofΩ due tobaryons,cold dark matter anddark energy. The spatial geometry of theuniverse has been measured by theWMAP spacecraft to be nearly flat. This means that the universe can be well approximated by a model where the spatial curvature parameterk 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.

The first Friedmann equation is often seen in terms of the present values of the density parameters, that is[11]H2H02=Ω0,Ra−4+Ω0,Ma−3+Ω0,ka−2+Ω0,Λ.{\displaystyle {\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 }.}HereΩ0,R is the radiation density today (whena = 1),Ω0,M is the matter (dark plusbaryonic) density today,Ω0,k = 1 − Ω0 is the "spatial curvature density" today, andΩ0,Λ is the cosmological constant or vacuum density today.

Other forms

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The Hubble parameter can change over time if other parts of the equation are time dependent (in particular the mass density, the vacuum energy, or the spatial curvature). Evaluating the Hubble parameter at the present time yields Hubble's constant which is the proportionality constant ofHubble's law. Applied to a fluid with a givenequation of state, the Friedmann equations yield the time evolution and geometry of the universe as a function of the fluid density.

FLRW models

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Relativisitic cosmology models based on the FLRW metric and obeying the Friedmann equations are calledFRW models.[4]: 73 Direct observation of stars has shown their velocities to be dominated by radial recession, validating these assumptions for cosmological models.[4]: 65 These models are the basis of the standard model[12] ofBig Bang cosmological including the currentΛCDM model.[3]: 25.1.3 

To apply the metric to cosmology and predict its time evolution via the scale factora(t){\displaystyle a(t)} requires Einstein's field equations together with a way of calculating the density,ρ(t),{\displaystyle \rho (t),} such as acosmological equation of state.This process allows an approximate analytic solutionEinstein's field equationsGμν+Λgμν=κTμν{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }=\kappa T_{\mu \nu }} giving the Friedmann equations when theenergy–momentum tensor is similarly assumed to be isotropic and homogeneous. The resulting equations are:[13](a˙a)2+kc2a2−Λc23=κc43ρ2a¨a+(a˙a)2+kc2a2−Λc2=−κc2p.{\displaystyle {\begin{aligned}{\left({\frac {\dot {a}}{a}}\right)}^{2}+{\frac {kc^{2}}{a^{2}}}-{\frac {\Lambda c^{2}}{3}}&={\frac {\kappa c^{4}}{3}}\rho \\[4pt]2{\frac {\ddot {a}}{a}}+{\left({\frac {\dot {a}}{a}}\right)}^{2}+{\frac {kc^{2}}{a^{2}}}-\Lambda c^{2}&=-\kappa c^{2}p.\end{aligned}}}

Because the FLRW model assumes homogeneity, some popular accounts mistakenly assert that the Big Bang model cannot account for the observed lumpiness of the universe. In a strictly FLRW model, there are no clusters of galaxies or stars, since these are objects much denser than a typical part of the universe. Nonetheless, the FLRW model is used as a first approximation for the evolution of the real, lumpy universe because it is simple to calculate, and models that calculate the lumpiness in the universe are added onto the FLRW models as extensions. Most cosmologists agree that theobservable universe is well approximated by analmost FLRW model, i.e., a model that follows the FLRW metric apart fromprimordial density fluctuations. As of 2003[update], the theoretical implications of the various extensions to the FLRW model appear to be well understood, and the goal is to make these consistent with observations fromCOBE andWMAP.

Interpretation

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The pair of equations given above is equivalent to the following pair of equationsρ˙=−3a˙a(ρ+pc2)a¨a=−κc46(ρ+3pc2)+Λc23{\displaystyle {\begin{aligned}{\dot {\rho }}&=-3{\frac {\dot {a}}{a}}\left(\rho +{\frac {p}{c^{2}}}\right)\\[1ex]{\frac {\ddot {a}}{a}}&=-{\frac {\kappa c^{4}}{6}}\left(\rho +{\frac {3p}{c^{2}}}\right)+{\frac {\Lambda c^{2}}{3}}\end{aligned}}}withk{\displaystyle k}, the spatial curvature index, serving as aconstant of integration for the first equation.

The first equation can be derived also from thermodynamical considerations and is equivalent to thefirst law of thermodynamics, assuming the expansion of the universe is anadiabatic process (which is implicitly assumed in the derivation of the Friedmann–Lemaître–Robertson–Walker metric).

The second equation states that both the energy density and the pressure cause the expansion rate of the universea˙{\displaystyle {\dot {a}}} to decrease, i.e., both cause a deceleration in the expansion of the universe. This is a consequence ofgravitation, with pressure playing a similar role to that of energy (or mass) density, according to the principles ofgeneral relativity. Thecosmological constant, on the other hand,causes an acceleration in the expansion of the universe.

Cosmological constant

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Thecosmological constant term can be omitted if we make the following replacementsρ→ρ−Λκc2,p→p+Λκ.{\displaystyle {\begin{aligned}\rho &\to \rho -{\frac {\Lambda }{\kappa c^{2}}},&p&\to p+{\frac {\Lambda }{\kappa }}.\end{aligned}}}

Therefore, thecosmological constant can be interpreted as arising from a form of energy that has negative pressure, equal in magnitude to its (positive) mass-energy density:p=−ρc2,{\displaystyle p=-\rho c^{2}\,,}which is an equation of state of vacuum withdark energy.

An attempt to generalize this top=wρc2{\displaystyle p=w\rho c^{2}}would not havegeneral invariance without further modification.

In fact, in order to get a term that causes an acceleration of the universe expansion, it is enough to have ascalar field that satisfiesp<−ρc23.{\displaystyle p<-{\frac {\rho c^{2}}{3}}.}Such a field is sometimes calledquintessence.

Dust models

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Setting the pressure of the perfect fluid in the Friedmann equations to zero (p=0{\displaystyle p=0}) gives a cosmologicaldust model.[14]: 231 

Newtonian analog

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In 1934 McCrea and Milne[15] showed that the Friedmann equations in the case of a pressureless fluid can be derived with non-relativistic Newtonian dynamics.[14]: 231 −a3ρ˙=3a2a˙ρ+3a2pa˙c2a˙22−κc4a3ρ6a=−kc22.{\displaystyle {\begin{aligned}-a^{3}{\dot {\rho }}=3a^{2}{\dot {a}}\rho +{\frac {3a^{2}p{\dot {a}}}{c^{2}}}\,\\[1ex]{\frac {{\dot {a}}^{2}}{2}}-{\frac {\kappa c^{4}a^{3}\rho }{6a}}=-{\frac {kc^{2}}{2}}\,.\end{aligned}}}

The first equation says that the decrease in the mass contained in a fixed cube (whose side is momentarilya) is the amount that leaves through the sides due to the expansion of the universe plus the mass equivalent of the work done by pressure against the material being expelled. This is the conservation of mass–energy (first law of thermodynamics) contained within a part of the universe.

The second equation says that the kinetic energy (seen from the origin) of a particle of unit mass moving with the expansion plus its (negative)gravitational potential energy (relative to the mass contained in the sphere of matter closer to the origin) is equal to a constant related to the curvature of the universe. In other words, the energy (relative to the origin) of a co-moving particle in free-fall is conserved. General relativity merely adds a connection between the spatial curvature of the universe and the energy of such a particle: positive total energy implies negative curvature and negative total energy implies positive curvature.

Useful solutions

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The Friedmann equations can be solved exactly in presence of aperfect fluid with equation of statep=wρc2,{\displaystyle p=w\rho c^{2},}wherep is thepressure,ρ is the mass density of the fluid in the comoving frame andw is some constant.

In spatially flat case (k = 0), the solution for the scale factor isa(t)=a0t23(w+1){\displaystyle a(t)=a_{0}\,t^{\frac {2}{3(w+1)}}}wherea0 is some integration constant to be fixed by the choice of initial conditions. This family of solutions labelled byw is extremely important for cosmology. For example,w = 0 describes amatter-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 asa(t)∝t2/3matter-dominated{\displaystyle a(t)\propto t^{2/3}\qquad {\text{matter-dominated}}}

Another important example is the case of aradiation-dominated universe, namely whenw =⁠1/3⁠. This leads toa(t)∝t1/2radiation-dominated{\displaystyle a(t)\propto t^{1/2}\qquad {\text{radiation-dominated}}}

Note that this solution is not valid for domination of the cosmological constant, which corresponds to anw = −1. In this case the energy density is constant and the scale factor grows exponentially.

Solutions for other values ofk can be found atTersic, Balsa."Lecture Notes on Astrophysics". Retrieved24 February 2022.

Mixtures

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If the matter is a mixture of two or more non-interacting fluids each with such an equation of state, thenρ˙f=−3H(ρf+pfc2){\displaystyle {\dot {\rho }}_{f}=-3H\left(\rho _{f}+{\frac {p_{f}}{c^{2}}}\right)}holds separately for each such fluidf. In each case,ρ˙f=−3H(ρf+wfρf){\displaystyle {\dot {\rho }}_{f}=-3H\left(\rho _{f}+w_{f}\rho _{f}\right)\,}from which we getρf∝a−3(1+wf).{\displaystyle {\rho }_{f}\propto a^{-3\left(1+w_{f}\right)}\,.}

For example, one can form a linear combination of such termsρ=Aa−3+Ba−4+Ca0{\displaystyle \rho =Aa^{-3}+Ba^{-4}+Ca^{0}\,}whereA is the density of "dust" (ordinary matter,w = 0) whena = 1;B is the density of radiation (w =⁠1/3⁠) whena = 1; andC is the density of "dark energy" (w = −1). One then substitutes this into(a˙a)2=8πG3ρ−kc2a2{\displaystyle \left({\frac {\dot {a}}{a}}\right)^{2}={\frac {8\pi G}{3}}\rho -{\frac {kc^{2}}{a^{2}}}}and solves fora as a function of time.

History

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Alexander Friedmann

Friedmann published two cosmology papers in the 1922-1923 time frame. He adopted the same homogeneity and isotropy assumptions used byAlbert Einstein and byWillem de Sitter in their papers, both published in 1917. Both of the earlier works also assumed the universe was static, eternally unchanging. Einstein postulated an additional term to his equations of general relativity to ensure this stability. In his paper, de Sitter showed that spacetime had curvature even in the absence of matter: the new equations of general relativity implied that a vacuum had properties that altered spacetime.[16]: 152 

The universe being static was a fundamental assumption of philosophy and science. However, Friedmann abandoned the idea in his first paper "On the Curvature of Space". Starting with Einstein's 10 equations of relativity, Friedmann applies the symmetry of an isotropic universe and a simple model for mass-energy density to derive a relationship between that density and the curvature of spacetime. He demonstrates that in addition to a single static solution, many time dependent solutions also exist.[16]: 157 

Friedmann's second paper, "On the Possibility of a World With Constant Negative Curvature," published in 1924 explored more complex geometrical ideas. This paper established the idea that the finiteness of spacetime was not a property that could be established based on the equations of general relativity alone: both finite and infinite geometries could be used to give solutions. Friedmann used two concepts of a three dimensional sphere as analogy: a trip at constant latitude could return to the starting point or the sphere might have an infinite number of sheets and the trip never repeats.[16]: 167 

Friedmann's papers were largely ignored except – initially – by Einstein who actively dismissed them. However onceEdwin Hubble published astronomicalevidence that the universe was expanding, Einstein became convinced. Unfortunately for Friedmann,Georges Lemaître discovered some aspects of the same solutions and wrote persuasively about the concept of a universe born from a "primordial atom". Thus historians give these two scientists equal billing for the discovery.[17]

In popular culture

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Several students atTsinghua University (CCPleaderXi Jinping'salma mater) participating in the2022 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.[18]

See also

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Sources

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  1. ^Friedman, A (29 May 1922)."Über die Krümmung des Raumes"(PDF).Z. Phys. (in German).10 (1).Petrograd:Springer-Verlag GmbH & Co. KG:377–386.Bibcode:1922ZPhy...10..377F.doi:10.1007/BF01332580.S2CID 125190902 – via Michael Kachelrieß:Norges Teknisk-Naturvitenskapelige Universitet. (English translation:Friedman, A (1999). "On the Curvature of Space".General Relativity and Gravitation.31 (12). Translated by F.R. Ellis; H. van Elst.Springer-Verlag GmbH & Co. KG:1991–2000.Bibcode:1999GReGr..31.1991F.doi:10.1023/A:1026751225741.S2CID 122950995.). The original Russian manuscript of this paper is preserved in theEhrenfest archive.
  2. ^Friedmann, A (1924). "Über die Möglichkeit einer Welt mit konstanter negativer Krümmung des Raumes".Z. Phys. (in German).21 (1):326–332.Bibcode:1924ZPhy...21..326F.doi:10.1007/BF01328280.S2CID 120551579. (English translation:Friedmann, A (1999). "On the Possibility of a World with Constant Negative Curvature of Space".General Relativity and Gravitation.31 (12):2001–2008.Bibcode:1999GReGr..31.2001F.doi:10.1023/A:1026755309811.S2CID 123512351.)
  3. ^abcdefgNavas, S.; et al. (Particle Data Group) (2024). "Review of Particle Physics".Physical Review D.110 (3):1–708.doi:10.1103/PhysRevD.110.030001.hdl:20.500.11850/695340. 22.1.3 The Friedmann equations of motion
  4. ^abcdefghPeacock, J. A. (1998-12-28).Cosmological Physics (1 ed.). Cambridge University Press.doi:10.1017/cbo9780511804533.ISBN 978-0-521-41072-4.
  5. ^abcD'Inverno, Ray (2008).Introducing Einstein's relativity (Repr ed.). Oxford: Clarendon Press.ISBN 978-0-19-859686-8.
  6. ^abcJohn. A. Peacock (1999)."3. THE ISOTROPIC UNIVERSE 3.1 The Robertson-Walker Metric".Cosmological Physics.Cambridge University Press:NASA/IPAC Extragalactic Database -California Institute of Technology.Archived 8 September 2025 at theWayback Machine
  7. ^Dodelson, Scott (2003).Modern cosmology. San Diego, Calif: Academic Press.ISBN 978-0-12-219141-1.
  8. ^Rees, Martin (2001).Just six numbers: the deep forces that shape the universe. Astronomy/science (Repr. ed.). New York, NY: Basic Books.ISBN 978-0-465-03673-8.
  9. ^"Universe 101".NASA. RetrievedSeptember 9, 2015.The actual density of atoms is equivalent to roughly 1 proton per 4 cubic meters.
  10. ^Scolnic, Daniel; Riess, Adam G.; Murakami, Yukei S.; Peterson, Erik R.; Brout, Dillon; Acevedo, Maria; Carreres, Bastien; Jones, David O.; Said, Khaled; Howlett, Cullan; Anand, Gagandeep S. (2025-01-15)."The Hubble Tension in Our Own Backyard: DESI and the Nearness of the Coma Cluster".The Astrophysical Journal Letters.979 (1): L9.arXiv:2409.14546.Bibcode:2025ApJ...979L...9S.doi:10.3847/2041-8213/ada0bd.ISSN 2041-8205.
  11. ^Nemiroff, Robert J.; Patla, Bijunath (2008). "Adventures in Friedmann cosmology: A detailed expansion of the cosmological Friedmann equations".American Journal of Physics.76 (3):265–276.arXiv:astro-ph/0703739.Bibcode:2008AmJPh..76..265N.doi:10.1119/1.2830536.S2CID 51782808.
  12. ^Bergström, Lars; Goobar, Ariel (2008).Cosmology and particle astrophysics. Springer Praxis books in astronomy and planetary science (2. ed., reprinted ed.). Chichester, UK: Praxis Publ. p. 61.ISBN 978-3-540-32924-4.
  13. ^Rosu, H. C.; Ojeda-May, P. (June 2006). "Supersymmetry of FRW Barotropic Cosmologies".International Journal of Theoretical Physics.45 (6):1152–1157.arXiv:gr-qc/0510004.Bibcode:2006IJTP...45.1152R.doi:10.1007/s10773-006-9123-2.ISSN 0020-7748.S2CID 119496918.
  14. ^abLongair, Malcolm S. (2023).Galaxy Formation. Astronomy and Astrophysics Library. Berlin, Heidelberg: Springer Berlin Heidelberg.doi:10.1007/978-3-662-65891-8.ISBN 978-3-662-65890-1.
  15. ^McCrea, W. H.; Milne, E. A. (1934). "Newtonian universes and the curvature of space".Quarterly Journal of Mathematics.5:73–80.Bibcode:1934QJMat...5...73M.doi:10.1093/qmath/os-5.1.73.
  16. ^abcTropp, Eduard A.; Frenkel, Viktor Ya.; Chernin, Artur D. (1993-06-03).Alexander A Friedmann: The Man who Made the Universe Expand. Translated by Dron, Alexander; Burov, Michael (1 ed.). Cambridge University Press.doi:10.1017/cbo9780511608131.ISBN 978-0-521-38470-4.
  17. ^Belenkiy, Ari (2012-10-01)."Alexander Friedmann and the origins of modern cosmology".Physics Today.65 (10):38–43.Bibcode:2012PhT....65j..38B.doi:10.1063/PT.3.1750.ISSN 0031-9228.
  18. ^Murphy, Matt (November 28, 2022)."China's protests: Blank paper becomes the symbol of rare demonstrations".BBC News.

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