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itical density 5 density parameter toggle density parameter subsection 5 1 other forms 6 flrw models toggle flrw models subsection 6 1 interpretation 6 2 cosmological constant 6 3 dust models 6 4 newtonian analog 7 useful solutions toggle useful solutions subsection 7 1 mixtures 8 history 9 in popular culture 10 see also 11 sources 12 further reading toggle the table of contents friedmann equations 32 languages العربية azərbaycanca বাংলা català čeština deutsch español eesti euskara فارسی suomi français עברית հայերեն bahasa indonesia italiano 日本語 ქართული 한국어 nederlands polski português русский simple english slovenščina svenska ไทย türkçe українська اردو tiếng việt 中文 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia equations in physical cosmology part of a series on physical cosmology big bang universe age of the universe chronology of the universe early universe inflation baryogenesis nucleosynthesis backgrounds gravitational wave gwb microwave cmb neutrino cnb expansion and future hubble s law redshift expansion of the universe flrw metric friedmann equations lambda cdm model future of an expanding universe ultimate fate of the universe components and structure components dark energy dark matter photons baryons structure shape of the universe galaxy filament galaxy formation large quasar group large scale structure reionization structure formation experiments black hole initiative bhi boomerang cosmic background explorer cobe dark energy survey planck space observatory sloan digital sky survey sdss 2df galaxy redshift survey 2df wilkinson microwave anisotropy probe wmap scientists aaronson alfvén alpher copernicus de sitter dicke ehlers einstein ellis friedmann galileo gamow guth hawking hubble huygens kepler lemaître mather newton penrose penzias rubin schmidt smoot suntzeff sunyaev tolman wilson zeldovich list of cosmologists subject history discovery of cosmic microwave background radiation history of the big bang theory timeline of cosmological theories category astronomy portal v t e the friedmann equations also known as the friedmann lemaître fl equations are a set of equations in physical cosmology that govern cosmic expansion in homogeneous and isotropic models of the universe within the context of general relativity they were first derived by alexander friedmann in 1922 from einstein s field equations of gravitation for the friedmann lemaître robertson walker metric and a perfect fluid with a given mass density ρ and pressure p 1 the equations for negative 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 the standard model of modern cosmology although such a description is also associated with the further developed lambda cdm model the flrw model was developed independently by the named authors in the 1920s and 1930s assumptions edit main article friedmann lemaître robertson walker metric the friedmann equations use three assumptions 3 22 1 3 the friedmann lemaître robertson walker metric einstein s equations for general relativity and a perfect fluid source the metric in turn starts with the simplifying assumption that the universe is spatially homogeneous and isotropic that is the cosmological principle empirically this is justified on scales larger than the order of 100 mpc the metric can be written as 4 65 c 2 d τ 2 c 2 d t 2 r 2 t d r 2 s k 2 r d ψ 2 displaystyle c 2 d tau 2 c 2 dt 2 r 2 t left dr 2 s_ k 2 r d psi 2 right where s 1 r sinh r s 0 1 s 1 sin r displaystyle s_ 1 r sinh r s_ 0 1 s_ 1 sin r these three possibilities correspond to parameter k 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 relates the evolution of this scale factor to the stress energy tensor using the tensor for a perfect fluid results in the equations described below 4 73 equations edit general relativity g μ ν λ g μ ν κ t μ ν displaystyle g_ mu nu lambda g_ mu nu kappa t_ mu nu introduction history timeline tests mathematical formulation fundamental concepts equivalence principle special relativity world line pseudo riemannian manifold phenomena kepler problem gravitational lensing gravitational redshift gravitational time dilation gravitational waves frame dragging geodetic effect event horizon singularity black hole spacetime spacetime diagrams minkowski spacetime metric tensor equations formalisms equations linearized gravity einstein field equations friedmann geodesics mathisson papapetrou dixon hamilton jacobi einstein raychaudhuri teukolsky formalisms adm np bssn post newtonian advanced theory kaluza klein theory quantum gravity quantum field theory in curved spacetime solutions schwarzschild interior reissner nordström einstein rosen waves wormhole gödel kerr kerr newman kerr newman de sitter kasner kantowski sachs lemaître tolman wahlquist taub nut milne robertson walker oppenheimer snyder pp wave van stockum dust hartle thorne vaidya peres de sitter schwarzschild mcvittie weyl scientists einstein lorentz hilbert poincaré schwarzschild de sitter reissner nordström weyl eddington friedmann milne zwicky lemaître oppenheimer gödel wheeler robertson bardeen walker kerr chandrasekhar ehlers penrose hawking raychaudhuri taylor hulse van stockum taub newman yau thorne others physics portal category v t e there are two independent friedmann equations for modelling a homogeneous isotropic universe the first is 3 h 2 r r 2 8 π g ρ 3 k r 2 λ 3 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 λ 3 4 π g 3 ρ 3 p displaystyle frac ddot r r frac lambda 3 frac 4 pi g 3 left rho 3p right the term friedmann equation sometimes is used only for the first equation 3 in these equations h is the hubble parameter r t is the cosmological scale factor g displaystyle g is the newtonian constant of gravitation λ is the cosmological constant with dimension length 2 ρ is the energy density and p is the isotropic pressure k is constant throughout a particular solution but may vary from one solution to another the units set the speed of light in vacuum to one in previous equations r ρ and p are functions of time if the cosmological constant λ is ignored the term k r 2 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 8 π g ρ 3 displaystyle 8 pi g rho 3 against kinetic energy r r displaystyle dot r r the winner depends upon the k value in the total energy if k is 1 gravity eventually causes the universe to contract these conclusions will be altered if the λ is not zero 3 using the first equation the second equation can be re expressed as 3 ρ 3 h ρ p c 2 displaystyle dot rho 3h left rho frac p c 2 right which eliminates λ alternatively the conservation of mass energy t α β β 0 displaystyle t alpha beta _ beta 0 leads to the same result 3 spatial curvature edit the first friedmann equation contains the discrete parameter k the value of which determines the shape of the universe 1 is a 3 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 is flat 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 density ρ c 3 h 2 8 π g displaystyle rho _ c frac 3h 2 8 pi g where h displaystyle h is the hubble parameter and g 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 edit a dimensionless scale factor can be defined a t r t r 0 displaystyle a t equiv frac r t r_ 0 using the present day value r 0 r now displaystyle r_ 0 r text now the friedmann equations can be written in terms of this dimensionless scale factor h 2 t a a 2 8 π g 3 ρ t ρ c ρ 0 a 2 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 where a d a d t displaystyle dot a da dt ρ c 3 h 0 2 8 π g displaystyle rho _ c 3h_ 0 2 8 pi g and ρ 0 ρ t now displaystyle rho _ 0 rho t text now 7 3 critical density edit that value of the mass energy density ρ displaystyle rho that gives k 0 displaystyle k 0 when λ 0 displaystyle lambda 0 is called the critical density ρ c 3 h 2 8 π g displaystyle rho _ c equiv frac 3h 2 8 pi g if the universe has higher density ρ ρ c 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 ρ c 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 edit the density parameter ω is defined as the ratio of the actual or observed density ρ to the critical density ρ c of the friedmann universe 4 74 ω ρ ρ c 8 π g ρ 3 h 2 displaystyle omega frac rho rho _ c frac 8 pi g rho 3h 2 both the density ρ t displaystyle rho t and the hubble parameter h 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 of monatomic hydrogen per cubic metre whereas the average density of ordinary 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 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 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 although 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 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 assuming λ 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 new h 0 displaystyle h_ 0 value we find 10 ρ 3 h 0 2 8 π g 1 10 10 26 k g m 3 1 88 10 26 h 2 k g m 3 2 78 10 11 h 2 m m p c 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 h 0 76 5 2 2 k m s 1 m p c 1 2 48 10 18 s 1 textstyle h_ 0 76 5 pm 2 2 mathrm km s 1 mpc 1 approx 2 48 times 10 18 mathrm s 1 h h 0 100 k m s m p c textstyle h frac h_ 0 100 mathrm km s mpc ρ c 8 5 10 27 k g m 3 displaystyle rho _ c 8 5 times 10 27 mathrm kg m 3 given the value of dark energy to be ω λ 0 647 displaystyle omega _ lambda 0 647 this term originally was used as a means to determine the spatial geometry of the universe where ρ c is the critical density for which the spatial geometry is flat or euclidean assuming a zero vacuum energy density if ω is larger than unity the space sections of the universe are closed the universe will eventually stop expanding then collapse if 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 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 λcdm model there are important components of ω due to baryons cold dark matter and dark energy the spatial geometry of the universe has been measured by the wmap spacecraft to be nearly flat this means that the universe can be well approximated by a model where the spatial curvature parameter k 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 h 2 h 0 2 ω 0 r a 4 ω 0 m a 3 ω 0 k a 2 ω 0 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 here ω 0 r is the radiation density today when a 1 ω 0 m is the matter dark plus baryonic density today ω 0 k 1 ω 0 is the spatial curvature density today and ω 0 λ is the cosmological constant or vacuum density today other forms edit this section does not cite any sources please help improve this section by adding citations to reliable sources unsourced material may be challenged and removed september 2024 learn how and when to remove this message 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 of hubble s law applied to a fluid with a given equation of state the friedmann equations yield the time evolution and geometry of the universe as a function of the fluid density flrw models edit relativisitic cosmology models based on the flrw metric and obeying the friedmann equations are called frw 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 of big bang cosmological including the current λcdm model 3 25 1 3 to apply the metric to cosmology and predict its time evolution via the scale factor a t displaystyle a t requires einstein s field equations together with a way of calculating the density ρ t displaystyle rho t such as a cosmological eq...
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