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id 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 equation of state this process allows an approximate analytic solution einstein s field equations g μ ν λ g μ ν κ t μ ν displaystyle g_ mu nu lambda g_ mu nu kappa t_ mu nu giving the friedmann equations when the energy momentum tensor is similarly assumed to be isotropic and homogeneous the resulting equations are 13 a a 2 k c 2 a 2 λ c 2 3 κ c 4 3 ρ 2 a a a a 2 k c 2 a 2 λ c 2 κ c 2 p 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 the observable universe is well approximated by an almost flrw model i e a model that follows the flrw metric apart from primordial 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 from cobe and wmap interpretation edit the pair of equations given above is equivalent to the following pair of equations ρ 3 a a ρ p c 2 a a κ c 4 6 ρ 3 p c 2 λ c 2 3 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 with k displaystyle k the spatial curvature index serving as a constant of integration for the first equation the first equation can be derived also from thermodynamical considerations and is equivalent to the first law of thermodynamics assuming the expansion of the universe is an adiabatic 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 universe a displaystyle dot a to decrease i e both cause a deceleration in the expansion of the universe this is a consequence of gravitation with pressure playing a similar role to that of energy or mass density according to the principles of general relativity the cosmological constant on the other hand causes an acceleration in the expansion of the universe cosmological constant edit the cosmological constant term can be omitted if we make the following replacements ρ ρ λ κ c 2 p p λ κ displaystyle begin aligned rho to rho frac lambda kappa c 2 p to p frac lambda kappa end aligned therefore the cosmological 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 ρ c 2 displaystyle p rho c 2 which is an equation of state of vacuum with dark energy an attempt to generalize this to p w ρ c 2 displaystyle p w rho c 2 would not have general 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 a scalar field that satisfies p ρ c 2 3 displaystyle p frac rho c 2 3 such a field is sometimes called quintessence dust models edit setting the pressure of the perfect fluid in the friedmann equations to zero p 0 displaystyle p 0 gives a cosmological dust model 14 231 newtonian analog edit 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 a 3 ρ 3 a 2 a ρ 3 a 2 p a c 2 a 2 2 κ c 4 a 3 ρ 6 a k c 2 2 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 momentarily a 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 ...
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