Friedmann equations
The Friedmann equations, also known as the Friedmann–Lemaître '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 and pressure. The equations for negative spatial curvature were given by Friedmann in 1924.
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
The Friedmann equations use three assumptions:- the Friedmann–Lemaître–Robertson–Walker metric,
- Einstein's equations for general relativity, and
- a perfect fluid source.
The metric can be written as:
where
These three possibilities correspond to parameter of ' flat space, ' a sphere of constant positive curvature or a hyperbolic space with constant negative curvature.
Here the radial position has been decomposed into a time-dependent scale factor,, and a comoving coordinate,.
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 the stress–energy tensor for a perfect fluid, results in the equations are described below.
Equations
There are two independent Friedmann equations for modelling a homogeneous, isotropic universe.The first is:
and second is:
The term Friedmann equation sometimes is used only for the first equation.
In these equations,
is the Hubble parameter, is the cosmological scale factor, is the Newtonian constant of gravitation, is the cosmological constant with dimension length−2, is the energy density and is the isotropic pressure. 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,,, and are functions of time. If the cosmological constant,, is ignored, the term in the first Friedmann equation can be interpreted as a Newtonian total energy, so the evolution of the universe pits gravitational potential energy, against kinetic energy,. The winner depends upon the value in the total energy: if k is +1, gravity eventually causes the universe to contract. These conclusions will be altered if the is not zero.
Using the first equation, the second equation can be re-expressed as:
which eliminates. Alternatively the conservation of mass–energy:
leads to the same result.
Spatial curvature
The first Friedmann equation contains the discrete parameter, the value of which determines the shape of the universe:- is a 3-sphere, the universe is "closed": starting off on some paths through the universe return to the starting point - analogous to a sphere: finite but unbounded.
- is flat Euclidean space and infinite.
- is a 3-hyperboloid the universe is "open": infinite and no paths return.
where is the Hubble parameter and is the gravitational constant. A universe at the critical density is spatially flat, while higher density gives a closed universe and lower density gives an open one.
Dimensionless scale factor
A dimensionless scale factor can be defined:using the present day value
The Friedmann equations can be written in terms of this dimensionless scale factor:
where,, and.
Critical density
That value of the mass-energy density, that gives when is called the critical density:If the universe has higher density,, then it is called "spatially closed": in this simple approximation the universe would eventually contract. On the other hand, if has lower density,, then it is called "spatially open" and expands forever. Therefore the geometry of the universe is directly connected to its density.
Density parameter
The density parameter is defined as the ratio of the actual density to the critical density of the Friedmann universe:Both the density and the Hubble parameter depend upon time and thus the density parameter varies with time.
The critical density is equivalent to approximately five atoms 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.
File:UniverseComposition.svg|thumb|right|375px|Estimated relative distribution for components of the energy density of the universe. Dark energy dominates the total energy while dark matter constitutes most of the mass. Of the remaining baryonic matter, 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, 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, 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 to be zero and setting the normalised spatial curvature,, equal to zero. When the substitutions are applied to the first of the Friedmann equations given the new value we find:
where:
This term originally was used as a means to determine the spatial geometry of the universe, where is the critical density for which the spatial geometry is flat. Assuming a zero vacuum energy density, if is larger than unity, the space sections of the universe are closed; the universe will eventually stop expanding, then collapse. If 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 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 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 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
Here is the radiation density today, is the matter density today, is the "spatial curvature density" today, and is the cosmological constant or vacuum density today.
Other forms
The Hubble parameter can change over time if other parts of the equation are time dependent. 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
Relativisitic cosmology models based on the FLRW metric and obeying the Friedmann equations are called FRW models.Direct observation of stars has shown their velocities to be dominated by radial recession, validating these assumptions for cosmological models.
These models are the basis of the standard model of Big Bang cosmological including the current ΛCDM model.
To apply the metric to cosmology and predict its time evolution via the scale factor requires Einstein's field equations together with a way of calculating the density, such as a cosmological equation of state.
This process allows an approximate analytic solution Einstein's field equations giving the Friedmann equations when the energy–momentum tensor is similarly assumed to be isotropic and homogeneous. The resulting equations are:
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., 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
The pair of equations given above is equivalent to the following pair of equationswith, 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.
The second equation states that both the energy density and the pressure cause the expansion rate of the universe 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 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
The cosmological constant term can be omitted if we make the following replacementsTherefore, the cosmological constant can be interpreted as arising from a form of energy that has negative pressure, equal in magnitude to its mass-energy density:
which is an equation of state of vacuum with dark energy.
An attempt to generalize this to
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
Such a field is sometimes called quintessence.
Dust models
Setting the pressure of the perfect fluid in the Friedmann equations to zero gives a cosmological dust model.Newtonian analog
In 1934 McCrea and Milne showed that the Friedmann equations in the case of a pressureless fluid can be derived with non-relativistic Newtonian dynamics.The first equation says that the decrease in the mass contained in a fixed cube 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 contained within a part of the universe.
The second equation says that the kinetic energy of a particle of unit mass moving with the expansion plus its gravitational potential energy is equal to a constant related to the curvature of the universe. In other words, the energy 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
The Friedmann equations can be solved exactly in presence of a perfect fluid with equation of statewhere is the pressure, is the mass density of the fluid in the comoving frame and is some constant.
In spatially flat case, the solution for the scale factor is
where is some integration constant to be fixed by the choice of initial conditions. This family of solutions labelled by is extremely important for cosmology. For example, describes a 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
Another important example is the case of a radiation-dominated universe, namely when. This leads to
Note that this solution is not valid for domination of the cosmological constant, which corresponds to an. In this case the energy density is constant and the scale factor grows exponentially.
Solutions for other values of can be found at
Mixtures
If the matter is a mixture of two or more non-interacting fluids each with such an equation of state, thenholds separately for each such fluid. In each case,
from which we get
For example, one can form a linear combination of such terms
where is the density of "dust" when ; is the density of radiation when ; and is the density of "dark energy". One then substitutes this into
and solves for as a function of time.
History
Friedmann published two cosmology papers in the 1922-1923 time frame. He adopted the same homogeneity and isotropy assumptions used by Albert Einstein and by Willem 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.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.
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.
Friedmann's papers were largely ignored except – initially – by Einstein who actively dismissed them. However once Edwin Hubble published astronomical evidence 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.