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 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.