Friedmann equations
The Friedmann equation describes the evolution of the universe. The equation predictions about the expansion or contraction can thereby be derived depending on the energy content of the universe. Equation is obtained from the requirement of constant curvature and by applying the cosmological principle ( " the universe is spatially homogeneous and isotropic ").
Substituting the Robertson -Walker metric in the Einstein field equations of General Relativity (GR ), then we obtain the Friedmann equation as 0-0 component
As well as a trace of the i- i components
The combination of these two formulas, the acceleration equation follows
Herein, the total energy density (including the cosmological constant ), the pressure, the scale factor, is the gravitational constant, the bending parameters (0, 1, -1) from the Robertson Walker metric and the Hubble scalar.
Understanding
Albert Einstein was initially from a static universe that neither expands nor contracts. To this end, he had to introduce into his equations of general relativity, a corresponding constant, which he called the cosmological constant ( Λ ).
The Russian mathematician and physicist Alexander Friedmann rejected the assumption of a static universe and set the cosmological constant equal to zero. Instead, he proposed three models of an expanding universe with the Friedmann equations named after him. This influenced the result considerably the physical concepts and models of Einstein.
Equations say, depending on the total energy density values for the different curvature of the pre- space-time ( according to the values -1, 0 or 1 is in the above equations):
Depending on the equation of state of matter contained in the universe also arise in three different ways for the further development of the universe:
The various possibilities for the curvature and the expansion history of the universe are initially independent of each other. Only by various restrictive assumptions about the occurring forms of matter arise dependencies.
The expansion of the universe was discovered in 1929 by Edwin Hubble by astronomical observations. These observations thus confirmed Friedmann's assumption of an expanding universe.
The expansion rate is given by the Hubble constant H0. For H0, the age of the universe can be determined, each of the three models provides a different value.
From recent measurements of the expansion rate of the background radiation of the universe is currently (March 2004) is as follows:
- The Hubble constant is 71 km / (s * Megaparsec ), where 1 parsec = 3.26 light years. This results in an age of the universe 13.7 billion years ago.
- The universe is flat within the measurement accuracy.
- The expansion is accelerating.
The total energy density of the universe is made up according to the latest findings consist of:
- 73 % vacuum energy density ( dark energy )
- 23% cold dark matter
- 4% baryonic matter, i.e., the "normal" elements
- If at all, less than 1% hot dark matter.
Derivation
The field equations of general relativity
Although gravity is the weakest of the four known interactions, it is on larger scales is the dominant force in the universe and determine its development and dynamics. The current best description of gravity is general relativity theory (ART). This links the distribution and dynamics of matter with the geometry of space-time according to:
Here, the Einstein tensor G describing the geometry of space-time, while the energy -momentum tensor T includes all matter and energy fields. The (0,2) - tensor is called Einstein metric and represents the relativistic generalization of the metric tensor
Dar. for static and flat Minkowski spacetime to curved space-times is the cosmological constant. The latter is often interpreted as a vacuum energy which arises from the zoo in the virtual particles, but their true nature is still unclear.
Exact solutions for the field equations have been found only for highly symmetric matter distributions. The problem is to find for a matter and energy distribution T is a suitable metric g, from which the Einstein tensor G composed in a complex manner.
The metric can be represented through the so-called line element,
Where a summation is implied on identical high - and subscript indices ( Einstein summation convention).
Solution of the field equations for a symmetric universe
The distribution of matter in the universe is at small distances very irregular, however, appears increasingly isotropic from several hundred megaparsecs, that is the same looking in all directions. Assuming that an observer is privileged in the universe in any way ( the Copernican principle), it derives directly that the universe looks like from every point of isotropic and homogeneous. This is also known as a cosmological principle.
Howard Percy Robertson ( 1935) and Arthur Geoffrey Walker ( 1936) independently found a solution to the field equations for the case of such an idealized universe with constant curvature. The line element of this metric which was used already in 1922 by Friedmann is,
Here, the " comoving " radial coordinate represents the proper time of a " comoving observer ", the expansion factor of the universe, and and identify the two angular coordinates, analogous to a spherical coordinate system. A comoving observer follows the expansion of the universe, its comoving radial coordinate in this case retains its numerical value.
The function distinguishes between three-dimensional spacelike hypersurfaces of constant time with positive, vanishing, or negative curvature. Under such a hypersurface is understood all the events that take place at the same cosmological time. For example, our Milky Way and all other galaxies form a spacelike hypersurface today. Only we see these galaxies due to the light travel time is not in this state today, but in an individual and already past state. The spacelike hypersurface which they span, so no observation is available.
Is given by
By rescaling the radial coordinate and redefinition of the scale factor is possible to define the curvature parameter to one of the values -1, 0 or 1. With the Robertson - Walker metric, the Friedmann equations can be derived from the Einstein's field equations. Details can be found among others in Gravitation ( Misner, Thorne and Wheeler, 1973).
The Friedmann equation
The requirement of isotropy of the distribution of matter in the universe also follows that the spatial component of the energy-momentum tensor must be a multiple of Einheitstensors:
While standing for the homogeneous density and the pressure. Both functions depend only on the time-like parameter.
Substituting this energy -momentum tensor and the Robertson -Walker metric in the field equations, we obtain the Friedmann equation and the acceleration equation
On passing the Friedmann equation ( 1) with respect to time and sets them into the second equation, we obtain an equation that describes the conservation of energy in a clear manner
Therefore, the Friedmann equation is sufficient to describe together with the energy conservation law, the global evolution of the universe. The equations at the beginning of the article result by expressing the cosmological constant by the energy density and the pressure.
Special solutions
We now have two equations for the three unknowns, and. To obtain a clear solution, is therefore a further equation, the equation of state of matter necessary. Ordinary ( baryonic ) matter, radiation and cosmological constant are the main sources of gravity on the right side of the field equations of ART. The matter may in this case be regarded as a pressureless " dust ", ie the particles move without colliding with non- relativistic velocities. So that the following three equations of state are valid for the three unknown functions:
From the conservation of energy, this results in the relationship between density and scale factor
As an initial value for the first Friedmann equation ( also known as the Friedmann - Lemaître equation, named after Georges Lemaître in addition, they also discovered independently by Friedmann 1927) is used, where the cosmological time is in the now. With the constants
Which parameterize the density of matter and vacuum energy density, the Friedmann - Lemaître equation can then also as
Be written. Hubble function is in this case, as above, in accordance with
Defined. This describes the rate of expansion of the universe, with the present time. The radiation density has been neglected, as it is with drops and therefore over the matter density rapidly insignificant.
Solving the Friedmann - Lemaître equation for the specific time you can see that the constants are not independent, but that applies
Substituting this into the Friedmann - Lemaître equation, one obtains the well-known representation:
For a flat universe () like ours, you can specify an explicit solution of this equation for the scale factor. With the method of separation of variables can transform the differential equation into an integral. calculated to be so
This expression can be inverted. If you select such a way that the universe and thus has a compact beginning, and one uses, one obtains
This expression describes the expansion behavior for a flat universe with a cosmological constant. Peacock ( 2001) and Carroll ( 1992) have derived an identical expression in other analytical form. The measured on the spacecraft WMAP fluctuations in the background radiation allow conclusions on the geometry of our universe. Thus, this is flat, with a matter density parameter, a vacuum density parameter and a Hubble constant of.
Cosmological redshift and distance measurements
In dynamic and curved space-times, there are, in contrast to Euclidean spaces, no unique measure of distance more. There are several, equal distance definitions, based on the line element of a photon and the cosmological redshift as a common denominator.