Friedmann equation

The Friedmann equation describes the development of the universe . It is sometimes referred to as the Friedmann-Lemaître equation because it was discovered independently by Alexander Friedmann and also by Georges Lemaître . It is a simplification of Einstein's field equations of general relativity under the assumption of a homogeneous and isotropic universe. Depending on the energy content of the universe, the equation can be used to predict its development over time, i.e. H. the special form of expansion or contraction.

The distribution of matter in the universe is very irregular at short distances, but from several hundred megaparsecs it appears increasingly isotropic , i.e. it looks the same in all directions. Assuming that an observer is in no way privileged in the universe ( Copernican principle ), it immediately follows that the universe looks isotropic and homogeneous from every point of view . This is also known as the Cosmological Principle . If one takes this isotropy of the matter distribution into account, it follows that the spatial part of the energy-momentum tensor has a relatively simple form and must be a multiple of the unit tensor. The energy-momentum tensor thus has the following form:

${\displaystyle (T^{\mu }{}_{\nu })=(g^{\mu \kappa }T_{\kappa \nu })=\operatorname {diag} (-\rho c^{2},p,p,p)\,.}$

${\displaystyle \rho =\rho (t)}$stands for the spatially homogeneous mass density and for the pressure . Both functions only depend on the time-like parameter . ${\displaystyle p=p(t)}$ ${\displaystyle t}$

The cosmological principle now makes the further assumption necessary that the spatial curvature of the room should be independent of the position in the room. This assumption leads to a relatively special form of the metric tensor . If this tensor and the form of the energy-momentum tensor just shown are inserted into Einstein's field equations of general relativity theory (GTR) with cosmological constants , the Robertson-Walker metric can be derived from it, which is described in more detail below. With this derivation you also get the Friedmann equation in its modern version with cosmological constant: ${\displaystyle \Lambda }$

${\displaystyle H^{2}=\left({\frac {\dot {a}}{a}}\right)^{2}={\frac {8\pi G}{3}}\rho -{\frac {kc^{2}}{a^{2}}}\ +{\frac {\Lambda c^{2}}{3}},}$

as well as the acceleration equation

${\displaystyle {\dot {H}}+H^{2}={\frac {\ddot {a}}{a}}=-{\frac {4\pi G}{3c^{2}}}\left(\rho c^{2}+3p\right)+{\frac {\Lambda c^{2}}{3}}\,.}$

Here the scale factor , the gravitational constant and the curvature parameter (0, +1, −1) from the Robertson-Walker metric denotes . denotes the Hubble parameter . ${\displaystyle a(t)}$ ${\displaystyle G}$ ${\displaystyle k}$ ${\displaystyle H}$

Basics

Albert Einstein initially assumed a static universe that neither expands nor contracts. To do this, he had to introduce a corresponding constant in his equations of general relativity , which he called the cosmological constant (Λ).

The Russian mathematician and physicist Alexander Friedmann rejected this assumption of a static universe and set the cosmological constant equal to zero. Instead, he set up three models of an expanding universe with the Friedmann equations named after him . These subsequently influenced Einstein's physical conceptions and models considerably.

The equations predict different values ​​for the curvature of space-time depending on the total energy density (corresponding to the values ​​−1, 0 or +1 for in the above equations): ${\displaystyle k}$

1. Model: The energy density of the universe is greater than the critical energy density (see below). Then the curvature of spacetime is positive , the universe is “spherical” (a two-dimensional analog would be the surface of a sphere). Incidentally, such a spherical universe is also closed: although unlimited, it would only be finite. If you walk long enough in one direction, you will eventually come back to where you started. ${\displaystyle (k=1)}$
2. Model: The energy density is exactly as great as the critical energy density. The spacetime has vanishing curvature , the universe is "flat" (would correspond to a plane in two dimensions ). ${\displaystyle (k=0)}$
3. Model: The energy density is less than the critical value. The curvature of spacetime is negative , the universe “ hyperbolic ”. ${\displaystyle (k=-1)}$

Depending on the equation of state of the matter contained in the universe, there are also three different possibilities for the further development of the universe:

1. Possibility: Gravitation is able to slow down the expansion so far that it comes to a standstill and reverses. The universe contracts to a single point ( Big Crunch ). One can only speculate about the further development "after" this event. Some scenarios envisage the possibility of a “pulsating” universe.
2. Possibility: Gravitation slows down the expansion further and further, but does not bring it to a standstill.
3. Possibility: The expansion accelerates and the ordinary matter in the universe is thinned out more and more.

The different possibilities for the curvature and the expansion behavior of the universe are initially independent of each other. Dependencies arise only through various limiting assumptions about the forms of matter occurring.

The expansion of the universe described by the Friedmann equation provides an explanation for the linear relationship between redshift and distance discovered by Edwin Hubble in 1929 . Hubble himself initially interpreted his observations as an optical Doppler effect. Models of static universes, which were previously popular, cannot explain the observed redshift and thus continued to lose importance.

The expansion rate is given with the Hubble constant H ₀ . The age of the universe can be determined from H ₀ , with each of the three models providing a different value.

From the latest measurements of the expansion rate via the background radiation of the universe, the following picture is currently (August 2012):

• The Hubble constant is 74.3 km / (s · Megaparsec), where: 1 parsec = 3.26 light years. This results in an age of the universe of 13.82 billion years.
• The universe is flat within the scope of the measurement accuracy.
• The expansion is accelerating.

According to the latest findings, the entire energy density of the universe is made up of:

• 68% vacuum energy density (dark energy)
• 28% cold dark matter
• 4% baryonic matter, i.e. H. the "normal" elements
• if anything, less than 1% hot dark matter .

Derivation

The field equations of general relativity

Although gravity is the weakest of the four known interactions, on a larger scale it represents the dominant force in the universe and determines its development and dynamics. The best description of gravity at present is general relativity (GTR). This links the distribution and dynamics of matter with the geometry of space-time according to:

${\displaystyle G_{\mu \nu }={\frac {8\pi G}{c^{4}}}T_{\mu \nu }-g_{\mu \nu }\Lambda \qquad \mu ,\nu \in \{0,1,2,3\}}$

Here the Einstein tensor G describes the geometry of space-time, while the energy-momentum tensor T includes all matter and energy fields. The (0,2) - tensor is called the Einstein metric and represents the general relativistic generalization of the metric tensor ${\displaystyle g}$

${\displaystyle (\eta _{\mu \nu })=\operatorname {diag} (-1,1,1,1)}$

represents the static and flat Minkowski spacetime on curved spacetime. represents the cosmological constant . The latter is interpreted, among other things, as vacuum energy , which can be calculated with the help of virtual particles , but gives unsatisfactory values. Their real nature is not yet sufficiently understood. ${\displaystyle \Lambda }$

Exact solutions for the field equations have so far only been found for highly symmetrical material distributions. The problem is to find a suitable metric tensor g for the idealized matter and energy distribution T described above , from which the Einstein tensor G is composed.

The metric tensor can be represented using the so-called line element ,

${\displaystyle {\mathrm {d} }s^{2}=g_{\mu \nu }\,{\mathrm {d} }x^{\mu }\,{\mathrm {d} }x^{\nu }}$

With identical, superscript and subscript indices, all possible values ​​of the index are to be added up. This abbreviated notation is also called Einstein's summation convention .

Metric tensor for a symmetrical universe

Howard P. Robertson (1935) and Arthur Geoffrey Walker (1936) found, as already indicated above, a solution for the field equations for the case of an idealized cosmos with constant curvature. The line element of this geometry, which was already used by Friedmann in 1922, is

${\displaystyle \mathrm {d} s^{2}=-c^{2}\mathrm {d} t^{2}+a^{2}(t)\left[\mathrm {d} r^{2}+f_{k}^{\,2}(r)\left(\mathrm {d} \theta ^{2}+\sin ^{2}\theta \,\mathrm {d} \phi ^{2}\right)\right]\,.}$

Here, the “moving” radial coordinate represents the proper time of a “moving observer”, the expansion factor of the universe. and identify the two angular coordinates, analogous to a spherical coordinate system. A moving observer follows the expansion of the universe. Its moving radial coordinate retains its numerical value. ${\displaystyle r}$ ${\displaystyle t}$ ${\displaystyle a(t)}$ ${\displaystyle \theta }$ ${\displaystyle \phi }$

The function differentiates between three-dimensional space-like hypersurfaces of constant time with positive, vanishing, or negative curvature . Such a hypersurface is understood to mean all events that take place at the same cosmological time. For example, our Milky Way and all other galaxies today form a space-like hypersurface. Only because of the time of flight we do not see these galaxies in their current state, but in an individual and already past state. The space-like hypersurface they span is therefore not accessible to any observation. ${\displaystyle f_{k}(r)}$ ${\displaystyle t}$ ${\displaystyle k}$

${\displaystyle f_{k}(r)}$ is given by

${\displaystyle f_{k}(r)={\begin{cases}{\frac {1}{\sqrt {k}}}\sin({\sqrt {k}}r)&k>0\\r&k=0\;.\\{\frac {1}{\sqrt {-k}}}\sinh({\sqrt {-k}}r)&k<0\end{cases}}}$

By rescaling the radial coordinate and redefining the scale factor , the curvature parameter can be set to one of the values ​​−1, 0 or 1. With the Robertson-Walker metric and the form of the energy-momentum tensor shown above, the Friedmann equations can be derived from Einstein's field equations. Details can be found in Gravitation (Misner, Thorne and Wheeler, 1973). ${\displaystyle r}$ ${\displaystyle a}$ ${\displaystyle k}$

Energy conservation

The Friedmann equation and the acceleration equation can be combined to form another equation which clearly describes the conservation of mass and energy

${\displaystyle {\frac {\text{d}}{{\text{d}}t}}\,(\rho \,a^{3})=-{\frac {p}{c^{2}}}\,{\frac {\text{d}}{{\text{d}}t}}\,(a^{3})\,.}$

The Friedmann equation is therefore sufficient to describe the global development of the universe together with the law of conservation of energy.

Special solutions

The Friedmann equation and the acceleration equation contain the three unknown functions , and . In order to obtain an unambiguous solution, a further equation, the equation of state of matter, is necessary. Ordinary ( baryonic ) matter, radiation and the cosmological constant form the main sources of gravity on the right-hand side of the GTR field equations. The matter can be seen as pressureless "dust", i. H. the particles move without collision at non-relativistic velocities. The following three equations of state apply to the three unknown functions: ${\displaystyle a(t)}$ ${\displaystyle \rho (t)}$ ${\displaystyle p(t)}$

${\displaystyle p_{\mathrm {mat} }=0\ }$

${\displaystyle p_{\mathrm {str} }=c^{2}\rho _{\mathrm {str} }/3\ }$

${\displaystyle p_{\Lambda }=-c^{2}\rho _{\Lambda }\ }$.

The relationship between density and scale factor results from the conservation of energy ${\displaystyle \rho }$ ${\displaystyle a}$

${\displaystyle \rho _{\mathrm {mat} }\propto a^{-3}\ }$

${\displaystyle \rho _{\mathrm {str} }\propto a^{-4}\ }$

${\displaystyle \rho _{\Lambda }={\text{const.}}\ }$

The starting value for the Friedmann equation is used, where the cosmological time represents in the now. With the constants ${\displaystyle a(t_{0})=a_{0}}$ ${\displaystyle t_{0}}$

${\displaystyle \Omega _{\rm {M}}:={\frac {8\pi G}{3H_{0}^{2}}}\,\rho _{0},\qquad \Omega _{\Lambda }:={\frac {\Lambda c^{2}}{3H_{0}^{2}}}\;,}$

which parameterize the matter density and vacuum energy density can then also use the Friedmann-Lemaître equation as

${\displaystyle H^{2}(t)=H_{0}^{2}\,\left(\Omega _{\rm {M}}{\frac {a_{0}^{3}}{a^{3}}}-{\frac {kc^{2}}{a^{2}H_{0}^{2}}}+\Omega _{\Lambda }\right)\,}$

to be written. The Hubble function is, as above, according to

${\displaystyle H(t):={\dot {a}}(t)/a(t)\,}$

Are defined. This describes the rate of expansion of the universe at the present time. The radiation density was neglected because it falls with it and therefore quickly becomes insignificant compared to the matter density. ${\displaystyle H_{0}=H(t_{0})}$ ${\displaystyle a^{-4}}$

If you solve the Friedmann equation for the specific point in time , you can see that the constants are not independent, but that it applies ${\displaystyle t=t_{0}}$

${\displaystyle {\frac {kc^{2}}{a_{0}^{2}H_{0}^{2}}}=\Omega _{\rm {M}}+\Omega _{\Lambda }-1\,.}$

If you put this in the Friedmann equation, you get the best-known representation:

${\displaystyle H^{2}(t)=H_{0}^{2}\,\left(\Omega _{\rm {M}}{\frac {a_{0}^{3}}{a^{3}}}+\left(1-\Omega _{\rm {M}}-\Omega _{\Lambda }\right){\frac {a_{0}^{2}}{a^{2}}}+\Omega _{\Lambda }\right)\,.}$

For a flat universe like ours, one can give an explicit solution to this equation for the scale factor. This differential equation can be converted into an integral using the method of variable separation. If one chooses the constant of integration so that besides also applies, then it follows: ${\displaystyle \Omega _{\rm {M}}+\Omega _{\Lambda }=1}$ ${\displaystyle t_{0}}$ ${\displaystyle a_{0}=1}$ ${\displaystyle a(t_{0})=1}$

${\displaystyle t(a)-t_{0}={\frac {2}{3H_{0}{\sqrt {\Omega _{\Lambda }}}}}\,\ln \left({\frac {(a/a_{0})^{3/2}\Omega _{\Lambda }+{\sqrt {\Omega _{\rm {M}}\Omega _{\Lambda }+(a/a_{0})^{3}\Omega _{\Lambda }^{2}}}}{\Omega _{\Lambda }+{\sqrt {\Omega _{\rm {M}}\Omega _{\Lambda }+\Omega _{\Lambda }^{2}}}}}\right)\;,}$

If one then chooses so that the universe has a singular beginning, the age of the world is calculated in this simplified model, i. H. neglecting the radiation era according to: ${\displaystyle a(0)=0}$

${\displaystyle t_{0}={\frac {1}{3H_{0}{\sqrt {\Omega _{\Lambda }}}}}\,\ln \left({\frac {1+{\sqrt {\Omega _{\Lambda }}}}{1-{\sqrt {\Omega _{\Lambda }}}}}\right).}$

The formula for goes on ${\displaystyle t(a)}$

${\displaystyle t(a)={\frac {2}{3H_{0}{\sqrt {\Omega _{\Lambda }}}}}\;{\rm {arsinh\left({\sqrt {{\frac {\Omega _{\Lambda }}{\Omega _{\rm {M}}}}\left({\frac {a}{a_{0}}}\right)^{3}}}\right)}}}$

simplify. A simple transformation gives the following formula for the time dependency of the scale factor:

${\displaystyle a(t)=a_{0}\;{\sqrt[{3}]{\frac {\Omega _{\rm {M}}}{\Omega _{\Lambda }}}}\;\sinh ^{2/3}(\omega t),\quad {\mbox{mit}}\quad \omega :={\frac {3\,H_{0}{\sqrt {\Omega _{\Lambda }}}}{2}}\,.}$

This expression describes the expansion behavior for a flat universe with cosmological constant. Peacock (2001) and Carroll (1992) have derived an identical expression in a different analytical form. It continues:

${\displaystyle H(t)=H_{0}\,{\sqrt {\Omega _{\Lambda }}}\,\coth(\omega t).}$

The fluctuations in the background radiation measured by the Planck space telescope allow conclusions to be drawn about the geometry of our universe. So this is flat, with a matter density parameter , a vacuum density parameter and a Hubble constant of . ${\displaystyle \Omega _{\rm {M}}=0{,}32}$ ${\displaystyle \Omega _{\Lambda }=0{,}68}$ ${\displaystyle H_{0}=67{,}11~{{\text{km s}}^{-1}}{\text{/Mpc}}}$

Cosmological redshift and distance measures

In contrast to Euclidean spaces , there is no longer a clear measure of distance in dynamic and curved spacetime . Rather, there are different, equally valid definitions of distance that can be justified or derived from the line element of a photon and the cosmological redshift .

Individual evidence

1. ↑ ^{a b} Torsten Fließbach : General Theory of Relativity . 4th edition, Elsevier - Spektrum Akademischer Verlag, 2003, ISBN 3-8274-1356-7

See also

• cosmology
• Big Bang
• Expansion of the universe
• Supernova Cosmology Project

Web links

• Friedmann models
• Video: Cosmology of Friedmann's Equations . Jörn Loviscach 2014, made available by the Technical Information Library (TIB), doi : 10.5446 / 19913 .

literature

• SM Caroll, WH Press, EL Turner: The Cosmolocial Constant , Ann. Rev. Astr. Astrophys., Vol. 30, 1992, pp. 499-542
• A. Friedmann: About the curvature of space . In: Zeitschrift für Physik, Volume 10, No. 1, 1922, pp. 377-386
• C. Misner, K. Thorne, JA Wheeler: Gravitation , WH Freeman, San Francisco, 1973. ISBN 0-7167-0344-0 .
• JA Peacock: Cosmological Physics , Cambridge University Press, 2001, ISBN 0-521-42270-1 .
• HP Robertson: Kinematics and world structure , Astrophysical Journal, Vol 82, 1935, pp 284-301, Vol 83, 1936, pp 187-201, pp 257-271
• AG Walker: On Milne's theory of world-structure , Proc. Lond. Math. Soc. (2), Vol. 42, 1936, pp. 90-127