Cosmological constant

The cosmological constant (usually abbreviated by the great Greek lambda ) is a physical constant in Albert Einstein's equations of general relativity , which describes gravitational force as the geometric curvature of space-time . In SI units , the dimension has 1 / L ² (unit: m ⁻² ). Its value can be positive, negative or zero a priori . ${\displaystyle \Lambda \!\,}$ ${\displaystyle \Lambda }$

definition

While the prevailing opinion in physics for a long time was that the value of the cosmological constant was zero, recent observations have come to a very small, positive value. The cosmological constant is no longer interpreted as a parameter of the general theory of relativity (as introduced by Einstein), but as the constant energy density (here mass density, unit: kg m ⁻³ ) of the vacuum: ${\displaystyle \rho _{\mathrm {vac} }}$

${\displaystyle \Lambda ={\frac {8\pi G}{c^{2}}}\rho _{\mathrm {vac} }\approx {1{,}9\cdot 10^{-26}}~\mathrm {m/kg} \cdot \rho _{\mathrm {vac} }}$ ,

where is the circle number Pi , the gravitational constant and the speed of light in a vacuum . ${\displaystyle \pi \,}$ ${\displaystyle G\,}$ ${\displaystyle c\,}$

In modern cosmology the following is usually used instead of the dimensionless density parameter : ${\displaystyle \Lambda }$ ${\displaystyle \Omega _{\Lambda }}$

${\displaystyle \Omega _{\Lambda }={\frac {\Lambda c^{2}}{3H_{0}^{2}}}={\frac {8\pi G}{3H_{0}^{2}}}\rho _{\mathrm {vac} }={\frac {\rho _{\mathrm {vac} }}{\rho _{\mathrm {c} }}}}$

with the critical mass density

${\displaystyle \rho _{\mathrm {c} }\equiv {\frac {3H_{0}^{2}}{8\pi G}}\approx 8{,}5\cdot 10^{-27}~\mathrm {kg/m^{3}} }$.

It is

${\displaystyle H_{0}\approx 67~{\frac {\mathrm {km} }{\mathrm {s} \cdot \mathrm {Mpc} }}}$

the Hubble constant .

The assumption that the vacuum energy density remains constant even with expansion of the universe leads to the equation of state

${\displaystyle \rho _{\mathrm {vac} }=-{\frac {p}{c^{2}}}}$,

that is, a positive vacuum energy density leads to negative pressure , which drives the accelerated expansion of the universe. Every form of energy has this effect (with light quantum gases it is ), but in general the energy density is no longer constant over time. The generalization of the cosmological constant to time-variable energy densities of this kind is called dark energy . ${\displaystyle p}$ ${\displaystyle p<-{\tfrac {1}{3}}\rho c^{2}}$ ${\displaystyle p=+{\tfrac {1}{3}}\rho c^{2}\,}$

From a number of different observations, the value of the cosmological constant is estimated today , that is, about 70% of the energy density in the universe is in the form of the cosmological constant or dark energy. ${\displaystyle \Omega _{\Lambda }\approx 0{,}7}$

history

The Einstein field equations of general relativity can be formulated with or without cosmological constant. However, a matter-filled universe, the development of which is described by equations without the constant, cannot be static, but must necessarily expand or collapse. However, when Einstein set up his equations, the universe was considered static. So that the equations with matter describe a static universe (and not one that collapses due to gravitational attraction), Einstein introduced the constant in an ad hoc hypothesis . It works (if it is positive) like an "expansion force" opposite to gravitational attraction.

However, this static solution is unstable, and the smallest deviations from the ideal distribution of matter cause the universe to collapse or expand again depending on the sign of the disturbance. When Edwin Hubble discovered the expansion of the universe based on the galaxy escape and Alexander Alexandrowitsch Friedmann (1922, 1924) and Georges Lemaître (1927) discovered cosmological expanding solutions of the field equations, Einstein rejected the idea of ​​the cosmological constant and allegedly called it the "greatest Donkey of my life ”. The abandonment of cosmological constants did not happen immediately, however, but only prevailed in the early 1930s.

Modern context

After the cosmological constant became less important with the discovery of the expansion of the universe, it was of more academic interest. It regained importance through attempts to establish a unified theory of all natural forces . These are described by quantum field theories , and the vacuum fluctuations of the fields of these quantum field theories would make a contribution to the cosmological constant that is many orders of magnitude too high. This is known as the cosmological constant problem. The problem remains unsolved to this day. For example, today's popular theories with supersymmetry have the advantage that the contributions of the fermions and bosons in the vacuum fluctuations to the cosmological constant cancel each other out with exact supersymmetry, but the symmetry is broken in nature.

Another starting point for understanding the cosmological constant lies in the theory of the inflationary universe . This can be explained well by a positive cosmological constant.

From 1998 the cosmological constant experienced a renaissance: Based on the brightness or redshift of distant supernovae of type Ia , one can determine that the universe is expanding at an accelerated rate. This accelerated expansion can be described very well with a cosmological constant and is part of the successful lambda CDM model , the standard model of cosmology.

See also

• Dark energy

literature

• Torsten Fließbach : General Theory of Relativity . 4th edition. Elsevier - Spektrum Akademischer Verlag, 2003, ISBN 3-8274-1356-7 .
• Steven Weinberg The Cosmological Constant Problem . In: Reviews of Modern Physics , Volume 61, 1989, pp. 1-23 (older review article).
• Norbert Straumann : On the Cosmological Constant Problems and the Astronomical Evidence for a Homogeneous Energy Density with Negative Pressure . Lecture, Institut Poincaré, Paris 2002, arxiv : astro-ph / 0203330
• Norbert Straumann: The history of the cosmological constant problem . 2002, arxiv : gr-qc / 0208027
• Sean Carroll : The cosmological constant . Living Reviews in Relativity, 2001, arxiv : astro-ph / 0004075

Web links

• T. Davis, B. Griffen: Cosmological constant . In: Scholarpedia (English)
• Steven Weinberg: The Cosmological Constant Problems . caltech.edu; Retrieved August 4, 2011
• JP Leahy: Einstein's Greatest Blunder. The Cosmological Constant .

Individual evidence

1. ↑ Albert Einstein: Cosmological considerations on the general theory of relativity . In: Session reports of the Royal Prussian Academy of Sciences (Berlin) . 1917, p. 142-152 ( ECHO ).
2. ↑ ..the biggest blunder he ever made in his life , George Gamow My World Line , Viking Press 1970, p. 44. Einstein had said this after Gamow in discussions with him. Einstein himself describes in Meaning of Relativity (Appendix 1, edition Routledge 2003, p. 115) the cosmological constant more prosaically as a complication of the theory, which impairs the logical simplicity of the theory and only because of the problem of the constant density of matter approach, which also occurs in Newtonian theory was necessary in the field equations for a static universe. According to Friedman's solution (which he presents in the appendix) this would no longer be necessary. Einstein and de Sitter make a similar statement in Proc.Nat.Acad.Sci. , Volume 18, 1932, p. 213
3. ↑ Einstein in the reports of the meetings of the Prussian Academy of Sciences , 1931, p. 235
4. ^ Adam G. Riess et al .: The Farthest Known Supernova: Support for an Accelerating Universe and a Glimpse of the Epoch of Deceleration . In: Astroph. Journ. tape 560 , 2001, p. 49-71 , bibcode : 2001ApJ ... 560 ... 49R .