Flatness problem

The local curvature of the space-time of the universe depends on whether the relative density is greater, equal, or less than one. From top to bottom: a spherical universe with great density ; a hyperbolic , small-density universe ; a flat, critical-density universe . In contrast to the illustration shown, spacetime is four-dimensional. ${\displaystyle \Omega }$ ${\displaystyle (\Omega >1,k>0)}$ ${\displaystyle (\Omega <1,k<0)}$ ${\displaystyle (\Omega =1,k=0)}$

The flatness problem is a cosmological problem of the Lambda CDM model , which describes the evolution of the universe . It stems from the observation that in the lambda CDM model, the density parameter must be fine- tuned to a special, critical value, since a slight deviation from this value would have an extreme influence on today's universe.

In the case of the flatness problem, the parameter that requires such fine tuning is the density parameter of mass and energy in the Friedmann equation . For a flat universe, as it is observed, a density of must be assumed at Planck time . Any deviation from this critical value would increase sharply over time, so that today's observed density, which is required for a flat universe, would not be possible. Although this fine-tuning does not contradict the Lambda CDM model, the need for such precise definition of the initial conditions seems unnatural. The question of why the density is so close to the critical value is unanswered within the framework of the Lambda CDM model. ${\displaystyle \Omega }$ ${\displaystyle \textstyle |1-\Omega |/\Omega <10^{-58}}$

The problem was first mentioned by Robert Dicke in 1969. The most widely accepted solution to the problem in cosmology at the moment is cosmological inflation , during which there was a phase of very strong expansion in the first fractions of a second after the Big Bang . Along with the horizon problem and the problem of stable magnetic monopoles , the flatness problem is one of the three main problems of the lambda CDM model, which are the motivation for inflation theories.

Energy density and Friedmann equation

According to Einstein's field equations of the general theory of relativity , spacetime is influenced by the presence of matter and energy. Spacetime appears flat on small scales. On larger scales, space-time is curved by gravity . Since the theory of relativity assumes the equivalence of mass and energy , this is also the case in the presence of energy (such as light or other electromagnetic radiation ). The strength of the curvature depends on the density of matter or energy.

This behavior is described by the first Friedmann equation . In a universe without a cosmological constant it reads:

${\displaystyle H^{2}={\frac {8\pi G}{3}}\rho -{\frac {kc^{2}}{a^{2}}}}$

where the Hubble parameter is a measure of the rate at which the universe is expanding and the total density of mass and energy in the universe. The scale factor determines the “size” of the universe and is the curvature parameter that determines how flat or curved spacetime is. The cases , and correspond to a closed, flat or open universe. The constants and are the gravitational constant and the speed of light . ${\displaystyle H}$ ${\displaystyle \rho }$ ${\displaystyle a}$ ${\displaystyle k}$ ${\displaystyle k>0}$ ${\displaystyle k=0}$ ${\displaystyle k<0}$ ${\displaystyle G}$ ${\displaystyle c}$

The above equation can be simplified by defining a critical density . For a given value of , the critical density is defined as that which is required for a flat universe with : ${\displaystyle \rho _{\mathrm {c} }}$ ${\displaystyle H}$ ${\displaystyle k=0}$

${\displaystyle \rho _{\mathrm {c} }={\frac {3H^{2}}{8\pi G}}}$.

The gravitational constant is known, and the rate of expansion can be determined by measuring the speed at which distant galaxies are moving away from us. This allows a value of about 10 ⁻²⁶ kg m ^{−3 to} be calculated for the critical density . The ratio of this actual density to this critical value is denoted by. The deviation from from the value therefore determines the geometry of the universe: is an open universe with low density, a flat universe with precisely critical density and a closed universe with high density. ${\displaystyle G}$ ${\displaystyle H}$ ${\displaystyle \rho _{\mathrm {c} }}$ ${\displaystyle \Omega }$ ${\displaystyle \Omega }$ ${\displaystyle 1}$ ${\displaystyle \Omega <1}$ ${\displaystyle \Omega =1}$ ${\displaystyle \Omega >1}$

The Friedmann equation

${\displaystyle {\frac {3a^{2}}{8\pi G}}H^{2}=\rho a^{2}-{\frac {3kc^{2}}{8\pi G}},}$

admitted

${\displaystyle \rho _{\mathrm {c} }a^{2}-\rho a^{2}=-{\frac {3kc^{2}}{8\pi G}},}$

as well as by removing and inserting transform to ${\displaystyle \rho a^{2}}$ ${\displaystyle \Omega =\rho /\rho _{\mathrm {c} }}$

${\displaystyle (\Omega ^{-1}-1)\rho a^{2}={\frac {-3kc^{2}}{8\pi G}}.}$

The right hand side of the last equation contains only constants, which is why the right and left hand sides of the equation must be constant throughout the evolution of the universe.

As the universe expands, the scale factor increases and the density decreases as the matter (or energy) spreads across the larger universe. For the Lambda CDM model, in which the universe contains mostly matter and radiation most of the time, the value of decreases more than increases, which is why the factor decreases over time. Since the Planck era , it has decreased by a factor of around , which is why it must have increased by the same factor in order for your product to remain constant. ${\displaystyle a}$ ${\displaystyle \rho }$ ${\displaystyle \rho }$ ${\displaystyle a^{2}}$ ${\displaystyle \rho a^{2}}$ ${\displaystyle \rho a^{2}}$ ${\displaystyle 10^{60}}$ ${\displaystyle (\Omega ^{-1}-1)}$

Individual evidence

1. ↑ ^{a b} flatness problem. In: Lexicon of Physics. Spektrum Akademischer Verlag, accessed on November 26, 2017 . 
2. ^ JA Peacock: Cosmological Physics . Cambridge University Press, Cambridge 1998, ISBN 978-0-521-42270-3 .
3. ^ Robert H. Dicke: Gravitation and the Universe: Jayne Lectures for 1969 . American Philosophical Society, 1970, ISBN 978-0871690784 , p. 62.
4. ^ Alan P. Lightman: Ancient Light: Our Changing View of the Universe . Harvard University Press, January 1, 1993, ISBN 978-0-674-03363-4 , p. 61.
5. ↑ Barbara Ryden: Introduction to Cosmology . Addison-Wesley, San Francisco 2002, ISBN 0-8053-8912-1 .
6. ↑ Peter Coles, Francesco Lucchin: Cosmology . Wiley, Chichester 1997, ISBN 0-471-95473-X .