User:Olaf Davis/Flatness Problem

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The flatness problem is a cosmological fine-tuning problem within the Big Bang model. Along with the monopole problem and the horizon problem, it is one of the three primary motivations for the theory of an inflationary universe[1]. The flatness problem arises because of the observation that the density of the universe today is very close to the critical density required for spatial flatness.[2]. Since the total energy density of the universe departs rapidly from the critical value over cosmic time,[3] the early universe must have had a density even closer to the critical density, leading cosmologists to question how the density of the early universe came to be fine-tuned to this 'special' value.

Contents

[edit] Energy Density and Spacetime Curvature

The local geometry of the universe is determined by whether Omega is less than, equal to or greater than 1. From top to bottom: a spherical universe, a hyperbolic universe, and a flat universe.
The local geometry of the universe is determined by whether Omega is less than, equal to or greater than 1. From top to bottom: a spherical universe, a hyperbolic universe, and a flat universe.

According to Einstein's field equations of general relativity, the structure of spacetime is affected by the presence of matter and energy. On small scales space appears flat - as does the surface of the Earth if one looks at a small area. On large scales however, space is bent by the gravitational effect of matter. The amount of bending (or curvature) of the universe depends on the density of matter/energy in the universe.

[edit] Friedmann equation

This relationship can be expressed by the first Friedmann Equation. Ignoring dark energy, this is:

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

Here H is the Hubble parameter, a measure of the rate at which the universe is expanding. ρ is the total density of mass and energy in the universe, a is the scale factor of the universe, and k is the curvature parameter - that is, a measure of how curved spacetime is. The constants G and c are Newton's gravitational constant and the speed of light, respectively.

Astronomers often simplify this equation by defining a critical density, ρc. For a given value of H, this is defined as the density required for a flat universe, i.e. k = 0. Thus the above equation implies \rho_c = \tfrac{3H^2}{8\pi G}. The ratio of the actual density to this critical value is called Ω, and it's difference from 1 determines the geometry of the universe: Ω > 1 corresponds to a greater than critical density, ρ > ρc, and hence a closed universe. Ω < 1 gives a low density open universe, and Ω equal to exactly 1 gives a flat universe.

The Friedmann Equation above can now be rearranged into the form

(\Omega^{-1} - 1)\rho a^2 = \frac{-3kc^2}{8\pi G}.[4]

The right hand side of this expression contains only constants, and therefore the left hand side must remain constant throughout the evolution of the universe.

As the universe expands the scale factor a increases, but the density ρ decreases. For the standard model of the universe containing mainly matter and radiation, ρ decreases more quickly than a2 increases, and so the factor ρa2 will decrease. Since the time of the Plank era, shortly after the Big Bang, this term has decreased by a factor of around 1060,[5] and so (Ω − 1 − 1) must have increased by a similar amount to retain the constant value of their product.

[edit] Current Value of Ω

The value of Ω at the present time is denoted Ω0. Data from the Wilkinson Microwave Anisotropy Probe's three year results, combined with that from the Sloan Digital Sky Survey, constrain Ω0 to be 1 within 1%.[2] In other words the term | Ω − 1 | is currently less than 0.01, and therefore must have been less than 10 − 62 at the Plank era.

This tiny value is the crux of the flatness problem. If the initial density of the universe could take any value, it would seem extremely surprising to find it so 'finely tuned' to the critical value ρc. This therefore motivates a search for some reason the density should take such a specific value.

[edit] Early Universe

In the early universe Ω is the ratio of the energy density to the critical density at that time. In the Lambda-CDM cosmology favoured by astronomers, the early universe is dominated by radiation, then by matter. In this case, if Ω is much greater than 1, the universe quickly recollapses in a Big crunch. If Ω is much less than one, the universe expands so quickly that matter cannot collapse under gravity to form galaxies or stars. If the current value of Ω is extrapolated back to the Planck time the value of Ω is such that \Omega~=~1~\pm~10^{-60}. That this value is so close to the critical value when it could take on any value at all is regarded as a highly improbable coincidence.

[edit] Inflation

The problem is that a simple big bang theory cannot explain how an Ω so close to unity could arise. The problem is solved by the hypothesis of an inflationary universe, in which very shortly after the Big Bang, the universe increased in size by an enormous factor. Such an inflation would have smoothed out any non-flatness originally present and resulted in a universe with a density extremely close to the critical density.

[edit] References

  1. ^ Barbara Ryden. Introduction to Cosmology. Addison Wesley. 
  2. ^ a b D. N. Spergel et al. (June 2007). "Wilkinson Microwave Anisotropy Probe (WMAP) Three Year Results: Implications for Cosmology". ApJS 170: 337-408. doi:10.1086/513700. 
  3. ^ Peacock, J. A. (1998). Cosmological Physics. Cambridge: Cambridge University Press. ISBN 978-0521422703. 
  4. ^ Peter Coles and Francesco Lucchin. Cosmology. Wiley. 
  5. ^ Peter Coles and Francesco Lucchin. Cosmology. Wiley.