Flatness Problem - Energy Density and The Friedmann Equation

Energy Density and The Friedmann Equation

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. Since relativity indicates that matter and energy are equivalent, this effect is also produced by the presence of energy (such as light and other electromagnetic radiation) in addition to matter. The amount of bending (or curvature) of the universe depends on the density of matter/energy present.

This relationship can be expressed by the first Friedmann equation. In a universe without a cosmological constant, this is:

Here is the Hubble parameter, a measure of the rate at which the universe is expanding. is the total density of mass and energy in the universe, is the scale factor (essentially the 'size' of the universe), and is the curvature parameter — that is, a measure of how curved spacetime is. A positive, zero or negative value of corresponds to a respectively closed, flat or open universe. The constants and are Newton's gravitational constant and the speed of light, respectively.

Cosmologists often simplify this equation by defining a critical density, . For a given value of, this is defined as the density required for a flat universe, i.e. . Thus the above equation implies

.

Since the constant is known and the expansion rate can be measured by observing the speed at which distant galaxies are receding from us, can be determined. Its value is currently around 10−26 kg m−3. The ratio of the actual density to this critical value is called Ω, and its difference from 1 determines the geometry of the universe: Ω > 1 corresponds to a greater than critical density, 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 as follows:

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 increases, but the density decreases as matter (or energy) becomes spread out. For the standard model of the universe which contains mainly matter and radiation for most of its history, decreases more quickly than increases, and so the factor will decrease. Since the time of the Planck era, shortly after the Big Bang, this term has decreased by a factor of around and so must have increased by a similar amount to retain the constant value of their product.