The flatness problem (also known as the oldness problem) is a cosmological fine-tuning problem within the Big Bang model of the universe. Measurements find the current universe close to perfectly flat and expansion of the universe increases flatness. Consequently the early universe must have been exceptionally close to flat. In standard cosmology based on the Friedmann equations the density of matter and energy in the universe affects the curvature of space-time, with a very specific critical value being required for a flat universe. The current density of the universe is observed to be very close to this critical value. Since any departure of the total density from the critical value would increase rapidly over cosmic time, the early universe must have had a density even closer to the critical density, departing from it by one part in 1062 or less. This leads cosmologists to question how the initial density came to be so closely fine-tuned to this 'special' value.
The problem was first mentioned by Robert Dicke in 1969. The most commonly accepted solution among cosmologists is cosmic inflation, the idea that the universe went through a brief period of extremely rapid expansion in the first fraction of a second after the Big Bang; along with the monopole problem and the horizon problem, the flatness problem is one of the three primary motivations for inflationary theory.
Flatness Flatness in cosmology is a curved spacetime geometry with zero curvature. Curvature can be measured by comparing the radius of a circle around any point to the circumference:
Any small region in spacetime is locally flat. By analogy, the Earth appears flat in small regions.
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 large scales space is curved 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 . The ratio of the actual density to this critical value is called Ω, and its difference from 1 determines the geometry of the universe: corresponds to a greater than critical density, , and hence a closed universe. gives a low density open universe, and Ω equal to exactly 1 gives a flat universe.
The Friedmann equation,
can be re-arranged into
which after factoring , and using , leads to
The right hand side of the last expression above contains constants only 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.
Current value of Ω
Measurement The value of Ω at the present time is denoted Ω0. This value can be deduced by measuring the curvature of spacetime (since , or , is defined as the density for which the curvature ). The curvature can be inferred from a number of observations.
One such observation is that of anisotropies (that is, variations with direction - see below) in the Cosmic Microwave Background (CMB) radiation. The CMB is electromagnetic radiation which fills the universe, left over from an early stage in its history when it was filled with photons and a hot, dense plasma. This plasma cooled as the universe expanded, and when it cooled enough to form stable atoms it no longer absorbed the photons. The photons present at that stage have been propagating ever since, growing fainter and less energetic as they spread through the ever-expanding universe.
The temperature of this radiation is almost the same at all points on the sky, but there is a slight variation (around one part in 100,000) between the temperature received from different directions. The angular scale of these fluctuations - the typical angle between a hot patch and a cold patch on the skySince there are fluctuations on many scales, not a single angular separation between hot and cold spots, the necessary measure is the angular scale of the first peak in the anisotropies' power spectrum. See Cosmic Microwave Background#Primary anisotropy. - depends on the curvature of the universe which in turn depends on its density as described above. Thus, measurements of this angular scale allow an estimation of Ω0.
Another probe of Ω0 is the frequency of Type-Ia supernovae at different distances from Earth.Ryden p. 168 These supernovae, the explosions of degenerate white dwarf stars, are a type of standard candle; this means that the processes governing their intrinsic brightness are well understood so that a measure of apparent brightness when seen from Earth can be used to derive accurate distance measures for them (the apparent brightness decreasing in proportion to the square of the distance - see luminosity distance). Comparing this distance to the redshift of the supernovae gives a measure of the rate at which the universe has been expanding at different points in history. Since the expansion rate evolves differently over time in cosmologies with different total densities, Ω0 can be inferred from the supernovae data.
Data from the Wilkinson Microwave Anisotropy Probe (WMAP, measuring CMB anisotropies) combined with that from the Sloan Digital Sky Survey and observations of type-Ia supernovae constrain Ω0 to be 1 within 1%. In other words, the term |Ω − 1| is currently less than 0.01, and therefore must have been less than 10−62 at the Planck era. The cosmological parameters measured by the Planck spacecraft mission reaffirmed previous results by WMAP.
Implication 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 . Indeed, a very small departure of Ω from 1 in the early universe would have been magnified during billions of years of expansion to create a current density very far from critical. In the case of an overdensity this would lead to a universe so dense it would cease expanding and collapse into a Big Crunch (an opposite to the Big Bang in which all matter and energy falls back into an extremely dense state) in a few years or less; in the case of an underdensity it would expand so quickly and become so sparse it would soon seem essentially empty, and gravity would not be strong enough by comparison to cause matter to collapse and form galaxies resulting in a big freeze. In either case the universe would contain no complex structures such as galaxies, stars, planets and any form of life.Ryden p. 193
This problem with the Big Bang model was first pointed out by Robert Dicke in 1969, and it motivated a search for some reason the density should take such a specific value.
Solutions to the problem Some cosmologists agreed with Dicke that the flatness problem was a serious one, in need of a fundamental reason for the closeness of the density to criticality. But there was also a school of thought which denied that there was a problem to solve, arguing instead that since the universe must have some density it may as well have one close to as far from it, and that speculating on a reason for any particular value was "beyond the domain of science". That, however, is a minority viewpoint, even among those sceptical of the existence of the flatness problem. Several cosmologists have argued that, for a variety of reasons, the flatness problem is based on a misunderstanding.
Anthropic principle
One solution to the problem is to invoke the anthropic principle, which states that humans should take into account the conditions necessary for them to exist when speculating about causes of the universe's properties. If two types of universe seem equally likely but only one is suitable for the evolution of intelligent life, the anthropic principle suggests that finding ourselves in that universe is no surprise: if the other universe had existed instead, there would be no observers to notice the fact.
The principle can be applied to solve the flatness problem in two somewhat different ways. The first (an application of the 'strong anthropic principle') was suggested by C. B. Collins and Stephen Hawking, who in 1973 considered the existence of an infinite number of universes such that every possible combination of initial properties was held by some universe. In such a situation, they argued, only those universes with exactly the correct density for forming galaxies and stars would give rise to intelligent observers such as humans: therefore, the fact that we observe Ω to be so close to 1 would be "simply a reflection of our own existence".
An alternative approach, which makes use of the 'weak anthropic principle', is to suppose that the universe is infinite in size, but with the density varying in different places (i.e. an inhomogeneous universe). Thus some regions will be over-dense and some under-dense . These regions may be extremely far apart - perhaps so far that light has not had time to travel from one to another during the age of the universe (that is, they lie outside one another's cosmological horizons). Therefore, each region would behave essentially as a separate universe: if we happened to live in a large patch of almost-critical density we would have no way of knowing of the existence of far-off under- or over-dense patches since no light or other signal has reached us from them. An appeal to the anthropic principle can then be made, arguing that intelligent life would only arise in those patches with Ω very close to 1, and that therefore our living in such a patch is unsurprising.
This latter argument makes use of a version of the anthropic principle which is 'weaker' in the sense that it requires no speculation on multiple universes, or on the probabilities of various different universes existing instead of the current one. It requires only a single universe which is infinite - or merely large enough that many disconnected patches can form - and that the density varies in different regions (which is certainly the case on smaller scales, giving rise to galactic clusters and voids).