Flatness problem

The flatness problem is a cosmological fine-tuning problem within the Big Bang model. The flatness problem arises because of the effect which the density of the universe has on its geometry, and the observation that the density today is very close to the critical density required for spatial flatness.. Since the total energy density of the universe departs rapidly from the critical value over cosmic time, 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.

The problem was first mentioned by Robert Dicke in 1969. The most commonly-accepted solution among cosmologists is cosmic inflation; along with the monopole problem and the horizon problem, the flatness problem is one of the three primary motivations for inflationary theory.

Energy Density and Spacetime Curvature

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.

Friedmann equation

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 with greater than critical density; a hyperbolic, underdense universe; and a flat universe with exactly the critical density.

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. $\rho$ is the total density of mass and energy in the universe, $a$ is the scale factor (essentially the 'size' 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, $\rho _{c}$. 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 $\Omega$, and its difference from 1 determines the geometry of the universe: $\Omega >1$ corresponds to a greater than critical density, $\rho >\rho _{c}$, and hence a closed universe. $\Omega <1$ gives a low density open universe, and $\Omega$ equal to exactly 1 gives a flat universe.

The Friedmann Equation above can now be rearranged into the form

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

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 $\rho$ 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, $\rho$ decreases more quickly than $a^{2}$ increases, and so the factor $\rho a^{2}$ will decrease. Since the time of the Plank era, shortly after the Big Bang, this term has decreased by a factor of around $10^{60},$ and so $(\Omega ^{-1}-1)$ must have increased by a similar amount to retain the constant value of their product.

Current Value of $\Omega$

The relative density $\Omega$ against cosmic time t (neither axis to scale). Each curve represents a possible universe: note that $\Omega$ diverges rapidly from 1. The blue curve is a universe similar to our own, which at the present time (right of the graph) has a small $|\Omega -1|$ and therefore must have begun with $\Omega$ very close to 1 indeed. The red curve is a hypothetical different universe in which the initial value of $\Omega$ differed slightly too much from 1: by the present day it has diverged massively and would not be able to support galaxies, stars or planets.

The value of $\Omega$ at the present time is denoted $\Omega _{0}$. This value can be measured in a number of ways since signals from distant objects will be warped by the curvature of the space through which they pass, giving clues as to the current density. One such observation is that of anisotropies in the Cosmic Microwave Background (CMB) radiation; another is the frequency of Type-Ia supernovae at different distances from Earth.

Data from the Wilkinson Microwave Anisotropy Probe (measuring CMB anisotropies) combined with that from the Sloan Digital Sky Survey (observing Ia supernovae) constrain $\Omega _{0}$ to be 1 within 1%. In other words the term $|\Omega -1|$ 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 $\rho _{c}$. Indeed, a very small departure of $\Omega$ 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 ($\rho >\rho _{c}$) this would lead to a universe so dense it would collapse into a Big Crunch in a few years or less; in the case of an underdensity ($\rho <\rho _{c}$) 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. In either case the universe would contain no complex structures such as galaxies, stars, planets and people.

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 $\rho _{crit}$ as far from it, and that speculating on a reason for any particular value was "beyond the domain of science". Enough cosmologists saw the problem as a real one, however, for various reasons to be proposed.

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 $\Omega$ 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 ($\Omega >1$) and some under-dense ($\Omega <1$). 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 such 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 $\Omega$ 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).

Criticisms of the Anthropic Principle

However, the Anthropic Principle has been criticised by many scientists. For example, in 1979 Bernard Carr and Martin Rees argued that the principle “is entirely post hoc: it has not yet been used to predict any feature of the Universe.” . Others have taken objection to its philosophical basis, with Ernan McMullin writing in 1994 that "the weak Anthropic principle is trivial ... and the strong Anthropic principle is indefensible." Since many physicists and philosophers of science do not consider the principle to be compatible with the scientific method, another explanation for the flatness problem was needed.

Inflation

Main article: Cosmic inflation

The standard solution to the Flatness Problem invokes cosmic inflation, a process whereby the universe expands exponentially quickly during a short period in its early history. The idea of inflation was first thought of in 1979, and published in 1981, by Alan Guth. His two main motivations for doing so were the flatness problem and the horizon problem, another fine-tuning problem of physical cosmology.

The proposed cause of inflation is a field which permeates space and drives the expansion. The field contains a certain energy density, but unlike the density of matter or radiation, which dominate the late universe, the density of the inflationary field remains roughly constant. Therefore the term $\rho a^{2}$ increases extremely rapidly as the scale factor $a$ grows exponentially. Recalling the Friedman Equation

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

and the fact that the right-hand side of this expression is constant, the term $|\Omega -1|$ must therefore decrease with time.

Thus if $|\Omega -1|$ initially takes any arbitrary value, a period of inflation can force it down towards 0 and leave it extremely small - around $10^{-62}$ as required above, for example. Subsequent evolution of the universe will cause the value to grow, bringing it to the currently observed value of around 0.01. Thus the sensitive dependence on the initial value of $\Omega$ has been removed: a large and therefore 'unsurprising' starting value need not become massively amplified and lead to a very curved universe with no opportunity to form galaxies and other structures.

This solving of the flatness problem is considered one of the major motivations for the existence of an inflationary epoch.