Flatness problem
The flatness problem (also known as the oldness problem) is a cosmological fine-tuning problem within the Big Bang model of the universe. Such problems arise from the observation that some of the initial conditions of the universe appear to be fine-tuned to very 'special' values, and that small deviations from these values would have extreme effects on the appearance of the universe at the current time.
In the case of the
The problem was first mentioned by
Energy density and the Friedmann equation
According to
This relationship can be expressed by the first
Here is the
- .
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
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
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 Ω = 1, or , is defined as the density for which the curvature k = 0). The curvature can be inferred from a number of observations.
One such observation is that of
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 sky[nb 1] - 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.[6][nb 2]
Another probe of Ω0 is the frequency of
Data from the
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
This problem with the Big Bang model was first pointed out by
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".[14] 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,[15] but that seems to be widely ignored by many. Enough cosmologists saw the problem as a real one, however, for various solutions to be proposed.
Anthropic principle
One solution to the problem is to invoke the
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
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 (Ω > 1) and some under-dense (Ω < 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 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.[17]
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
However, the anthropic principle has been criticised by many scientists.[18] 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.”[18][19] 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,[18] another explanation for the flatness problem was needed.
Inflation
The standard solution to the flatness problem invokes cosmic inflation, a process whereby the universe
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 the matter or radiation present in the late universe, which decrease over time, the density of the inflationary field remains roughly constant as space expands. Therefore, the term increases extremely rapidly as the scale factor grows exponentially. Recalling the Friedmann Equation
- ,
and the fact that the right-hand side of this expression is constant, the term must therefore decrease with time.
Thus if initially takes any arbitrary value, a period of inflation can force it down towards 0 and leave it extremely small - around 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 Ω has been removed: a large and therefore 'unsurprising' starting value need not become amplified and lead to a very curved universe with no opportunity to form galaxies and other structures.
This success in solving the flatness problem is considered one of the major motivations for inflationary theory.[4][23]
Post inflation
Although inflationary theory is regarded as having had much success, and the evidence for it is compelling, it is not universally accepted: cosmologists recognize that there are still gaps in the theory and are open to the possibility that future observations will disprove it.[24][25] In particular, in the absence of any firm evidence for what the field driving inflation should be, many different versions of the theory have been proposed.[26] Many of these contain parameters or initial conditions which themselves require fine-tuning[26] in much the way that the early density does without inflation.
For these reasons work is still being done on alternative solutions to the flatness problem. These have included non-standard interpretations of the effect of dark energy[27] and gravity,[28] particle production in an oscillating universe,[29] and use of a Bayesian statistical approach to argue that the problem is non-existent. The latter argument, suggested for example by Evrard and Coles, maintains that the idea that Ω being close to 1 is 'unlikely' is based on assumptions about the likely distribution of the parameter which are not necessarily justified.[30] Despite this ongoing work, inflation remains by far the dominant explanation for the flatness problem.[1][4] The question arises, however, whether it is still the dominant explanation because it is the best explanation, or because the community is unaware of progress on this problem.[31] In particular, in addition to the idea that Ω is not a suitable parameter in this context, other arguments against the flatness problem have been presented: if the universe collapses in the future, then the flatness problem "exists", but only for a relatively short time, so a typical observer would not expect to measure Ω appreciably different from 1;[32] in the case of a universe which expands forever with a positive cosmological constant, fine-tuning is needed not to achieve a (nearly) flat universe, but also to avoid it.[33]
Einstein–Cartan theory
The flatness problem is naturally solved by the
See also
Notes
- Cosmic Microwave Background#Primary anisotropy.
- ^ Liddle[6] uses an alternative notation in which Ω0 is the current density of matter alone, excluding any contribution from dark energy; his Ω0+ΩΛ corresponds to Ω0 in this article.
References
- ^ ISBN 978-0-521-42270-3.
- ISBN 978-0871690784.
- ISBN 978-0-674-03363-4.
- ^ ISBN 978-0-8053-8912-8.
- ^ ISBN 978-0-471-95473-6.
- ^ ISBN 978-0-470-84835-7.
- ^ Ryden p. 168
- S2CID 119352299.
- S2CID 1386346.
- ^ Cain, Fraser; Today, Universe. "How do we know the universe is flat? Discovering the topology of the universe". phys.org. Retrieved 2023-03-26.
- ^ darkmatterdarkenergy (2015-03-06). "Planck Mission Full Results Confirm Canonical Cosmology Model". Dark Matter, Dark Energy, Dark Gravity. Retrieved 2023-03-26.
- ISSN 0004-6361.
- ^ Ryden p. 193
- ^ ISBN 978-0-7923-6311-8.
- S2CID 233403196.
- ^ doi:10.1086/151965.
- ISBN 978-0-19-851949-2.
- ^ a b c Mosterín, Jesús (2003). "Anthropic Explanations in Cosmology". Retrieved 2008-08-01.
- S2CID 4363262.
- .
- .
- ^ Brawer, Roberta (February 1996). "Inflationary Cosmology and the Horizon and Flatness Problems: The Mutual Constitution of Explanation and Questions".
- ISBN 978-0-521-56689-6.
- ISBN 978-1-4020-0155-0.
- ^ Guth, Alan (1997). "Was Cosmic Inflation the 'Bang' of the Big Bang?". The Beamline. 27. Retrieved 2008-09-07.
- ^ S2CID 118432957.
- S2CID 15885200.
- S2CID 1113031.
- Bibcode:1997AAS...190.3806A.
- S2CID 14096945..
- S2CID 119066780.
- S2CID 85526633.
- S2CID 40500958.
- .
- S2CID 118434253.