View source for Flatness problem

[[File:End of universe.jpg|thumb|275px|The local geometry of the universe is determined by whether the relative density Ω is less than, equal to or greater than 1. From top to bottom: a [[spherical]] universe with greater than critical density (Ω>1, k>0); a [[hyperbolic 3-manifold|hyperbolic]], underdense universe (Ω<1, k<0); and a flat universe with exactly the critical density (Ω=1, k=0). The spacetime of the universe is, unlike the diagrams, four-dimensional.]]

The '''flatness problem''' (also known as the '''oldness problem''') is a [[physical cosmology|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 [[Flatness (cosmology)|flatness]] problem, the parameter which appears fine-tuned is the [[density of the universe|density of matter and energy in the universe]]. This value affects the curvature of space-time, with a very specific [[Critical density (cosmology)|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 the total density departs rapidly from the critical value over [[cosmic time]],<ref name="peacock">{{cite book |last= Peacock|first=J. A. |title= Cosmological Physics |url= https://archive.org/details/cosmologicalphys0000peac|url-access= registration|date= 1998|publisher=Cambridge University Press |location=Cambridge |isbn= 978-0-521-42270-3}}</ref> the early universe must have had a density even closer to the critical density, departing from it by one part in 10<sup>62</sup> 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.<ref name="Dicke1970">{{cite book|author=Robert H. Dicke|title=Gravitation and the Universe: Jayne Lectures for 1969|date=1970|publisher=American Philosophical Society|isbn=978-0871690784}}</ref>{{rp|62,}}<ref name="Lightman1993">{{cite book|author=Alan P. Lightman|title=Ancient Light: Our Changing View of the Universe|url=https://books.google.com/books?id=nvk9sqbFe3UC|date=1 January 1993|publisher=Harvard University Press|isbn=978-0-674-03363-4}}</ref>{{rp|61}} 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.<ref name = Ryden>{{cite book|author = Barbara Ryden|title = Introduction to Cosmology|publisher = Addison Wesley|location = San Francisco|isbn = 978-0-8053-8912-8|date = 2002}}</ref>

==Energy density and the Friedmann equation==
According to [[Albert Einstein|Einstein]]'s [[Einstein field equations|field equations]] of [[general relativity]], the structure of [[spacetime]] is affected by the presence of [[matter]] and energy. On small scales space appears flat &ndash; 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 [[Mass–energy equivalence|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 [[Shape of the universe|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:

:<math>H^2 = \frac{8 \pi G}{3} \rho - \frac{kc^2}{a^2}</math>

Here <math>H</math> is the [[Hubble parameter]], a measure of the rate at which the universe is expanding. <math>\rho</math> is the total density of mass and energy in the universe, <math>a</math> is the [[Scale factor (cosmology)|scale factor]] (essentially the 'size' of the universe), and <math>k</math> is the curvature parameter &mdash; that is, a measure of how curved spacetime is. A positive, zero or negative value of <math>k</math> corresponds to a respectively closed, flat or open universe. The constants <math>G</math> and <math>c</math> are Newton's [[gravitational constant]] and the [[speed of light]], respectively.

[[Cosmologists]] often simplify this equation by defining a critical density, <math>\rho_c</math>. For a given value of <math>H</math>, this is defined as the density required for a flat universe, i.e. {{nowrap|<math>k = 0</math>}}. Thus the above equation implies

:<math>\rho_c = \frac{3H^2}{8\pi G}</math>.

Since the constant <math>G</math> is known and the expansion rate <math>H</math> can be measured by observing the speed at which distant galaxies are receding from us,
<math>\rho_c</math> can be determined. Its value is currently around {{nowrap|10<sup>&minus;26</sup> kg m<sup>&minus;3</sup>}}. The ratio of the actual density to this critical value is called Ω, and its difference from 1 determines the geometry of the universe: {{nowrap|Ω > 1}} corresponds to a greater than critical density, {{nowrap|<math>\rho > \rho_c</math>}}, and hence a [[closed universe]]. {{nowrap|Ω < 1}} gives a low density [[open universe]], and Ω equal to exactly 1 gives a [[flat universe]].

The Friedmann equation,

:<math>\frac{3a^2}{8\pi G}H^2 = \rho a^2 - \frac{3kc^2}{8 \pi G},</math>

can be re-arranged into

:<math>\rho_c a^2 - \rho a^2 = - \frac{3kc^2}{8 \pi G},</math>

which after factoring <math>\rho a^2</math>, and using <math>\Omega=\rho/\rho_c</math>, leads to

:<math>(\Omega^{-1} - 1)\rho a^2 = \frac{-3kc^2}{8 \pi G}.</math><ref name=Coles>{{cite book|author1=Peter Coles |author2=Francesco Lucchin |title = Cosmology|publisher = Wiley |location = Chichester |isbn = 978-0-471-95473-6|date = 1997}}</ref>

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 <math>a</math> increases, but the density <math>\rho</math> decreases as matter (or energy) becomes spread out. For the [[concordance cosmology|standard model of the universe]] which contains mainly matter and radiation for most of its history, <math>\rho</math> decreases more quickly than <math>a^2</math> increases, and so the factor {{nowrap|<math>\rho a^2</math>}} will decrease. Since the time of the [[Planck era]], shortly after the Big Bang, this term has decreased by a factor of around <math>10^{60},</math><ref name=Coles/> and so {{nowrap|<math>(\Omega^{-1} - 1)</math>}} must have increased by a similar amount to retain the constant value of their product.

==Current value of Ω==
[[File:Flatness problem density graph.svg|thumb|275px|The relative density Ω against [[cosmic time]] ''t'' (neither axis to scale). Each curve represents a possible universe: note that Ω 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 &#124;Ω&nbsp;&minus;&nbsp;1&#124; and therefore must have begun with Ω very close to 1 indeed. The red curve is a hypothetical different universe in which the initial value of Ω differed slightly too much from 1: by the present day it has diverged extremely and would not be able to support galaxies, stars or planets.]]

===Measurement===
The value of Ω at the present time is denoted Ω<sub>0</sub>. This value can be deduced by measuring the curvature of spacetime (since {{nowrap|1=Ω = 1}}, or <math>\rho=\rho_c</math>, is defined as the density for which the curvature {{nowrap|1=''k'' = 0}}). 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 (physics)|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 sky<ref group="nb">Since 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]].</ref> - 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 Ω<sub>0</sub>.<ref name=Liddle>{{cite book |last=Liddle |first=Andrew |title=An Introduction to Modern Cosmology |edition=2nd |date=2007 |publisher=Wiley |location=Chichester; Hoboken, NJ |page=157 |isbn=978-0-470-84835-7}}</ref>{{refn|Liddle<ref name=Liddle/> uses an alternative notation in which Ω<sub>0</sub> is the current density of [[matter]] alone, excluding any contribution from [[dark energy]]; his Ω<sub>0</sub>+Ω<sub>Λ</sub> corresponds to Ω<sub>0</sub> in this article.|group=nb}}

Another probe of Ω<sub>0</sub> is the frequency of [[Type Ia supernova|Type-Ia]] [[supernovae]] at different distances from Earth.<ref>Ryden p. 168</ref><ref>{{cite journal |title=Cosmological Implications of the MAXIMA-1 High-Resolution Cosmic Microwave Background Anisotropy Measurement |last=Stompor |first=Radek |bibcode=2001ApJ...561L...7S |date=2001 |display-authors=etal |journal=The Astrophysical Journal |volume=561 |issue=1 |page=L7–L10 |doi=10.1086/324438|arxiv = astro-ph/0105062 }}</ref> 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, Ω<sub>0</sub> can be inferred from the supernovae data.

Data from the [[Wilkinson Microwave Anisotropy Probe]] (measuring CMB anisotropies) combined with that from the [[Sloan Digital Sky Survey]] and observations of type-Ia supernovae constrain Ω<sub>0</sub> to be 1 within 1%.<ref name="wmap3">{{cite journal|author = D. N. Spergel|title = Wilkinson Microwave Anisotropy Probe (WMAP) Three Year Results: Implications for Cosmology|date=June 2007|journal=Astrophysical Journal Supplement Series|volume=170|issue = 2|pages=337–408|bibcode=2007ApJS..170..377S|doi = 10.1086/513700|arxiv = astro-ph/0603449|name-list-format=vanc|display-authors = 1|last2 = Bean|first2 = R.|last3 = Dore|first3 = O.|last4 = Nolta|first4 = M. R.|last5 = Bennett|first5 = C. L.|last6 = Dunkley|first6 = J.|last7 = Hinshaw|first7 = G.|last8 = Jarosik|first8 = N.|last9 = Komatsu|first9 = E. }}</ref> In other words, the term {{nowrap begin}}|Ω &minus; 1|{{nowrap end}} is currently less than 0.01, and therefore must have been less than 10<sup>&minus;62</sup> at the [[Planck era]].

===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 <math>\rho_c</math>. 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 {{nowrap|(<math>\rho > \rho_c</math>)}} 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 {{nowrap|(<math>\rho < \rho_c</math>)}} 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 [[galaxy formation|form galaxies]]. In either case the universe would contain no complex structures such as galaxies, stars, planets and any form of life.<ref>Ryden p. 193</ref>

This problem with the Big Bang model was first pointed out by [[Robert Dicke]] in 1969,<ref name=Reality>{{cite book |url=https://books.google.com/?id=OIG0F37QrmQC&pg=PT237&lpg=PT237&dq=%22flatness+problem+was%22 |title=The Reality of the Unobservable: Observability, Unobservability and Their Impact on the Issue of Scientific Realism |first=Evandro |last=Agazzi |author2=Massimo Pauri |isbn=978-0-7923-6311-8 |publisher=Springer |date=2000 |page=226|bibcode=2000ruou.book.....A }}</ref> 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 <math>\rho_{crit}</math> as far from it, and that speculating on a reason for any particular value was "beyond the domain of science".<ref name=Reality /> 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 [[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 [[Sapience|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]],<ref name="Collins Hawking">{{cite journal |bibcode=1973ApJ...180..317C |title=Why is the Universe Isotropic? |last=Collins |first=C. B. |author2=Hawking, S. |journal=Astrophysical Journal |pages=317–334 |volume=180 |date=1973 |doi=10.1086/151965 }}</ref> who in 1973 considered the existence of an [[multiple universes|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."<ref name="Collins Hawking" />

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 [[Homogeneity (physics)|inhomogeneous]] universe). Thus some regions will be over-dense {{nowrap|(Ω > 1)}} and some under-dense  {{nowrap|(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 [[Observable universe#Cosmological horizon|cosmological horizon]]s). 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.<ref>{{cite book |last=Barrow |first=John D. |author2=Tiple, Frank J. |title=The Anthropic Cosmological Principle |date=1986 |publisher=Clarendon Press |location=Oxford |isbn=978-0-19-851949-2 |page=411 |url-access=registration |url=https://archive.org/details/anthropiccosmolo00barr_0 }}</ref>

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 cluster]]s and [[void (astronomy)|voids]]).

However, the anthropic principle has been [[Anthropic principle#Criticisms|criticised]] by many scientists.<ref name="Anthropic Explanations">{{cite web | url=http://philsci-archive.pitt.edu/archive/00001658/ | last = Mosterín | first = Jesús | title = Anthropic Explanations in Cosmology | date = 2003 | accessdate = 2008-08-01 }}</ref> 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.”<ref name="Anthropic Explanations" /><ref>{{cite journal|last = Carr |first = Bernard J. |author2=Rees, Martin |title = The anthropic principle and the structure of the physical world |date=April 1979 |journal=Nature |volume=278|issue = 5705 |pages=605–612 |bibcode=1979Natur.278..605C|doi = 10.1038/278605a0}}</ref> 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]],<ref name="Anthropic Explanations" /> another explanation for the flatness problem was needed.

===Inflation===
{{Main|Cosmic inflation}}
The standard solution to the flatness problem invokes cosmic inflation, a process whereby the universe [[expanding universe|expands]] [[exponential growth|exponentially]] quickly (i.e. <math>a</math> grows as <math>e^{\lambda t}</math> with time <math>t</math>, for some constant <math>\lambda</math>) during a short period in its early history. The theory of inflation was first proposed in 1979, and published in 1981, by [[Alan Guth]].<ref>{{cite journal  |journal=[[Physical Review D]] |volume=23 |issue=2 |page=347 |doi= 10.1103/PhysRevD.23.347 |title=The Growth of Inflation |last=Castelvecchi  |first=Davide|bibcode = 1981PhRvD..23..347G  |date=1981 }}</ref><ref>{{cite journal |doi= 10.1103/PhysRevD.23.347 |title=Inflationary universe: A possible solution to the horizon and flatness problems |last=Guth |first=Alan |date=January 1981 |journal=[[Physical Review D]] | volume = 23 | issue = 2 | pages = 347–356|bibcode = 1981PhRvD..23..347G }}</ref> 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 (physics)|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 <math>\rho a^2</math> increases extremely rapidly as the scale factor <math>a</math> grows exponentially. Recalling the Friedmann Equation

:<math>(\Omega^{-1} - 1)\rho a^2 = \frac{-3kc^2}{8\pi G}</math>,

and the fact that the right-hand side of this expression is constant, the term <math> | \Omega^{-1} - 1 | </math> must therefore decrease with time.

Thus if  <math> | \Omega^{-1} - 1 | </math> initially takes any arbitrary value, a period of inflation can force it down towards 0 and leave it extremely small - around <math>10^{-62}</math> 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.<ref name=Ryden /><ref>{{cite book |last= Coles |first= Peter |author2=Ellis, George F. R. |title= Is the Universe Open or Closed? The Density of Matter in the Universe |publisher= [[Cambridge University Press]] |location= Cambridge |date= 1997 |isbn= 978-0-521-56689-6 }}</ref>

===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.<ref>{{cite book |last=Albrecht |first=Andreas |title=Proceedings of the NATO Advanced Study Institute on Structure Formation in the Universe, Cambridge 1999 |journal=Nato Asic Proc. 565: Structure Formation in the Universe |volume=565 |pages=17 |date=August 2000 |arxiv=astro-ph/0007247 |bibcode=2001sfu..conf...17A |isbn=978-1-4020-0155-0}}</ref><ref>{{cite journal|last=Guth |first=Alan |title=Was Cosmic Inflation the 'Bang' of the Big Bang? |date=1997 |journal=The Beamline |volume=27 |url=http://nedwww.ipac.caltech.edu/level5/Guth/Guth_contents.html |accessdate=2008-09-07}}</ref> 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.<ref name="Bird et al.">{{cite journal|author1=Bird, Simeon |author2=[[Hiranya Peiris|Peiris, Hiranya V.]] |author3=Easther, Richard |title=Fine-tuning criteria for inflation and the search for primordial gravitational waves |date=July 2008 |journal=Physical Review D|doi=10.1103/PhysRevD.78.083518|volume=78|issue=8|pages=083518|bibcode = 2008PhRvD..78h3518B |arxiv = 0807.3745 }}</ref> Many of these contain parameters or initial conditions which themselves require fine-tuning<ref name="Bird et al."/> 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<ref>{{cite journal|last=Chernin |first=Arthur D. |title=Cosmic vacuum and the 'flatness problem' in the concordant model |date=January 2003 |journal=New Astronomy |volume=8|issue=1 |pages=79–83 |bibcode=2003NewA....8...79C|doi=10.1016/S1384-1076(02)00180-X|arxiv = astro-ph/0211489 }}</ref> and gravity,<ref>{{cite journal|last=Nikolic |first=Hrvoje |title=Some Remarks on a Nongeometrical Interpretation of Gravity and the Flatness Problem |date=August 1999 |journal=General Relativity and Gravitation |volume=31|issue=8 |page=1211 |bibcode=1999GReGr..31.1211N|doi=10.1023/A:1026760304901|arxiv = gr-qc/9901057 }}</ref> particle production in an oscillating universe,<ref>{{cite journal|last=Anderson |first=P. R. |author2=R. Schokman |author3=M. Zaramensky |title=A Solution to the Flatness Problem via Particle Production in an Oscillating Universe |date=May 1997 |journal=Bulletin of the American Astronomical Society |volume=29 |page=828 |bibcode=1997AAS...190.3806A}}</ref> and use of a [[Bayesian statistics|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.<ref>{{cite journal|last=Evrard |first=G |author2=P. Coles |title=Getting the measure of the flatness problem |date=October 1995 |journal=Classical and Quantum Gravity |volume=12|issue=10 |pages=L93–L97 |bibcode=1995CQGra..12L..93E|doi=10.1088/0264-9381/12/10/001|arxiv = astro-ph/9507020 }}.</ref> Despite this ongoing work, inflation remains by far the dominant explanation for the flatness problem.<ref name="peacock"/><ref name=Ryden />

===Einstein–Cartan theory===
{{Main|Einstein–Cartan theory}}
The flatness problem is naturally solved by the [[Einstein–Cartan theory|Einstein–Cartan–Sciama–Kibble theory of gravity]], without an exotic form of matter required in inflationary theory.<ref>{{cite journal |author=Poplawski, N. J. |date=2010 |title=Cosmology with torsion: An alternative to cosmic inflation| journal=Phys. Lett. B |volume=694 |issue=3 |pages=181–185 |doi=10.1016/j.physletb.2010.09.056|arxiv = 1007.0587 |bibcode = 2010PhLB..694..181P }}</ref><ref>{{cite journal |author=Poplawski, N. |date=2012 |title=Nonsingular, big-bounce cosmology from spinor-torsion coupling |journal=Phys. Rev. D |volume=85 |issue=10 |pages=107502 |doi=10.1103/PhysRevD.85.107502|arxiv = 1111.4595 |bibcode = 2012PhRvD..85j7502P }}</ref> This theory extends general relativity by removing a constraint of the symmetry of the affine connection and regarding its antisymmetric part, the [[torsion tensor]], as a dynamical variable. It has no free parameters. Including torsion gives the correct conservation law for the total (orbital plus intrinsic) [[angular momentum]] of matter in the presence of gravity. The minimal coupling between torsion and Dirac spinors obeying the [[nonlinear Dirac equation]] generates a spin-spin interaction which is significant in [[fermion]]ic matter at extremely high densities. Such an interaction averts the unphysical big bang singularity, replacing it with a bounce at a finite minimum scale factor, before which the Universe was contracting. The rapid expansion immediately after the [[big bounce]] explains why the present Universe at largest scales appears spatially flat, homogeneous and isotropic. As the density of the Universe decreases, the effects of torsion weaken and the Universe smoothly enters the radiation-dominated era.

==See also==
* [[Magnetic monopole]]
* [[Horizon problem]]

==Notes==
{{reflist|group=nb}}

==References==
{{reflist|2}}

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