Editing Flatness problem (section)

==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_{c}</math> as far from it, and that speculating on a reason for any particular value was "beyond the domain of science".<ref name=Reality /> 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.<ref>{{cite journal|last = Helbig |first = Phillip|title = Arguments against the flatness problem in classical cosmology: a review |date=December 2021 |journal=European Physical Journal H |volume=46|issue = 1 |pages=10 |bibcode= 2021EPJH...46...10H |doi = 10.1140/epjh/s13129-021-00006-9| s2cid=233403196 | url=https://orbi.uliege.be/bitstream/2268/296452/1/flatness_history.pdf }}</ref>

===Anthropic principle===
{{main|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 |doi-access=free }}</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#Horizons|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=Tipler, Frank J. |title=The Anthropic Cosmological Principle |date=1986 |publisher=Clarendon Press |location=Oxford |isbn=978-0-19-851949-2 |page=[https://archive.org/details/anthropiccosmolo00barr_0/page/411 411] |url-access=registration |url=https://archive.org/details/anthropiccosmolo00barr_0/page/411 }}</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#Reception and controversies|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 | access-date = 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|s2cid = 4363262 }}</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 |doi-access=free }}</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 |doi-access=free }}</ref> His two main motivations for doing so were the flatness problem and the [[horizon problem]], another fine-tuning problem of physical cosmology. However, "In December, 1980 when Guth was developing his inflation model, he was not trying to solve either the flatness or horizon problems. Indeed, at that time, he knew nothing of the horizon problem and had never quantitatively calculated the flatness problem".<ref>{{Cite web|last=Brawer|first=Roberta|date=February 1996|title=Inflationary Cosmology and the Horizon and Flatness Problems: The Mutual Constitution of Explanation and Questions|url=https://s3.cern.ch/inspire-prod-files-b/b11715bd3ecff0e22e0fdf99d5005ca0}}</ref> He was a particle physicist trying to solve the magnetic monopole problem."

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>

However, some physicists deny that inflationary theory resolves the flatness problem, arguing that it merely moves the fine-tuning from the probability distribution to the potential of a field,<ref>{{Cite web |last=Hossenfelder |first=Sabine |date=2017-10-17 |title=Sabine Hossenfelder: Backreaction: I totally mean it: Inflation never solved the flatness problem. |url=https://backreaction.blogspot.com/2017/10/i-totally-mean-it-inflation-never.html |access-date=2024-09-29 |website=Sabine Hossenfelder}}</ref> or even deny that it is a scientific theory.<ref>{{Cite web |last=Ijjas |first=Anna |last2=Steinhardt |first2=Paul |author-link2=Paul Steinhardt |last3=Loeb |first3=Abraham |author-link3=Avi Loeb |date=2017-02-01 |title=Cosmic Inflation Theory Faces Challenges |url=https://www.scientificamerican.com/article/cosmic-inflation-theory-faces-challenges/ |access-date=2024-09-29 |website=Scientific American |language=en}}</ref><ref>{{Cite web |last=Hossenfelder |first=Sabine |date=2017-10-13 |title=Sabine Hossenfelder: Backreaction: Is the inflationary universe a scientific theory? Not anymore. |url=https://backreaction.blogspot.com/2017/10/is-inflationary-universe-scientific.html |access-date=2024-09-29 |website=Sabine Hossenfelder}}</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 |page=17 |date=August 2000 |arxiv=astro-ph/0007247 |bibcode=2001ASIC..565...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 |access-date=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-link=Hiranya Peiris |author2=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 |s2cid=118432957 }}</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 |s2cid=15885200 }}</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 |s2cid=1113031 }}</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 |s2cid=14096945 }}.</ref> Despite this ongoing work, inflation remains by far the dominant explanation for the flatness problem.<ref name="peacock"/><ref name=Ryden />  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.<ref>{{cite journal|last=Holman |first=Marc |title=How Problematic is the Near-Euclidean Spatial Geometry of the Large-Scale Universe?  |date=November 2018 |journal=Foundations of Physics |volume=48|issue=11 |pages=1617–1647 |bibcode=2018FoPh...48.1617H|doi=10.1007/s10701-018-0218-4|arxiv=1803.05148 |s2cid=119066780 }}</ref>  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;<ref>{{cite journal|last=Helbig |first=Phillip |title=Is there a flatness problem in classical cosmology? |date=March 2012 |journal=Monthly Notices of the Royal Astronomical Society |volume=421|issue=1 |pages=561–569 |bibcode=2012MNRAS.421..561H|doi=10.1111/j.1365-2966.2011.20334.x|doi-access=free |arxiv=1112.1666 |s2cid=85526633 }}</ref> 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.<ref>{{cite journal|last=Lake |first=Kayll |title=The Flatness Problem and Λ  |date=May 2005 |journal=Physical Review Letters |volume=94|issue=20 |page=201102 |bibcode=2005PhRvL..94t1102L|doi=10.1103/PhysRevLett.94.201102|pmid=16090234 |arxiv=astro-ph/0404319 |s2cid=40500958 }}</ref>

===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 |s2cid=118434253 }}</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.