Editing Flatness problem (section)

==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 {{abs|Ω &minus; 1}} 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 |url=https://archive.org/details/introductiontomo00lidd_717 |url-access=limited |edition=2nd |date=2007 |publisher=Wiley |location=Chichester; Hoboken, NJ |page=[https://archive.org/details/introductiontomo00lidd_717/page/n173 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 |s2cid=119352299 }}</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]] (WMAP, 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-style=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. |s2cid = 1386346}}</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]]. The cosmological parameters measured by [[Planck (spacecraft)|Planck spacecraft mission]] reaffirmed previous results by WMAP.<ref>{{Cite web |last1=Cain |first1=Fraser |last2=Today |first2=Universe |title=How do we know the universe is flat? Discovering the topology of the universe |url=https://phys.org/news/2017-06-universe-flat-topology.html |access-date=2023-03-26 |website=phys.org |language=en}}</ref><ref>{{Cite web |last=darkmatterdarkenergy |date=2015-03-06 |title=Planck Mission Full Results Confirm Canonical Cosmology Model |url=https://darkmatterdarkenergy.com/2015/03/07/planck-mission-full-results-confirm-canonical-cosmology-model/ |access-date=2023-03-26 |website=Dark Matter, Dark Energy, Dark Gravity |language=en}}</ref><ref>{{Cite journal |last1=Planck Collaboration |last2=Aghanim |first2=N. |last3=Akrami |first3=Y. |last4=Ashdown |first4=M. |last5=Aumont |first5=J. |last6=Baccigalupi |first6=C. |last7=Ballardini |first7=M. |last8=Banday |first8=A. J. |last9=Barreiro |first9=R. B. |last10=Bartolo |first10=N. |last11=Basak |first11=S. |last12=Battye |first12=R. |last13=Benabed |first13=K. |last14=Bernard |first14=J.-P. |last15=Bersanelli |first15=M. |date=August 2021 |title=Planck 2018 results: VI. Cosmological parameters (Corrigendum) |url=https://www.aanda.org/10.1051/0004-6361/201833910e |journal=Astronomy & Astrophysics |volume=652 |pages=C4 |doi=10.1051/0004-6361/201833910e |bibcode=2021A&A...652C...4P |issn=0004-6361|hdl=10902/24951 |hdl-access=free }}</ref>

===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]] resulting in a [[big Freeze|big freeze]]. 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/books?id=OIG0F37QrmQC&q=%22flatness+problem+was%22&pg=PT237 |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.