Editing Flatness problem (section)

===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>