View source for Scale factor (cosmology)

{{Short description|Expansion of the universe parameter}}
The [[expansion of the universe]] is parametrized by a [[dimensionless]] '''scale factor''' <math>a </math>. Also known as the '''cosmic scale factor''' or sometimes the '''Robertson–Walker scale factor''',<ref name=Weinberg_Cosmology>{{cite book |url=https://books.google.com/books?id=48C-ym2EmZkC&pg=PA3 |author=Steven Weinberg |title=Cosmology |publisher=[[Oxford University Press]] |page=3 |date=2008 |isbn=978-0-19-852682-7}}</ref> this is a key parameter of the [[Friedmann equations]].

In the early stages of the [[Big Bang]], most of the energy was in the form of radiation, and that radiation was the dominant influence on the expansion of the universe. Later, with cooling from the expansion the roles of matter and radiation changed and the universe entered a matter-dominated era. Recent results suggest that we have already entered an era dominated by [[dark energy]], but examination of the roles of matter and radiation are most important for understanding the early universe.

Using the dimensionless scale factor to characterize the expansion of the universe, the effective energy densities of radiation and matter scale differently. This leads to a '''radiation-dominated era''' in the very early universe but a transition to a '''matter-dominated era''' at a later time and, since about 4 billion years ago, a subsequent '''dark-energy-dominated era'''.<ref name="Frieman">{{Cite journal|last1=Frieman|first1=Joshua A.|last2=Turner|first2=Michael S.|last3=Huterer|first3=Dragan|date=2008-01-01|title=Dark Energy and the Accelerating Universe|journal=Annual Review of Astronomy and Astrophysics|volume=46|issue=1|pages=385–432|arxiv=0803.0982|bibcode=2008ARA&A..46..385F|doi=10.1146/annurev.astro.46.060407.145243|s2cid=15117520}}</ref>{{refn|1= <ref name="Frieman" /> p. 6: "The Universe has gone through three distinct eras: radiation-dominated, ''z'' ≳ 3000; matter-dominated, 3000 ≳ ''z'' ≳ 0.5; and dark-energy dominated, ''z'' ≲ 0.5. The evolution of the scale factor is controlled by the dominant energy form: a(t) ∝ t<sup>2/3(1+w)</sup> (for constant ''w''). During the radiation-dominated era, a(t) ∝ t<sup>1/2</sup>; during the matter-dominated era, a(t) ∝ t<sup>2/3</sup>; and for the dark-energy-dominated era, assuming ''w'' = −1, asymptotically a(t) ∝ exp(Ht)."<br />
p. 44: "Taken together, all the current data provide strong evidence for the existence of dark energy; they constrain the fraction of critical density contributed by dark energy, 0.76 ± 0.02, and the equation-of-state parameter, ''w'' ≈ −1 ± 0.1 (stat) ±0.1 (sys), assuming that ''w'' is constant. This implies that the Universe began accelerating at redshift ''z'' ∼ 0.4 and age ''t'' ∼ 10 Gyr. These results are robust – data from any one method can be removed without compromising the constraints – and they are not substantially weakened by dropping the assumption of spatial flatness."|group="notes"}}

==Detail==
Some insight into the expansion can be obtained from a Newtonian expansion model which leads to a simplified version of the Friedmann equation. It relates the proper distance (which can change over time, unlike the [[comoving distance]] <math>d_C</math> which is constant and set to today's distance) between a pair of objects, e.g. two galaxy clusters, moving with the Hubble flow in an expanding or contracting [[Friedmann–Lemaître–Robertson–Walker metric|FLRW universe]] at any arbitrary time <math>t</math> to their distance at some reference time <math>t_0</math>. The formula for this is:
<math display="block">d(t) = a(t) d_0,\,</math>
where <math>d(t)</math> is the proper distance at epoch <math>t</math>, <math>d_0</math> is the distance at the reference time <math>t_0</math>, usually also referred to as comoving distance, and <math>a(t)</math> is the scale factor.<ref>{{cite book|last=Schutz|first=Bernard|title=Gravity from the Ground Up: An Introductory Guide to Gravity and General Relativity|publisher =[[Cambridge University Press]]|date= 2003|page=[https://books.google.com/books?id=iEZNXvYwyNwC&lpg=PP1&pg=PA363 363]|isbn=978-0-521-45506-0 }}</ref> Thus, by definition, <math>d_0 = d(t_0)</math> and <math>a(t_0) = 1</math>.

The scale factor is dimensionless, with <math>t</math> counted from the birth of the universe and <math>t_0</math> set to the present [[age of the universe]]: {{val|13.799|0.021|u=Gyr}}<ref name="Planck 2015">{{cite journal|author=Planck Collaboration|year=2016|title=Planck 2015 results. XIII. Cosmological parameters (See Table 4 on page 31 of pdf).|url=https://www.research.manchester.ac.uk/portal/en/publications/planck-2015-results(491d214e-7255-415e-97b5-96d8ae621eaa).html|journal=Astronomy & Astrophysics|volume=594|pages=A13|arxiv=1502.01589|bibcode=2016A&A...594A..13P|doi=10.1051/0004-6361/201525830|s2cid=119262962}}</ref> giving the current value of <math>a</math> as <math>a(t_0)</math> or <math>1</math>.

The evolution of the scale factor is a dynamical question, determined by the equations of [[general relativity]], which are presented in the case of a locally isotropic, locally homogeneous universe by the [[Friedmann equations]].

The [[Hubble's law#Interpretation|Hubble parameter]] is defined as:

<math display="block">H(t) \equiv \frac{\dot{a}(t)}{a(t)}</math>

where the dot represents a time [[derivative]]. The Hubble parameter varies with time, not with space, with the Hubble constant <math>H_0</math> being its current value.

From the previous equation <math>d(t) = d_0 a(t)</math> one can see that <math>\dot{d}(t) = d_0 \dot{a}(t)</math>, and also that <math>d_0 = \frac{d(t)}{a(t)}</math>, so combining these gives <math>\dot{d}(t) = \frac{d(t) \dot{a}(t)}{a(t)}</math>, and substituting the above definition of the Hubble parameter gives <math>\dot{d}(t) = H(t) d(t)</math> which is just [[Hubble's law]].

Current evidence suggests that [[Accelerating universe|the expansion of the universe is accelerating]], which means that the second derivative of the scale factor <math>\ddot{a}(t)</math> is positive, or equivalently that the first derivative <math>\dot{a}(t)</math> is increasing over time.<ref>{{cite book|last=Jones|first=Mark H. |author2=Robert J. Lambourne|title=An Introduction to Galaxies and Cosmology|publisher =Cambridge University Press|date= 2004|page=[https://books.google.com/books?id=36K1PfetZegC&lpg=PP1&pg=PA244 244]|isbn=978-0-521-83738-5 }}</ref> This also implies that any given galaxy recedes from us with increasing speed over time, i.e. for that galaxy <math>\dot{d}(t)</math> is increasing with time. In contrast, the Hubble parameter seems to be decreasing with time, meaning that if we were to look at some fixed distance d and watch a series of different galaxies pass that distance, later galaxies would pass that distance at a smaller velocity than earlier ones.<ref>[http://curious.astro.cornell.edu/question.php?number=575 Is the universe expanding faster than the speed of light?] (see final paragraph) {{webarchive |url=https://web.archive.org/web/20101128035752/http://curious.astro.cornell.edu/question.php?number=575 |date=November 28, 2010 }}</ref>

According to the [[Friedmann–Lemaître–Robertson–Walker metric]] which is used to model the expanding universe, if at present time we receive light from a distant object with a [[redshift]] of ''z'', then the scale factor at the time the object originally emitted that light is <math>a(t) = \frac{1}{1 + z}</math>.<ref>Davies, Paul (1992), ''The New Physics'', [https://books.google.com/books?id=akb2FpZSGnMC&lpg=PP1&pg=PA187 p. 187].</ref><ref>Mukhanov, V. F. (2005), ''Physical Foundations of Cosmology'', [https://books.google.com/books?id=1TXO7GmwZFgC&lpg=PP1&pg=PA58 p. 58].</ref>

==Chronology==
{{see|Chronology of the universe}}

===Radiation-dominated era===
After [[Inflation (cosmology)|Inflation]], and until about 47,000 years [[cosmic time|after the Big Bang]], the dynamics of the [[early universe]] were set by [[radiation]] (referring generally to the constituents of the universe which moved [[special relativity|relativistically]], principally [[photon]]s and [[neutrino]]s).<ref>Ryden, Barbara, "Introduction to Cosmology", 2006, eqn. 5.25, 6.41</ref>

For a radiation-dominated universe the evolution of the scale factor in the [[Friedmann–Lemaître–Robertson–Walker metric]] is obtained solving the [[Friedmann equations]]:<ref>{{harvtxt|Padmanabhan|1993|p=64}}</ref>
<math display="block">a(t) \propto t^{1/2}. </math>

===Matter-dominated era===
Between about 47,000 years and 9.8 billion years [[cosmic time|after the Big Bang]],<ref>Ryden, Barbara, "Introduction to Cosmology", 2006, eqn. 6.33, 6.41</ref> the [[energy density]] of matter exceeded both the energy density of radiation and the vacuum energy density.<ref>Zelik, M and Gregory, S: "Introductory Astronomy & Astrophysics", page 497. Thompson Learning, Inc. 1998</ref>

When the [[early universe]] was about 47,000 years old (redshift 3600), [[mass–energy equivalence|mass–energy]] density surpassed the [[radiant energy|radiation energy]], although the universe remained [[optical depth|optically thick]] to radiation until the universe was about 378,000 years old (redshift 1100). This second moment in time (close to the time of [[Recombination (cosmology)|recombination]]), at which the photons which compose the [[cosmic microwave background radiation]] were last scattered, is often mistaken{{pov inline|date=October 2020}} as marking the end of the radiation era.

For a matter-dominated universe the evolution of the scale factor in the [[Friedmann–Lemaître–Robertson–Walker metric]] is easily obtained solving the [[Friedmann equations]]:
<math display="block">a(t) \propto t^{2/3}</math>

===Dark-energy-dominated era===
In [[physical cosmology]], the '''dark-energy-dominated era''' is proposed as the last of the three phases of the known universe, beginning when the Universe was about 9.8 billion years old.<ref>Ryden, Barbara, "Introduction to Cosmology", 2006, eqn. 6.33</ref> In the era of [[Inflation_(cosmology)|cosmic inflation]], the Hubble parameter is also thought to be constant, so the expansion law of the dark-energy-dominated era also holds for the inflationary prequel of the big bang.

The [[cosmological constant]] is given the symbol Λ, and, considered as a source term in the [[Einstein field equations|Einstein field equation]], can be viewed as equivalent to a "mass" of empty space, or [[dark energy]]. Since this increases with the volume of the universe, the expansion pressure is effectively constant, independent of the scale of the universe, while the other terms decrease with time. Thus, as the density of other forms of matter – dust and radiation – drops to very low concentrations, the cosmological constant (or "dark energy") term will eventually dominate the energy density of the Universe. Recent measurements of the change in Hubble constant with time, based on observations of distant [[supernova]]e, show this acceleration in expansion rate,<ref>[https://www.nobelprize.org/nobel_prizes/physics/laureates/2011/press.html The Nobel Prize in Physics 2011]. Retrieved 18 May 2017.</ref> indicating the presence of such dark energy.

For a dark-energy-dominated universe, the evolution of the scale factor in the [[Friedmann–Lemaître–Robertson–Walker metric]] is easily obtained solving the [[Friedmann equations]]:
<math display="block">a(t) \propto \exp(H_0t)</math>
Here, the coefficient <math>H_0</math>in the exponential, the [[Hubble constant]], is
<math display="block">H_0 = \sqrt{8\pi G \rho_\mathrm{full} / 3} = \sqrt{\Lambda / 3}.</math>
This exponential dependence on time makes the spacetime geometry identical to the [[de Sitter universe]], and only holds for a positive sign of the cosmological constant, which is the case according to the currently accepted value of the [[Cosmological_constant#Positive_value|cosmological constant]], Λ, that is approximately {{val|2|e=−35|u=s-2}}.
The current density of the [[observable universe]] is of the order of {{val|9.44|e=−27|u=kg/m3}} and the age of the universe is of the order of 13.8&nbsp;billion years, or {{val|4.358|e=17|u=s}}. The Hubble constant, <math>H_0</math>, is ≈{{val|70.88|u=km/s/Mpc}} (The Hubble time is 13.79&nbsp;billion years).

==See also==
*[[Cosmological principle]]
*[[Lambda-CDM model]]
*[[Redshift]]

==Notes==
{{reflist|group=notes}}

==References==
{{Reflist}}
*{{cite book|last=Padmanabhan|first=Thanu|title=Structure formation in the universe|publisher=Cambridge University Press|location=Cambridge|year=1993|isbn=978-0-521-42486-8|url=https://books.google.com/books?id=AJlOVBRZJtIC}}
*{{cite journal|title=First-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Determination of Cosmological Parameters|last=Spergel|first=D. N. |display-authors=etal|journal=Astrophysical Journal Supplement|volume=148|issue=1|pages=175–194|year=2003|url=http://adsabs.harvard.edu/cgi-bin/bib_query?arXiv:astro-ph/0302209|doi=10.1086/377226|bibcode=2003ApJS..148..175S|arxiv=astro-ph/0302209|citeseerx=10.1.1.985.6441|s2cid=10794058 }}

==External links==
*[http://www.astro.ucla.edu/~wright/cosmo_03.htm Relation of the scale factor with the cosmological constant and the Hubble constant]
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[[Category:Physical cosmology]]