Editing Scale factor (cosmology) (section)

==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>