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The '''scale factor''', '''cosmic scale factor''' or sometimes the '''[[Friedmann-Lemaître-Robertson-Walker metric|Robertson-Walker scale factor]]'''<ref name=Weinberg_Cosmology>{{cite book |url=http://books.google.com/books?id=48C-ym2EmZkC&pg=PA3 |author=Steven Weinberg |title=Cosmology |publisher=Oxford University Press |page=3 |year=2008 |isbn=978-0-19-852682-7}}</ref> parameter of the [[Friedmann equations]] is a function of time which represents the [[metric expansion of space|relative expansion]] of the [[universe]]. It relates the proper distance (which can change over time, unlike the [[comoving distance]] which is constant) 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>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> 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 | year= 2003 | page = [http://books.google.com/books?id=iEZNXvYwyNwC&lpg=PP1&pg=PA363#v=onepage&q&f=false 363] | isbn = 978-0-521-45506-0 }}</ref> Thus, by definition, <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]]: <math>13.798\pm0.037\,\mathrm{Gyr}</math><ref name='planck_overview'>{{cite journal | arxiv=1303.5062 | title=Planck 2013 results. I. Overview of products and scientific results | author=Planck collaboration | journal=Submitted to Astronomy & Astrophysics | year=2013|bibcode = 2013arXiv1303.5062P }}</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: | |||
:<math>H \equiv {\dot{a}(t) \over a(t)}</math> | |||
where the dot represents a time [[derivative]]. 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 d(t)</math> which is just [[Hubble's law]]. | |||
Current evidence suggests that [[Accelerating universe|the expansion rate 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. | coauthors = Robert J. Lambourne | title = An Introduction to Galaxies and Cosmology | publisher =Cambridge University Press | year= 2004 | page = [http://books.google.com/books?id=36K1PfetZegC&lpg=PP1&pg=PA244#v=onepage&q&f=false 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)</ref> | |||
According to the [[Friedmann–Lemaître–Robertson–Walker metric]] which is used to model the expanding universe, if at the 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'', [http://books.google.com/books?id=akb2FpZSGnMC&lpg=PP1&pg=PA187#v=onepage&q&f=false p. 187].</ref><ref>Mukhanov, V. F. (2005), ''Physical Foundations of Cosmology'', [http://books.google.com/books?id=1TXO7GmwZFgC&lpg=PP1&pg=PA58#v=onepage&q&f=false p. 58].</ref> | |||
==See also== | |||
*[[Friedmann equations]] | |||
*[[Friedmann-Lemaître-Robertson-Walker metric]] | |||
*[[Redshift]] | |||
*[[Cosmological principle]] | |||
*[[Lambda-CDM model]] | |||
*[[Hubble's law]] | |||
==References== | |||
<references/> | |||
==External links== | |||
*[http://www.astro.ucla.edu/~wright/cosmo_03.htm Relation of the scale factor with the cosmological constant and the Hubble constant] | |||
{{DEFAULTSORT:Scale Factor (Cosmology)}} | |||
[[Category:Physical cosmology]] | |||
[[eo:Universa krusta faktoro]] | |||
Latest revision as of 08:24, 2 October 2013
The scale factor, cosmic scale factor or sometimes the Robertson-Walker scale factor[1] parameter of the Friedmann equations is a function of time which represents the relative expansion of the universe. It relates the proper distance (which can change over time, unlike the comoving distance which is constant) between a pair of objects, e.g. two galaxy clusters, moving with the Hubble flow in an expanding or contracting FLRW universe at any arbitrary time to their distance at some reference time . The formula for this is:
where is the proper distance at epoch , is the distance at the reference time and is the scale factor.[2] Thus, by definition, .
The scale factor is dimensionless, with counted from the birth of the universe and set to the present age of the universe: [3] giving the current value of as or .
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 parameter is defined:
where the dot represents a time derivative. From the previous equation one can see that , and also that , so combining these gives , and substituting the above definition of the Hubble parameter gives which is just Hubble's law.
Current evidence suggests that the expansion rate of the universe is accelerating, which means that the second derivative of the scale factor is positive, or equivalently that the first derivative is increasing over time.[4] This also implies that any given galaxy recedes from us with increasing speed over time, i.e. for that galaxy 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.[5]
According to the Friedmann–Lemaître–Robertson–Walker metric which is used to model the expanding universe, if at the 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 .[6][7]
See also
- Friedmann equations
- Friedmann-Lemaître-Robertson-Walker metric
- Redshift
- Cosmological principle
- Lambda-CDM model
- Hubble's law
References
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
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There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ Is the universe expanding faster than the speed of light? (see final paragraph)
- ↑ Davies, Paul (1992), The New Physics, p. 187.
- ↑ Mukhanov, V. F. (2005), Physical Foundations of Cosmology, p. 58.