Scale factor (cosmology): Difference between revisions

Content deleted Content added

Inline

Revision as of 18:30, 28 February 2016

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 $t$ to their distance at some reference time $t_{0}$ . The formula for this is:

d(t)=a(t)d_{0},\,

where $d(t)$ is the proper distance at epoch $t$ , $d_{0}$ is the distance at the reference time $t_{0}$ and $a(t)$ is the scale factor.^[2] Thus, by definition, $a(t_{0})=1$ .

The scale factor is dimensionless, with $t$ counted from the birth of the universe and $t_{0}$ set to the present age of the universe: $13.799\pm 0.021\,\mathrm {Gyr}$ ^[3] giving the current value of $a$ as $a(t_{0})$ or $1$ .

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:

H\equiv {{\dot {a}}(t) \over a(t)}

where the dot represents a time derivative. From the previous equation $d(t)=d_{0}a(t)$ one can see that ${\dot {d}}(t)=d_{0}{\dot {a}}(t)$ , and also that $d_{0}={\frac {d(t)}{a(t)}}$ , so combining these gives ${\dot {d}}(t)={\frac {d(t){\dot {a}}(t)}{a(t)}}$ , and substituting the above definition of the Hubble parameter gives ${\dot {d}}(t)=Hd(t)$ 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 ${\ddot {a}}(t)$ is positive, or equivalently that the first derivative ${\dot {a}}(t)$ 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 ${\dot {d}}(t)$ 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 $a(t)={\frac {1}{1+z}}$ .^[6]^[7]

References

^ Steven Weinberg (2008). Cosmology. Oxford University Press. p. 3. ISBN 978-0-19-852682-7.
^ Schutz, Bernard (2003). Gravity from the Ground Up: An Introductory Guide to Gravity and General Relativity. Cambridge University Press. p. 363. ISBN 978-0-521-45506-0.
^ Planck Collaboration (2015). "Planck 2015 results. XIII. Cosmological parameters (See Table 4 on page 31 of pfd)". arXiv:1502.01589. {{cite journal}}: Cite journal requires |journal= (help)
^ Jones, Mark H.; Robert J. Lambourne (2004). An Introduction to Galaxies and Cosmology. Cambridge University Press. p. 244. ISBN 978-0-521-83738-5.
^ Is the universe expanding faster than the speed of light? (see final paragraph) Template:Wayback
^ Davies, Paul (1992), The New Physics, p. 187.
^ Mukhanov, V. F. (2005), Physical Foundations of Cosmology, p. 58.

External links

Relation of the scale factor with the cosmological constant and the Hubble constant

[Weinberg_Cosmology-1] Steven Weinberg (2008). Cosmology. Oxford University Press. p. 3. ISBN 978-0-19-852682-7.

[2] Schutz, Bernard (2003). Gravity from the Ground Up: An Introductory Guide to Gravity and General Relativity. Cambridge University Press. p. 363. ISBN 978-0-521-45506-0.

[Planck_2015-3] Planck Collaboration (2015). "Planck 2015 results. XIII. Cosmological parameters (See Table 4 on page 31 of pfd)". arXiv:1502.01589. {{cite journal}}: Cite journal requires |journal= (help)

[4] Jones, Mark H.; Robert J. Lambourne (2004). An Introduction to Galaxies and Cosmology. Cambridge University Press. p. 244. ISBN 978-0-521-83738-5.

[5] Is the universe expanding faster than the speed of light? (see final paragraph) Template:Wayback

[6] Davies, Paul (1992), The New Physics, p. 187.

[7] Mukhanov, V. F. (2005), Physical Foundations of Cosmology, p. 58.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

@@ Line 18: / Line 18: @@
 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. |author2=Robert J. Lambourne  | title = An Introduction to Galaxies and Cosmology | publisher =Cambridge University Press | date=  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>{{dead link|date=February 2016}}
+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. |author2=Robert J. Lambourne  | title = An Introduction to Galaxies and Cosmology | publisher =Cambridge University Press | date=  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) {{wayback|url=http://curious.astro.cornell.edu/question.php?number=575 |date=20101128035752 }}</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>

Revision as of 18:30, 28 February 2016

See also

References

External links