Editing Redshift (section)

===Cosmic expansion===
{{Main|Expansion of the universe}}

The observations of increasing redshifts from more and more distant galaxies can be modeled assuming a [[cosmological principle|homogeneous and isotropic universe]] combined with [[general relativity]].  This cosmological redshift can be written as a function of {{math|''a''}}, the time-dependent cosmic [[Scale factor (cosmology)|scale factor]]:<ref>{{Cite book |last=Peacock |first=J. A. |url=https://www.cambridge.org/core/product/identifier/9780511804533/type/book |title=Cosmological Physics |date=1998-12-28 |publisher=Cambridge University Press |isbn=978-0-521-41072-4 |edition=1 |doi=10.1017/cbo9780511804533}}</ref>{{rp|72}}
:<math>1+z = \frac{a_\mathrm{now}}{a_\mathrm{then}} = \frac{a_0}{a(t)}</math>

The scale factor is [[monotonic function|monotonically increasing]] as time passes. Thus {{math|''z''}} is positive, close to zero for local stars, and increasing for distant galaxies that appear redshifted.

Using a [[Friedmann–Robertson–Walker model]] of the expansion of the universe, redshift can be related to the age of an observed object, the so-called ''[[cosmic time]]–redshift relation''. Denote a density ratio as {{math|Ω<sub>0</sub>}}:

:<math>\Omega_0 = \frac {\rho}{ \rho_\text{crit}} \ , </math>

with {{math|''ρ''<sub>crit</sub>}} the critical density demarcating a universe that eventually crunches from one that simply expands. This density is about three hydrogen atoms per cubic meter of space.<ref Name=Weinberg>{{cite book |first=Steven | last=Weinberg |edition=2nd |title=The First Three Minutes: A Modern View of the Origin of the Universe | page=34 |isbn=9780-465-02437-7 |date=1993 |publisher=Basic Books|title-link=The First Three Minutes: A Modern View of the Origin of the Universe }}</ref> At large redshifts, {{math| ''1 + z'' > Ω<sub>0</sub><sup>−1</sup>}}, one finds:

:<math> t(z) \approx \frac {2}{3 H_0 {\Omega_0}^{1/2} } z^{-3/2}\ , </math>

where {{math|''H''<sub>0</sub>}} is the present-day [[Hubble constant]], and {{math|''z''}} is the redshift.<ref name="Bergström">{{cite book |title=Cosmology and Particle Astrophysics |url=https://books.google.com/books?id=CQYu_sutWAoC&pg=PA77 |page=77, Eq.4.79 |isbn=978-3-540-32924-4 |publisher=Springer |edition=2nd|date=2006|first1 = Lars |last1=Bergström|first2 = Ariel |last2=Goobar|author-link1=Lars Bergström (physicist) |author-link2=Ariel Goobar }}</ref><ref name = Longair>{{cite book |title=Galaxy Formation |first=M. S. |last=Longair |url=https://books.google.com/books?id=2ARuLT-tk5EC&pg=PA161 |page=161 |isbn=978-3-540-63785-1 |publisher=Springer |date=1998}}</ref>

The [[cosmological redshift]] is commonly attributed to stretching of the wavelengths of photons due to the stretching of space. This interpretation can be misleading. 
As required by [[general relativity]], the cosmological expansion of space has no effect on local physics. There is no term related to expansion in [[Maxwell's equations]] that govern light propagation. The cosmological redshift can be interpreted as an accumulation of infinitesimal Doppler shifts along the trajectory of the light.<ref name="Hogg">{{cite journal |author=Bunn |first1=E. F. |last2=Hogg |first2=D. W. |year=2009 |title=The kinematic origin of the cosmological redshift |journal=American Journal of Physics |volume=77 |issue=8 |pages=688–694 |arxiv=0808.1081 |bibcode=2009AmJPh..77..688B |doi=10.1119/1.3129103 |s2cid=1365918}}</ref>

There are several websites for calculating various times and distances from redshift, as the precise calculations require numerical integrals for most values of the parameters.<ref name="UCLA-2018">{{cite web |last=Wright |first=Edward L. |title=UCLA Cosmological Calculator |url=http://www.astro.ucla.edu/~wright/ACC.html |date=2018 |work=[[UCLA]] |access-date=6 August 2022 }} For parameter values as of 2018, H<sub>0</sub>=67.4 and Omega<sub>M</sub>=0.315, see the table at [[Lambda-CDM model#Parameters|Lambda-CDM model § Parameters]].</ref><ref name="ICRAR-2022">{{cite web |author=Staff |title=ICRAR Cosmology Calculator |url=https://cosmocalc.icrar.org/ |date=2022 |work=[[International Centre for Radio Astronomy Research]] |access-date=6 August 2022 }}</ref>

====Distinguishing between cosmological and local effects====

The redshift of a galaxy includes both a component related to [[recessional velocity]] from expansion of the universe, and a component related to the [[peculiar motion]] of the galaxy with respect to its local universe.<ref>{{cite journal
 | title=A comparison between the Doppler and cosmological redshifts
 | last=Bedran | first=M. L. | year=2002
 | journal=American Journal of Physics
 | volume=70 | issue=4 | pages=406–408
 | doi=10.1119/1.1446856 | bibcode=2002AmJPh..70..406B
 | url=http://www.df.uba.ar/users/sgil/physics_paper_doc/papers_phys/cosmo/doppler_redshift.pdf
 | access-date=2023-03-16
}}</ref> The redshift due to expansion of the universe depends upon the recessional velocity in a fashion determined by the cosmological model chosen to describe the expansion of the universe, which is very different from how Doppler redshift depends upon local velocity.<ref name="Harrison2">{{cite journal |last=Harrison |first=Edward |date=1992 |title=The redshift-distance and velocity-distance laws |journal=Astrophysical Journal, Part 1 |volume=403 |pages=28–31 |bibcode=1993ApJ...403...28H |doi=10.1086/172179 |doi-access=free}}. A pdf file can be found here [http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1993ApJ...403...28H&data_type=PDF_HIGH&whole_paper=YES&type=PRINTER&filetype=.pdf].</ref> Describing the cosmological expansion origin of redshift, cosmologist [[Edward Robert Harrison]] said, "Light leaves a galaxy, which is stationary in its local region of space, and is eventually received by observers who are stationary in their own local region of space. Between the galaxy and the observer, light travels through vast regions of expanding space. As a result, all wavelengths of the light are stretched by the expansion of space. It is as simple as that..."<ref>{{Harvnb|Harrison|2000|p=302}}.</ref> [[Steven Weinberg]] clarified, "The increase of wavelength from emission to absorption of light does not depend on the rate of change of {{math|''a''(''t'')}} [the [[Scale factor (cosmology)|scale factor]]] at the times of emission or absorption, but on the increase of {{math|''a''(''t'')}} in the whole period from emission to absorption."<ref name=Weinberg_Cosmology>{{cite book |url=https://books.google.com/books?id=48C-ym2EmZkC&pg=PA11 |first=Steven | last=Weinberg |title=Cosmology |publisher=Oxford University Press |page=11 |date=2008 |isbn=978-0-19-852682-7}}</ref>

If the universe were contracting instead of expanding, we would see distant galaxies blueshifted by an amount proportional to their distance instead of redshifted.<ref>This is only true in a universe where there are no [[peculiar velocity|peculiar velocities]]. Otherwise, redshifts combine as

:<math>1+z=(1+z_{\mathrm{Doppler}})(1+z_{\mathrm{expansion}})</math>
which yields solutions where certain objects that "recede" are blueshifted and other objects that "approach" are redshifted. For more on this bizarre result see: {{cite journal
 | last1=Davis | first1=T. M. | last2=Lineweaver | first2=C. H. | last3=Webb | first3=J. K.
 | title=Solutions to the tethered galaxy problem in an expanding universe and the observation of receding blueshifted objects
 | journal=American Journal of Physics
 | volume=71 | issue=4 | pages=358–364
 | date=April 2003 | doi=10.1119/1.1528916 
 | arxiv=astro-ph/0104349 | bibcode=2003AmJPh..71..358D | s2cid=3219383 }}</ref>