redshift#Effects due to physical optics and radiative transfer

File:High-redshift galaxy candidates in the Hubble Ultra Deep Field 2012.jpg, 2012{{cite news|title=Hubble census finds galaxies at redshifts 9 to 12|url=https://esahubble.org/news/heic1219/|access-date=13 December 2012|newspaper=ESA/Hubble Press Release}} ]]

Using a telescope and a spectrometer, the variation in intensity of star light with frequency can be measured. The resulting spectrum can be compared to the spectrum from hot gases expected in stars, such as hydrogen, in a laboratory on Earth. As illustrated with the idealized spectrum in the top-right, to determine the redshift, features in the two spectra such as absorption lines, emission lines, or other variations in light intensity may be shifted.

Redshift (and blueshift) may be characterized by the relative difference between the observed and emitted wavelengths (or frequency) of an object. In astronomy, it is customary to refer to this change using a dimensionless quantity called {{math|z}}. If {{math|λ}} represents wavelength and {{math|f}} represents frequency (note, {{math|λf {{=}} c}} where {{math|c}} is the speed of light), then {{math|z}} is defined by the equations:For a tutorial on how to define and interpret large redshift measurements, see:
{{cite web

| title=Extragalactic Redshifts

| first=John

| last=Huchra

| publisher=Harvard-Smithsonian Center for Astrophysics

| website=NASA/IPAC Extragalactic Database

| url=http://ned.ipac.caltech.edu/help/zdef.html

| access-date=2023-03-16

| archive-date=2013-12-22

| archive-url=https://web.archive.org/web/20131222052715/http://ned.ipac.caltech.edu/help/zdef.html

}}

Doppler effect blueshifts ({{math|z < 0}}) are associated with objects approaching (moving closer to) the observer with the light shifting to greater energies. Conversely, Doppler effect redshifts ({{math|z > 0}}) are associated with objects receding (moving away) from the observer with the light shifting to lower energies. Likewise, gravitational blueshifts are associated with light emitted from a source residing within a weaker gravitational field as observed from within a stronger gravitational field, while gravitational redshifting implies the opposite conditions.

The history of the subject began in the 19th century, with the development of classical wave mechanics and the exploration of phenomena which are associated with the Doppler effect. The effect is named after the Austrian mathematician, Christian Doppler, who offered the first known physical explanation for the phenomenon in 1842.

{{cite book

|last=Doppler | first=Christian

|date=1846

|title=Beiträge zur fixsternenkunde

|location=Prague |publisher=G. Haase Söhne

|bibcode=1846befi.book.....D

|volume=69

}}{{Cite book |last=Becker |first=Barbara J. |url=https://www.cambridge.org/core/product/identifier/9780511751417/type/book |title=Unravelling Starlight: William and Margaret Huggins and the Rise of the New Astronomy |date=2011-02-17 |publisher=Cambridge University Press |isbn=978-1-107-00229-6 |edition=1 |doi=10.1017/cbo9780511751417}}{{rp|107}} In 1845, the hypothesis was tested and confirmed for sound waves by the Dutch scientist Christophorus Buys Ballot.

{{cite book

|last=Maulik | first=Dev

|chapter=Doppler Sonography: A Brief History

|chapter-url=https://books.google.com/books?id=HedeGJms0n4C&q=%22Ballot%22&pg=PA3

|editor1-last=Maulik | editor1-first=Dev

|editor2-last=Zalud | editor2-first=Ivica

|date=2005

|title=Doppler Ultrasound in Obstetrics And Gynecology

|url= https://www.springer.com/west/home/medicine/gynecology?SGWID=4-10066-22-46625046-0

|isbn=978-3-540-23088-5

|publisher=Springer

}} Doppler correctly predicted that the phenomenon would apply to all waves and, in particular, suggested that the varying colors of stars could be attributed to their motion with respect to the Earth.

{{cite web

|last1=O'Connor | first1=John J.

|last2=Robertson | first2=Edmund F.

|date=1998

|url=http://www-history.mcs.st-andrews.ac.uk/Biographies/Doppler.html

|title=Christian Andreas Doppler

|work=MacTutor History of Mathematics archive

|publisher=University of St Andrews

}}

Unaware of Doppler's work, French physicist Hippolyte Fizeau in 1848, suggested that a shift in spectral lines from stars might be used to measure their motion relative to Earth.{{rp|109}} In 1850 François-Napoléon-Marie Moigno analyzed about both Doppler's and Fizeau's ideas in a publication read by both James Clerk Maxwell and William Huggins, who initially stuck to the idea that the color of stars related to their chemistry, however by 1868, Huggins was the first to determine the velocity of a star moving away from the Earth by the analysis of spectral shifts.

{{cite journal

|last=Huggins | first=William

|date=1868

|title=Further Observations on the Spectra of Some of the Stars and Nebulae, with an Attempt to Determine Therefrom Whether These Bodies are Moving towards or from the Earth, Also Observations on the Spectra of the Sun and of Comet II

|journal=Philosophical Transactions of the Royal Society of London

|volume= 158 |pages=529–564

|bibcode=1868RSPT..158..529H

|doi=10.1098/rstl.1868.0022

}}{{rp|111}}

In 1871, optical redshift was confirmed when the phenomenon was observed in Fraunhofer lines, using solar rotation, about 0.1 Å in the red.{{cite journal |last1=Nolte |first1=David D. |title=The fall and rise of the Doppler effect |journal=Physics Today |date=1 March 2020 |volume=73 |issue=3 |pages=30–35 |doi=10.1063/PT.3.4429 |doi-access=free |bibcode=2020PhT....73c..30N }} In 1887, Vogel and Scheiner discovered the "annual Doppler effect", the yearly change in the Doppler shift of stars located near the ecliptic, due to the orbital velocity of the Earth.{{cite book|last=Pannekoek|first=A.|title=A History of Astronomy |date=1961|publisher=Dover|page=451|isbn=978-0-486-65994-7}} In 1901, Aristarkh Belopolsky verified optical redshift in the laboratory using a system of rotating mirrors.

{{cite journal

|last=Bélopolsky | first=A.

|date=1901

|bibcode=1901ApJ....13...15B

|title=On an Apparatus for the Laboratory Demonstration of the Doppler-Fizeau Principle

|journal=Astrophysical Journal

|volume=13 |page=15

|doi=10.1086/140786

|doi-access=free

}}

Beginning with observations in 1912, Vesto Slipher discovered that the Andromeda Galaxy had a blue shift, indicating that it was moving towards the Earth.{{Cite book |last=Robert |first=Smith |title=The Oxford handbook of the history of modern cosmology |date=2019 |publisher=Oxford University Press |isbn=978-0-19-881766-6 |editor-last=Kragh |editor-first=Helge |chapter=Observations and the universe |oclc=1052868704 |editor-last2=Longair |editor-first2=Malcolm S.}} Slipher first reported on his measurement in the inaugural volume of the Lowell Observatory Bulletin.

{{cite journal

|last=Slipher | first=Vesto

|date=1912

|title=The radial velocity of the Andromeda Nebula

|journal=Lowell Observatory Bulletin

|volume=1 |issue=8

|pages=2.56–2.57

|bibcode=1913LowOB...2...56S

|quote=The magnitude of this velocity, which is the greatest hitherto observed, raises the question whether the velocity-like displacement might not be due to some other cause, but I believe we have at present no other interpretation for it

}} Three years later, he wrote a review in the journal Popular Astronomy.

{{cite journal

|last=Slipher | first=Vesto

|title=Spectrographic Observations of Nebulae

|journal=Popular Astronomy

|volume=23 |pages=21–24 |date=1915

|bibcode=1915PA.....23...21S

}} In it he stated that "the early discovery that the great Andromeda spiral had the quite exceptional velocity of –300 km(/s) showed the means then available, capable of investigating not only the spectra of the spirals but their velocities as well."

{{cite journal |last=Slipher | first=Vesto |date=1915 |title=Spectrographic Observations of Nebulae |journal=Popular Astronomy |volume=23 |page=22 |bibcode=1915PA.....23...21S}} Slipher reported the velocities for 15 spiral nebulae spread across the entire celestial sphere, all but three having observable "positive" (that is recessional) velocities.

Until 1923 the nature of the nebulae was unclear. By that year

Edwin Hubble had established that these were galaxies and worked out a procedure to measure distance based on the period-luminosity relation of variable Cepheids stars. This make it possible to test a prediction by Willem de Sitter in 1917 that redshift would be correlated with distance.

In 1929 Hubble combined his distance estimates with redshift data from Slipher's reports and measurements by Milton Humason to report an approximate relationship between the redshift and distance, a result now called Hubble's law.{{rp|64}}

{{cite journal

|doi=10.1073/pnas.15.3.168

|last=Hubble |first=Edwin

|date=1929

|bibcode=1929PNAS...15..168H

|title=A Relation between Distance and Radial Velocity among Extra-Galactic Nebulae

|journal=Proceedings of the National Academy of Sciences of the United States of America

|volume=15 |issue=3 |pages=168–173

|pmid=16577160

|pmc=522427

|doi-access=free

}}{{Cite web|url=https://imagine.gsfc.nasa.gov/educators/programs/cosmictimes/online_edition/1929/expanding.html|title=Universe is Expanding|date=2017-12-08|access-date=2023-09-06 |publisher=Goddard Space Flight Center}}

Theories relating to the redshift-distance relation also evolved during the decade of the 1920s.

The solution to the equations of general relativity described by de Sitter contained no matter, but in 1922 Alexander Friedmann's derived dynamic solutions, now called the Friedmann–equations, based on frictionless fluid models.{{cite journal

|last=Friedman |first=A. A.

|date=1922

|title=Über die Krümmung des Raumes

|journal=Zeitschrift für Physik

|volume=10

|issue=1 |pages=377–386

|doi=10.1007/BF01332580

|bibcode = 1922ZPhy...10..377F |s2cid=125190902

}} English translation in {{cite journal |title=On the Curvature of Space|doi=10.1023/A:1026751225741 |last=Friedman |first=A. |date=1999 |journal=General Relativity and Gravitation |volume=31 |issue=12 |pages=1991–2000 |bibcode=1999GReGr..31.1991F|s2cid=122950995 }}) Independently Georges Lemaître derived similar equations in 1927 and his analysis became widely known around the time of Hubble's key publication.{{rp|77}}

By early 1930 the combination of the redshift measurements and theoretical models established a major breakthrough in the new science of cosmology: the universe had a history and its expansion could be investigated with physical models backed up with observational astronomy.{{rp|99}}

Arthur Eddington used the term "red-shift" as early as 1923,{{Cite book |last=Eddington |first=Arthur Stanley |url=https://books.google.com/books?id=errkj2WXGzIC&pg=PA164 |title=The Mathematical Theory of Relativity |date=1923 |publisher=The University Press |page=164 |language=en |author-link=Arthur Eddington}}{{Cite OED|term=redshift|id=160477|access-date=2023-03-17}} although the word does not appear unhyphenated until about 1934, when Willem de Sitter used it.

{{cite journal

|last=de Sitter | first=W.

|date=1934

|title=On distance, magnitude, and related quantities in an expanding universe

|journal=Bulletin of the Astronomical Institutes of the Netherlands

|volume=7 |page=205

|bibcode=1934BAN.....7..205D

|quote=It thus becomes urgent to investigate the effect of the redshift and of the metric of the universe on the apparent magnitude and observed numbers of nebulae of given magnitude

}}

In the 1960s the discovery of quasars, which appear as very blue point sources and thus were initially thought to be unusual stars, lead to the idea that they were as bright as they were because they were closer than their redshift data indicated. A flurry of theoretical and observational work concluded that these objects were very powerful but distant astronomical objects.{{rp|261}}

Redshifts are differences between two wavelength measurements and wavelengths are a property of both the photons and the measuring equipment. Thus redshifts characterize differences between two measurement locations. These differences are

commonly organized in three groups, attributed to relative motion between the source and the observer, to the expansion of the universe, and to gravity.{{cite journal |last1=Lewis |first1=Geraint F. |title=On The Relativity of Redshifts: Does Space Really "Expand"? |journal=Australian Physics |date=2016 |volume=53 |page=95 |arxiv=1605.08634 }} The following sections explain these groups.

{{Main|Doppler effect|Relativistic Doppler effect}}

Image:Suzredshift.gif, yellow (~575 nm wavelength) ball appears greenish (blueshift to ~565 nm wavelength) approaching observer, turns orange (redshift to ~585 nm wavelength) as it passes, and returns to yellow when motion stops. To observe such a change in color, the object would have to be traveling at approximately 5,200 km/s, or about 32 times faster than the speed record for the fastest space probe.]]

File:Redshift blueshift.svg

If a source of the light is moving away from an observer, then redshift ({{math|z > 0}}) occurs; if the source moves towards the observer, then blueshift ({{math|z < 0}}) occurs. This is true for all electromagnetic waves and is explained by the Doppler effect. Consequently, this type of redshift is called the Doppler redshift. If the source moves away from the observer with velocity {{math|v}}, which is much less than the speed of light ({{math|v ≪ c}}), the redshift is given by

:$z\; \backslash approx\; \backslash frac\{v\}\{c\}$ (since $\backslash gamma\; \backslash approx\; 1$)

where {{math|c}} is the speed of light. In the classical Doppler effect, the frequency of the source is not modified, but the recessional motion causes the illusion of a lower frequency.

A more complete treatment of the Doppler redshift requires considering relativistic effects associated with motion of sources close to the speed of light. A complete derivation of the effect can be found in the article on the relativistic Doppler effect. In brief, objects moving close to the speed of light will experience deviations from the above formula due to the time dilation of special relativity which can be corrected for by introducing the Lorentz factor {{math|γ}} into the classical Doppler formula as follows (for motion solely in the line of sight):

:$1\; +\; z\; =\; \backslash left(1\; +\; \backslash frac\{v\}\{c\}\backslash right)\; \backslash gamma.$

This phenomenon was first observed in a 1938 experiment performed by Herbert E. Ives and G.R. Stilwell, called the Ives–Stilwell experiment.{{cite journal | last1 = Ives | first1 = H. | last2 = Stilwell | first2 = G. | year = 1938 | title = An Experimental study of the rate of a moving atomic clock | journal = Journal of the Optical Society of America | volume = 28 | issue = 7| pages = 215–226 | doi=10.1364/josa.28.000215 | bibcode = 1938JOSA...28..215I}}

Since the Lorentz factor is dependent only on the magnitude of the velocity, this causes the redshift associated with the relativistic correction to be independent of the orientation of the source movement. In contrast, the classical part of the formula is dependent on the projection of the movement of the source into the line-of-sight which yields different results for different orientations. If {{math|θ}} is the angle between the direction of relative motion and the direction of emission in the observer's frame{{cite book|last=Freund|first=Jurgen|title=Special Relativity for Beginners|date=2008|publisher=World Scientific|page=120|isbn=978-981-277-160-5}} (zero angle is directly away from the observer), the full form for the relativistic Doppler effect becomes:

:$1+\; z\; =\; \backslash frac\{1\; +\; v\; \backslash cos\; (\backslash theta)/c\}\{\backslash sqrt\{1-v^2/c^2\}\}$

and for motion solely in the line of sight ({{math|θ {{=}} 0°}}), this equation reduces to:

:$1\; +\; z\; =\; \backslash sqrt\{\backslash frac\{1+v/c\}\{1-v/c\}\}$

For the special case that the light is moving at right angle ({{math|θ {{=}} 90°}}) to the direction of relative motion in the observer's frame,{{cite book|last=Ditchburn|first=R. |title=Light|date=1961|publisher=Dover|page=329|isbn=978-0-12-218101-6}} the relativistic redshift is known as the transverse redshift, and a redshift:

:$1\; +\; z\; =\; \backslash frac\{1\}\{\backslash sqrt\{1-v^2/c^2\}\}$

is measured, even though the object is not moving away from the observer. Even when the source is moving towards the observer, if there is a transverse component to the motion then there is some speed at which the dilation just cancels the expected blueshift and at higher speed the approaching source will be redshifted.

See "[http://www.physics.uq.edu.au/people/ross/phys2100/doppler.htm Photons, Relativity, Doppler shift] {{Webarchive|url=https://web.archive.org/web/20060827063802/http://www.physics.uq.edu.au/people/ross/phys2100/doppler.htm |date=2006-08-27 }} " at the University of Queensland

{{Main|Expansion of the universe}}

The observations of increasing redshifts from more and more distant galaxies can be modeled assuming a 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:{{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}}{{rp|72}}

:$1+z\; =\; \backslash frac\{a\_\backslash mathrm\{now\}\}\{a\_\backslash mathrm\{then\}\}\; =\; \backslash frac\{a\_0\}\{a(t)\}$

The scale factor is 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|Ω₀}}:

:$\backslash Omega\_0\; =\; \backslash frac\; \{\backslash rho\}\{\; \backslash rho\_\backslash text\{crit\}\}\; \backslash \; ,$

with {{math|ρ_crit}} 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.{{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 }} At large redshifts, {{math| 1 + z > Ω₀⁻¹}}, one finds:

:$t(z)\; \backslash approx\; \backslash frac\; \{2\}\{3\; H\_0\; \{\backslash Omega\_0\}^\{1/2\}\; \}\; z^\{-3/2\}\backslash \; ,$

where {{math|H₀}} is the present-day Hubble constant, and {{math|z}} is the redshift.{{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 }}{{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}}

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.{{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}}

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.{{cite web |author=Staff |title=UCLA Cosmological Calculator |url=http://www.astro.ucla.edu/~wright/ACC.html |date=2015 |work=UCLA |access-date=6 August 2022 }} Light travel distance was calculated from redshift value using the UCLA Cosmological Calculator, with parameters values as of 2015: H₀=67.74 and Omega_M=0.3089 (see Table/Planck2015 at "Lambda-CDM model#Parameters" ){{cite web |author=Staff |title=UCLA Cosmological Calculator |url=http://www.astro.ucla.edu/~wright/ACC.html |date=2018 |work=UCLA |access-date=6 August 2022 }} Light travel distance was calculated from redshift value using the UCLA Cosmological Calculator, with parameters values as of 2018: H₀=67.4 and Omega_M=0.315 (see Table/Planck2018 at "Lambda-CDM model#Parameters" ){{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 }} ICRAR Cosmology Calculator - Set H₀=67.4 and Omega_M=0.315 (see Table/Planck2018 at "Lambda-CDM model#Parameters"){{cite web |last=Kempner |first=Joshua |title=KEMPNER Cosmology Calculator |url=https://www.kempner.net/cosmic.php |date=2022 |work=Kempner.net |access-date=6 August 2022 }} KEMP Cosmology Calculator - Set H₀=67.4, Omega_M=0.315, and Omega_Λ=0.6847 (see Table/Planck2018 at "Lambda-CDM model#Parameters")

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.{{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

}} 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.{{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]. 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..."{{Harvnb|Harrison|2000|p=302}}. 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] at the times of emission or absorption, but on the increase of {{math|a(t)}} in the whole period from emission to absorption."{{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}}

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

:$1+z=(1+z\_\{\backslash mathrm\{Doppler\}\})(1+z\_\{\backslash mathrm\{expansion\}\})$

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 }}

{{Main|Gravitational redshift}}

In the theory of general relativity, there is time dilation within a gravitational well. Light emitted within the well will appear to have fewer cycles per second when measured outside of the well, due to differences in the two clocks.{{Cite book |last=Zee |first=Anthony |title=Einstein Gravity in a Nutshell |date=2013 |publisher=Princeton University Press |isbn=978-0-691-14558-7 |edition=1st |series=In a Nutshell Series |location=Princeton}}{{rp|284}} This is known as the gravitational redshift or Einstein Shift.{{cite journal | last=Chant | first=C. A. | bibcode = 1930JRASC..24..390C | title = Notes and Queries (Telescopes and Observatory Equipment – The Einstein Shift of Solar Lines) | date = 1930 | journal = Journal of the Royal Astronomical Society of Canada | volume = 24 | page = 390 }} The theoretical derivation of this effect follows from the Schwarzschild solution of the Einstein equations which yields the following formula for redshift associated with a photon traveling in the gravitational field of an uncharged, nonrotating, spherically symmetric mass:

:$1+z=\backslash frac\{1\}\{\backslash sqrt\{1-\backslash frac\{2GM\}\{rc^2\}\}\},$

where

• {{math|G}} is the gravitational constant,
• {{math|M}} is the mass of the object creating the gravitational field,
• {{math|r}} is the radial coordinate of the source (which is analogous to the classical distance from the center of the object, but is actually a Schwarzschild coordinate), and
• {{math|c}} is the speed of light.

This gravitational redshift result can be derived from the assumptions of special relativity and the equivalence principle; the full theory of general relativity is not required.{{cite journal | last = Einstein | first = A. | author-link = Albert Einstein | date = 1907 | title = Über das Relativitätsprinzip und die aus demselben gezogenen Folgerungen | journal = Jahrbuch der Radioaktivität und Elektronik | volume = 4 | pages = 411–462 | bibcode=1908JRE.....4..411E}} See p. 458 The influence of a gravitational field on clocks

The effect is very small but measurable on Earth using the Mössbauer effect and was first observed in the Pound–Rebka experiment.{{cite journal | doi = 10.1103/PhysRevLett.4.337 | title = Apparent Weight of Photons | date = 1960 | last1 = Pound | first1 = R. | last2 = Rebka | first2 = G. | journal = Physical Review Letters | volume = 4 | issue = 7 | pages = 337–341 | bibcode=1960PhRvL...4..337P| doi-access = free }}. This paper was the first measurement. However, it is significant near a black hole, and as an object approaches the event horizon the red shift becomes infinite. It is also the dominant cause of large angular-scale temperature fluctuations in the cosmic microwave background radiation (see Sachs–Wolfe effect).{{cite journal | last1=Sachs | first1=R. K. | author-link=Rainer K. Sachs | last2=Wolfe | first2=A. M. | author-link2=Arthur M. Wolfe | date=1967 | title=Perturbations of a cosmological model and angular variations of the cosmic microwave background | journal=Astrophysical Journal | volume=147 | issue=73 | doi=10.1086/148982 | page=73 | bibcode=1967ApJ...147...73S }}

Several important special-case formulae for redshift in certain special spacetime geometries, as summarized in the following table. In all cases the magnitude of the shift (the value of {{math|z}}) is independent of the wavelength.See Binney and Merrifeld (1998), Carroll and Ostlie (1996), Kutner (2003) for applications in astronomy.

File:Look-back time by redshift.png of extragalactic observations by their redshift up to z=20.{{cite arXiv |eprint=1303.5961 |last1=Pilipenko |first1=Sergey V. |title=Paper-and-pencil cosmological calculator |date=2013 |class=astro-ph.CO }} Including [https://code.google.com/archive/p/cosmonom/downloads Fortran-90 code] upon which the citing charts and formulae are based. There are websites for calculating many such physical measures from redshift.]]

The redshift observed in astronomy can be measured because the emission and absorption spectra for atoms are distinctive and well known, calibrated from spectroscopic experiments in laboratories on Earth. When the redshifts of various absorption and emission lines from a single astronomical object are measured, {{math|z}} is found to be remarkably constant. Although distant objects may be slightly blurred and lines broadened, it is by no more than can be explained by thermal or mechanical motion of the source. For these reasons and others, the consensus among astronomers is that the redshifts they observe are due to some combination of the three established forms of Doppler-like redshifts. Alternative hypotheses and explanations for redshift such as tired light are not generally considered plausible.When cosmological redshifts were first discovered, Fritz Zwicky proposed an effect known as tired light. While usually considered for historical interests, it is sometimes, along with intrinsic redshift suggestions, utilized by nonstandard cosmologies. In 1981, H. J. Reboul summarised many [http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1981A%26AS...45..129R&db_key=AST&data_type=HTML&format=&high=42ca922c9c23806 alternative redshift mechanisms] that had been discussed in the literature since the 1930s. In 2001, Geoffrey Burbidge remarked in a [http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=2001PASP..113..899B&db_key=AST&data_type=HTML review] that the wider astronomical community has marginalized such discussions since the 1960s. Burbidge and Halton Arp, while investigating the mystery of the nature of quasars, tried to develop alternative redshift mechanisms, and very few of their fellow scientists acknowledged let alone accepted their work. Moreover, {{cite journal | title=Timescale Stretch Parameterization of Type Ia Supernova B-Band Lightcurves | first1=G. | last1=Goldhaber | first2=D. E. | last2=Groom | first3=A. | last3=Kim | first4=G. | last4=Aldering | first5=P. | last5=Astier | first6=A. | last6=Conley | first7=S. E. | last7=Deustua | first8=R. | last8=Ellis | first9=S. | last9=Fabbro | first10=A. S. | last10=Fruchter | first11=A. | last11=Goobar | first12=I. | last12=Hook | first13=M. | last13=Irwin | first14=M. | last14=Kim | first15=R. A. | last15=Knop | first16=C. | last16=Lidman | first17=R. | last17=McMahon | first18=P. E. | last18=Nugent | first19=R. | last19=Pain | first20=N. | last20=Panagia | first21=C. R. | last21=Pennypacker | first22=S. | last22=Perlmutter | first23=P. | last23=Ruiz-Lapuente | first24=B. | last24=Schaefe | first25=N. A. | last25=Walton | first26=T. | last26=York | display-authors=1 | year=2001 | journal=Astrophysical Journal | volume=558 | issue=1 | pages=359–386 | doi=10.1086/322460 | arxiv=astro-ph/0104382 | bibcode=2001ApJ...558..359G | s2cid=17237531| doi-access=free }} pointed out that alternative theories are unable to account for timescale stretch observed in type Ia supernovae

Spectroscopy, as a measurement, is considerably more difficult than simple photometry, which measures the brightness of astronomical objects through certain filters.For a review of the subject of photometry, consider: {{cite book | last=Budding | first=E. | title=Introduction to Astronomical Photometry | publisher=Cambridge University Press | date=September 24, 1993 | isbn=0-521-41867-4 }} When photometric data is all that is available (for example, the Hubble Deep Field and the Hubble Ultra Deep Field), astronomers rely on a technique for measuring photometric redshifts.The technique was first described by: {{cite conference | last=Baum | first=W. A. | year=1962 | editor-first=G. C. | editor-last=McVittie | title=Problems of extra-galactic research | page=390 | conference=IAU Symposium No. 15 }} Due to the broad wavelength ranges in photometric filters and the necessary assumptions about the nature of the spectrum at the light-source, errors for these sorts of measurements can range up to {{math|δz {{=}} 0.5}}, and are much less reliable than spectroscopic determinations.{{cite journal | last1=Bolzonella | first1=M. | last2=Miralles | first2=J.-M. | last3=Pelló | first3=R. | title=Photometric redshifts based on standard SED fitting procedures | journal=Astronomy and Astrophysics | volume=363 | pages=476–492 | year=2000 | arxiv=astro-ph/0003380 | bibcode=2000A&A...363..476B }}

However, photometry does at least allow a qualitative characterization of a redshift. For example, if a Sun-like spectrum had a redshift of {{math|z {{=}} 1}}, it would be brightest in the infrared (1000nm) rather than at the blue-green (500nm) color associated with the peak of its blackbody spectrum, and the light intensity will be reduced in the filter by a factor of four, {{math|(1 + z){{sup|2}}}}. Both the photon count rate and the photon energy are redshifted. (See K correction for more details on the photometric consequences of redshift.)A pedagogical overview of the K-correction by David Hogg and other members of the SDSS collaboration can be found at: {{cite arXiv | title=The K correction | last1=Hogg | first1=David W. | last2=Baldry | first2=Ivan K. | last3=Blanton | first3=Michael R. | last4=Eisenstein | first4=Daniel J. | display-authors=1 | date=October 2002 | eprint=astro-ph/0210394}}

Determining the redshift of an object with spectroscopy requires the wavelength of the emitted light in the rest frame of the source. Astronomical applications rely on distinct spectral lines. Redshifts cannot be calculated by looking at unidentified features whose rest-frame frequency is unknown, or with a spectrum that is featureless or white noise (random fluctuations in a spectrum). Thus gamma-ray bursts themselves cannot be used for reliable redshift measurements, but optical afterglow associated with the burst can be analyzed for redshifts.{{Cite web |title=Swift: About Swift |url=https://swift.gsfc.nasa.gov/about_swift/redshift.html |access-date=2025-04-07 |website=swift.gsfc.nasa.gov}}

In nearby objects (within our Milky Way galaxy) observed redshifts are almost always related to the line-of-sight velocities associated with the objects being observed. Observations of such redshifts and blueshifts enable astronomers to measure velocities and parametrize the masses of the orbiting stars in spectroscopic binaries. Similarly, small redshifts and blueshifts detected in the spectroscopic measurements of individual stars are one way astronomers have been able to diagnose and measure the presence and characteristics of planetary systems around other stars and have even made very detailed differential measurements of redshifts during planetary transits to determine precise orbital parameters. Some approaches are able to track the redshift variations in multiple objects at once.{{cite journal |last1=Ge |first1=Jian |last2=Van Eyken |first2=Julian |last3=Mahadevan |first3=Suvrath |author3-link=Suvrath Mahadevan |last4=Dewitt |first4=Curtis |last5=Kane |first5=Stephen R. |last6=Cohen |first6=Roger |last7=Vanden Heuvel |first7=Andrew |last8=Fleming |first8=Scott W. |last9=Guo |first9=Pengcheng |last10=Henry |first10=Gregory W. |last11=Schneider |first11=Donald P. |last12=Ramsey |first12=Lawrence W. |last13=Wittenmyer |first13=Robert A. |last14=Endl |first14=Michael |last15=Cochran |first15=William D. |display-authors=4 |date=2006 |title=The First Extrasolar Planet Discovered with a New-Generation High-Throughput Doppler Instrument |journal=The Astrophysical Journal |volume=648 |issue=1 |pages=683–695 |arxiv=astro-ph/0605247 |bibcode=2006ApJ...648..683G |doi=10.1086/505699 |s2cid=13879217 |last16=Ford |first16=Eric B. |last17=Martin |first17=Eduardo L. |last18=Israelian |first18=Garik |last19=Valenti |first19=Jeff |last20=Montes |first20=David}}

Finely detailed measurements of redshifts are used in helioseismology to determine the precise movements of the photosphere of the Sun.{{cite journal | doi = 10.1007/BF00243557 | title = Solar and stellar seismology | date = 1988 | last1 = Libbrecht | first1 = Keng | journal = Space Science Reviews | volume = 47 | issue = 3–4 |bibcode=1988SSRv...47..275L | pages=275–301| s2cid = 120897051 | url = https://authors.library.caltech.edu/104214/1/1988SSRv___47__275L.pdf }} Redshifts have also been used to make the first measurements of the rotation rates of planets,In 1871 Hermann Carl Vogel measured the rotation rate of Venus. Vesto Slipher was working on such measurements when he turned his attention to spiral nebulae. velocities of interstellar clouds,An early review by Oort, J. H. on the subject: {{cite journal | title=The formation of galaxies and the origin of the high-velocity hydrogen | journal=Astronomy and Astrophysics | volume=7 | page=381 | date=1970 | bibcode=1970A&A.....7..381O | last= Oort | first= J. H. }} the rotation of galaxies, and the dynamics of accretion onto neutron stars and black holes which exhibit both Doppler and gravitational redshifts.{{cite journal| last=Asaoka | first=Ikuko | bibcode=1989PASJ...41..763A | title=X-ray spectra at infinity from a relativistic accretion disk around a Kerr black hole | journal=Publications of the Astronomical Society of Japan | volume=41 | issue=4 | date=1989 | pages=763–778 }} The temperatures of various emitting and absorbing objects can be obtained by measuring Doppler broadening—effectively redshifts and blueshifts over a single emission or absorption line.{{cite book | last1=Rybicki | first1=G. B. | first2=A. R. | last2=Lightman | title=Radiative Processes in Astrophysics | publisher=John Wiley & Sons | year=1979 | page=288 | isbn=0-471-82759-2 }} By measuring the broadening and shifts of the 21-centimeter hydrogen line in different directions, astronomers have been able to measure the recessional velocities of interstellar gas, which in turn reveals the rotation curve of our Milky Way. Similar measurements have been performed on other galaxies, such as Andromeda. As a diagnostic tool, redshift measurements are one of the most important spectroscopic measurements made in astronomy.

The most distant objects exhibit larger redshifts corresponding to the Hubble flow of the universe. The largest-observed redshift, corresponding to the greatest distance and furthest back in time, is that of the cosmic microwave background radiation; the numerical value of its redshift is about {{math|z {{=}} 1089}} ({{math|z {{=}} 0}} corresponds to present time), and it shows the state of the universe about 13.8 billion years ago,{{cite web

| title=Cosmic Detectives

| url=http://www.esa.int/Our_Activities/Space_Science/Cosmic_detectives

| publisher=The European Space Agency (ESA)

| date=2013-04-02

| access-date=2013-04-25

}} and 379,000 years after the initial moments of the Big Bang.An accurate measurement of the cosmic microwave background was achieved by the COBE experiment. The final published temperature of 2.73 K was reported in this paper: {{cite journal | last1=Fixsen | first1=D. J. | last2=Cheng | first2=E. S. | last3=Cottingham | first3=D. A. | last4=Eplee | first4=R. E. Jr. | last5=Isaacman | first5=R. B. | last6=Mather | first6=J. C. | last7=Meyer | first7=S. S. | last8=Noerdlinger | first8=P. D. | last9=Shafer | first9=R. A. | last10=Weiss | first10=R. | last11=Wright | first11=E. L. | last12=Bennett | first12=C. L. | last13=Boggess | first13=N. W. | author-link13 = Nancy Boggess|last14=Kelsall | first14=T. | last15=Moseley | first15=S. H. | last16=Silverberg | first16=R. F. | last17=Smoot | first17=G. F. | last18=Wilkinson | first18=D. T. | date=January 1994 | title=Cosmic microwave background dipole spectrum measured by the COBE FIRAS instrument | journal=Astrophysical Journal | volume=420 | page=445 | doi=10.1086/173575 | bibcode=1994ApJ...420..445F }}. The most accurate measurement as of 2006 was achieved by the WMAP experiment.

The luminous point-like cores of quasars were the first "high-redshift" ({{math|z > 0.1}}) objects discovered before the improvement of telescopes allowed for the discovery of other high-redshift galaxies.{{cite journal |last1=Kellermann |first1=K.I. |title=The Discovery of Quasars and its Aftermath |journal=Journal of Astronomical History and Heritage |date=2014 |volume=17 |issue=3 |pages=267–282 |doi=10.3724/SP.J.1440-2807.2014.03.03 |arxiv=1304.3627 }}

For galaxies more distant than the Local Group and the nearby Virgo Cluster, but within a thousand megaparsecs or so, the redshift is approximately proportional to the galaxy's distance. This correlation was first observed by Edwin Hubble and has come to be known as Hubble's law. Vesto Slipher was the first to discover galactic redshifts, in about 1912, while Hubble correlated Slipher's measurements with distances he measured by other means to formulate his Law. Hubble's law follows in part from the Copernican principle.Peebles (1993). Because it is usually not known how luminous objects are, measuring the redshift is easier than more direct distance measurements, so redshift is sometimes in practice converted to a crude distance measurement using Hubble's law.{{Cite web |last=Halstead |first=Evan |date=2021-08-16 |title=Introduction to General Relativity: 7.3: Redshift |url=https://phys.libretexts.org/Courses/Skidmore_College/Introduction_to_General_Relativity/07:_Cosmology/7.03:_Redshift |access-date=2025-03-06 |website=Physics LibreTexts |language=en}}

Gravitational interactions of galaxies with each other and clusters cause a significant scatter in the normal plot of the Hubble diagram. The peculiar velocities associated with galaxies superimpose a rough trace of the mass of virialized objects in the universe. This effect leads to such phenomena as nearby galaxies (such as the Andromeda Galaxy) exhibiting blueshifts as we fall towards a common barycenter, and redshift maps of clusters showing a fingers of god effect due to the scatter of peculiar velocities in a roughly spherical distribution. This added component gives cosmologists a chance to measure the masses of objects independent of the mass-to-light ratio (the ratio of a galaxy's mass in solar masses to its brightness in solar luminosities), an important tool for measuring dark matter.{{cite book | first1=James | last1=Binney | first2=Scott | last2=Treimane | title=Galactic dynamics|publisher=Princeton University Press | isbn=978-0-691-08445-9 | date=1994 }}{{Page needed|date=March 2023}}

The Hubble law's linear relationship between distance and redshift assumes that the rate of expansion of the universe is constant. However, when the universe was much younger, the expansion rate, and thus the Hubble "constant", was larger than it is today. For more distant galaxies, then, whose light has been travelling to us for much longer times, the approximation of constant expansion rate fails, and the Hubble law becomes a non-linear integral relationship and dependent on the history of the expansion rate since the emission of the light from the galaxy in question. Observations of the redshift-distance relationship can be used, then, to determine the expansion history of the universe and thus the matter and energy content.{{Cite web |last=Knox |first=Lloyd |date=2016-12-22 |title=Physics 156: A Cosmology Workbook: 1.7: The Distance-Redshift Relation |url=https://phys.libretexts.org/Courses/University_of_California_Davis/Physics_156:_A_Cosmology_Workbook/01:_Workbook/1.07:_The_Distance-Redshift_Relation |access-date=2025-03-06 |website=Physics LibreTexts |language=en}}

While it was long believed that the expansion rate has been continuously decreasing since the Big Bang, observations beginning in 1988 of the redshift-distance relationship using Type Ia supernovae have suggested that in comparatively recent times the expansion rate of the universe has begun to accelerate.{{cite web|url=https://www.nobelprize.org/uploads/2019/05/popular-physicsprize2011.pdf |title=The Nobel Prize in Physics 2011: Information for the Public |website=nobelprize.org |access-date=2023-06-13}}

{{see also|List of the most distant astronomical objects#List of most distant objects by type{{!}}List of most distant objects by type}}

File:Comoving distance and lookback time (Planck 2018).png and lookback time for the Planck 2018 cosmology parameters, from redshift 0 to 15, with distance (blue solid line) on the left axis, and time (orange dashed line) on the right. Note that the time that has passed (in giga years) from a given redshift until now is not the same as the distance (in giga light years) light would have traveled from that redshift, due to the expansion of the universe over the intervening period.]]

The most reliable redshifts are from spectroscopic data,{{Cite web |title=Redshift |url=https://lco.global/spacebook/light/redshift/ |access-date=2025-03-06 |website=lco.global |publisher=Las Cumbres Observatory |language=en}} and the highest-confirmed spectroscopic redshift of a galaxy is that of JADES-GS-z14-0 with a redshift of {{math|z {{=}} 14.32}}, corresponding to 290 million years after the Big Bang.{{Cite journal |last1=Carniani |first1=Stefano |last2=Hainline |first2=Kevin |last3=D'Eugenio |first3=Francesco |last4=Eisenstein |first4=Daniel J. |last5=Jakobsen |first5=Peter |last6=Witstok |first6=Joris |last7=Johnson |first7=Benjamin D. |last8=Chevallard |first8=Jacopo |last9=Maiolino |first9=Roberto |last10=Helton |first10=Jakob M. |last11=Willott |first11=Chris |last12=Robertson |first12=Brant |last13=Alberts |first13=Stacey |last14=Arribas |first14=Santiago |last15=Baker |first15=William M. |date=2024-07-29 |title=Spectroscopic confirmation of two luminous galaxies at a redshift of 14 |journal=Nature |volume=633 |issue=8029 |language=en |pages=318–322 |doi=10.1038/s41586-024-07860-9 |issn=1476-4687|doi-access=free |pmid=39074505 |pmc=11390484 |arxiv=2405.18485 |bibcode=2024Natur.633..318C }} The previous record was held by GN-z11,{{cite journal

| title=A Remarkably Luminous Galaxy at z=11.1 Measured with Hubble Space Telescope Grism Spectroscopy

| last1=Oesch | first1=P. A. | last2=Brammer | first2=G.

| last3=van Dokkum | first3=P. G. | last4=Illingworth | first4=G. D.

| last5=Bouwens | first5=R. J. | last6=Labbé | first6=I.

| last7=Franx | first7=M. | last8=Momcheva | first8=I.

| last9=Ashby | first9=M. L. N. | last10=Fazio | first10=G. G.

| last11=Gonzalez | first11=V. | last12=Holden | first12=B.

| last13=Magee | first13=D. | last14=Skelton | first14=R. E.

| last15=Smit | first15=R. | last16=Spitler | first16=L. R.

| last17=Trenti | first17=M. | last18=Willner | first18=S. P.

| display-authors=1 | journal=The Astrophysical Journal

| date=March 1, 2016 | volume=819 | issue=2 | page=129

| arxiv=1603.00461 | doi=10.3847/0004-637X/819/2/129

| bibcode=2016ApJ...819..129O | s2cid=119262750

| doi-access=free }} with a redshift of {{math|z {{=}} 11.1}}, corresponding to 400 million years after the Big Bang, and by UDFy-38135539

{{cite journal

| display-authors=4 | first1=M. D. | last1=Lehnert

| last2=Nesvadba | first2=N. P. | last3=Cuby | first3=J. G.

| last4=Swinbank | first4=A. M. | last5=Morris | first5=S.

| last6=Clément | first6=B. | last7=Evans | first7=C. J.

| last8=Bremer | first8=M. N. | last9=Basa | first9=S.

| title=Spectroscopic Confirmation of a galaxy at redshift z = 8.6

| journal=Nature | year=2010

| volume=467 | issue=7318 | pages=940–942

| doi=10.1038/nature09462 | pmid=20962840

| bibcode=2010Natur.467..940L | arxiv=1010.4312

| s2cid=4414781

}} at a redshift of {{math|z {{=}} 8.6}}, corresponding to 600 million years after the Big Bang.

Slightly less reliable are Lyman-break redshifts, the highest of which is the lensed galaxy A1689-zD1 at a redshift {{math|z {{=}} 7.5}}{{Cite journal|last1=Watson|first1=Darach|last2=Christensen|first2=Lise|last3=Knudsen|first3=Kirsten Kraiberg|last4=Richard|first4=Johan|last5=Gallazzi|first5=Anna|last6=Michałowski|first6=Michał Jerzy|title=A dusty, normal galaxy in the epoch of reionization|journal=Nature|volume=519|issue=7543|pages=327–330|doi=10.1038/nature14164|arxiv = 1503.00002 |bibcode = 2015Natur.519..327W|pmid=25731171|year=2015|s2cid=2514879}}{{cite journal

| title=Discovery of a Very Bright Strongly Lensed Galaxy Candidate at z ~ 7.6

| first1=L. D. | last1=Bradley | first2=R. J. | last2=Bouwens

| first3=H. C. | last3=Ford | first4=G. D. | last4=Illingworth

| first5=M. J. | last5=Jee | first6=N. | last6=Benítez

| first7=T. J. | last7=Broadhurst | first8=M. | last8=Franx

| first9=B. L. | last9=Frye | first10=L. | last10=Infante

| display-authors=1 | journal=The Astrophysical Journal

| volume=678 | issue=2 | pages=647–654 | year=2008

| bibcode=2008ApJ...678..647B | s2cid=15574239

| doi=10.1086/533519 | arxiv=0802.2506

}} and the next highest being {{math|z {{=}} 7.0}}.{{cite journal

| display-authors=1 | first1=E. | last1=Egami

| first2=J.-P. | last2=Kneib | first3=G. H. | last3=Rieke

| first4=R. S. | last4=Ellis | first5=J. | last5=Richard

| first6=J. | last6=Rigby | first7=C. | last7=Papovich

| first8=D. | last8=Stark | first9=M. R. | last9=Santos

| first10=J.-S. | last10=Huang | first11=H. | last11=Dole

| first12=E. Le | last12=Floc'H | first13=P. G. | last13=Pérez-González

| title=Spitzer and Hubble Space Telescope Constraints on the Physical Properties of the z~7 Galaxy Strongly Lensed by A2218

| journal=The Astrophysical Journal

| volume=618 | issue=1 | pages=L5–L8 | year=2005

| bibcode=2005ApJ...618L...5E | doi=10.1086/427550

| arxiv=astro-ph/0411117 | s2cid=15920310 }} The most distant-observed gamma-ray burst with a spectroscopic redshift measurement was GRB 090423, which had a redshift of {{math|z {{=}} 8.2}}.{{cite journal

| title=GRB 090423 reveals an exploding star at the epoch of re-ionization

| last1=Salvaterra | first1=R. | first2=M. Della | last2=Valle

| last3=Campana | first3=S. |author-link3=Sergio Campana (astrophysicist)| last4=Chincarini | first4=G.

| last5=Covino | first5=S. | last6=d'Avanzo | first6=P.

| last7=Fernández-Soto | first7=A. | last8=Guidorzi | first8=C.

| last9=Mannucci | first9=F. | last10=Margutti | first10=R.

| last11=Thöne | first11=C. C. | last12=Antonelli | first12=L. A.

| last13=Barthelmy | first13=S. D. | last14=De Pasquale | first14=M.

| last15=d'Elia | first15=V. | last16=Fiore | first16=F.

| last17=Fugazza | first17=D. | last18=Hunt | first18=L. K.

| last19=Maiorano | first19=E. | last20=Marinoni | first20=S.

| last21=Marshall | first21=F. E. | last22=Molinari | first22=E.

| last23=Nousek | first23=J. | last24=Pian | first24=E.

| last25=Racusin | first25=J. L. | last26=Stella | first26=L.

| last27=Amati | first27=L. | last28=Andreuzzi | first28=G.

| last29=Cusumano | first29=G. | last30=Fenimore | first30=E. E.

| display-authors=4 | journal=Nature

| volume=461 | issue=7268 | pages=1258–60

| doi=10.1038/nature08445 | date=2009 | pmid=19865166

| s2cid=205218263 | bibcode=2009Natur.461.1258S |arxiv=0906.1578

}} The most distant-known quasar, ULAS J1342+0928, is at {{math|z {{=}} 7.54}}.{{cite web|url=https://news.mit.edu/2017/scientists-observe-supermassive-black-hole-infant-universe-1206|title=Scientists observe supermassive black hole in infant universe|website=MIT News |publisher=Massachusetts Institute of Technology |date=2017-12-06 |first=Jennifer |last=Chu}}{{cite journal |last1=Bañados |first1=Eduardo |last2=Venemans |first2=Bram P. |last3=Mazzucchelli |first3=Chiara |last4=Farina |first4=Emanuele P. |last5=Walter |first5=Fabian |last6=Wang |first6=Feige |last7=Decarli |first7=Roberto |last8=Stern |first8=Daniel |last9=Fan |first9=Xiaohui |last10=Davies |first10=Frederick B. |last11=Hennawi |first11=Joseph F. |last12=Simcoe |first12=Robert A. |last13=Turner |first13=Monica L. |last14=Rix |first14=Hans-Walter |last15=Yang |first15=Jinyi |last16=Kelson |first16=Daniel D. |last17=Rudie |first17=Gwen C. |last18=Winters |first18=Jan Martin |title=An 800-million-solar-mass black hole in a significantly neutral Universe at a redshift of 7.5 |journal=Nature |date=January 2018 |volume=553 |issue=7689 |pages=473–476 |doi=10.1038/nature25180 |pmid=29211709 |arxiv=1712.01860 |bibcode=2018Natur.553..473B |s2cid=205263326 }} The highest-known redshift radio galaxy (TGSS1530) is at a redshift {{math|z {{=}} 5.72}}{{cite journal|last1=Saxena|first1=A.|date=2018|title=Discovery of a radio galaxy at z = 5.72|journal=Monthly Notices of the Royal Astronomical Society|volume=480|issue=2|pages=2733–2742|arxiv=1806.01191|bibcode=2018MNRAS.480.2733S|doi=10.1093/mnras/sty1996|doi-access=free |s2cid=118830412}} and the highest-known redshift molecular material is the detection of emission from the CO molecule from the quasar SDSS J1148+5251 at {{math|z {{=}} 6.42}}.{{cite journal | doi = 10.1038/nature01821 | title = Molecular gas in the host galaxy of a quasar at redshift z = 6.42 | date = 2003 | last1 = Walter | first1 = Fabian | last2 = Bertoldi | first2 = Frank | last3 = Carilli | first3 = Chris | last4 = Cox | first4 = Pierre | last5 = Lo | first5 = K. Y. | last6 = Neri | first6 = Roberto | last7 = Fan | first7 = Xiaohui | last8 = Omont | first8 = Alain | last9 = Strauss | first9 = Michael A. | last10 = Menten | first10 = Karl M. | journal = Nature | volume = 424 | issue = 6947 | pages = 406–8 | pmid = 12879063 |bibcode=2003Natur.424..406W|arxiv = astro-ph/0307410 |s2cid = 4419009| display-authors = 4 }}

Extremely red objects (EROs) are astronomical sources of radiation that radiate energy in the red and near infrared part of the electromagnetic spectrum. These may be starburst galaxies that have a high redshift accompanied by reddening from intervening dust, or they could be highly redshifted elliptical galaxies with an older (and therefore redder) stellar population.

{{cite journal

| display-authors=4

| author=Smail, Ian

| author2=Owen, F. N.

| author3=Morrison, G. E.

| author4=Keel, W. C.

| author5=Ivison, R. J.

| author6=Ledlow, M. J.

| journal=The Astrophysical Journal | volume=581 | issue=2

| pages=844–864 | doi=10.1086/344440 | bibcode=2002ApJ...581..844S

| title=The Diversity of Extremely Red Objects

| date=2002

|arxiv = astro-ph/0208434 | s2cid=51737034

}} Objects that are even redder than EROs are termed hyper extremely red objects (HEROs).

{{cite journal

| display-authors=4

| author=Totani, Tomonori

| author2=Yoshii, Yuzuru

| author3=Iwamuro, Fumihide

| author4=Maihara, Toshinori

| author5=Motohara, Kentaro

| title=Hyper Extremely Red Objects in the Subaru Deep Field: Evidence for Primordial Elliptical Galaxies in the Dusty Starburst Phase

| journal=The Astrophysical Journal | volume=558 | issue=2

| date=2001 | pages=L87–L91 | doi=10.1086/323619

| bibcode=2001ApJ...558L..87T

|arxiv = astro-ph/0108145 | s2cid=119511017

}}

The cosmic microwave background has a redshift of {{math|z {{=}} 1089}}, corresponding to an age of approximately 379,000 years after the Big Bang and a proper distance of more than 46 billion light-years.

{{cite journal | last1 = Lineweaver | first1 = Charles | first2=Tamara M. | last2=Davis | date = 2005 | title = Misconceptions about the Big Bang | journal = Scientific American | volume = 292 | issue = 3 | pages = 36–45 | doi = 10.1038/scientificamerican0305-36 | bibcode = 2005SciAm.292c..36L }} This redshift corresponds to a shift in average temperature from 3000K down to 3K.{{cite journal|last1=Gawiser|first1=E.|last2=Silk|first2=J.|date=2000|title=The cosmic microwave background radiation|journal=Physics Reports|volume=333–334|issue=2000|pages=245–267|doi=10.1016/S0370-1573(00)00025-9|arxiv=astro-ph/0002044|bibcode = 2000PhR...333..245G |citeseerx=10.1.1.588.3349|s2cid=15398837}}

The yet-to-be-observed first light from the oldest Population III stars, not long after atoms first formed and the CMB ceased to be absorbed almost completely, may have redshifts in the range of {{math|20 < z < 100}}.{{cite journal|bibcode=2006MNRAS.373L..98N|arxiv = astro-ph/0604050 |doi = 10.1111/j.1745-3933.2006.00251.x|title=The first stars in the Universe|date=2006|last1=Naoz|first1=S.|last2=Noter|first2=S.|last3=Barkana|first3=R.|journal=Monthly Notices of the Royal Astronomical Society: Letters|volume=373|issue = 1 |pages=L98–L102 |doi-access = free |s2cid = 14454275 }} Other high-redshift events predicted by physics but not presently observable are the cosmic neutrino background from about two seconds after the Big Bang (and a redshift in excess of {{math|z > 10{{sup|10}}}}){{cite journal|bibcode=2006PhR...429..307L|arxiv = astro-ph/0603494 |doi = 10.1016/j.physrep.2006.04.001|title=Massive neutrinos and cosmology|date=2006|last1=Lesgourgues|first1=J|last2=Pastor|first2=S|journal=Physics Reports|volume=429|issue=6|pages=307–379 |s2cid = 5955312 }} and the cosmic gravitational wave background emitted directly from inflation at a redshift in excess of {{math|z > 10{{sup|25}}}}.{{cite journal|bibcode=2005PhyU...48.1235G|arxiv = gr-qc/0504018 |doi = 10.1070/PU2005v048n12ABEH005795|title=Relic gravitational waves and cosmology|date=2005|last1=Grishchuk|first1=Leonid P|journal=Physics-Uspekhi|volume=48|issue=12|pages=1235–1247 |s2cid = 11957123 }}

In June 2015, astronomers reported evidence for Population III stars in the Cosmos Redshift 7 galaxy at {{math|z {{=}} 6.60}}. Such stars are likely to have existed in the very early universe (i.e., at high redshift), and may have started the production of chemical elements heavier than hydrogen that are needed for the later formation of planets and life as we know it.{{cite journal |last1=Sobral |first1=David |last2=Matthee |first2=Jorryt |last3=Darvish |first3=Behnam |last4=Schaerer |first4=Daniel |last5=Mobasher |first5=Bahram |last6=Röttgering |first6=Huub J. A. |last7=Santos |first7=Sérgio |last8=Hemmati |first8=Shoubaneh |title=Evidence For POPIII-Like Stellar Populations In The Most Luminous LYMAN-α Emitters At The Epoch Of Re-Ionisation: Spectroscopic Confirmation |date=4 June 2015 |journal=The Astrophysical Journal |doi=10.1088/0004-637x/808/2/139 |bibcode=2015ApJ...808..139S |volume=808 |issue=2 |page=139|arxiv=1504.01734|s2cid=18471887 }}{{cite news |last=Overbye |first=Dennis |author-link=Dennis Overbye |title=Astronomers Report Finding Earliest Stars That Enriched Cosmos |url=https://www.nytimes.com/2015/06/18/science/space/astronomers-report-finding-earliest-stars-that-enriched-cosmos.html |date=17 June 2015 |work=The New York Times |access-date=17 June 2015 }}

{{Main|Redshift survey}}

File:2dfgrs.png

With advent of automated telescopes and improvements in spectroscopes, a number of collaborations have been made to map the universe in redshift space. By combining redshift with angular position data, a redshift survey maps the 3D distribution of matter within a field of the sky. These observations are used to measure properties of the large-scale structure of the universe. The Great Wall, a vast supercluster of galaxies over 500 million light-years wide, provides a dramatic example of a large-scale structure that redshift surveys can detect.{{cite journal | title=Mapping the Universe | first1=M. J. | last1=Geller | first2=J. P. | last2=Huchra | journal=Science | volume=246 | issue=4932 | pages=897–903 | year=1989 | doi=10.1126/science.246.4932.897 | pmid=17812575 | bibcode=1989Sci...246..897G | s2cid=31328798 }}

The first redshift survey was the CfA Redshift Survey, started in 1977 with the initial data collection completed in 1982.See the CfA website for more details: {{cite web

| title=The CfA Redshift Survey

| first=John P. | last=Huchra | author-link=John Huchra

| publisher=Harvard & Smithsonian Center for Astrophysics

| url=https://lweb.cfa.harvard.edu/~dfabricant/huchra/zcat/

| access-date=2023-03-20

}} More recently, the 2dF Galaxy Redshift Survey determined the large-scale structure of one section of the universe, measuring redshifts for over 220,000 galaxies; data collection was completed in 2002, and the final data set was released 30 June 2003.{{cite journal

|title=The 2dF galaxy redshift survey: Power-spectrum analysis of the final dataset and cosmological implications

| first1=Shaun | last1=Cole | author-link=Shaun Cole

| last2=Percival | first2=Will J. | last3=Peacock | first3=John A.

| last4=Norberg | first4=Peder | last5=Baugh | first5=Carlton M.

| last6=Frenk | first6=Carlos S. | last7=Baldry | first7=Ivan

| last8=Bland-Hawthorn | first8=Joss | last9=Bridges | first9=Terry

| last10=Cannon | first10=Russell | last11=Colless | first11=Matthew

| last12=Collins | first12=Chris | last13=Couch | first13=Warrick

| last14=Cross | first14=Nicholas J. G. | last15=Dalton | first15=Gavin

| last16=Eke | first16=Vincent R. | last17=De Propris | first17=Roberto

| last18=Driver | first18=Simon P. | last19=Efstathiou | first19=George

| last20=Ellis | first20=Richard S. | last21=Glazebrook | first21=Karl

| last22=Jackson | first22=Carole | last23=Jenkins | first23=Adrian

| last24=Lahav | first24=Ofer | last25=Lewis | first25=Ian

| last26=Lumsden | first26=Stuart | last27=Maddox | first27=Steve

| last28=Madgwick | first28=Darren | last29=Peterson | first29=Bruce A.

| last30=Sutherland | first30=Will | last31=Taylor | first31=Keith

| journal=Monthly Notices of the Royal Astronomical Society

| volume=362 | issue=2 | pages=505–34 | date=2005

| bibcode=2005MNRAS.362..505C | arxiv=astro-ph/0501174

| doi=10.1111/j.1365-2966.2005.09318.x

| doi-access=free | s2cid=6906627| display-authors=4

}} [http://msowww.anu.edu.au/2dFGRS/ 2dF Galaxy Redshift Survey homepage] {{Webarchive|url=https://web.archive.org/web/20070205010241/http://msowww.anu.edu.au/2dFGRS/ |date=2007-02-05 }} The Sloan Digital Sky Survey (SDSS), is ongoing as of 2013 and aims to measure the redshifts of around 3 million objects.{{cite web | url=https://www.sdss3.org/ | access-date=2023-03-20 | title=SDSS-III | website=www.sdss3.org }} SDSS has recorded redshifts for galaxies as high as 0.8, and has been involved in the detection of quasars beyond {{math|z {{=}} 6}}. The DEEP2 Redshift Survey uses the Keck telescopes with the new "DEIMOS" spectrograph; a follow-up to the pilot program DEEP1, DEEP2 is designed to measure faint galaxies with redshifts 0.7 and above, and it is therefore planned to provide a high-redshift complement to SDSS and 2dF.{{cite conference | title=Science objectives and early results of the DEEP2 redshift survey| first1=Marc | last1=Davis |collaboration=DEEP2 collaboration |date=2002 | conference=Conference on Astronomical Telescopes and Instrumentation, Waikoloa, Hawaii, 22–28 Aug 2002 | arxiv=astro-ph/0209419 | bibcode=2003SPIE.4834..161D | doi=10.1117/12.457897 }}

The interactions and phenomena summarized in the subjects of radiative transfer and physical optics can result in shifts in the wavelength and frequency of electromagnetic radiation. In such cases, the shifts correspond to a physical energy transfer to matter or other photons rather than being by a transformation between reference frames. Such shifts can be from such physical phenomena as coherence effects or the scattering of electromagnetic radiation whether from charged elementary particles, from particulates, or from fluctuations of the index of refraction in a dielectric medium as occurs in the radio phenomenon of radio whistlers. While such phenomena are sometimes referred to as "redshifts" and "blueshifts", in astrophysics light-matter interactions that result in energy shifts in the radiation field are generally referred to as "reddening" rather than "redshifting" which, as a term, is normally reserved for the effects discussed above.

In many circumstances scattering causes radiation to redden because entropy results in the predominance of many low-energy photons over few high-energy ones (while conserving total energy). Except possibly under carefully controlled conditions, scattering does not produce the same relative change in wavelength across the whole spectrum; that is, any calculated {{math|z}} is generally a function of wavelength. Furthermore, scattering from random media generally occurs at many angles, and {{math|z}} is a function of the scattering angle. If multiple scattering occurs, or the scattering particles have relative motion, then there is generally distortion of spectral lines as well.

In interstellar astronomy, visible spectra can appear redder due to scattering processes in a phenomenon referred to as interstellar reddening—similarly Rayleigh scattering causes the atmospheric reddening of the Sun seen in the sunrise or sunset and causes the rest of the sky to have a blue color. This phenomenon is distinct from redshifting because the spectroscopic lines are not shifted to other wavelengths in reddened objects and there is an additional dimming and distortion associated with the phenomenon due to photons being scattered in and out of the line of sight.{{Cite web |last=Impey |first=Chris |editor-last=Gay |editor-first=Pamela |title=Dust Extinction and Reddening |url=https://www.teachastronomy.com/textbook/The-Milky-Way/Dust-Extinction-and-Reddening/ |access-date=2025-03-06 |website=Teach Astronomy - Dust Extinction and Reddening |publisher=Teach Astronomy |language=en}}

{{redirect|Blueshift|the term as used in photochemistry|hypsochromic shift|the political phenomenon|blue shift (politics)|other uses of "blueshift" or "blue shift"}}

The opposite of a redshift is a blueshift. A blueshift is any decrease in wavelength (increase in energy), with a corresponding increase in frequency, of an electromagnetic wave. In visible light, this shifts a color towards the blue end of the spectrum.

File:Redshift blueshift.svg

Doppler blueshift is caused by movement of a source towards the observer. The term applies to any decrease in wavelength and increase in frequency caused by relative motion, even outside the visible spectrum. Only objects moving at near-relativistic speeds toward the observer are noticeably bluer to the naked eye, but the wavelength of any reflected or emitted photon or other particle is shortened in the direction of travel.{{cite book|title=In Quest of the Universe | first1=Karl F. | last1=Kuhn | first2=Theo | last2=Koupelis |year= 2004|publisher=Jones & Bartlett Publishers|isbn=978-0-7637-0810-8|pages=122–3}}

Doppler blueshift is used in astronomy to determine relative motion:

• The Andromeda Galaxy is moving toward our own Milky Way galaxy within the Local Group; thus, when observed from Earth, its light is undergoing a blueshift.{{cite book |last=Woodhouse |first=Chris |chapter=M31 (Andromeda Galaxy) |date=2017-12-04 |title=The Astrophotography Manual |pages=308–313 |edition=2nd |publisher=Routledge |language=en |doi=10.4324/9781315159225-42 |isbn=978-1-315-15922-5}}
• Components of a binary star system will be blueshifted when moving towards Earth
• When observing spiral galaxies, the side spinning toward us will have a slight blueshift relative to the side spinning away from us (see Tully–Fisher relation).
• Blazars are known to propel relativistic jets toward us, emitting synchrotron radiation and bremsstrahlung that appears blueshifted.{{Cite book |arxiv=2412.11565 |doi=10.69646/aob104p029 |chapter=Monitoring Blazar Variability to Understand Extragalactic Jets |title=Publications of the Astronomical Observatory of Belgrade |date=2024 |last1=Maria Raiteri |first1=Claudia |volume=104 |pages=29–38 |isbn=978-86-82296-11-9 }}
• Nearby stars such as Barnard's Star are moving toward us, resulting in a very small blueshift.
• Doppler blueshift of distant objects with a high z can be subtracted from the much larger cosmological redshift to determine relative motion in the expanding universe.{{cite journal | title = The Largest Blueshifts of the [O III] Emission Line in Two Narrow-Line Quasars | journal = Astrophysical Journal | date = January 2005 | first1=Kentaro | last1=Aoki | first2=Toshihiro | last2=Kawaguchi | first3=Kouji | last3=Ohta | volume = 618 | issue = 2 | pages = 601–608 |arxiv = astro-ph/0409546 |bibcode = 2005ApJ...618..601A |doi = 10.1086/426075 | s2cid = 17680991 }}

Image:Gravitional well.jpg (protons, electrons, photons, etc.) falling into a gravity well become more energetic and undergo observer-independent blueshifting.]]

Unlike the relative Doppler blueshift, caused by movement of a source towards the observer and thus dependent on the received angle of the photon, gravitational blueshift is absolute and does not depend on the received angle of the photon:

{{Blockquote|Photons climbing out of a gravitating object become less energetic. This loss of energy is known as a "redshifting", as photons in the visible spectrum would appear more red. Similarly, photons falling into a gravitational field become more energetic and exhibit a blueshifting. ... Note that the magnitude of the redshifting (blueshifting) effect is not a function of the emitted angle or the received angle of the photon—it depends only on how far radially the photon had to climb out of (fall into) the potential well.{{cite web| first=R. J. | last=Nemiroff| title=Gravitational Principles and Mathematics| url=http://antwrp.gsfc.nasa.gov/htmltest/gifcity/nslens_math.html| date=1993| publisher=NASA}}{{cite journal| first=R. J. | last=Nemiroff| title=Visual distortions near a neutron star and black hole| date=1993| journal=American Journal of Physics| volume=61| issue=7| pages=619–632| bibcode=1993AmJPh..61..619N| doi=10.1119/1.17224| arxiv=astro-ph/9312003v1| s2cid=16640860}}}}

It is a natural consequence of conservation of energy and mass–energy equivalence, and was confirmed experimentally in 1959 with the Pound–Rebka experiment. Gravitational blueshift contributes to cosmic microwave background (CMB) anisotropy via the Sachs–Wolfe effect: when a gravitational well evolves while a photon is passing, the amount of blueshift on approach will differ from the amount of gravitational redshift as it leaves the region.{{cite book | last1 = Bonometto | first1 = Silvio | last2 = Gorini | first2 = Vittorio | last3 = Moschella | first3 = Ugo | title = Modern Cosmology | publisher = CRC Press | date = 2002 | isbn = 978-0-7503-0810-6 }}

There are faraway active galaxies that show a blueshift in their [O III] emission lines. One of the largest blueshifts is found in the narrow-line quasar, PG 1543+489, which has a relative velocity of −1150 km/s. These types of galaxies are called "blue outliers".

In a hypothetical universe undergoing a runaway Big Crunch contraction, a cosmological blueshift would be observed, with galaxies further away being increasingly blueshifted—the exact opposite of the actually observed cosmological redshift in the present expanding universe.{{Cite web |last=Miller |first=Cole |title=Cosmology |url=https://www.astro.umd.edu/~miller/teaching/questions/cosmology.html |access-date=2025-03-06 |website=www.astro.umd.edu}}

• Gravitational potential
• Mattig formula
• Relativistic Doppler effect

{{Reflist|30em}}

• Odenwald, S. & Fienberg, RT. 1993; "Galaxy Redshifts Reconsidered" in Sky & Telescope Feb. 2003; pp31–35 (This article is useful further reading in distinguishing between the 3 types of redshift and their causes.)
• Lineweaver, Charles H. and Tamara M. Davis, "[https://web.archive.org/web/20070715030354/http://www.sciam.com/article.cfm?chanID=sa006&colID=1&articleID=0009F0CA-C523-1213-852383414B7F0147 Misconceptions about the Big Bang]", Scientific American, March 2005. (This article is useful for explaining the cosmological redshift mechanism as well as clearing up misconceptions regarding the physics of the expansion of space.)

• {{cite book | last=Nussbaumer|first=Harry|author2=Lydia Bieri |author2-link=Lydia Bieri|title=Discovering the Expanding Universe|publisher=Cambridge University Press|date=2009|isbn=978-0-521-51484-2}}
• {{cite book | last=Binney|first=James|author2=Michael Merrifeld |title=Galactic Astronomy|publisher=Princeton University Press|date=1998|isbn=978-0-691-02565-0}}
• {{cite book | author=Carroll, Bradley W. | author2=Dale A. Ostlie | name-list-style=amp| title=An Introduction to Modern Astrophysics| publisher=Addison-Wesley Publishing Company, Inc.| date=1996| isbn=978-0-201-54730-6}}
• {{cite book | author=Feynman, Richard | author2=Leighton, Robert | author3=Sands, Matthew | title=Feynman Lectures on Physics. Vol. 1 | publisher=Addison-Wesley | date=1989 | isbn=978-0-201-51003-4| title-link=The Feynman Lectures on Physics }}
• {{cite book | last = Grøn | first = Øyvind |author-link=Øyvind Grøn|author2=Hervik, Sigbjørn | title = Einstein's General Theory of Relativity | location = New York | publisher = Springer | date = 2007 | isbn = 978-0-387-69199-2}}
• {{cite book |last=Harrison |first=Edward |date=2000 |title=Cosmology: The Science of the Universe |edition=2nd |publisher=Cambridge University Press |isbn=978-0-521-66148-5}}
• {{cite book | author=Kutner, Marc | title=Astronomy: A Physical Perspective | url=https://archive.org/details/astronomyphysica00kutn | url-access=registration | publisher=Cambridge University Press | date=2003 | isbn=978-0-521-52927-3}}
• {{cite book | last = Misner | first = Charles | author2 = Thorne, Kip S. | author3 = Wheeler, John Archibald | title = Gravitation | location = San Francisco | publisher = W. H. Freeman | date = 1973 | isbn = 978-0-7167-0344-0}}
• {{cite book | first = P. J. E. | last = Peebles | title = Principles of Physical Cosmology | publisher = Princeton University Press | date = 1993 | isbn = 978-0-691-01933-8 | url = https://archive.org/details/principlesofphys00pjep }}
• {{cite book | title=Spacetime Physics: Introduction to Special Relativity | edition=2nd | publisher=W.H. Freeman | date=1992 | isbn=978-0-7167-2327-1 | last1=Taylor | first1=Edwin F. | last2=Wheeler | first2=John Archibald | author-link2=John Archibald Wheeler | url=https://archive.org/details/spacetimephysics00edwi_0 }}
• {{cite book | first = Steven | last = Weinberg | title = Gravitation and Cosmology | publisher = John Wiley | date = 1971 | isbn = 978-0-471-92567-5 | url = https://archive.org/details/gravitationcosmo00stev_0 }}
• See also physical cosmology textbooks for applications of the cosmological and gravitational redshifts.

{{Commons|Redshift}}

{{wiktionary}}

• [http://www.astro.ucla.edu/~wright/doppler.htm Ned Wright's Cosmology tutorial]
• [https://web.archive.org/web/20051203093117/http://coolcosmos.ipac.caltech.edu/cosmic_classroom/cosmic_reference/redshift.html Cosmic reference guide entry on redshift]
• [https://web.archive.org/web/20051121214031/http://www.asterism.org/tutorials/tut29-1.htm Mike Luciuk's Astronomical Redshift tutorial]
• [http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/Images/hu_animexp.gif Animated GIF of Cosmological Redshift] by Wayne Hu
• {{cite web|last1=Merrifield|first1=Michael|last2=Hill|first2=Richard|title=Z Redshift|url=http://www.sixtysymbols.com/videos/redshift.htm|work=SIXTψ SYMBΦLS|date=2009|publisher=Brady Haran for the University of Nottingham}}

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