Absorption lines in thevisible spectrum of asupercluster of distant galaxies (right), as compared to absorption lines in the visible spectrum of theSun (left). Arrows indicate redshift. Wavelength increases up towards the red and beyond (frequency decreases).
Three forms of redshift occur inastronomy andcosmology:Doppler redshifts due to the relative motions of radiation sources,gravitational redshift as radiation escapes fromgravitational potentials, and cosmological redshifts caused by theuniverse expanding. Inastronomy, the value of a redshift is often denoted by the letterz, corresponding to the fractional change in wavelength (positive for redshifts, negative for blueshifts), and by the wavelength ratio1 +z (which is greater than 1 for redshifts and less than 1 for blueshifts). Automated astronomical redshift surveys are an important tool for learning about the large-scale structure of the universe. Redshift and blueshift can also be related tophoton energy and, viaPlanck's law, to a correspondingblackbody temperature.
Other physical processes exist that can lead to a shift in the frequency of electromagnetic radiation, includingscattering andoptical effects; however, the resulting changes are distinguishable from (astronomical) redshift and are not generally referred to as such.
Using a telescope and aspectrometer, 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 ashydrogen, in a laboratory on Earth. As illustrated with the idealised spectrum in the top-right, to determine the redshift, features in the two spectra such asabsorption lines,emission lines, or other variations in light intensity may be shifted.
Redshift (and blueshift) may be characterised 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 adimensionless quantity calledz. Ifλ represents wavelength andf represents frequency (note,λf =c wherec is thespeed of light), thenz is defined by the equations:[3]
Calculation of redshift,
Based on wavelength
Based on frequency
Doppler effect blueshifts (z < 0) are associated with objects approaching (moving closer to) the observer with the light shifting to greaterenergies. Conversely, Doppler effect redshifts (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 weakergravitational 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 ofclassical wave mechanics and the exploration of phenomena which are associated with theDoppler effect. The effect is named after the Austrian mathematicianChristian Doppler, who offered the first known physical explanation for the phenomenon in 1842.[4][5]: 107 In 1845, the hypothesis was tested and confirmed forsound waves by the Dutch scientistChristophorus Buys Ballot.[6] Doppler correctly predicted that the phenomenon would apply to allwaves and, in particular, suggested that the varyingcolors of stars could be attributed to their motion with respect to the Earth.[7]
Unaware of Doppler's work, French physicistHippolyte Fizeau suggested in 1848 that a shift inspectral lines from stars might be used to measure their motion relative to Earth.[5]: 109 In 1850,François-Napoléon-Marie Moigno analysed both Doppler's and Fizeau's ideas in a publication read by bothJames Clerk Maxwell andWilliam 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.[8][5]: 111
In 1871, optical redshift was confirmed when the phenomenon was observed inFraunhofer lines, using solar rotation, about 0.1 Å in the red.[9] In 1887,Hermann Carl Vogel andJulius 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.[10] In 1901,Aristarkh Belopolsky verified optical redshift in the laboratory using a system of rotating mirrors.[11][9]
Beginning with observations in 1912,Vesto Slipher discovered that theAndromeda Galaxy had a blue shift, indicating that it was moving towards the Earth.[12] Slipher first reported his measurement in the inaugural volume of theLowell Observatory Bulletin.[13] Three years later, he wrote a review in the journalPopular Astronomy.[14] 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."[14] Slipher reported the velocities for 15spiral nebulae spread across the entirecelestial sphere, all but three having observable "positive" (that is recessional) velocities.[12]
Until 1923 the nature of the nebulae was unclear. By that yearEdwin Hubble had established that these weregalaxies and worked out a procedure to measure distance based on the period-luminosity relation of variableCepheids stars. This made it possible to test a prediction byWillem 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 byMilton Humason to report an approximate relationship between the redshift and distance, a result now calledHubble's law.[12]: 64 [15][16]
Theories relating to the redshift-distance relation also evolved during the 1920s. The solution to the equations of general relativity described by de Sitter contained no matter, but in 1922Alexander Friedmann derived dynamic solutions, now called theFriedmann equations, based on frictionless fluid models.[17] IndependentlyGeorges Lemaître derived similar equations in 1927 and his analysis became widely known around the time of Hubble's key publication.[12]: 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.[12]: 99
When cosmological redshifts were first discovered,Fritz Zwicky proposed an effect known astired light. However this model has largely been ruled out by timescale stretch observations intype Ia supernovae.[18]
Arthur Eddington used the term "red shift" as early as 1923, which is the oldest example of the term reported by theOxford English Dictionary.[19][20] Willem de Sitter used the single-word versionredshift in 1934.[21]
In the 1960s the discovery ofquasars, which appear as very blue point sources and thus were initially thought to be unusual stars, led 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.[12]: 261
Redshifts are differences between two wavelength measurements and wavelengths are a property of both the photons and the measuring equipment. Thus redshifts characterise differences between two measurement locations. These differences are commonly organised in three groups, attributed to relative motion between the source and the observer, to the expansion of the universe, and to gravity.[22] The following sections explain these groups.
Doppler effect, yellow (c. 575nm wavelength) ball appears greenish (blueshift to c. 565 nm wavelength) approaching observer, turnsorange (redshift to c. 585 nm wavelength) as it passes, and returns to yellow when motion stops. To observe such a change in colour, the object would have to be travelling at approximately 5,200km/s, or about 32 times faster than the speed record for thefastest space probe.Redshift and blueshift
If a source of the light is moving away from an observer, then redshift (z > 0) occurs; if the source moves towards the observer, thenblueshift (z < 0) occurs. This is true for all electromagnetic waves and is explained by theDoppler effect. Consequently, this type of redshift is called theDoppler redshift. If the source moves away from the observer withvelocityv, which is much less than the speed of light (v ≪c), the redshift is given by
wherec is thespeed of light (since). 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 therelativistic Doppler effect. In brief, objects moving close to the speed of light will experience deviations from the above formula due to thetime dilation ofspecial relativity which can be corrected for by introducing theLorentz factorγ into the classical Doppler formula as follows (for motion solely in the line of sight):
Since the Lorentz factor is dependent only on themagnitude 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 theprojection of the movement of the source into theline-of-sight which yields different results for different orientations. Ifθ is the angle between the direction of relative motion and the direction of emission in the observer's frame[24] (zero angle is directly away from the observer), the full form for the relativistic Doppler effect becomes:
and for motion solely in the line of sight (θ = 0°), this equation reduces to:
For the special case that the light is moving atright angle (θ = 90°) to the direction of relative motion in the observer's frame,[25] the relativistic redshift is known as thetransverse redshift, and a redshift:
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.[26]
The observations of increasing redshifts from more and more distant galaxies can be modelled assuming ahomogeneous and isotropic universe combined withgeneral relativity. This cosmological redshift can be written as a function ofa, the time-dependent cosmicscale factor:[27]: 72
The scale factor ismonotonically increasing as time passes. Thusz is positive, close to zero for local stars, and increasing for distant galaxies that appear redshifted.
Using aFriedmann–Robertson–Walker model of the expansion of the universe, redshift can be related to the age of an observed object, the so-calledcosmic time–redshift relation. Denote a density ratio asΩ0:
withρcrit 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.[28] At large redshifts,1 + z > Ω0−1, one finds:
Thecosmological redshift is commonly attributed to stretching of the wavelengths of photons due to the stretching of space. This interpretation can be misleading. As required bygeneral relativity, the cosmological expansion of space has no effect on local physics. There is no term related to expansion inMaxwell'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.[31]
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.[32][33]
Distinguishing between cosmological and local effects
The redshift of a galaxy includes both a component related torecessional velocity from expansion of the universe, and a component related to thepeculiar motion of the galaxy with respect to its local universe.[34] 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.[35] Describing the cosmological expansion origin of redshift, cosmologistEdward 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..."[36]Steven Weinberg clarified, "The increase of wavelength from emission to absorption of light does not depend on the rate of change ofa(t) [thescale factor] at the times of emission or absorption, but on the increase ofa(t) in the whole period from emission to absorption."[37]
M is themass of the object creating the gravitational field,
r is the radial coordinate of the source (which is analogous to the classical distance from the center of the object, but is actually aSchwarzschild coordinate), and
This gravitational redshift result can be derived from the assumptions ofspecial relativity and theequivalence principle; the full theory of general relativity is not required.[40]
Several important special-case formulae for redshift in certain special spacetime geometries are summarised in the following table. In all cases the magnitude of the shift (the value ofz) is independent of the wavelength.[43]
The redshift observed in astronomy can be measured because theemission andabsorption spectra foratoms are distinctive and well known, calibrated fromspectroscopic experiments inlaboratories on Earth. When the redshifts of various absorption and emission lines from a single astronomical object are measured,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 bythermal or mechanicalmotion 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.
Spectroscopy, as a measurement, is considerably more difficult than simplephotometry, which measures thebrightness of astronomical objects through certainfilters. When photometric data is all that is available (for example, theHubble Deep Field and theHubble Ultra Deep Field), astronomers rely on a technique for measuringphotometric redshifts.[45] 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 toδz = 0.5, and are much less reliable than spectroscopic determinations.[46]
However, photometry does at least allow a qualitative characterisation of a redshift. For example, if a Sun-like spectrum had a redshift ofz = 1, it would be brightest in theinfrared (1000 nm) rather than at the blue-green (500 nm) color associated with the peak of itsblackbody spectrum, and the light intensity will be reduced in the filter by a factor of four,(1 +z)2. Both the photon count rate and the photon energy are redshifted. (SeeK correction for more details on the photometric consequences of redshift.)
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 orwhite noise (random fluctuations in a spectrum). Thusgamma-ray bursts themselves cannot be used for reliable redshift measurements, but optical afterglow associated with the burst can be analysed for redshifts.[47]
In nearby objects (within ourMilky Way galaxy) observed redshifts are almost always related to theline-of-sight velocities associated with the objects being observed. Observations of such redshifts and blueshifts enable astronomers to measurevelocities and parametrise themasses of theorbitingstars inspectroscopic binaries. Similarly, small redshifts and blueshifts detected in the spectroscopic measurements of individual stars are one way astronomers have been able todiagnose and measure the presence and characteristics ofplanetary systems around other stars and have even made verydetailed differential measurements of redshifts duringplanetary transits to determine precise orbital parameters. Some approaches are able to track the redshift variations in multiple objects at once.[48]
The most distant objects exhibit larger redshifts corresponding to theHubble flow of theuniverse. The largest-observed redshift, corresponding to the greatest distance and furthest back in time, is that of thecosmic microwave background radiation; thenumerical value of its redshift is aboutz = 1089 (z = 0 corresponds to present time), and it shows the state of the universe about 13.8 billion years ago,[54] and 379,000 years after the initial moments of theBig Bang.
The luminous point-like cores ofquasars were the first "high-redshift" (z > 0.1) objects discovered before the improvement of telescopes allowed for the discovery of other high-redshift galaxies.[55]
For galaxies more distant than theLocal Group and the nearbyVirgo Cluster, but within a thousand megaparsecs or so, the redshift is approximately proportional to the galaxy's distance. This correlation was first observed byEdwin Hubble and has come to be known asHubble's law.Vesto Slipher was the first to discover galactic redshifts, in about 1912, while Hubble correlated Slipher's measurements with distances hemeasured by other means to formulate his law.[56] Because it is usually not known howluminous 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.[57]
Gravitational interactions of galaxies with each other and clusters cause a significantscatter in the normal plot of the Hubble diagram. Thepeculiar velocities associated with galaxies superimpose a rough trace of themass ofvirialised objects in the universe. This effect leads to such phenomena as nearby galaxies (such as theAndromeda Galaxy) exhibiting blueshifts as we fall towards a commonbarycenter, and redshift maps of clusters showing afingers of god effect due to the scatter of peculiar velocities in a roughly spherical distribution.[58] These "redshift-space distortions" can be used as a cosmological probe in their own right, providing information on how structure formed in the universe,[59] and how gravity behaves on large scales.[60]
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.[61]
It was long believed that the expansion rate has been continuously decreasing since the Big Bang, but observations beginning in 1988 of the redshift-distance relationship usingType Ia supernovae have suggested that in comparatively recent times the expansion rate of the universe hasbegun to accelerate.[62]
Comoving distance andlookback 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 billions of years) from a given redshift until now is not the same as the distance (in giga light years) light would have travelled from that redshift, due to the expansion of the universe over the intervening period.
The most reliable redshifts are fromspectroscopic data,[63] and the highest-confirmed spectroscopic redshift of a galaxy is that ofJADES-GS-z14-0 with a redshift ofz = 14.32, corresponding to 290 million years after the Big Bang.[64] The previous record was held byGN-z11,[65] with a redshift ofz = 11.1, corresponding to 400 million years after the Big Bang.
Slightly less reliable areLyman-break redshifts, the highest of which is the lensed galaxy A1689-zD1 at a redshiftz = 7.5[66][67] and the next highest beingz = 7.0.[68] The most distant-observedgamma-ray burst with a spectroscopic redshift measurement wasGRB 090423, which had a redshift ofz = 8.2.[69] The most distant-known quasar,ULAS J1342+0928, is atz = 7.54.[70][71] The highest-known redshift radio galaxy (TGSS1530) is at a redshiftz = 5.72[72] and the highest-known redshift molecular material is the detection of emission from the CO molecule from the quasar SDSS J1148+5251 atz = 6.42.[73]
Extremely red objects (EROs) areastronomical 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.[74] Objects that are even redder than EROs are termedhyper extremely red objects (HEROs).[75]
Thecosmic microwave background has a redshift ofz = 1089, corresponding to an age of approximately 379,000 years after the Big Bang and aproper distance of more than 46 billion light-years.[76] This redshift corresponds to a shift in average temperature from 3000 K down to 3 K.[77] The yet-to-be-observed first light from the oldestPopulation III stars, not long after atoms first formed and the CMB ceased to be absorbed almost completely, may have redshifts in the range of20 <z < 100.[78] Other high-redshift events predicted by physics but not presently observable are thecosmic neutrino background from about two seconds after the Big Bang (and a redshift in excess ofz > 1010)[79] and the cosmicgravitational wave background emitted directly frominflation at a redshift in excess ofz > 1025.[80]
In June 2015, astronomers reported evidence forPopulation III stars in theCosmos Redshift 7galaxy atz = 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 ofchemical elements heavier thanhydrogen that are needed for the later formation ofplanets andlife as we know it.[81][82]
With advent ofautomated telescopes and improvements inspectroscopes, 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 thelarge-scale structure of the universe. TheGreat Wall, a vastsupercluster of galaxies over 500 millionlight-years wide, provides a dramatic example of a large-scale structure that redshift surveys can detect.[83]
The first redshift survey was theCfA Redshift Survey, started in 1977 with the initial data collection completed in 1982.[84] More recently, the2dF 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 finaldata set was released 30 June 2003.[85][86] TheSloan Digital Sky Survey (SDSS) began collecting data in 1998[87] and published its eighteenth data release in 2023.[88] SSDS has measured redshifts for galaxies as high as 0.8, and has recorded over 100,000quasars atz = 3 and beyond.[89] TheDEEP2 Redshift Survey used theKeck telescopes with the "DEIMOS"spectrograph; a follow-up to the pilot program DEEP1, DEEP2 was designed to measure faint galaxies with redshifts 0.7 and above, and it recorded redshifts of over 38,000 objects by its conclusion in 2013.[90][91]
Effects from physical optics or radiative transfer
The interactions and phenomena summarised in the subjects ofradiative transfer andphysical 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 ascoherence effects or thescattering ofelectromagnetic radiation whether fromchargedelementary particles, fromparticulates, or from fluctuations of theindex of refraction in adielectric medium as occurs in the radio phenomenon ofradio whistlers.[43] 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 theeffects discussed above.[43]
In many circumstances scattering causes radiation to redden becauseentropy results in the predominance of many low-energy photons over few high-energy ones (whileconserving total energy).[43] Except possibly under carefully controlled conditions, scattering does not produce the same relative change in wavelength across the whole spectrum; that is, any calculatedz is generally afunction of wavelength. Furthermore, scattering fromrandommedia generally occurs at manyangles, andz is a function of the scattering angle. If multiple scattering occurs, or the scattering particles have relative motion, then there is generally distortion ofspectral lines as well.[43]
Ininterstellar astronomy,visible spectra can appear redder due to scattering processes in a phenomenon referred to asinterstellar reddening[43]—similarlyRayleigh scattering causes theatmospheric reddening of the Sun seen in the sunrise or sunset and causes the rest of the sky to have a blue colour. This phenomenon is distinct from redshifting because thespectroscopic lines are not shifted to other wavelengths in reddened objects and there is an additionaldimming and distortion associated with the phenomenon due to photons being scattered in and out of theline of sight.[92]
The opposite of a redshift is ablueshift. A blueshift is any decrease inwavelength (increase inenergy), with a corresponding increase in frequency, of anelectromagnetic wave. Invisible light, this shifts a color towards the blue end of the spectrum.
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 thevisible spectrum. Only objects moving at near-relativistic speeds toward the observer are noticeably bluer to thenaked eye, but the wavelength of any reflected or emitted photon or other particle is shortened in the direction of travel.[93]
Doppler blueshift is used inastronomy to determine relative motion:
Components of abinary star system will be blueshifted when moving towards Earth
When observing spiral galaxies, the side spinning toward us will have a slight blueshiftrelative to the side spinning away from us (seeTully–Fisher relation).
Matter waves (protons, electrons, photons, etc.) falling into agravity well become more energetic and undergo observer-independent blueshifting.
Unlike therelative Doppler blueshift, caused by movement of a source towards the observer and thus dependent on the received angle of the photon, gravitational blueshift isabsolute and does not depend on the received angle of the photon:
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.[97][98]
There are far-awayactive galaxies that show a blueshift in their[O III] emissionlines. One of the largest blueshifts is found in the narrow-linequasar,PG 1543+489, which has a relative velocity of −1150 km/s.[96] These types of galaxies are called "blue outliers".[96]
In a hypothetical universe undergoing a runawayBig Crunch contraction, a cosmological blueshift would be observed, with galaxies further away being increasingly blueshifted—the exact opposite of the actually observedcosmological redshift in the presentexpanding universe.[100]
^Huchra, John."Extragalactic Redshifts".NASA/IPAC Extragalactic Database. Harvard-Smithsonian Center for Astrophysics. Archived fromthe original on 2013-12-22. Retrieved2023-03-16.
^Doppler, Christian (1846).Beiträge zur Fixsternenkunde [Contributions to fixed-star science] (in German). Vol. 69. Prague: G. Haase Söhne.Bibcode:1846befi.book.....D.
^abcdefRobert, Smith (2019). "Observations and the universe". In Kragh, Helge; Longair, Malcolm S. (eds.).The Oxford handbook of the history of modern cosmology. Oxford University Press.ISBN978-0-19-881766-6.OCLC1052868704.
^Slipher, Vesto (1912). "The radial velocity of the Andromeda Nebula".Lowell Observatory Bulletin.1 (8):2.56 –2.57.Bibcode:1913LowOB...2...56S.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
^de Sitter, W. (1934). "On distance, magnitude, and related quantities in an expanding universe".Bulletin of the Astronomical Institutes of the Netherlands.7: 205.Bibcode:1934BAN.....7..205D.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
^Lewis, Geraint F. (2016). "On The Relativity of Redshifts: Does Space Really 'Expand'?".Australian Physics.53: 95.arXiv:1605.08634.
^Einstein, A. (1907). "Über das Relativitätsprinzip und die aus demselben gezogenen Folgerungen" [On the relativity principle and the consequences drawn from it].Jahrbuch der Radioaktivität und Elektronik (in German).4: 458.Bibcode:1908JRE.....4..411E.
^Pilipenko, Sergey V. (2013). "Paper-and-pencil cosmological calculator".arXiv:1303.5961 [astro-ph.CO]. Includes algorithm used to draw the graph.
^The technique was first described by:Baum, W. A. (1962). McVittie, G. C. (ed.).Problems of extra-galactic research. IAU Symposium No. 15. p. 390.
^Bolzonella, M.; Miralles, J.-M.; Pelló, R. (2000). "Photometric redshifts based on standard SED fitting procedures".Astronomy and Astrophysics.363:476–492.arXiv:astro-ph/0003380.Bibcode:2000A&A...363..476B.
^Asaoka, Ikuko (1989). "X-ray spectra at infinity from a relativistic accretion disk around a Kerr black hole".Publications of the Astronomical Society of Japan.41 (4):763–778.Bibcode:1989PASJ...41..763A.doi:10.1093/pasj/41.4.763.
^Rybicki, G. B.; Lightman, A. R. (1979).Radiative Processes in Astrophysics. John Wiley & Sons. p. 288.ISBN0-471-82759-2.
^"Cosmic Detectives". The European Space Agency (ESA). 2013-04-02. Retrieved2013-04-25.
^Totani, Tomonori; Yoshii, Yuzuru; Iwamuro, Fumihide; Maihara, Toshinori; Motohara, Kentaro (2001). "Hyper Extremely Red Objects in the Subaru Deep Field: Evidence for Primordial Elliptical Galaxies in the Dusty Starburst Phase".The Astrophysical Journal.558 (2):L87 –L91.arXiv:astro-ph/0108145.Bibcode:2001ApJ...558L..87T.doi:10.1086/323619.S2CID119511017.
^Davis, Marc; et al. (DEEP2 collaboration) (2002).Science objectives and early results of the DEEP2 redshift survey. Conference on Astronomical Telescopes and Instrumentation, Waikoloa, Hawaii, 22–28 August 2002.arXiv:astro-ph/0209419.Bibcode:2003SPIE.4834..161D.doi:10.1117/12.457897.
^Maria Raiteri, Claudia (2024). "Monitoring Blazar Variability to Understand Extragalactic Jets".Publications of the Astronomical Observatory of Belgrade. Vol. 104. pp. 29–38.arXiv:2412.11565.doi:10.69646/aob104p029.ISBN978-86-82296-11-9.
Odenwald, S. & Fienberg, RT. 1993; "Galaxy Redshifts Reconsidered" inSky & 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, "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.)
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