# Missing Terms in Weinberg's treatment of perturbations on Newtonian Cosmology

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I was reading Appendix F of Steven Weingberg's book "Cosmology". In this Appendix he works out the perturbations to a cosmological fluid described by non-relativistic hydrodynamics and Newtonian gravity.

It turns out that the first order perturbations satisfy,

$$frac{partial delta ho }{partial t } + 3 H delta ho + H vec{X} cdot abla delta ho + ar{ ho} abla cdot vec{v} = 0, qquad ag{1}$$

$$frac{partial delta vec{v}}{partial t } + H vec{X} cdot abla delta vec{v} + H delta vec{v} = - abla delta phi, qquad ag{2}$$

$$abla^2 delta phi = 4pi G delta ho. qquad ag{3}$$

Weinberg applies the following Fourier transform to these equations,

$$f(vec{X},t) = int exp left( frac{i vec{q} cdot vec{X}}{a} ight) f_{vec{q}}(t) mathrm{d}^3vec{q}$$,

where $f(vec{X},t)$ is a place holder for $delta vec{v}, delta ho,$ and $delta phi$.

The resulting equations he gets are,

$$frac{mathrm d delta ho_{vec{q}}}{mathrm d t } + 3 H delta ho_{vec{q}} + frac{iar{ ho}}{a} vec{q} cdot delta vec{v}_{vec{q}} = 0 qquad ag{1'}$$

$$frac{mathrm d delta vec{v}_{vec{q}}}{mathrm d t } + H delta vec{v}_{vec{q}} = -frac{i}{a} vec{q} delta phi_{vec{q}} qquad ag{2'}$$

$$vec{q}^2 delta phi_{vec{q}} = -4pi G a^2 delta ho_{vec{q}} qquad ag{3'}$$.

For the most part these new equations can be obtained by making the substitution $abla ightarrow i vec{q}/a$.

My question : There doesn't seem to be any terms in the transformed equations which correspond to the terms $H vec{X} cdot abla delta ho$ and $H vec{X} cdot abla delta vec{v}$. Weinberg makes no comment about their absence. Is anyone aware of a legitimate mathematical reason for these terms to disappear in the transformed equations?

The answer turns out to be embarrassingly simple, and I suspect nobody came up with it because of my poor communication in the question.

The $a$ is the occurring in the Fourier transform is the scale factor which has a time dependence. So if we look at the term $frac{partial}{partial t} ho$ we will get,

$$frac{partial}{partial t} ho = frac{partial}{partial t} int exp left( frac{ i vec{q} cdot vec{X}}{a(t)} ight) ho_{vec{q}}(t) mathrm{d}^3 vec{q}$$

$$= int frac{partial}{partial t}left( frac{ i vec{q} cdot vec{X}}{a(t)} ight) exp left( frac{ i vec{q} cdot vec{X}}{a(t)} ight) ho_{vec{q}}(t) + exp left( frac{ i vec{q} cdot vec{X}}{a(t)} ight) frac{ mathrm{d} ho_{vec{q}}}{mathrm{d} t } mathrm{d}^3 vec{q}$$

$$= int left(-frac{dot{a}}{a} frac{ i vec{q} cdot vec{X}}{a(t)} ight) exp left( frac{ i vec{q} cdot vec{X}}{a(t)} ight) ho_{vec{q}}(t) + exp left( frac{ i vec{q} cdot vec{X}}{a(t)} ight) frac{ mathrm{d} ho_{vec{q}}}{mathrm{d} t } mathrm{d}^3 vec{q}$$

$$= int left(-Hfrac{ i vec{q} cdot vec{X}}{a(t)} ight) exp left( frac{ i vec{q} cdot vec{X}}{a(t)} ight) ho_{vec{q}}(t) + exp left( frac{ i vec{q} cdot vec{X}}{a(t)} ight) frac{ mathrm{d} ho_{vec{q}}}{mathrm{d} t } mathrm{d}^3 vec{q}$$

The left hand term in the integrand matches the Fourier transform of $H (vec{X} cdot abla) ho$. Which is why the terms cancel.

## Missing Terms in Weinberg's treatment of perturbations on Newtonian Cosmology - Astronomy

We present a systematic treatment of the linear theory of scalar gravitational perturbations in the synchronous gauge and the conformal Newtonian (or longitudinal) gauge. We first derive the transformation law relating the two gauges. We then write down in parallel in both gauges the coupled, linearized Boltzmann, Einstein and fluid equations that govern the evolution of the metric perturbations and the density fluctuations of the particle species. The particle species considered include cold dark matter (CDM), baryons, photons, massless neutrinos, and massive neutrinos (a hot dark matter or HDM candidate), where the CDM and baryon components are treated as fluids while a detailed phase-space description is given to the photons and neutrinos. The linear evolution equations presented are applicable to any $Omega=1$ model with CDM or a mixture of CDM and HDM. Isentropic initial conditions on super-horizon scales are derived. The equations are solved numerically in both gauges for a CDM+HDM model with $Omega_< m cold>=0.65,$ $Omega_< m hot>=0.3$, and $Omega_< m baryon>=0.05$. We discuss the evolution of the metric and the density perturbations and compare their different behaviors outside the horizon in the two gauges. In a companion paper we integrate the geodesic equations for the neutrino particles in the perturbed conformal Newtonian background metric computed here. The purpose is to obtain an accurate sampling of the neutrino phase space for the HDM initial conditions in $N$-body simulations of the CDM+HDM models.

## Missing Terms in Weinberg's treatment of perturbations on Newtonian Cosmology - Astronomy

One of the fundamental questions of astronomy is "How did this all get here?" (Kleinmann 1988 private communication). This is deeply tied to observations bearing on the formation, evolution, and distribution of galaxies. First we develop some basic consideration on the "standard" big-bang picture, then deal with the relevant observations of the extragalactic universe.

The Hubble expansion (conventionally interpreted as an expansion of spacetime with galaxies carried along for the ride) and cosmological principle together imply one of two kinds of universe:

There are several immediately relevant data to help us choose a best bet scheme:

The most useful description of cosmological models involves general relativity. Its geometric basis allows a natural treatment of light propagation along geodisics, which is how we get most of our information. Static solutions to Einstein's equations are unstable unless there is some repulsive force giving rise to nonzero cosmological constant &Lambda. A description of this kind has two locally observable parameters, in terms of the behavior of a scale factor R. This is defined so that the evolution of the distance between two comoving points (that is with no peculiar velocities superimposed on those from the cosmological expansion) evolves as r12 = r0 R(t). These parameters are the Hubble ratio (not quite constant since it will in general change with cosmic time)

and the density or deceleration parameter

For both of these, a subscript 0 indicates their evaluation at the present epoch. q is related to the open or closed nature of the universe. The critical closure density is &rho0 = 3 H² / 8 &rhoG as may be easily derived from Newtonian physics. Consider a volume element of gas in a uniform medium of density &rho at distance r from some arbitrary central point. Using a theorem of Newton, the gravitational attraction of a uniform medium outside r has no net effect, so the gravitational force due to the material within r generates a potential - GM(< r)&rho dV/r = -4G &rho r² &rho² dV/3 for the matter in the volume element dV. Its kinetic energy for a Hubble-like flow is mv²/2 or &rho dV H² r² /2. At critical density, the net energy is zero or alternately the material in dV is marginally bound: 4 G &pi r² &rho>² dV /3 = &rho dVr² /2 This is satisfied for the value of &rho0 given above. The density in units of &rho0 is often denoted by &Omega. For &Lambda=0, q = &Omega/2 so that q0 = 1/2 is the critical value.

Enormous effort has gone into the determination of H0 and q0 via "classical" tests using galaxies as standard candles. Measurement of H0 has been discussed in the distance-scale lecture. The problem with measuring the deceleration parameter is that the universe is close to the critical density, so that departures from this critical value become apparent only over such large path lengths that galaxy evolution becomes the dominant factor. What little we know about galaxy evolution says that galaxies won't be standard candles. These tests are well documented in the review by Sandage 1988 (ARA&A 26, 561) in an eloquent testimonial to the ultimate failure of most of them. These rely on the propagation of radiation in an expanding metric and the consequent breakdown of the inverse-square law for intensities and the inverse angular diameter-distance relation (more later on what distance means in this context). Further discussion of these tests may be found in Weinberg, Gravitation and Cosmology, Peebles, Physical Cosmology, and ch. 3 of Weedman, Quasar Astronomy.

Hubble diagram or magnitude-redshift test. Assume we have a set of galaxies with unchanging and known intrinsic brightness. It has been popular to take radio galaxies or third-brightest cluster members for this radio galaxies in the K-band are especially well-behaved. At small redshifts, the Hubble expansion makes the magnitude-log z relation linear, with some scatter due to galaxy properties and our ignorance of just how to select a perfectly uniform set of galaxies. At high z, the relation may curve for various q0. To see how, consider various measures of distance to an object of redshift z. The proper distance is that traversed by a photon in its own frame from there and then to here and now, which is to say DP = 1/c × light travel time. The luminosity distance is that which makes the inverse-square law work as it does in a flat universe: luminosity/flux = 4 &pi DL². Finally, the angular diameter distance is that for which the linear size-angular size relation has its familiar small-angle form &theta = length/DA.

For the standard (Friedman) model,

This is one of a set of closed forms worked out by Mattig (1958 Astron. Nachr. 284, 109) in a demonstration of analytical virtuosity starting from the Robertson-Walker metric and curvature scale in a Friedmann model. [I tried to reproduce this derivation as an inquisitive grad student, and gave up after two days when it wasn't getting any closer. George Blumenthal said it took him three days.] Then we have DL = (1+z) DP and DA = DL/(1+z)² which simplify in various ways for the special cases q0=0,1/2,1. For example, if q0=1, the lookback time &tauLB = DP/c = (H0) -1 (z/<1+z>).

In principle, the curvature in the extended Hubble diagram for standard candles can give q0 as shown by Weinberg p. 448 quoting Sandage. Big news has been the finding that high-redshift type Ia supernovae show evidence of upward curvature in the analogous magnitude-redshift relation, implying a nonzero cosmological constant. Type Ia objects are] useful because they have a narrow absolute-magnitude dispersion to begin with, and much of this dispersion is corralated with independently measurable fading timescale, plus the fact that they should all have the same kind of progenitor which formed its own heavy elements so that initial-metallicity effects should be small. The evidence is shown in Fig. 4 of Riess et al. (1998 AJ 116, 1009, reproduced by permission of the AAS):

The differences among various values of q0 become important only for z > 1. Applying this test to real data on galaxies is further complicated by various effects, some of whioch don't enter for the supernova test. The K-correction accounts for the fact that one is no longer observing the same part of the spectrum at various redshifts, and for the decrease in photon arrival rate (by a factor 1+z) even if one follows the same emitted-frame wavelength with z. Galaxy evolution, both stellar and dynamical, turns out to be so strong that it's much easier to measure than q0. For cluster galaxies, richness effects enter - at high redshift it's easier to pick out rich clusters whose nth brightest members are then brighter than expected. It is not clear that this is the path to q0 because of the dominant role of galaxy luminosity and spectral evolution.

The angular diameter - redshift test looks for a breakdown of the inversion relation between distance and angular diameter of some set of standard measuring rods (say galaxy isophotes or radio-galaxy lobe separations). Surface brightnesses must be corrected for dimming by (1+z) 4 due to expansion of space plus photon energy and arrival-time decreases. The form of this has been used as a test (the Tolman test) that redshifts really correspond to an expansion, and not to once-popular "tired-light" phenomena (Sandage and Perelmuter 1990 ApJ 350, 481 361, 1 1991 ApJ 370, 455, Wirth 1997 PASP 109, 344). Here again, one might have to deal with evolutionary effects in such objects as radio sources - have they always been the same size? At least with galaxy structures one has some control over how much dynamical evolution might have gone on.

It is interesting that, for any given positive value of q0, there exists a minimum in the &theta(z) relation for any linear size things at higher redshift look bigger again because they were quite nearby when the light was emitted. For q0=1/2, the angular diameter goes through a minimum at z=5/4. Using the analytical form listed, for example, in Lang, Astrophys. Formulae, eqn. 5-314, we have:

(If you want to experiment with this, here's some simple IDL code which generated the plot). This effect may be in part responsible for the large apparent sizes of very-high-redshift radio galaxies (see Djorgovski in Nearly Normal Galaxies, for example). In interpreting sizes of high-redshift objects, though, surface-brightness dimming can be a dominant effect.

Galaxy number counts may be used to trace the history of the volume per galaxy, so that the N(z) relation implies R(t) since the scale factor R scales as (1+z) -1 . If we could count a conserved population over a wide redshift range, we could learn q0 directly from its definition. An early attempt used six-color mapping to estimate photometric redshifts (Loh and Spillar 1986 ApJLett 307, L1). Modelling with a standard luminosity function, they get &Omega= 0.9 +0.7 -0.5 from 1000 "field" galaxies. This is independent of color evolution, but is dependent on luminosity evolution and merging both these processes could change the number of galaxies in a given luminosity range with redshift. As discussed by Sandage 1988, most applications of this test are more sensitive to galaxy evolution than to cosmology (because the universe is old enough for us to be here talking about it).

There are additional constraints from local (or indeed laboratory) observations. Most prominent is the cosmic microwave background. This is the radiation field at the epoch of (re)combination, when the universe first became transparent enough for radiation to propagat freely over cosmologically interesting distances. At the high densities then, the radiation field was fully themalized (blackbody). Since then the observed temperature has dropped due to redshifting (a redshifted blackbody is another lower-temperature blackbody). See Peebles, chapter 7, for some complications. Recombination was somewhere around z

10 4 set by the temperature at which hydrogen ionizes and the observed CMB temperature (which we know from COBE to be 2.785 K, Mather et al 1990 APJLett 354, L37 and Smoot et al 1991 ApJLett 371, L1). The CMB is highly isotropic (except for a dipole term thought to reflect our motion with respect to the large-scale velocity defined by the CMB emission surface) and embarrassingly smooth. Its spectrum is as perfect a blackbody as can be measured: from Fig. 4 of Fixsen et al. 1996 (ApJ 473, 576, by permission of the AAS),

We would expect some lumps corresponding to protogalaxies and -clusters, since it is hard enough to understand how galaxies form by z

5 from inhomogeneities at recombination. These were finally detected in a convincing way by the COBE group, with subsequent confirmation by ground-based observations from Tenerife and Antarctica as well as balloon-borne instruments. The fluctuations, as measured at a resolution of a few degrees, are at the level &Delta T /

3 × 10 -6 , which is (to factors "of order unity") the density contrast &Delta &rho / &rho of fluctuations. Linear development of perturbations won't cut it to clump matter fast enough there is something major here that we don't know about making galaxies. I am holding out for nonbaryonic matter already clumped at recombination and providing seeds for forming baryonic structures, but then I could be convinced otherwise if anything resembling evidence shows up. The CMB was one of the major downfalls of a steady-state picture something has clearly changed since the time when space was uniformly filled with 4000-K plasma.

Absorption of the CMB by hot gas in cluster is expected (the Sunyaev-Zeldovich effect) and has been observed after years of upper limits - this is interesting as direct confirmation that we are not seeing some unknown local effect, since the CMB comes from behind clusters at substantial redshift. Note that the CMB temperature should scale as 1+z measurements of low-energy fine-structure levels (specifically C II * ) in QSO absorption-line systems indeed show this effect. As shown by Ge, Bechtold, and Black 1997 ApJ 474, 67, (courtesy of the AAS),

In retrospect, the first hint of the CMB was the excitation temperature of interstellar CN seen in absorption at 3874 Å against galactic stars, with observations tracing back to Adams 1941 (ApJ 93, 11 see Thaddeus 1972 ARA&A 10, 305 for a fuller history). As discussed by Roth, Meyer, and Hawkins (1993 ApJL 413, L67), these lines are excited by absorbing radiation at 1.3 and 2.6mm. These observations have some philosophical interest in showing the uniformity of this radiation on galactic scales.

Ehlers, J.: In: Proceedings of the mathematical-natural science of the Mainz academy of Science and Literature. Nr. 11, 792 (1961) Gen. Rel. Grav. 25, 1225 translated in (1993) Ellis, G.F.R.: General relativity and cosmology. In: Sachs, R.K. (ed.) Proceedings of the International Summer School of Physics Enrico Fermi Course, vol. 47, p. 104. Academic Press, New York (1971)

Hwang, J., Noh, H.: Phys. Rev. D 72, 044011 (2004) Noh, H., Hwang, J.: Class. Quant. Grav. 22, 3181 (2005)

## Contents

Modern cosmology developed along tandem tracks of theory and observation. In 1916, Albert Einstein published his theory of general relativity, which provided a unified description of gravity as a geometric property of space and time. [4] At the time, Einstein believed in a static universe, but found that his original formulation of the theory did not permit it. [5] This is because masses distributed throughout the universe gravitationally attract, and move toward each other over time. [6] However, he realized that his equations permitted the introduction of a constant term which could counteract the attractive force of gravity on the cosmic scale. Einstein published his first paper on relativistic cosmology in 1917, in which he added this cosmological constant to his field equations in order to force them to model a static universe. [7] The Einstein model describes a static universe space is finite and unbounded (analogous to the surface of a sphere, which has a finite area but no edges). However, this so-called Einstein model is unstable to small perturbations—it will eventually start to expand or contract. [5] It was later realized that Einstein's model was just one of a larger set of possibilities, all of which were consistent with general relativity and the cosmological principle. The cosmological solutions of general relativity were found by Alexander Friedmann in the early 1920s. [8] His equations describe the Friedmann–Lemaître–Robertson–Walker universe, which may expand or contract, and whose geometry may be open, flat, or closed.

In the 1910s, Vesto Slipher (and later Carl Wilhelm Wirtz) interpreted the red shift of spiral nebulae as a Doppler shift that indicated they were receding from Earth. [12] [13] However, it is difficult to determine the distance to astronomical objects. One way is to compare the physical size of an object to its angular size, but a physical size must be assumed to do this. Another method is to measure the brightness of an object and assume an intrinsic luminosity, from which the distance may be determined using the inverse-square law. Due to the difficulty of using these methods, they did not realize that the nebulae were actually galaxies outside our own Milky Way, nor did they speculate about the cosmological implications. In 1927, the Belgian Roman Catholic priest Georges Lemaître independently derived the Friedmann–Lemaître–Robertson–Walker equations and proposed, on the basis of the recession of spiral nebulae, that the universe began with the "explosion" of a "primeval atom" [14] —which was later called the Big Bang. In 1929, Edwin Hubble provided an observational basis for Lemaître's theory. Hubble showed that the spiral nebulae were galaxies by determining their distances using measurements of the brightness of Cepheid variable stars. He discovered a relationship between the redshift of a galaxy and its distance. He interpreted this as evidence that the galaxies are receding from Earth in every direction at speeds proportional to their distance. [15] This fact is now known as Hubble's law, though the numerical factor Hubble found relating recessional velocity and distance was off by a factor of ten, due to not knowing about the types of Cepheid variables.

Given the cosmological principle, Hubble's law suggested that the universe was expanding. Two primary explanations were proposed for the expansion. One was Lemaître's Big Bang theory, advocated and developed by George Gamow. The other explanation was Fred Hoyle's steady state model in which new matter is created as the galaxies move away from each other. In this model, the universe is roughly the same at any point in time. [16] [17]

For a number of years, support for these theories was evenly divided. However, the observational evidence began to support the idea that the universe evolved from a hot dense state. The discovery of the cosmic microwave background in 1965 lent strong support to the Big Bang model, [17] and since the precise measurements of the cosmic microwave background by the Cosmic Background Explorer in the early 1990s, few cosmologists have seriously proposed other theories of the origin and evolution of the cosmos. One consequence of this is that in standard general relativity, the universe began with a singularity, as demonstrated by Roger Penrose and Stephen Hawking in the 1960s. [18]

An alternative view to extend the Big Bang model, suggesting the universe had no beginning or singularity and the age of the universe is infinite, has been presented. [19] [20] [21]

The lightest chemical elements, primarily hydrogen and helium, were created during the Big Bang through the process of nucleosynthesis. [22] In a sequence of stellar nucleosynthesis reactions, smaller atomic nuclei are then combined into larger atomic nuclei, ultimately forming stable iron group elements such as iron and nickel, which have the highest nuclear binding energies. [23] The net process results in a later energy release, meaning subsequent to the Big Bang. [24] Such reactions of nuclear particles can lead to sudden energy releases from cataclysmic variable stars such as novae. Gravitational collapse of matter into black holes also powers the most energetic processes, generally seen in the nuclear regions of galaxies, forming quasars and active galaxies.

Cosmologists cannot explain all cosmic phenomena exactly, such as those related to the accelerating expansion of the universe, using conventional forms of energy. Instead, cosmologists propose a new form of energy called dark energy that permeates all space. [25] One hypothesis is that dark energy is just the vacuum energy, a component of empty space that is associated with the virtual particles that exist due to the uncertainty principle. [26]

There is no clear way to define the total energy in the universe using the most widely accepted theory of gravity, general relativity. Therefore, it remains controversial whether the total energy is conserved in an expanding universe. For instance, each photon that travels through intergalactic space loses energy due to the redshift effect. This energy is not obviously transferred to any other system, so seems to be permanently lost. On the other hand, some cosmologists insist that energy is conserved in some sense this follows the law of conservation of energy. [27]

Different forms of energy may dominate the cosmos – relativistic particles which are referred to as radiation, or non-relativistic particles referred to as matter. Relativistic particles are particles whose rest mass is zero or negligible compared to their kinetic energy, and so move at the speed of light or very close to it non-relativistic particles have much higher rest mass than their energy and so move much slower than the speed of light.

As the universe expands, both matter and radiation become diluted. However, the energy densities of radiation and matter dilute at different rates. As a particular volume expands, mass-energy density is changed only by the increase in volume, but the energy density of radiation is changed both by the increase in volume and by the increase in the wavelength of the photons that make it up. Thus the energy of radiation becomes a smaller part of the universe's total energy than that of matter as it expands. The very early universe is said to have been 'radiation dominated' and radiation controlled the deceleration of expansion. Later, as the average energy per photon becomes roughly 10 eV and lower, matter dictates the rate of deceleration and the universe is said to be 'matter dominated'. The intermediate case is not treated well analytically. As the expansion of the universe continues, matter dilutes even further and the cosmological constant becomes dominant, leading to an acceleration in the universe's expansion.

The history of the universe is a central issue in cosmology. The history of the universe is divided into different periods called epochs, according to the dominant forces and processes in each period. The standard cosmological model is known as the Lambda-CDM model.

### Equations of motion Edit

Within the standard cosmological model, the equations of motion governing the universe as a whole are derived from general relativity with a small, positive cosmological constant. [28] The solution is an expanding universe due to this expansion, the radiation and matter in the universe cool down and become diluted. At first, the expansion is slowed down by gravitation attracting the radiation and matter in the universe. However, as these become diluted, the cosmological constant becomes more dominant and the expansion of the universe starts to accelerate rather than decelerate. In our universe this happened billions of years ago. [29]

### Particle physics in cosmology Edit

During the earliest moments of the universe, the average energy density was very high, making knowledge of particle physics critical to understanding this environment. Hence, scattering processes and decay of unstable elementary particles are important for cosmological models of this period.

As a rule of thumb, a scattering or a decay process is cosmologically important in a certain epoch if the time scale describing that process is smaller than, or comparable to, the time scale of the expansion of the universe. [ clarification needed ] The time scale that describes the expansion of the universe is 1 / H with H being the Hubble parameter, which varies with time. The expansion timescale 1 / H is roughly equal to the age of the universe at each point in time.

### Timeline of the Big Bang Edit

Observations suggest that the universe began around 13.8 billion years ago. [30] Since then, the evolution of the universe has passed through three phases. The very early universe, which is still poorly understood, was the split second in which the universe was so hot that particles had energies higher than those currently accessible in particle accelerators on Earth. Therefore, while the basic features of this epoch have been worked out in the Big Bang theory, the details are largely based on educated guesses. Following this, in the early universe, the evolution of the universe proceeded according to known high energy physics. This is when the first protons, electrons and neutrons formed, then nuclei and finally atoms. With the formation of neutral hydrogen, the cosmic microwave background was emitted. Finally, the epoch of structure formation began, when matter started to aggregate into the first stars and quasars, and ultimately galaxies, clusters of galaxies and superclusters formed. The future of the universe is not yet firmly known, but according to the ΛCDM model it will continue expanding forever.

Below, some of the most active areas of inquiry in cosmology are described, in roughly chronological order. This does not include all of the Big Bang cosmology, which is presented in Timeline of the Big Bang.

### Very early universe Edit

The early, hot universe appears to be well explained by the Big Bang from roughly 10 −33 seconds onwards, but there are several problems. One is that there is no compelling reason, using current particle physics, for the universe to be flat, homogeneous, and isotropic (see the cosmological principle). Moreover, grand unified theories of particle physics suggest that there should be magnetic monopoles in the universe, which have not been found. These problems are resolved by a brief period of cosmic inflation, which drives the universe to flatness, smooths out anisotropies and inhomogeneities to the observed level, and exponentially dilutes the monopoles. [31] The physical model behind cosmic inflation is extremely simple, but it has not yet been confirmed by particle physics, and there are difficult problems reconciling inflation and quantum field theory. [ vague ] Some cosmologists think that string theory and brane cosmology will provide an alternative to inflation. [32]

Another major problem in cosmology is what caused the universe to contain far more matter than antimatter. Cosmologists can observationally deduce that the universe is not split into regions of matter and antimatter. If it were, there would be X-rays and gamma rays produced as a result of annihilation, but this is not observed. Therefore, some process in the early universe must have created a small excess of matter over antimatter, and this (currently not understood) process is called baryogenesis. Three required conditions for baryogenesis were derived by Andrei Sakharov in 1967, and requires a violation of the particle physics symmetry, called CP-symmetry, between matter and antimatter. [33] However, particle accelerators measure too small a violation of CP-symmetry to account for the baryon asymmetry. Cosmologists and particle physicists look for additional violations of the CP-symmetry in the early universe that might account for the baryon asymmetry. [34]

Both the problems of baryogenesis and cosmic inflation are very closely related to particle physics, and their resolution might come from high energy theory and experiment, rather than through observations of the universe. [ speculation? ]

### Big Bang Theory Edit

Big Bang nucleosynthesis is the theory of the formation of the elements in the early universe. It finished when the universe was about three minutes old and its temperature dropped below that at which nuclear fusion could occur. Big Bang nucleosynthesis had a brief period during which it could operate, so only the very lightest elements were produced. Starting from hydrogen ions (protons), it principally produced deuterium, helium-4, and lithium. Other elements were produced in only trace abundances. The basic theory of nucleosynthesis was developed in 1948 by George Gamow, Ralph Asher Alpher, and Robert Herman. [35] It was used for many years as a probe of physics at the time of the Big Bang, as the theory of Big Bang nucleosynthesis connects the abundances of primordial light elements with the features of the early universe. [22] Specifically, it can be used to test the equivalence principle, [36] to probe dark matter, and test neutrino physics. [37] Some cosmologists have proposed that Big Bang nucleosynthesis suggests there is a fourth "sterile" species of neutrino. [38]

#### Standard model of Big Bang cosmology Edit

The ΛCDM (Lambda cold dark matter) or Lambda-CDM model is a parametrization of the Big Bang cosmological model in which the universe contains a cosmological constant, denoted by Lambda (Greek Λ), associated with dark energy, and cold dark matter (abbreviated CDM). It is frequently referred to as the standard model of Big Bang cosmology. [39] [40]

### Cosmic microwave background Edit

The cosmic microwave background is radiation left over from decoupling after the epoch of recombination when neutral atoms first formed. At this point, radiation produced in the Big Bang stopped Thomson scattering from charged ions. The radiation, first observed in 1965 by Arno Penzias and Robert Woodrow Wilson, has a perfect thermal black-body spectrum. It has a temperature of 2.7 kelvins today and is isotropic to one part in 10 5 . Cosmological perturbation theory, which describes the evolution of slight inhomogeneities in the early universe, has allowed cosmologists to precisely calculate the angular power spectrum of the radiation, and it has been measured by the recent satellite experiments (COBE and WMAP) [41] and many ground and balloon-based experiments (such as Degree Angular Scale Interferometer, Cosmic Background Imager, and Boomerang). [42] One of the goals of these efforts is to measure the basic parameters of the Lambda-CDM model with increasing accuracy, as well as to test the predictions of the Big Bang model and look for new physics. The results of measurements made by WMAP, for example, have placed limits on the neutrino masses. [43]

Newer experiments, such as QUIET and the Atacama Cosmology Telescope, are trying to measure the polarization of the cosmic microwave background. [44] These measurements are expected to provide further confirmation of the theory as well as information about cosmic inflation, and the so-called secondary anisotropies, [45] such as the Sunyaev-Zel'dovich effect and Sachs-Wolfe effect, which are caused by interaction between galaxies and clusters with the cosmic microwave background. [46] [47]

On 17 March 2014, astronomers of the BICEP2 Collaboration announced the apparent detection of B-mode polarization of the CMB, considered to be evidence of primordial gravitational waves that are predicted by the theory of inflation to occur during the earliest phase of the Big Bang. [9] [10] [11] [48] However, later that year the Planck collaboration provided a more accurate measurement of cosmic dust, concluding that the B-mode signal from dust is the same strength as that reported from BICEP2. [49] [50] On 30 January 2015, a joint analysis of BICEP2 and Planck data was published and the European Space Agency announced that the signal can be entirely attributed to interstellar dust in the Milky Way. [51]

### Formation and evolution of large-scale structure Edit

Understanding the formation and evolution of the largest and earliest structures (i.e., quasars, galaxies, clusters and superclusters) is one of the largest efforts in cosmology. Cosmologists study a model of hierarchical structure formation in which structures form from the bottom up, with smaller objects forming first, while the largest objects, such as superclusters, are still assembling. [52] One way to study structure in the universe is to survey the visible galaxies, in order to construct a three-dimensional picture of the galaxies in the universe and measure the matter power spectrum. This is the approach of the Sloan Digital Sky Survey and the 2dF Galaxy Redshift Survey. [53] [54]

Another tool for understanding structure formation is simulations, which cosmologists use to study the gravitational aggregation of matter in the universe, as it clusters into filaments, superclusters and voids. Most simulations contain only non-baryonic cold dark matter, which should suffice to understand the universe on the largest scales, as there is much more dark matter in the universe than visible, baryonic matter. More advanced simulations are starting to include baryons and study the formation of individual galaxies. Cosmologists study these simulations to see if they agree with the galaxy surveys, and to understand any discrepancy. [55]

Other, complementary observations to measure the distribution of matter in the distant universe and to probe reionization include:

• The Lyman-alpha forest, which allows cosmologists to measure the distribution of neutral atomic hydrogen gas in the early universe, by measuring the absorption of light from distant quasars by the gas. [56]
• The 21 centimeter absorption line of neutral atomic hydrogen also provides a sensitive test of cosmology. [57] , the distortion of a distant image by gravitational lensing due to dark matter. [58]

These will help cosmologists settle the question of when and how structure formed in the universe.

### Dark matter Edit

Evidence from Big Bang nucleosynthesis, the cosmic microwave background, structure formation, and galaxy rotation curves suggests that about 23% of the mass of the universe consists of non-baryonic dark matter, whereas only 4% consists of visible, baryonic matter. The gravitational effects of dark matter are well understood, as it behaves like a cold, non-radiative fluid that forms haloes around galaxies. Dark matter has never been detected in the laboratory, and the particle physics nature of dark matter remains completely unknown. Without observational constraints, there are a number of candidates, such as a stable supersymmetric particle, a weakly interacting massive particle, a gravitationally-interacting massive particle, an axion, and a massive compact halo object. Alternatives to the dark matter hypothesis include a modification of gravity at small accelerations (MOND) or an effect from brane cosmology. [59]

### Dark energy Edit

If the universe is flat, there must be an additional component making up 73% (in addition to the 23% dark matter and 4% baryons) of the energy density of the universe. This is called dark energy. In order not to interfere with Big Bang nucleosynthesis and the cosmic microwave background, it must not cluster in haloes like baryons and dark matter. There is strong observational evidence for dark energy, as the total energy density of the universe is known through constraints on the flatness of the universe, but the amount of clustering matter is tightly measured, and is much less than this. The case for dark energy was strengthened in 1999, when measurements demonstrated that the expansion of the universe has begun to gradually accelerate. [60]

Apart from its density and its clustering properties, nothing is known about dark energy. Quantum field theory predicts a cosmological constant (CC) much like dark energy, but 120 orders of magnitude larger than that observed. [61] Steven Weinberg and a number of string theorists (see string landscape) have invoked the 'weak anthropic principle': i.e. the reason that physicists observe a universe with such a small cosmological constant is that no physicists (or any life) could exist in a universe with a larger cosmological constant. Many cosmologists find this an unsatisfying explanation: perhaps because while the weak anthropic principle is self-evident (given that living observers exist, there must be at least one universe with a cosmological constant which allows for life to exist) it does not attempt to explain the context of that universe. [62] For example, the weak anthropic principle alone does not distinguish between:

• Only one universe will ever exist and there is some underlying principle that constrains the CC to the value we observe.
• Only one universe will ever exist and although there is no underlying principle fixing the CC, we got lucky.
• Lots of universes exist (simultaneously or serially) with a range of CC values, and of course ours is one of the life-supporting ones.

Other possible explanations for dark energy include quintessence [63] or a modification of gravity on the largest scales. [64] The effect on cosmology of the dark energy that these models describe is given by the dark energy's equation of state, which varies depending upon the theory. The nature of dark energy is one of the most challenging problems in cosmology.

A better understanding of dark energy is likely to solve the problem of the ultimate fate of the universe. In the current cosmological epoch, the accelerated expansion due to dark energy is preventing structures larger than superclusters from forming. It is not known whether the acceleration will continue indefinitely, perhaps even increasing until a big rip, or whether it will eventually reverse, lead to a big freeze, or follow some other scenario. [65]

### Gravitational waves Edit

Gravitational waves are ripples in the curvature of spacetime that propagate as waves at the speed of light, generated in certain gravitational interactions that propagate outward from their source. Gravitational-wave astronomy is an emerging branch of observational astronomy which aims to use gravitational waves to collect observational data about sources of detectable gravitational waves such as binary star systems composed of white dwarfs, neutron stars, and black holes and events such as supernovae, and the formation of the early universe shortly after the Big Bang. [66]

In 2016, the LIGO Scientific Collaboration and Virgo Collaboration teams announced that they had made the first observation of gravitational waves, originating from a pair of merging black holes using the Advanced LIGO detectors. [67] [68] [69] On 15 June 2016, a second detection of gravitational waves from coalescing black holes was announced. [70] Besides LIGO, many other gravitational-wave observatories (detectors) are under construction. [71]

## Contents

The history of the subject began with the development in the 19th century of wave mechanics and the exploration of phenomena associated with the Doppler effect. The effect is named after Christian Doppler, who offered the first known physical explanation for the phenomenon in 1842. [1] The hypothesis was tested and confirmed for sound waves by the Dutch scientist Christophorus Buys Ballot in 1845. [2] Doppler correctly predicted that the phenomenon should 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. [3] Before this was verified, however, it was found that stellar colors were primarily due to a star's temperature, not motion. Only later was Doppler vindicated by verified redshift observations.

The first Doppler redshift was described by French physicist Hippolyte Fizeau in 1848, who pointed to the shift in spectral lines seen in stars as being due to the Doppler effect. The effect is sometimes called the "Doppler–Fizeau effect". In 1868, British astronomer William Huggins was the first to determine the velocity of a star moving away from the Earth by this method. [4] In 1871, optical redshift was confirmed when the phenomenon was observed in Fraunhofer lines using solar rotation, about 0.1 Å in the red. [5] 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. [6] In 1901, Aristarkh Belopolsky verified optical redshift in the laboratory using a system of rotating mirrors. [7]

The earliest occurrence of the term red-shift in print (in this hyphenated form) appears to be by American astronomer Walter S. Adams in 1908, in which he mentions "Two methods of investigating that nature of the nebular red-shift". [8] The word does not appear unhyphenated until about 1934 by Willem de Sitter, perhaps indicating that up to that point its German equivalent, Rotverschiebung, was more commonly used. [9]

Beginning with observations in 1912, Vesto Slipher discovered that most spiral galaxies, then mostly thought to be spiral nebulae, had considerable redshifts. Slipher first reports on his measurement in the inaugural volume of the Lowell Observatory Bulletin. [10] Three years later, he wrote a review in the journal Popular Astronomy. [11] In it he states 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." [12] Slipher reported the velocities for 15 spiral nebulae spread across the entire celestial sphere, all but three having observable "positive" (that is recessional) velocities. Subsequently, Edwin Hubble discovered an approximate relationship between the redshifts of such "nebulae" and the distances to them with the formulation of his eponymous Hubble's law. [13] These observations corroborated Alexander Friedmann's 1922 work, in which he derived the Friedmann–Lemaître equations. [14] They are today considered strong evidence for an expanding universe and the Big Bang theory. [15]

The spectrum of light that comes from a source (see idealized spectrum illustration top-right) can be measured. To determine the redshift, one searches for features in the spectrum such as absorption lines, emission lines, or other variations in light intensity. If found, these features can be compared with known features in the spectrum of various chemical compounds found in experiments where that compound is located on Earth. A very common atomic element in space is hydrogen. The spectrum of originally featureless light shone through hydrogen will show a signature spectrum specific to hydrogen that has features at regular intervals. If restricted to absorption lines it would look similar to the illustration (top right). If the same pattern of intervals is seen in an observed spectrum from a distant source but occurring at shifted wavelengths, it can be identified as hydrogen too. If the same spectral line is identified in both spectra—but at different wavelengths—then the redshift can be calculated using the table below. Determining the redshift of an object in this way requires a frequency or wavelength range. In order to calculate the redshift, one has to know the wavelength of the emitted light in the rest frame of the source: in other words, the wavelength that would be measured by an observer located adjacent to and comoving with the source. Since in astronomical applications this measurement cannot be done directly, because that would require traveling to the distant star of interest, the method using spectral lines described here is used instead. 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). [17]

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 z . If λ represents wavelength and f represents frequency (note, λf = c where c is the speed of light), then z is defined by the equations: [18]

Calculation of redshift, z
Based on wavelength Based on frequency
z = λ o b s v − λ e m i t λ e m i t >-lambda _ >> >>>> z = f e m i t − f o b s v f o b s v >-f_ >> >>>>
1 + z = λ o b s v λ e m i t >> >>>> 1 + z = f e m i t f o b s v >> >>>>

After z is measured, the distinction between redshift and blueshift is simply a matter of whether z is positive or negative. For example, Doppler effect blueshifts ( z < 0 ) are associated with objects approaching (moving closer to) the observer with the light shifting to greater energies. 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 weaker gravitational field as observed from within a stronger gravitational field, while gravitational redshifting implies the opposite conditions.

In general relativity one can derive 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 z ) is independent of the wavelength. [19]

For motion completely in the radial or line-of-sight direction:

For motion completely in the transverse direction:

### Doppler effect Edit

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 75 times faster than the speed record for the fastest man-made space probe.

If a source of the light is moving away from an observer, then redshift ( z > 0 ) occurs if the source moves towards the observer, then blueshift ( 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 v , which is much less than the speed of light ( vc ), the redshift is given by

where 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 γ into the classical Doppler formula as follows (for motion solely in the line of sight):

This phenomenon was first observed in a 1938 experiment performed by Herbert E. Ives and G.R. Stilwell, called the Ives–Stilwell experiment. [21]

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 θ is the angle between the direction of relative motion and the direction of emission in the observer's frame [22] (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 at right angle ( θ = 90° ) to the direction of relative motion in the observer's frame, [23] the relativistic redshift is known as the transverse 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. [24]

### Expansion of space Edit

In the earlier part of the twentieth century, Slipher, Wirtz and others made the first measurements of the redshifts and blueshifts of galaxies beyond the Milky Way. They initially interpreted these redshifts and blueshifts as being due to random motions, but later Lemaître (1927) and Hubble (1929), using previous data, discovered a roughly linear correlation between the increasing redshifts of, and distances to, galaxies. Lemaître realized that these observations could be explained by a mechanism of producing redshifts seen in Friedmann's solutions to Einstein's equations of general relativity. The correlation between redshifts and distances is required by all such models that have a metric expansion of space. [15] As a result, the wavelength of photons propagating through the expanding space is stretched, creating the cosmological redshift.

There is a distinction between a redshift in cosmological context as compared to that witnessed when nearby objects exhibit a local Doppler-effect redshift. Rather than cosmological redshifts being a consequence of the relative velocities that are subject to the laws of special relativity (and thus subject to the rule that no two locally separated objects can have relative velocities with respect to each other faster than the speed of light), the photons instead increase in wavelength and redshift because of a global feature of the spacetime through which they are traveling. One interpretation of this effect is the idea that space itself is expanding. [25] Due to the expansion increasing as distances increase, the distance between two remote galaxies can increase at more than 3 × 10 8 m/s, but this does not imply that the galaxies move faster than the speed of light at their present location (which is forbidden by Lorentz covariance).

#### Mathematical derivation Edit

The observational consequences of this effect can be derived using the equations from general relativity that describe a homogeneous and isotropic universe.

To derive the redshift effect, use the geodesic equation for a light wave, which is

• ds is the spacetime interval
• dt is the time interval
• dr is the spatial interval
• c is the speed of light
• a is the time-dependent cosmic scale factor
• k is the curvature per unit area.

For an observer observing the crest of a light wave at a position r = 0 and time t = tnow , the crest of the light wave was emitted at a time t = tthen in the past and a distant position r = R . Integrating over the path in both space and time that the light wave travels yields:

In general, the wavelength of light is not the same for the two positions and times considered due to the changing properties of the metric. When the wave was emitted, it had a wavelength λthen . The next crest of the light wave was emitted at a time

The observer sees the next crest of the observed light wave with a wavelength λnow to arrive at a time

Since the subsequent crest is again emitted from r = R and is observed at r = 0 , the following equation can be written:

The right-hand side of the two integral equations above are identical which means

Using the following manipulation:

For very small variations in time (over the period of one cycle of a light wave) the scale factor is essentially a constant ( a = anow today and a = athen previously). This yields

which can be rewritten as

Using the definition of redshift provided above, the equation

is obtained. In an expanding universe such as the one we inhabit, the scale factor is monotonically increasing as time passes, thus, z is positive and distant galaxies appear redshifted.

Using a 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 Ω0 :

with ρcrit 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. [26] At large redshifts, 1 + z > Ω0 −1 , one finds:

where H0 is the present-day Hubble constant, and z is the redshift. [27] [28] [29]

#### Distinguishing between cosmological and local effects Edit

For cosmological redshifts of z < 0.01 additional Doppler redshifts and blueshifts due to the peculiar motions of the galaxies relative to one another cause a wide scatter from the standard Hubble Law. [30] The resulting situation can be illustrated by the Expanding Rubber Sheet Universe, a common cosmological analogy used to describe the expansion of space. If two objects are represented by ball bearings and spacetime by a stretching rubber sheet, the Doppler effect is caused by rolling the balls across the sheet to create peculiar motion. The cosmological redshift occurs when the ball bearings are stuck to the sheet and the sheet is stretched. [31] [32] [33]

The redshifts of galaxies include both a component related to recessional velocity from expansion of the universe, and a component related to peculiar motion (Doppler shift). [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, 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. " [36] Steven Weinberg clarified, "The increase of wavelength from emission to absorption of light does not depend on the rate of change of a(t) [here a(t) is the Robertson–Walker scale factor] at the times of emission or absorption, but on the increase of a(t) in the whole period from emission to absorption." [37]

Popular literature often uses the expression "Doppler redshift" instead of "cosmological redshift" to describe the redshift of galaxies dominated by the expansion of spacetime, but the cosmological redshift is not found using the relativistic Doppler equation [38] which is instead characterized by special relativity thus v > c is impossible while, in contrast, v > c is possible for cosmological redshifts because the space which separates the objects (for example, a quasar from the Earth) can expand faster than the speed of light. [39] More mathematically, the viewpoint that "distant galaxies are receding" and the viewpoint that "the space between galaxies is expanding" are related by changing coordinate systems. Expressing this precisely requires working with the mathematics of the Friedmann–Robertson–Walker metric. [40]

If the universe were contracting instead of expanding, we would see distant galaxies blueshifted by an amount proportional to their distance instead of redshifted. [41]

### Gravitational redshift Edit

In the theory of general relativity, there is time dilation within a gravitational well. This is known as the gravitational redshift or Einstein Shift. [42] 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:

• G is the gravitational constant,
• M is the mass 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 a Schwarzchild coordinate), and
• 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. [43]

The effect is very small but measurable on Earth using the Mössbauer effect and was first observed in the Pound–Rebka experiment. [44] 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). [45]

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 redshift of various absorption and emission lines from a single astronomical object is 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 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. [46]

Spectroscopy, as a measurement, is considerably more difficult than simple photometry, which measures the brightness of astronomical objects through certain filters. [47] 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. [48] 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 δz = 0.5 , and are much less reliable than spectroscopic determinations. [49] However, photometry does at least allow a qualitative characterization of a redshift. For example, if a Sun-like spectrum had a redshift of z = 1 , it would be brightest in the infrared rather than at the yellow-green color associated with the peak of its blackbody 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. (See K correction for more details on the photometric consequences of redshift.) [50]

### Local observations Edit

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 have enabled astronomers to measure velocities and parametrize the masses of the orbiting stars in spectroscopic binaries, a method first employed in 1868 by British astronomer William Huggins. [4] 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. [51] Finely detailed measurements of redshifts are used in helioseismology to determine the precise movements of the photosphere of the Sun. [52] Redshifts have also been used to make the first measurements of the rotation rates of planets, [53] velocities of interstellar clouds, [54] the rotation of galaxies, [19] and the dynamics of accretion onto neutron stars and black holes which exhibit both Doppler and gravitational redshifts. [55] Additionally, 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. [56] 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. [19] Similar measurements have been performed on other galaxies, such as Andromeda. [19] As a diagnostic tool, redshift measurements are one of the most important spectroscopic measurements made in astronomy.

### Extragalactic observations Edit

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 z = 1089 ( z = 0 corresponds to present time), and it shows the state of the universe about 13.8 billion years ago, [57] and 379,000 years after the initial moments of the Big Bang. [58]

The luminous point-like cores of quasars were the first "high-redshift" ( z > 0.1 ) objects discovered before the improvement of telescopes allowed for the discovery of other high-redshift galaxies.

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 the year 1912, while Hubble correlated Slipher's measurements with distances he measured by other means to formulate his Law. In the widely accepted cosmological model based on general relativity, redshift is mainly a result of the expansion of space: this means that the farther away a galaxy is from us, the more the space has expanded in the time since the light left that galaxy, so the more the light has been stretched, the more redshifted the light is, and so the faster it appears to be moving away from us. Hubble's law follows in part from the Copernican principle. [59] 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.

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. [59] 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. [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.

While it was long believed that the expansion rate has been continuously decreasing since the Big Bang, recent observations 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.

### Highest redshifts Edit

Currently, the objects with the highest known redshifts are galaxies and the objects producing gamma ray bursts. The most reliable redshifts are from spectroscopic data, and the highest-confirmed spectroscopic redshift of a galaxy is that of GN-z11, [61] with a redshift of z = 11.1 , corresponding to 400 million years after the Big Bang. The previous record was held by UDFy-38135539 [62] at a redshift of 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 z = 7.5 [63] [64] and the next highest being z = 7.0 . [65] The most distant-observed gamma-ray burst with a spectroscopic redshift measurement was GRB 090423, which had a redshift of z = 8.2 . [66] The most distant-known quasar, ULAS J1342+0928, is at z = 7.54 . [67] [68] The highest-known redshift radio galaxy (TGSS1530) is at a redshift z = 5.72 [69] and the highest-known redshift molecular material is the detection of emission from the CO molecule from the quasar SDSS J1148+5251 at z = 6.42 . [70]

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. [71] Objects that are even redder than EROs are termed hyper extremely red objects (HEROs). [72]

The cosmic microwave background has a redshift of z = 1089 , corresponding to an age of approximately 379,000 years after the Big Bang and a comoving distance of more than 46 billion light-years. [73] 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 20 < z < 100 . [74] 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 z > 10 10 ) [75] and the cosmic gravitational wave background emitted directly from inflation at a redshift in excess of z > 10 25 . [76]

In June 2015, astronomers reported evidence for Population III stars in the Cosmos Redshift 7 galaxy at 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. [77] [78]

### Redshift surveys Edit

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. [79]

The first redshift survey was the CfA Redshift Survey, started in 1977 with the initial data collection completed in 1982. [80] 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. [81] The Sloan Digital Sky Survey (SDSS), is ongoing as of 2013 and aims to measure the redshifts of around 3 million objects. [82] SDSS has recorded redshifts for galaxies as high as 0.8, and has been involved in the detection of quasars beyond 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. [83]

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. [19] 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. [19]

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). [19] Except possibly under carefully controlled conditions, scattering does not produce the same relative change in wavelength across the whole spectrum that is, any calculated z is generally a function of wavelength. Furthermore, scattering from random media generally occurs at many angles, and 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. [19]

In interstellar astronomy, visible spectra can appear redder due to scattering processes in a phenomenon referred to as interstellar reddening [19] —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.

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5. O'Connor, John J. Robertson, Edmund F. (1998). "Christian Andreas Doppler". MacTutor History of Mathematics archive. University of St Andrews.
6. ^ ab
7. Huggins, William (1868). "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". Philosophical Transactions of the Royal Society of London. 158: 529–564. Bibcode:1868RSPT..158..529H. doi:10.1098/rstl.1868.0022.
8. ^
9. Reber, G. (1995). "Intergalactic Plasma". Astrophysics and Space Science. 227 (1–2): 93–96. Bibcode:1995Ap&SS.227. 93R. doi:10.1007/BF00678069. S2CID30000639.
10. ^
11. Pannekoek, A (1961). A History of Astronomy. Dover. p. 451. ISBN978-0-486-65994-7 .
12. ^
13. Bélopolsky, A. (1901). "On an Apparatus for the Laboratory Demonstration of the Doppler-Fizeau Principle". Astrophysical Journal. 13: 15. Bibcode:1901ApJ. 13. 15B. doi:10.1086/140786.
14. ^
15. Adams, Walter S. (1908). "Preliminary catalogue of lines affected in sun-spots". Contributions from the Mount Wilson Observatory / Carnegie Institution of Washington. Contributions from the Solar Observatory of the Carnegie Institution of Washington: Carnegie Institution of Washington. 22: 1–21. Bibcode:1908CMWCI..22. 1A. Reprinted in
16. Adams, Walter S. (1908). "Preliminary Catalogue of Lines Affected in Sun-Spots Region λ 4000 TO λ 4500". Astrophysical Journal. 27: 45. Bibcode:1908ApJ. 27. 45A. doi:10.1086/141524.
17. ^
18. 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
19. ^
20. Slipher, Vesto (1912). "The radial velocity of the Andromeda Nebula". Lowell Observatory Bulletin. 1: 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
21. ^
22. Slipher, Vesto (1915). "Spectrographic Observations of Nebulae". Popular Astronomy. 23: 21–24. Bibcode:1915PA. 23. 21S.
23. ^
24. Slipher, Vesto (1915). "Spectrographic Observations of Nebulae". Popular Astronomy. 23: 22. Bibcode:1915PA. 23. 21S.
25. ^
26. Hubble, Edwin (1929). "A Relation between Distance and Radial Velocity among Extra-Galactic Nebulae". Proceedings of the National Academy of Sciences of the United States of America. 15 (3): 168–173. Bibcode:1929PNAS. 15..168H. doi:10.1073/pnas.15.3.168. PMC522427 . PMID16577160.
27. ^
28. Friedman, A. A. (1922). "Über die Krümmung des Raumes". Zeitschrift für Physik. 10 (1): 377–386. Bibcode:1922ZPhy. 10..377F. doi:10.1007/BF01332580. S2CID125190902. English translation in
29. Friedman, A. (1999). "On the Curvature of Space". General Relativity and Gravitation. 31 (12): 1991–2000. Bibcode:1999GReGr..31.1991F. doi:10.1023/A:1026751225741. S2CID122950995. )
30. ^ ab This was recognized early on by physicists and astronomers working in cosmology in the 1930s. The earliest layman publication describing the details of this correspondence is
31. Eddington, Arthur (1933). The Expanding Universe: Astronomy's 'Great Debate', 1900–1931. Cambridge University Press. (Reprint: 978-0-521-34976-5)
32. ^
33. "Hubble census finds galaxies at redshifts 9 to 12". ESA/Hubble Press Release . Retrieved 13 December 2012 .
34. ^ See, for example, this 25 May 2004 press release from NASA's Swiftspace telescope that is researching gamma-ray bursts: "Measurements of the gamma-ray spectra obtained during the main outburst of the GRB have found little value as redshift indicators, due to the lack of well-defined features. However, optical observations of GRB afterglows have produced spectra with identifiable lines, leading to precise redshift measurements."
35. ^ See [1] for a tutorial on how to define and interpret large redshift measurements.
36. ^ abcdefghi See Binney and Merrifeld (1998), Carroll and Ostlie (1996), Kutner (2003) for applications in astronomy.
37. ^ Where z = redshift v|| = velocity parallel to line-of-sight (positive if moving away from receiver) c = speed of light γ = Lorentz factor a = scale factor G = gravitational constant M = object mass r = radial Schwarzschild coordinate, gtt = t,t component of the metric tensor
38. ^
39. Ives, H. Stilwell, G. (1938). "An Experimental study of the rate of a moving atomic clock". J. Opt. Soc. Am. 28 (7): 215–226. Bibcode:1938JOSA. 28..215I. doi:10.1364/josa.28.000215.
40. ^
41. Freund, Jurgen (2008). Special Relativity for Beginners. World Scientific. p. 120. ISBN978-981-277-160-5 .
42. ^
43. Ditchburn, R (1961). Light. Dover. p. 329. ISBN978-0-12-218101-6 .
44. ^ See "Photons, Relativity, Doppler shiftArchived 2006-08-27 at the Wayback Machine " at the University of Queensland
45. ^ The distinction is made clear in
46. Harrison, Edward Robert (2000). Cosmology: The Science of the Universe (2nd ed.). Cambridge University Press. pp. 306ff. ISBN978-0-521-66148-5 .
47. ^
48. Steven Weinberg (1993). The First Three Minutes: A Modern View of the Origin of the Universe (2nd ed.). Basic Books. p. 34. ISBN9780-465-02437-7 .
49. ^
50. Lars Bergström Ariel Goobar (2006). Cosmology and Particle Astrophysics (2nd ed.). Springer. p. 77, Eq.4.79. ISBN978-3-540-32924-4 .
51. ^
52. M.S. Longair (1998). Galaxy Formation. Springer. p. 161. ISBN978-3-540-63785-1 .
53. ^
54. Yu N Parijskij (2001). "The High Redshift Radio Universe". In Norma Sanchez (ed.). Current Topics in Astrofundamental Physics. Springer. p. 223. ISBN978-0-7923-6856-4 .
55. ^ Measurements of the peculiar velocities out to 5 Mpc using the Hubble Space Telescope were reported in 2003 by
56. Karachentsev et al. (2003). "Local galaxy flows within 5 Mpc". Astronomy and Astrophysics. 398 (2): 479–491. arXiv: astro-ph/0211011 . Bibcode:2003A&A. 398..479K. doi:10.1051/0004-6361:20021566. S2CID26822121.
57. ^
58. Theo Koupelis Karl F. Kuhn (2007). In Quest of the Universe (5th ed.). Jones & Bartlett Publishers. p. 557. ISBN978-0-7637-4387-1 .
59. ^ "It is perfectly valid to interpret the equations of relativity in terms of an expanding space. The mistake is to push analogies too far and imbue space with physical properties that are not consistent with the equations of relativity."
60. Geraint F. Lewis Francis, Matthew J. Barnes, Luke A. Kwan, Juliana et al. (2008). "Cosmological Radar Ranging in an Expanding Universe". Monthly Notices of the Royal Astronomical Society. 388 (3): 960–964. arXiv: 0805.2197 . Bibcode:2008MNRAS.388..960L. doi:10.1111/j.1365-2966.2008.13477.x. S2CID15147382.
61. ^
62. Michal Chodorowski (2007). "Is space really expanding? A counterexample". Concepts Phys. 4 (1): 17–34. arXiv: astro-ph/0601171 . Bibcode:2007ONCP. 4. 15C. doi:10.2478/v10005-007-0002-2. S2CID15931627.
63. ^ Bedran,M.L. (2002)"A comparison between the Doppler and cosmological redshifts"Am.J.Phys.70, 406–408
64. ^
65. Edward Harrison (1992). "The redshift-distance and velocity-distance laws". Astrophysical Journal, Part 1. 403: 28–31. Bibcode:1993ApJ. 403. 28H. doi:10.1086/172179. . A pdf file can be found here [2].
66. ^Harrison 2000, p. 315.
67. ^
68. Steven Weinberg (2008). Cosmology. Oxford University Press. p. 11. ISBN978-0-19-852682-7 .
69. ^ Odenwald & Fienberg 1993
70. ^ Speed faster than light is allowed because the expansion of the spacetimemetric is described by general relativity in terms of sequences of only locally valid inertial frames as opposed to a global Minkowski metric. Expansion faster than light is an integrated effect over many local inertial frames and is allowed because no single inertial frame is involved. The speed-of-light limitation applies only locally. See
71. Michal Chodorowski (2007). "Is space really expanding? A counterexample". Concepts Phys. 4: 17–34. arXiv: astro-ph/0601171 . Bibcode:2007ONCP. 4. 15C. doi:10.2478/v10005-007-0002-2. S2CID15931627.
72. ^ M. Weiss, What Causes the Hubble Redshift?, entry in the Physics FAQ (1994), available via John Baez's website
73. ^ This is only true in a universe where there are no peculiar velocities. Otherwise, redshifts combine as 1 + z = ( 1 + z D o p p l e r ) ( 1 + z e x p a n s i o n ) >)(1+z_ >)> which yields solutions where certain objects that "recede" are blueshifted and other objects that "approach" are redshifted. For more on this bizarre result see Davis, T. M., Lineweaver, C. H., and Webb, J. K. "Solutions to the tethered galaxy problem in an expanding universe and the observation of receding blueshifted objects", American Journal of Physics (2003), 71 358–364.
74. ^
75. Chant, C. A. (1930). "Notes and Queries (Telescopes and Observatory Equipment – The Einstein Shift of Solar Lines)". Journal of the Royal Astronomical Society of Canada. 24: 390. Bibcode:1930JRASC..24..390C.
76. ^
77. Einstein, A (1907). "Über das Relativitätsprinzip und die aus demselben gezogenen Folgerungen". Jahrbuch der Radioaktivität und Elektronik. 4: 411–462. See p. 458 The influence of a gravitational field on clocks
78. ^
79. Pound, R. Rebka, G. (1960). "Apparent Weight of Photons". Physical Review Letters. 4 (7): 337–341. Bibcode:1960PhRvL. 4..337P. doi: 10.1103/PhysRevLett.4.337 . . This paper was the first measurement.
80. ^
81. Sachs, R. K. Wolfe, A. M. (1967). "Perturbations of a cosmological model and angular variations of the cosmic microwave background". Astrophysical Journal. 147 (73): 73. Bibcode:1967ApJ. 147. 73S. doi:10.1086/148982.
82. ^ 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 alternative redshift mechanisms that had been discussed in the literature since the 1930s. In 2001, Geoffrey Burbidge remarked in a 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,
83. Goldhaber et al. (2001). "Timescale Stretch Parameterization of Type Ia Supernova B-Band Lightcurves". Astrophysical Journal. 558 (1): 359–386. arXiv: astro-ph/0104382 . Bibcode:2001ApJ. 558..359G. doi: 10.1086/322460 . S2CID17237531. pointed out that alternative theories are unable to account for timescale stretch observed in type Ia supernovae
84. ^ For a review of the subject of photometry, consider Budding, E., Introduction to Astronomical Photometry, Cambridge University Press (September 24, 1993), 0-521-41867-4
85. ^ The technique was first described by Baum, W. A.: 1962, in G. C. McVittie (ed.), Problems of extra-galactic research, p. 390, IAU Symposium No. 15
86. ^ Bolzonella, M. Miralles, J.-M. Pelló, R., Photometric redshifts based on standard SED fitting procedures, Astronomy and Astrophysics, 363, p.476–492 (2000).
87. ^ A pedagogical overview of the K-correction by David Hogg and other members of the SDSS collaboration can be found at astro-ph.
88. ^ The Exoplanet Tracker is the newest observing project to use this technique, able to track the redshift variations in multiple objects at once, as reported in
89. Ge, Jian Van Eyken, Julian Mahadevan, Suvrath Dewitt, Curtis et al. (2006). "The First Extrasolar Planet Discovered with a New‐Generation High‐Throughput Doppler Instrument". The Astrophysical Journal. 648 (1): 683–695. arXiv: astro-ph/0605247 . Bibcode:2006ApJ. 648..683G. doi:10.1086/505699. S2CID13879217.
90. ^
91. Libbrecht, Keng (1988). "Solar and stellar seismology" (PDF) . Space Science Reviews. 47 (3–4): 275–301. Bibcode:1988SSRv. 47..275L. doi:10.1007/BF00243557. S2CID120897051.
92. ^ 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.
93. ^ An early review by Oort, J. H. on the subject:
94. Oort, J. H. (1970). "The formation of galaxies and the origin of the high-velocity hydrogen". Astronomy and Astrophysics. 7: 381. Bibcode:1970A&A. 7..381O.
95. ^
96. 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.
97. ^ Rybicki, G. B. and A. R. Lightman, Radiative Processes in Astrophysics, John Wiley & Sons, 1979, p. 288 0-471-82759-2
98. ^
99. "Cosmic Detectives". The European Space Agency (ESA). 2013-04-02 . Retrieved 2013-04-25 .
100. ^ 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: Fixsen, D. J. Cheng, E. S. Cottingham, D. A. Eplee, R. E., Jr. Isaacman, R. B. Mather, J. C. Meyer, S. S. Noerdlinger, P. D. Shafer, R. A. Weiss, R. Wright, E. L. Bennett, C. L. Boggess, N. W. Kelsall, T. Moseley, S. H. Silverberg, R. F. Smoot, G. F. Wilkinson, D. T.. (1994). "Cosmic microwave background dipole spectrum measured by the COBE FIRAS instrument", Astrophysical Journal, 420, 445. The most accurate measurement as of 2006 was achieved by the WMAP experiment.
101. ^ ab Peebles (1993).
102. ^
103. Binney, James Scott Treimane (1994). Galactic dynamics. Princeton University Press. ISBN978-0-691-08445-9 .
104. ^
105. Oesch, P. A. Brammer, G. van Dokkum, P. et al. (March 1, 2016). "A Remarkably Luminous Galaxy at z=11.1 Measured with Hubble Space Telescope Grism Spectroscopy". The Astrophysical Journal. 819 (2): 129. arXiv: 1603.00461 . Bibcode:2016ApJ. 819..129O. doi:10.3847/0004-637X/819/2/129. S2CID119262750.
106. ^
107. M.D. Lehnert Nesvadba, NP Cuby, JG Swinbank, AM et al. (2010). "Spectroscopic Confirmation of a galaxy at redshift z = 8.6". Nature. 467 (7318): 940–942. arXiv: 1010.4312 . Bibcode:2010Natur.467..940L. doi:10.1038/nature09462. PMID20962840. S2CID4414781.
108. ^
109. Watson, Darach Christensen, Lise Knudsen, Kirsten Kraiberg Richard, Johan Gallazzi, Anna Michałowski, Michał Jerzy (2015). "A dusty, normal galaxy in the epoch of reionization". Nature. 519 (7543): 327–330. arXiv: 1503.00002 . Bibcode:2015Natur.519..327W. doi:10.1038/nature14164. PMID25731171. S2CID2514879.
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111. Bradley, L. et al. (2008). "Discovery of a Very Bright Strongly Lensed Galaxy Candidate at z

### Articles Edit

• 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, "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.)

### Books Edit

• Nussbaumer, Harry Lydia Bieri (2009). Discovering the Expanding Universe. Cambridge University Press. ISBN978-0-521-51484-2 .
• Binney, James Michael Merrifeld (1998). Galactic Astronomy. Princeton University Press. ISBN978-0-691-02565-0 .
• Carroll, Bradley W. & Dale A. Ostlie (1996). An Introduction to Modern Astrophysics. Addison-Wesley Publishing Company, Inc. ISBN978-0-201-54730-6 .
• Feynman, Richard Leighton, Robert Sands, Matthew (1989). Feynman Lectures on Physics. Vol. 1. Addison-Wesley. ISBN978-0-201-51003-4 .
• Grøn, Øyvind Hervik, Sigbjørn (2007). Einstein's General Theory of Relativity. New York: Springer. ISBN978-0-387-69199-2 .
• Kutner, Marc (2003). Astronomy: A Physical Perspective . Cambridge University Press. ISBN978-0-521-52927-3 .
• Misner, Charles Thorne, Kip S. Wheeler, John Archibald (1973). Gravitation. San Francisco: W. H. Freeman. ISBN978-0-7167-0344-0 .
• Peebles, P. J. E. (1993). Principles of Physical Cosmology. Princeton University Press. ISBN978-0-691-01933-8 .
• Taylor, Edwin F. Wheeler, John Archibald (1992). Spacetime Physics: Introduction to Special Relativity (2nd ed.). W.H. Freeman. ISBN978-0-7167-2327-1 .
• Weinberg, Steven (1971). Gravitation and Cosmology. John Wiley. ISBN978-0-471-92567-5 .
• See also physical cosmology textbooks for applications of the cosmological and gravitational redshifts.

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## Missing Terms in Weinberg's treatment of perturbations on Newtonian Cosmology - Astronomy

### Abstract

Both for the background world model and its linear perturbations Newtonian cosmology coincides with the zero-pressure limits of relativistic cosmology. However, such successes in Newtonian cosmology are not purely based on Newton's gravity, but are rather guided ones by previously known results in Einstein's theory. The action-at-a-distance nature of Newton's gravity requires further verification from Einstein's theory for its use in the large-scale nonlinear regimes. We study the domain of validity of the Newtonian cosmology by investigating weakly nonlinear regimes in relativistic cosmology assuming a zero-pressure and irrotational fluid. We show that, first, if we ignore the coupling with gravitational waves the Newtonian cosmology is exactly valid even to the second order in perturbation. Second, the pure relativistic correction terms start appearing from the third order. Third, the correction terms are independent of the horizon scale and are quite small in the large-scale near the horizon. These conclusions are based on our special (and proper) choice of variables and gauge conditions. In a complementary situation where the system is weakly relativistic but fully nonlinear (thus, far inside the horizon) we can employ the post-Newtonian approximation. We also show that in the large-scale structures the post-Newtonian effects are quite small. As a consequence, now we can rely on the Newtonian gravity in analyzing the evolution of nonlinear large-scale structures even near the horizon volume

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## Gauge transformation in cosmological perturbation

For a given coordinate system in the background spacetime, there are many possible coordinate systems in the perturbed spacetime, all close to each other, that we could use. As indicated in figure 2, the coordinate system #### associates point ##ar P## in the background with ##hat P##, whereas ##< ilde x^α>## associates the same background point ##ar P## with another point ## ilde P##.

1) My understanding here is that both coordinate #### and ##< ilde x^α>## give the same point in the background with different coordinate value, but what does equation (4.2) ( ## ilde x^α ( ilde P) = hat x^α (hat P) = ar x^α (ar P)##) mean? By this, does that mean there is an abuse of notation such that ##ar x^α (ar P)## is "THE" point?

The coordinate transformation relates the coordinates of the same point in the perturbed spacetime

## ilde x^α ( ilde P) = hat x^α ( ilde P) + ξ^α##
## ilde x^α (hat P) = hat x^α (hat P) + ξ^α##

2) What does he mean by this? Based on my search there is one statement saying,
A gauge transformation is different from a general coordinate transformation ##Q (x)## → ## ilde Q (x)##. In a gauge transformation, ## ilde Q## and ##Q## are both calculated at the same coordinate value corresponding to two different space-time points, while in a general coordinate transformation both Q and x are transformed so that we are dealing with the values of a quantity at the same space-time point observed in the two systems.

Based on my understanding ,in a gauge transformation we use ## ilde x^α## and ##hat x^α## to evaluate the same point in the perturbative spacetime, it yields different spacetime points in the "background spacetime"? For a general coordinate transformation if we transform ##Q → ilde Q## but evaluate at the same point ##x##, the "TRUE" location of the point is not changed, just the coordinate system/observer's perspective just like in the passive rotations. But if we also transform ##x → ilde x## then we not only change the perspective but also the "TRUE" position of the point, but in both observers's coordinate frame they see that the point has the same position with respect to their coordinate frame, is this correct? It seems to contradict the statement above that "we are dealing with the values of a quantity at the same space-time point observed in the two systems"

## Missing Terms in Weinberg's treatment of perturbations on Newtonian Cosmology - Astronomy

Post-Newtonian celestial dynamics is a relativistic theory of motion of massive bodies and test particles under the influence of relatively weak gravitational forces. The standard approach for development of this theory relies upon the key concept of the isolated astronomical system supplemented by the assumption that the background spacetime is flat. The standard post-Newtonian theory of motion was instrumental in the explanation of the existing experimental data on binary pulsars, satellite, and lunar laser ranging, and in building precise ephemerides of planets in the Solar System. Recent studies of the formation of large-scale structures in our Universe indicate that the standard post-Newtonian mechanics fails to describe more subtle dynamical effects in motion of the bodies comprising the astronomical systems of larger size—galaxies and clusters of galaxies—where the Riemann curvature of the expanding Friedmann-Lemaître-Robertson-Walker universe interacts with the local gravitational field of the astronomical system and, as such, cannot be ignored. The present paper outlines theoretical principles of the post-Newtonian mechanics in the expanding Universe. It is based upon the gauge-invariant theory of the Lagrangian perturbations of cosmological manifold caused by an isolated astronomical N -body system (the Solar System, a binary star, a galaxy, and a cluster of galaxies). We postulate that the geometric properties of the background manifold are described by a homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker metric governed by two primary components—the dark matter and the dark energy. The dark matter is treated as an ideal fluid with the Lagrangian taken in the form of pressure along with the scalar Clebsch potential as a dynamic variable. The dark energy is associated with a single scalar field with a potential which is hold unspecified as long as the theory permits. Both the Lagrangians of the dark matter and the scalar field are formulated in terms of the field variables which play a role of generalized coordinates in the Lagrangian formalism. It allows us to implement the powerful methods of variational calculus to derive the gauge-invariant field equations of the post-Newtonian celestial mechanics of an isolated astronomical system in an expanding universe. These equations generalize the field equations of the post-Newtonian theory in asymptotically flat spacetime by taking into account the cosmological effects explicitly and in a self-consistent manner without assuming the principle of liner superposition of the fields or a vacuole model of the isolated system, etc. The field equations for matter dynamic variables and gravitational field perturbations are coupled in the most general case of an arbitrary equation of state of matter of the background universe. We introduce a new cosmological gauge which generalizes the de Donder (harmonic) gauge of the post-Newtonian theory in asymptotically flat spacetime. This gauge significantly simplifies the gravitational field equations and allows one to find out the approximations where the field equations can be fully decoupled and solved analytically. The residual gauge freedom is explored and the residual gauge transformations are formulated in the form of the wave equations for the gauge functions. We demonstrate how the cosmological effects interfere with the local system and affect the local distribution of matter of the isolated system and its orbital dynamics. Finally, we worked out the precise mathematical definition of the Newtonian limit for an isolated system residing on the cosmological manifold. The results of the present paper can be useful in the Solar System for calculating more precise ephemerides of the Solar System bodies on extremely long time intervals, in galactic astronomy to study the dynamics of clusters of galaxies, and in gravitational wave astronomy for discussing the impact of cosmology on generation and propagation of gravitational waves emitted by coalescing binaries and/or merging galactic nuclei.

#### Authors & Affiliations

• Department of Physics and Astronomy, University of Missourì-Columbia, 322 Physics Building, Columbia, Missouri 65211, USA
• Sternberg Astronomical Institute, Moscow M. V. Lomonosov State University, Universitetskii Prospect 13, Moscow 119992, Russia

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## Cosmological Density Perturbations from Matter Domination to Recombination in Newtonian and MONDian Gravity Scenarios

Cosmological density perturbations governed by Newtonian and MONDian force laws scenarios for the period from matter domination to recombination have been investigated. Particularly, we find solutions for the density contrast equations obtained for both cases with respect to a homogeneous spatially flat Friedman-Lemaître-Robertson-Walker (FLRW) background using the Lie symmetry approach. Numerical solutions of the density contrast equations for both cases also have been provided in order to study the evolution of the density contrast. For the Newtonian case we find a limiting mass that dictates whether the growth of the density contrast is possible or not. Interestingly, in the Newtonian case, physical growth of the density contrast is not possible since the horizon mass is smaller than the limiting mass. On the other hand, growth of the density contrast is possible depending on the initial condition and the fluctuation mass of the MOND-dominated region despite the radiation pressure that leads to future structure formation.