Lambda-CDM model

The ΛCDM (Lambda cold dark matter) or Lambda-CDM model is a parametrization of the Big Bang cosmological model in which the universe contains three major components: first, a cosmological constant denoted by Lambda (Greek Λ) and associated with dark energy; second, the postulated cold dark matter (abbreviated CDM); and third, ordinary matter. It is frequently referred to as the standard model of Big Bang cosmology because it is the simplest model that provides a reasonably good account of the following properties of the cosmos:

The model assumes that general relativity is the correct theory of gravity on cosmological scales. It emerged in the late 1990s as a concordance cosmology, after a period of time when disparate observed properties of the universe appeared mutually inconsistent, and there was no consensus on the makeup of the energy density of the universe.

The ΛCDM model can be extended by adding cosmological inflation, quintessence and other elements that are current areas of speculation and research in cosmology.

Some alternative models challenge the assumptions of the ΛCDM model. Examples of these are modified Newtonian dynamics, entropic gravity, modified gravity, theories of large-scale variations in the matter density of the universe, bimetric gravity, and scale invariance of empty space.[1][2][3][4]


Lambda-Cold Dark Matter, Accelerated Expansion of the Universe, Big Bang-Inflation
Lambda-CDM, accelerated expansion of the universe. The time-line in this schematic diagram extends from the Big Bang/inflation era 13.7 Byr ago to the present cosmological time.

Most modern cosmological models are based on the cosmological principle, which states that our observational location in the universe is not unusual or special; on a large-enough scale, the universe looks the same in all directions (isotropy) and from every location (homogeneity).[5]

The model includes an expansion of metric space that is well documented both as the red shift of prominent spectral absorption or emission lines in the light from distant galaxies and as the time dilation in the light decay of supernova luminosity curves. Both effects are attributed to a Doppler shift in electromagnetic radiation as it travels across expanding space. Although this expansion increases the distance between objects that are not under shared gravitational influence, it does not increase the size of the objects (e.g. galaxies) in space. It also allows for distant galaxies to recede from each other at speeds greater than the speed of light; local expansion is less than the speed of light, but expansion summed across great distances can collectively exceed the speed of light.

The letter (lambda) represents the cosmological constant, which is currently associated with a vacuum energy or dark energy in empty space that is used to explain the contemporary accelerating expansion of space against the attractive effects of gravity. A cosmological constant has negative pressure, , which contributes to the stress-energy tensor that, according to the general theory of relativity, causes accelerating expansion. The fraction of the total energy density of our (flat or almost flat) universe that is dark energy, , is estimated to be 0.669 ± 0.038 based on the 2018 Dark Energy Survey results using Type Ia Supernovae[6] or 0.6847 ± 0.0073 based on the 2018 release of Planck satellite data, or more than 68.3% (2018 estimate) of the mass-energy density of the universe.[7]

Dark matter is postulated in order to account for gravitational effects observed in very large-scale structures (the "flat" rotation curves of galaxies; the gravitational lensing of light by galaxy clusters; and enhanced clustering of galaxies) that cannot be accounted for by the quantity of observed matter.

Cold dark matter as currently hypothesized is:

It consists of matter other than protons and neutrons (and electrons, by convention, although electrons are not baryons).
Its velocity is far less than the speed of light at the epoch of radiation-matter equality (thus neutrinos are excluded, being non-baryonic but not cold).
It cannot cool by radiating photons.
The dark matter particles interact with each other and other particles only through gravity and possibly the weak force.

Dark matter constitutes about 26.8% (2013 estimate) of the mass-energy density of the universe. The remaining 4.8% (2013 estimate) comprises all ordinary matter observed as atoms, chemical elements, gas and plasma, the stuff of which visible planets, stars and galaxies are made. The great majority of ordinary matter in the universe is unseen, since visible stars and gas inside galaxies and clusters account for less than 10% of the ordinary matter contribution to the mass-energy density of the universe.[8]

Also, the energy density includes a very small fraction (~ 0.01%) in cosmic microwave background radiation, and not more than 0.5% in relic neutrinos. Although very small today, these were much more important in the distant past, dominating the matter at redshift > 3200.

The model includes a single originating event, the "Big Bang", which was not an explosion but the abrupt appearance of expanding space-time containing radiation at temperatures of around 1015 K. This was immediately (within 10−29 seconds) followed by an exponential expansion of space by a scale multiplier of 1027 or more, known as cosmic inflation. The early universe remained hot (above 10,000 K) for several hundred thousand years, a state that is detectable as a residual cosmic microwave background, or CMB, a very low energy radiation emanating from all parts of the sky. The "Big Bang" scenario, with cosmic inflation and standard particle physics, is the only current cosmological model consistent with the observed continuing expansion of space, the observed distribution of lighter elements in the universe (hydrogen, helium, and lithium), and the spatial texture of minute irregularities (anisotropies) in the CMB radiation. Cosmic inflation also addresses the "horizon problem" in the CMB; indeed, it seems likely that the universe is larger than the observable particle horizon.

The model uses the Friedmann–Lemaître–Robertson–Walker metric, the Friedmann equations and the cosmological equations of state to describe the observable universe from right after the inflationary epoch to present and future.

Cosmic expansion history

The expansion of the universe is parameterized by a dimensionless scale factor (with time counted from the birth of the universe), defined relative to the present day, so ; the usual convention in cosmology is that subscript 0 denotes present-day values, so is the current age of the universe. The scale factor is related to the observed redshift[9] of the light emitted at time by

The expansion rate is described by the time-dependent Hubble parameter, , defined as

where is the time-derivative of the scale factor. The first Friedmann equation gives the expansion rate in terms of the matter+radiation density , the curvature , and the cosmological constant ,[9]

where as usual is the speed of light and is the gravitational constant. A critical density is the present-day density, which gives zero curvature , assuming the cosmological constant is zero, regardless of its actual value. Substituting these conditions to the Friedmann equation gives


where is the reduced Hubble constant. If the cosmological constant were actually zero, the critical density would also mark the dividing line between eventual recollapse of the universe to a Big Crunch, or unlimited expansion. For the Lambda-CDM model with a positive cosmological constant (as observed), the universe is predicted to expand forever regardless of whether the total density is slightly above or below the critical density; though other outcomes are possible in extended models where the dark energy is not constant but actually time-dependent.

It is standard to define the present-day density parameter for various species as the dimensionless ratio

where the subscript is one of for baryons, for cold dark matter, for radiation (photons plus relativistic neutrinos), and or for dark energy.

Since the densities of various species scale as different powers of , e.g. for matter etc., the Friedmann equation can be conveniently rewritten in terms of the various density parameters as

where w is the equation of state of dark energy, and assuming negligible neutrino mass (significant neutrino mass requires a more complex equation). The various parameters add up to by construction. In the general case this is integrated by computer to give the expansion history and also observable distance-redshift relations for any chosen values of the cosmological parameters, which can then be compared with observations such as supernovae and baryon acoustic oscillations.

In the minimal 6-parameter Lambda-CDM model, it is assumed that curvature is zero and , so this simplifies to

Observations show that the radiation density is very small today, ; if this term is neglected the above has an analytic solution[11]

where this is fairly accurate for or million years. Solving for gives the present age of the universe in terms of the other parameters.

It follows that the transition from decelerating to accelerating expansion (the second derivative crossing zero) occurred when

which evaluates to or for the best-fit parameters estimated from the Planck spacecraft.

Historical development

The discovery of the Cosmic Microwave Background (CMB) in 1964 confirmed a key prediction of the Big Bang cosmology. From that point on, it was generally accepted that the universe started in a hot, dense state and has been expanding over time. The rate of expansion depends on the types of matter and energy present in the universe, and in particular, whether the total density is above or below the so-called critical density.

During the 1970s, most attention focused on pure-baryonic models, but there were serious challenges explaining the formation of galaxies, given the small anisotropies in the CMB (upper limits at that time). In the early 1980s, it was realized that this could be resolved if cold dark matter dominated over the baryons, and the theory of cosmic inflation motivated models with critical density.

During the 1980s, most research focused on cold dark matter with critical density in matter, around 95% CDM and 5% baryons: these showed success at forming galaxies and clusters of galaxies, but problems remained; notably, the model required a Hubble constant lower than preferred by observations, and observations around 1988–1990 showed more large-scale galaxy clustering than predicted.

These difficulties sharpened with the discovery of CMB anisotropy by COBE in 1992, and several modified CDM models, including Λ-CDM and mixed cold and hot dark matter, came under active consideration through the mid-1990s. The Λ-CDM model then became the leading model following the observations of accelerating expansion in 1998, and was quickly supported by other observations: in 2000, the BOOMERanG microwave background experiment measured the total (matter–energy) density to be close to 100% of critical, whereas in 2001 the 2dFGRS galaxy redshift survey measured the matter density to be near 25%; the large difference between these values supports a positive Λ or dark energy. Much more precise spacecraft measurements of the microwave background from WMAP in 2003–2010 and Planck in 2013–2015 have continued to support the model and pin down the parameter values, most of which are now constrained below 1 percent uncertainty.

There is currently active research into many aspects of the Λ-CDM model, both to refine the parameters and possibly detect deviations. In addition, Λ-CDM has no explicit physical theory for the origin or physical nature of dark matter or dark energy; the nearly scale-invariant spectrum of the CMB perturbations, and their image across the celestial sphere, are believed to result from very small thermal and acoustic irregularities at the point of recombination.

A large majority of astronomers and astrophysicists support the ΛCDM model or close relatives of it, but Milgrom, McGaugh, and Kroupa are leading critics, attacking the dark matter portions of the theory from the perspective of galaxy formation models and supporting the alternative MOND theory, which requires a modification of the Einstein field equations and the Friedmann equations as seen in proposals such as MOG theory or TeVeS theory. Other proposals by theoretical astrophysicists of cosmological alternatives to Einstein's general relativity that attempt to account for dark energy or dark matter include f(R) gravity, scalar–tensor theories such as galileon theories, brane cosmologies, the DGP model, and massive gravity and its extensions such as bimetric gravity.


In addition to explaining pre-2000 observations, the model has made a number of successful predictions: notably the existence of the baryon acoustic oscillation feature, discovered in 2005 in the predicted location; and the statistics of weak gravitational lensing, first observed in 2000 by several teams. The polarization of the CMB, discovered in 2002 by DASI[12] is now a dramatic success: in the 2015 Planck data release,[13] there are seven observed peaks in the temperature (TT) power spectrum, six peaks in the temperature-polarization (TE) cross spectrum, and five peaks in the polarization (EE) spectrum. The six free parameters can be well constrained by the TT spectrum alone, and then the TE and EE spectra can be predicted theoretically to few-percent precision with no further adjustments allowed: comparison of theory and observations shows an excellent match.


Extensive searches for dark matter particles have so far shown no well-agreed detection; the dark energy may be almost impossible to detect in a laboratory, and its value is unnaturally small compared to naive theoretical predictions.

Comparison of the model with observations is very successful on large scales (larger than galaxies, up to the observable horizon), but may have some problems on sub-galaxy scales, possibly predicting too many dwarf galaxies and too much dark matter in the innermost regions of galaxies. This problem is called the "small scale crisis".[14] These small scales are harder to resolve in computer simulations, so it is not yet clear whether the problem is the simulations, non-standard properties of dark matter, or a more radical error in the model.

It has been argued that the ΛCDM model is built upon a foundation of conventionalist stratagems, rendering it unfalsifiable in the sense defined by Karl Popper.[15]


Planck Collaboration Cosmological parameters[17]
Description Symbol Value
Physical baryon density parameter[a] Ωb h2 0.02230±0.00014
Physical dark matter density parameter[a] Ωc h2 0.1188±0.0010
Age of the universe t0 13.799±0.021 × 109 years
Scalar spectral index ns 0.9667±0.0040
Curvature fluctuation amplitude,
k0 = 0.002 Mpc−1
Reionization optical depth τ 0.066±0.012
Total density parameter[b] Ωtot 1
Equation of state of dark energy w −1
Tensor/scalar ratio r 0
Running of spectral index 0
Sum of three neutrino masses 0.06 eV/c2[c][16]:40
Effective number of relativistic degrees
of freedom
Neff 3.046[d][16]:47
Hubble constant H0 67.74±0.46 km s−1 Mpc−1
Baryon density parameter[b] Ωb 0.0486±0.0010[e]
Dark matter density parameter[b] Ωc 0.2589±0.0057[f]
Matter density parameter[b] Ωm 0.3089±0.0062
Dark energy density parameter[b] ΩΛ 0.6911±0.0062
Critical density ρcrit (8.62±0.12)×10−27 kg/m3[g]
The present root-mean-square matter fluctuation

averaged over a sphere of radius 8h1 Mpc

σ8 0.8159±0.0086
Redshift at decoupling z 1089.90±0.23
Age at decoupling t 377700±3200 years[20]
Redshift of reionization (with uniform prior) zre 8.5+1.0

The simple ΛCDM model is based on six parameters: physical baryon density parameter; physical dark matter density parameter; the age of the universe; scalar spectral index; curvature fluctuation amplitude; and reionization optical depth.[22] In accordance with Occam's razor, six is the smallest number of parameters needed to give an acceptable fit to current observations; other possible parameters are fixed at "natural" values, e.g. total density parameter = 1.00, dark energy equation of state = −1. (See below for extended models that allow these to vary.)

The values of these six parameters are mostly not predicted by current theory (though, ideally, they may be related by a future "Theory of Everything"), except that most versions of cosmic inflation predict the scalar spectral index should be slightly smaller than 1, consistent with the estimated value 0.96. The parameter values, and uncertainties, are estimated using large computer searches to locate the region of parameter space providing an acceptable match to cosmological observations. From these six parameters, the other model values, such as the Hubble constant and the dark energy density, can be readily calculated.

Commonly, the set of observations fitted includes the cosmic microwave background anisotropy, the brightness/redshift relation for supernovae, and large-scale galaxy clustering including the baryon acoustic oscillation feature. Other observations, such as the Hubble constant, the abundance of galaxy clusters, weak gravitational lensing and globular cluster ages, are generally consistent with these, providing a check of the model, but are less precisely measured at present.

Parameter values listed below are from the Planck Collaboration Cosmological parameters 68% confidence limits for the base ΛCDM model from Planck CMB power spectra, in combination with lensing reconstruction and external data (BAO + JLA + H0).[16] See also Planck (spacecraft).

  1. ^ a b The "physical baryon density parameter" Ωb h2 is the "baryon density parameter" Ωb multiplied by the square of the reduced Hubble constant h = H0 / (100 km s−1 Mpc−1).[18][19] Likewise for the difference between "physical dark matter density parameter" and "dark matter density parameter".
  2. ^ a b c d e A density ρx = Ωxρcrit is expressed in terms of the critical density ρcrit, which is the total density of matter/energy needed for the universe to be spatially flat. Measurements indicate that the actual total density ρtot is very close if not equal to this value, see below.
  3. ^ This is the minimal value allowed by solar and terrestrial neutrino oscillation experiments.
  4. ^ from the Standard Model of particle physics
  5. ^ Calculated from Ωbh2 and h = H0 / (100 km s−1 Mpc−1).
  6. ^ Calculated from Ωch2 and h = H0 / (100 km s−1 Mpc−1).
  7. ^ Calculated from h = H0 / (100 km s−1 Mpc−1) per ρcrit = 1.87847×10−26 h2 kg m−3.[10]

Missing baryon problem

Massimo Persic and Paolo Salucci[23] firstly estimated the baryonic density today present in ellipticals, spirals, groups and clusters of galaxies. They performed an integration of the baryonic mass-to-light ratio over luminosity (in the following ), weighted with the luminosity function over the previously mentioned classes of astrophysical objects:

The result was:

where .

Note that this value is much lower than the prediction of standard cosmic nucleosynthesis , so that stars and gas in galaxies and in galaxy groups and clusters account for less than 10% of the primordially synthesized baryons. This issue is known as the problem of the "missing baryons".

Extended models

Extended model parameters
Description Symbol Value
Total density parameter 1.0023+0.0056
Equation of state of dark energy −0.980±0.053
Tensor-to-scalar ratio < 0.11, k0 = 0.002 Mpc−1 ()
Running of the spectral index −0.022±0.020, k0 = 0.002 Mpc−1
Sum of three neutrino masses < 0.58 eV/c2 ()
Physical neutrino density parameter < 0.0062

Extended models allow one or more of the "fixed" parameters above to vary, in addition to the basic six; so these models join smoothly to the basic six-parameter model in the limit that the additional parameter(s) approach the default values. For example, possible extensions of the simplest ΛCDM model allow for spatial curvature ( may be different from 1); or quintessence rather than a cosmological constant where the equation of state of dark energy is allowed to differ from −1. Cosmic inflation predicts tensor fluctuations (gravitational waves). Their amplitude is parameterized by the tensor-to-scalar ratio (denoted ), which is determined by the unknown energy scale of inflation. Other modifications allow hot dark matter in the form of neutrinos more massive than the minimal value, or a running spectral index; the latter is generally not favoured by simple cosmic inflation models.

Allowing additional variable parameter(s) will generally increase the uncertainties in the standard six parameters quoted above, and may also shift the central values slightly. The Table below shows results for each of the possible "6+1" scenarios with one additional variable parameter; this indicates that, as of 2015, there is no convincing evidence that any additional parameter is different from its default value.

Some researchers have suggested that there is a running spectral index, but no statistically significant study has revealed one. Theoretical expectations suggest that the tensor-to-scalar ratio should be between 0 and 0.3, and the latest results are now within those limits.

See also


  1. ^ Maeder, Andre (2017). "An Alternative to the ΛCDM Model: The Case of Scale Invariance". The Astrophysical Journal. 834 (2): 194. arXiv:1701.03964. Bibcode:2017ApJ...834..194M. doi:10.3847/1538-4357/834/2/194. ISSN 0004-637X.
  2. ^ Brouer, Margot (2017). "First test of Verlinde's theory of emergent gravity using weak gravitational lensing measurements". Monthly Notices of the Royal Astronomical Society. 466 (3): 2547–2559. arXiv:1612.03034. Bibcode:2017MNRAS.466.2547B. doi:10.1093/mnras/stw3192.
  3. ^ P. Kroupa, B. Famaey, K.S. de Boer, J. Dabringhausen, M. Pawlowski, C.M. Boily, H. Jerjen, D. Forbes, G. Hensler, M. Metz, "Local-Group tests of dark-matter concordance cosmology. Towards a new paradigm for structure formation" A&A 523, 32 (2010).
  4. ^ Petit, J. P.; D’Agostini, G. (2018-07-01). "Constraints on Janus Cosmological model from recent observations of supernovae type Ia". Astrophysics and Space Science. 363 (7): 139. Bibcode:2018Ap&SS.363..139D. doi:10.1007/s10509-018-3365-3. ISSN 1572-946X.
  5. ^ Andrew Liddle. An Introduction to Modern Cosmology (2nd ed.). London: Wiley, 2003.
  6. ^ Maeder, Andre; et al. (DES Collaboration) (2018). "First Cosmology Results using Type Ia Supernovae from the Dark Energy Survey: Constraints on Cosmological Parameters". arXiv:1811.02374 [astro-ph.CO].
  7. ^ Maeder, Andre; et al. (Planck Collaboration) (2018). "Planck 2018 results. VI. Cosmological parameters". arXiv:1807.06209 [astro-ph.CO].
  8. ^ Persic, Massimo; Salucci, Paolo (1992-09-01). "The baryon content of the Universe". Monthly Notices of the Royal Astronomical Society. 258 (1): 14P–18P. arXiv:astro-ph/0502178. Bibcode:1992MNRAS.258P..14P. doi:10.1093/mnras/258.1.14P. ISSN 0035-8711.
  9. ^ a b Dodelson, Scott (2008). Modern cosmology (4 ed.). San Diego, CA: Academic Press. ISBN 978-0122191411.
  10. ^ a b K.A. Olive; et al. (Particle Data Group) (2015). "The Review of Particle Physics. 2. Astrophysical constants and parameters" (PDF). Particle Data Group: Berkeley Lab. Archived from the original (PDF) on 3 December 2015. Retrieved 10 January 2016.
  11. ^ Frieman, Joshua A.; Turner, Michael S.; Huterer, Dragan (2008). "Dark Energy and the Accelerating Universe". Annual Review of Astronomy and Astrophysics. 46 (1): 385–432. arXiv:0803.0982. Bibcode:2008ARA&A..46..385F. doi:10.1146/annurev.astro.46.060407.145243.
  12. ^ Kovac, J. M.; Leitch, E. M.; Pryke, C.; Carlstrom, J. E.; Halverson, N. W.; Holzapfel, W. L. (2002). "Detection of polarization in the cosmic microwave background using DASI". Nature. 420 (6917): 772–787. arXiv:astro-ph/0209478. Bibcode:2002Natur.420..772K. doi:10.1038/nature01269. PMID 12490941.
  13. ^ Planck Collaboration (2016). "Planck 2015 Results. XIII. Cosmological Parameters". Astronomy & Astrophysics. 594 (13): A13. arXiv:1502.01589. Bibcode:2016A&A...594A..13P. doi:10.1051/0004-6361/201525830.
  14. ^ Rini, Matteo (2017). "Synopsis: Tackling the Small-Scale Crisis". Physical Review D. 95 (12): 121302. arXiv:1703.10559. Bibcode:2017PhRvD..95l1302N. doi:10.1103/PhysRevD.95.121302.
  15. ^ Merritt, David "Cosmology and Convention", Studies In History and Philosophy of Science Part B: Studies In History and Philosophy of Modern Physics, 57(1):41-52, February 2017.
  16. ^ a b c d Planck Collaboration (2016). "Planck 2015 results. XIII. Cosmological parameters". Astronomy & Astrophysics. 594 (13): A13. arXiv:1502.01589. Bibcode:2016A&A...594A..13P. doi:10.1051/0004-6361/201525830.
  17. ^ Planck 2015,[16] p. 32, table 4, last column.
  18. ^ Appendix A of the LSST Science Book Version 2.0
  19. ^ p. 7 of Findings of the Joint Dark Energy Mission Figure of Merit Science Working Group
  20. ^ a b Table 8 on p. 39 of Jarosik, N. et al. (WMAP Collaboration) (2011). "Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Sky Maps, Systematic Errors, and Basic Results" (PDF). The Astrophysical Journal Supplement Series. 192 (2): 14. arXiv:1001.4744. Bibcode:2011ApJS..192...14J. doi:10.1088/0067-0049/192/2/14. Retrieved 2010-12-04. (from NASA's WMAP Documents page)
  21. ^ Planck Collaboration; Adam, R.; Aghanim, N.; Ashdown, M.; Aumont, J.; Baccigalupi, C.; Ballardini, M.; Banday, A. J.; Barreiro, R. B. (2016-05-11). "Planck intermediate results. XLVII. Planck constraints on reionization history". Astronomy & Astrophysics. 596 (108): A108. arXiv:1605.03507. Bibcode:2016A&A...596A.108P. doi:10.1051/0004-6361/201628897.
  22. ^ Spergel, D. N. (2015). "The dark side of the cosmology: dark matter and dark energy". Science. 347 (6226): 1100–1102. Bibcode:2015Sci...347.1100S. doi:10.1126/science.aaa0980. PMID 25745164.
  23. ^ The baryon content of the universe, M. Persic and P. Salucci, Monthly Notices of the Royal Astronomical Society, 1992

Further reading

External links

Accelerating expansion of the universe

The accelerating expansion of the universe is the observation that the expansion of the universe is such that the velocity at which a distant galaxy is receding from the observer is continuously increasing with time.The accelerated expansion was discovered during 1998, by two independent projects, the Supernova Cosmology Project and the High-Z Supernova Search Team, which both used distant type Ia supernovae to measure the acceleration. The idea was that as type 1a supernovae have almost the same intrinsic brightness (a standard candle), and since objects that are further away appear dimmer, we can use the observed brightness of these supernovae to measure the distance to them. The distance can then be compared to the supernovae's cosmological redshift, which measures how much the universe has expanded since the supernova occurred. The unexpected result was that objects in the universe are moving away from one another at an accelerated rate. Cosmologists at the time expected that recession velocity would always be decelerating, due to the gravitational attraction of the matter in the universe. Three members of these two groups have subsequently been awarded Nobel Prizes for their discovery. Confirmatory evidence has been found in baryon acoustic oscillations, and in analyses of the clustering of galaxies.

The accelerated expansion of the universe is thought to have begun since the universe entered its dark-energy-dominated era roughly 5 billion years ago.

Within the framework of general relativity, an accelerated expansion can be accounted for by a positive value of the cosmological constant Λ, equivalent to the presence of a positive vacuum energy, dubbed "dark energy". While there are alternative possible explanations, the description assuming dark energy (positive Λ) is used in the current standard model of cosmology, which also includes cold dark matter (CDM) and is known as the Lambda-CDM model.

Cold dark matter

In cosmology and physics, cold dark matter (CDM) is a hypothetical type of dark matter. Observations indicate that approximately 85% of the matter in the universe is dark matter, with only a small fraction being the ordinary baryonic matter that composes stars, planets, and living organisms. Cold refers to the fact that the dark matter moves slowly compared to the speed of light, while dark indicates that it interacts very weakly with ordinary matter and electromagnetic radiation.

The physical nature of CDM is currently unknown, and there are a wide variety of possibilities. Among them are a new type of weakly interacting massive particle, primordial black holes, and axions.


Concordance may refer to:

Agreement (linguistics), a form of cross-reference between different parts of a sentence or phrase

Bible concordance, an alphabetical listing of terms in the Bible

Concordant coastline, in geology, where beds, or layers, of differing rock types form ridges that run parallel to the coast

Concordant pair, in statistics

Concordance (publishing), a list of words used in a body of work, with their immediate contexts

Concordance (genetics), the presence of the same trait in both members of a pair of twins (or set of individuals)

Concordance (medicine), involvement of patients in decision-making to improve patient compliance with medical advice

Concordance of evidence, in law

Concordance system, in Swiss politics, the presence of all major parties in the Federal Council

Concordance correlation coefficient, in statistics, a measurement of the agreement between two variables

Concordance database, a database tailored to legal applications and distributed by LexisNexis

Inter-rater reliability, in statistics, the degree to which multiple measurements of the same thing are similar

Lambda-CDM model of big-bang cosmology

Link concordance, a relation between mathematical links in knot theory

Conformal cyclic cosmology

The conformal cyclic cosmology (CCC) is a cosmological model in the framework of general relativity, advanced by the theoretical physicists Roger Penrose and Vahe Gurzadyan.

In CCC, the universe iterates through infinite cycles, with the future timelike infinity of each previous iteration being identified with the Big Bang singularity of the next. Penrose popularized this theory in his 2010 book Cycles of Time: An Extraordinary New View of the Universe.


Cosmology (from the Greek κόσμος, kosmos "world" and -λογία, -logia "study of") is a branch of astronomy concerned with the studies of the origin and evolution of the universe, from the Big Bang to today and on into the future. It is the scientific study of the origin, evolution, and eventual fate of the universe. Physical cosmology is the scientific study of the universe's origin, its large-scale structures and dynamics, and its ultimate fate, as well as the laws of science that govern these areas.The term cosmology was first used in English in 1656 in Thomas Blount's Glossographia, and in 1731 taken up in Latin by German philosopher Christian Wolff, in Cosmologia Generalis.Religious or mythological cosmology is a body of beliefs based on mythological, religious, and esoteric literature and traditions of creation myths and eschatology.

Physical cosmology is studied by scientists, such as astronomers and physicists, as well as philosophers, such as metaphysicians, philosophers of physics, and philosophers of space and time. Because of this shared scope with philosophy, theories in physical cosmology may include both scientific and non-scientific propositions, and may depend upon assumptions that cannot be tested. Cosmology differs from astronomy in that the former is concerned with the Universe as a whole while the latter deals with individual celestial objects. Modern physical cosmology is dominated by the Big Bang theory, which attempts to bring together observational astronomy and particle physics; more specifically, a standard parameterization of the Big Bang with dark matter and dark energy, known as the Lambda-CDM model.

Theoretical astrophysicist David N. Spergel has described cosmology as a "historical science" because "when we look out in space, we look back in time" due to the finite nature of the speed of light.

Dark energy

In physical cosmology and astronomy, dark energy is an unknown form of energy which is hypothesized to permeate all of space, tending to accelerate the expansion of the universe. Dark energy is the most accepted hypothesis to explain the observations since the 1990s indicating that the universe is expanding at an accelerating rate.Assuming that the standard model of cosmology is correct, the best current measurements indicate that dark energy contributes 68% of the total energy in the present-day observable universe. The mass–energy of dark matter and ordinary (baryonic) matter contribute 27% and 5%, respectively, and other components such as neutrinos and photons contribute a very small amount. The density of dark energy is very low (~ 7 × 10−30 g/cm3) much less than the density of ordinary matter or dark matter within galaxies. However, it dominates the mass–energy of the universe because it is uniform across space.Two proposed forms for dark energy are the cosmological constant, representing a constant energy density filling space homogeneously, and scalar fields such as quintessence or moduli, dynamic quantities whose energy density can vary in time and space. Contributions from scalar fields that are constant in space are usually also included in the cosmological constant. The cosmological constant can be formulated to be equivalent to the zero-point radiation of space i.e. the vacuum energy. Scalar fields that change in space can be difficult to distinguish from a cosmological constant because the change may be extremely slow.

Deceleration parameter

The deceleration parameter in cosmology is a dimensionless measure of the cosmic acceleration of the expansion of space in a Friedmann–Lemaître–Robertson–Walker universe. It is defined by:

where is the scale factor of the universe and the dots indicate derivatives by proper time. The expansion of the universe is said to be "accelerating" if (recent measurements suggest it is), and in this case the deceleration parameter will be negative. The minus sign and name "deceleration parameter" are historical; at the time of definition was expected to be negative, so a minus sign was inserted in the definition to make positive in that case. Since the evidence for the accelerating universe in the 1998–2003 era, it is now believed that is positive therefore the present-day value is negative (though was positive in the past before dark energy became dominant). In general varies with cosmic time, except in a few special cosmological models; the present-day value is denoted .

The Friedmann acceleration equation can be written as

where the sum extends over the different components, matter, radiation and dark energy, is the equivalent mass density of each component, is its pressure, and is the equation of state for each component. The value of is 0 for non-relativistic matter (baryons and dark matter), 1/3 for radiation, and −1 for a cosmological constant; for more general dark energy it may differ from −1, in which case it is denoted or simply .

Defining the critical density as

and the density parameters , substituting in the acceleration equation gives

where the density parameters are at the relevant cosmic epoch. At the present day is negligible, and if (cosmological constant) this simplifies to

where the density parameters are present-day values; this evaluates to for the parameters estimated from the Planck spacecraft data. (Note that the CMB, as a high-redshift measurement, does not directly measure ; but its value can be inferred by fitting cosmological models to the CMB data, then calculating from the other measured parameters as above).

The time derivative of the Hubble parameter can be written in terms of the deceleration parameter:

Except in the speculative case of phantom energy (which violates all the energy conditions), all postulated forms of mass-energy yield a deceleration parameter Thus, any non-phantom universe should have a decreasing Hubble parameter, except in the case of the distant future of a Lambda-CDM model, where will tend to −1 from above and the Hubble parameter will asymptote to a constant value of .

The above results imply that the universe would be decelerating for any cosmic fluid with equation of state greater than (any fluid satisfying the strong energy condition does so, as does any form of matter present in the Standard Model, but excluding inflation). However observations of distant type Ia supernovae indicate that is negative; the expansion of the universe is accelerating. This is an indication that the gravitational attraction of matter, on the cosmological scale, is more than counteracted by the negative pressure of dark energy, in the form of either quintessence or a positive cosmological constant.

Before the first indications of an accelerating universe, in 1998, it was thought that the universe was dominated by matter with negligible pressure, This implied that the deceleration parameter would be equal to , e.g. for a universe with or for a low-density zero-Lambda model. The experimental effort to discriminate these cases with supernovae actually revealed negative , evidence for cosmic acceleration, which has subsequently grown stronger.

Einstein–de Sitter universe

The Einstein–de Sitter universe is a model of the universe proposed by Albert Einstein and Willem de Sitter in 1932. On first learning of Edwin Hubble's discovery of a linear relation between the redshift of the galaxies and their distance, Einstein set the cosmological constant to zero in the Friedmann equations, resulting in a model of the expanding universe known as the Friedmann–Einstein universe. In 1932, Einstein and de Sitter proposed an even simpler cosmic model by assuming a vanishing spatial curvature as well as a vanishing cosmological constant. In modern parlance, the Einstein–de Sitter universe can be described as a cosmological model for a flat matter-only Friedmann–Lemaître–Robertson–Walker metric (FLRW) universe.

In the model, Einstein and de Sitter derived a simple relation between the average density of matter in the universe and its expansion according to H02= кρ/3 where H0 is the Hubble constant, ρ is the average density of matter and к is the Einstein constant. The size of the Einstein–de Sitter universe evolves with time as , making its current age 2/3 times the Hubble time. The Einstein–de Sitter universe became a standard model of the universe for many years because of its simplicity and because of a lack of empirical evidence for either spatial curvature or a cosmological constant. It also represented an important theoretical case of a universe of critical matter density poised just at the limit of eventually contracting. However, Einstein's later reviews of cosmology make it clear that he saw the model as only one of several possibilities for the expanding universe.

The Einstein–de Sitter universe was particularly popular in the 1980s, after the theory of cosmic inflation predicted that the curvature of the universe should be very close to zero. This case with zero cosmological constant implies the Einstein-de Sitter model, and the theory of cold dark matter was developed, initially with a cosmic matter budget around 95% cold dark matter and 5% baryons. However, in the 1990s various observations including galaxy clustering and measurements of the Hubble constant led to increasingly serious problems for this model. Following the discovery of the accelerating universe in 1998, and observations of the cosmic microwave background and galaxy redshift surveys in 2000-2003, it is now generally accepted that dark energy makes up around 70 percent of the present energy density while cold dark matter contributes around 25 percent, as in the modern Lambda-CDM model.

The Einstein-de Sitter model remains a good approximation to our universe in the past at redshifts between around 300 and 2, i.e. well after the radiation-dominated era but before dark energy became important.

Galaxy formation and evolution

The study of galaxy formation and evolution is concerned with the processes that formed a heterogeneous universe from a homogeneous beginning, the formation of the first galaxies, the way galaxies change over time, and the processes that have generated the variety of structures observed in nearby galaxies. Galaxy formation is hypothesized to occur from structure formation theories, as a result of tiny quantum fluctuations in the aftermath of the Big Bang. The simplest model in general agreement with observed phenomena is the Lambda-CDM model—that is, that clustering and merging allows galaxies to accumulate mass, determining both their shape and structure.

History of the Big Bang theory

The history of the Big Bang theory began with the Big Bang's development from observations and theoretical considerations. Much of the theoretical work in cosmology now involves extensions and refinements to the basic Big Bang model.

Institute for Computational Cosmology

The Institute for Computational Cosmology (ICC) is a Research Institute at Durham University, England. It was founded in November 2002 as part of the Ogden Centre for Fundamental Physics, which also includes the Institute for Particle Physics Phenomenology (IPPP). The ICC's primary mission is to advance fundamental knowledge in cosmology. Topics of active research include: the nature of dark matter and dark energy, the evolution of cosmic structure, the formation of galaxies, and the determination of fundamental parameters.

The current director of the ICC is Carlos Frenk; the deputy director is Shaun Cole. ICC researchers have played a central role in the development of the standard model of cosmology, Lambda-CDM model (ΛCDM). Because of the vast scale of questions in cosmology, advances often require supercomputer simulations in which a virtual Universe is allowed to evolve for 13.8 billion years from the Big Bang to the present day. The simulation is rerun with different ingredients or different physics, until it matches the observed Universe. This approach has required one of the most powerful supercomputers for academic research in the world, the “Cosmology Machine (COSMA)” as part of the DiRAC supercomputing consortium.

Mattig formula

Mattig's formula was an important formula in observational cosmology and extragalactic astronomy which gives relation between radial coordinate and redshift of a given source. It depends on the cosmological model being used and is used to calculate luminosity distance in terms of redshift.It assumes zero dark energy, and is therefore no longer applicable in modern cosmological models such as the Lambda-CDM model, (which require a numerical integration to get the distance-redshift relation). However, Mattig's formula was of considerable historical importance as the first analytic formula for the distance-redshift relationship for arbitrary matter density, and this spurred significant research in the 1960s and 1970s attempting to measure this relation.

Non-standard cosmology

A non-standard cosmology is any physical cosmological model of the universe that was, or still is, proposed as an alternative to the then-current standard model of cosmology. The term non-standard is applied to any theory that does not conform to the scientific consensus. Because the term depends on the prevailing consensus, the meaning of the term changes over time. For example, hot dark matter would not have been considered non-standard in 1990, but would be in 2010. Conversely, a non-zero cosmological constant resulting in an accelerating universe would have been considered non-standard in 1990, but is part of the standard cosmology in 2010.

Several major cosmological disputes have occurred throughout the history of cosmology. One of the earliest was the Copernican Revolution, which established the heliocentric model of the Solar System. More recent was the Great Debate of 1920, in the aftermath of which the Milky Way's status as but one of the Universe's many galaxies was established. From the 1940s to the 1960s, the astrophysical community was equally divided between supporters of the Big Bang theory and supporters of a rival steady state universe; this was eventually decided in favour of the Big Bang theory by advances in observational cosmology in the late 1960s. The current standard model of cosmology is the Lambda-CDM model, wherein the Universe is governed by General Relativity, began with a Big Bang and today is a nearly-flat universe that consists of approximately 5% baryons, 27% cold dark matter, and 68% dark energy.Lambda-CDM has been an extremely successful model, but retains some weaknesses (such as the dwarf galaxy problem). Research on extensions or modifications to Lambda-CDM, as well as fundamentally different models, is ongoing. Topics investigated include quintessence, Modified Newtonian Dynamics (MOND) and its relativistic generalization TeVeS, and warm dark matter.

Self-interacting dark matter

In astrophysics and particle physics, self-interacting dark matter (SIDM) assumes dark matter has self-interactions, in contrast to the collisionless dark matter assumed by the Lambda-CDM model. SIDM was postulated in 2000 to resolve a number of conflicts between observations and N-body simulations (of cold collisionless dark matter only) on the galactic scale and smaller. It was also used to explain the 2015 observations of ESO 146-5 the core of the Abell 3827 galaxy cluster. However, the latter finding has since been discounted based on further observations and modelling of the cluster.Self-interacting dark matter has also been postulated as an explanation for the DAMA annual modulation signal.

Standard model (disambiguation)

Standard model may refer to:

Standard Model of particle physics

The mathematical formulation of the Standard Model of particle physics

The Standard Solar Model of solar astrophysics

The Lambda-CDM model, the standard model of big bang cosmology

Standard model (cryptography)

Intended interpretation of a syntactical system, called standard model in mathematical logic

The standard models of set theory

The Standard Model (Exhibition) held in Stockholm, 2009

Structure formation

In physical cosmology, structure formation is the formation of galaxies, galaxy clusters and larger structures from small early density fluctuations. The universe, as is now known from observations of the cosmic microwave background radiation, began in a hot, dense, nearly uniform state approximately 13.8 billion years ago. However, looking in the sky today, we see structures on all scales, from stars and planets to galaxies and, on still larger scales, galaxy clusters and sheet-like structures of galaxies separated by enormous voids containing few galaxies. Structure formation attempts to model how these structures formed by gravitational instability of small early density ripples.The modern Lambda-CDM model is successful at predicting the observed large-scale distribution of galaxies, clusters and voids; but on the scale of individual galaxies there are many complications due to highly nonlinear processes involving baryonic physics, gas heating and cooling, star formation and feedback. Understanding the processes of galaxy formation is a major topic of modern cosmology research, both via observations such as the Hubble Ultra-Deep Field and via large computer simulations.

Timeline of cosmological theories

This timeline of cosmological theories and discoveries is a chronological record of the development of humanity's understanding of the cosmos over the last two-plus millennia. Modern cosmological ideas follow the development of the scientific discipline of physical cosmology.

Wilkinson Microwave Anisotropy Probe

The Wilkinson Microwave Anisotropy Probe (WMAP), originally known as the Microwave Anisotropy Probe (MAP), was a spacecraft operating from 2001 to 2010 which measured temperature differences across the sky in the cosmic microwave background (CMB) – the radiant heat remaining from the Big Bang. Headed by Professor Charles L. Bennett of Johns Hopkins University, the mission was developed in a joint partnership between the NASA Goddard Space Flight Center and Princeton University. The WMAP spacecraft was launched on June 30, 2001 from Florida. The WMAP mission succeeded the COBE space mission and was the second medium-class (MIDEX) spacecraft in the NASA Explorers program. In 2003, MAP was renamed WMAP in honor of cosmologist David Todd Wilkinson (1935–2002), who had been a member of the mission's science team. After nine years of operations, WMAP was switched off in 2010, following the launch of the more advanced Planck spacecraft by ESA in 2009.

WMAP's measurements played a key role in establishing the current Standard Model of Cosmology: the Lambda-CDM model. The WMAP data are very well fit by a universe that is dominated by dark energy in the form of a cosmological constant. Other cosmological data are also consistent, and together tightly constrain the Model. In the Lambda-CDM model of the universe, the age of the universe is 13.772±0.059 billion years. The WMAP mission's determination of the age of the universe is to better than 1% precision. The current expansion rate of the universe is (see Hubble constant) 69.32±0.80 km·s−1·Mpc−1. The content of the universe currently consists of 4.628%±0.093% ordinary baryonic matter; 24.02%+0.88%
cold dark matter (CDM) that neither emits nor absorbs light; and 71.35%+0.95%
of dark energy in the form of a cosmological constant that accelerates the expansion of the universe. Less than 1% of the current content of the universe is in neutrinos, but WMAP's measurements have found, for the first time in 2008, that the data prefer the existence of a cosmic neutrino background with an effective number of neutrino species of 3.26±0.35. The contents point to a Euclidean flat geometry, with curvature () of −0.0027+0.0039
. The WMAP measurements also support the cosmic inflation paradigm in several ways, including the flatness measurement.

The mission has won various awards: according to Science magazine, the WMAP was the Breakthrough of the Year for 2003. This mission's results papers were first and second in the "Super Hot Papers in Science Since 2003" list. Of the all-time most referenced papers in physics and astronomy in the INSPIRE-HEP database, only three have been published since 2000, and all three are WMAP publications. Bennett, Lyman A. Page, Jr., and David N. Spergel, the latter both of Princeton University, shared the 2010 Shaw Prize in astronomy for their work on WMAP. Bennett and the WMAP science team were awarded the 2012 Gruber Prize in cosmology. The 2018 Breakthrough Prize in Fundamental Physics was awarded to Bennett, Gary Hinshaw, Norman Jarosik, Page, Spergel and the WMAP science team.

As of October 2010, the WMAP spacecraft is derelict in a heliocentric graveyard orbit after 9 years of operations. All WMAP data are released to the public and have been subject to careful scrutiny. The final official data release was the nine-year release in 2012.

Some aspects of the data are statistically unusual for the Standard Model of Cosmology. For example, the largest angular-scale measurement, the quadrupole moment, is somewhat smaller than the Model would predict, but this discrepancy is not highly significant. A large cold spot and other features of the data are more statistically significant, and research continues into these.

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