Lambda-CDM model

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The Lambda-CDM, Lambda cold dark matter, or ΛCDM model is a mathematical model of the Big Bang theory with three major components:

1. a cosmological constant denoted by lambda (Λ) associated with dark energy
2. the postulated cold dark matter denoted by CDM
3. ordinary matter

It is referred to as the standard model of Big Bang cosmology^[1] because it is the simplest model that provides a reasonably good account of:

• the existence and structure of the cosmic microwave background
• the large-scale structure in the distribution of galaxies
• the observed abundances of hydrogen (including deuterium), helium, and lithium
• the accelerating expansion of the universe observed in the light from distant galaxies and supernovae

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

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, scale invariance of empty space, and decaying dark matter (DDM).^{[2][3][4][5][6]}

Overview

The ΛCDM model includes an expansion of metric space that is well documented both as the

The letter Λ (lambda) represents the cosmological constant, which is 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, ${\displaystyle p=-\rho c^{2}}$, 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, ${\displaystyle \Omega _{\Lambda }}$, is estimated to be 0.669 ± 0.038 based on the 2018 Dark Energy Survey results using Type Ia supernovae^[8] 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.^[9]

• Non-baryonic: Consists of matter other than protons and neutrons (and electrons, by convention, although electrons are not baryons)
• Cold: 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)
• Dissipationless: Cannot cool by radiating photons
• Collisionless: Dark matter particles interact with each other and other particles only through gravity and possibly the weak force

Dark matter constitutes about 26.5 %^[11] of the mass–energy density of the universe. The remaining 4.9 %^[11] 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.^[12]

The model includes a single originating event, the "

The model uses the Friedmann–Lemaître–Robertson–Walker metric, the Friedmann equations and the cosmological equations of state to describe the observable universe from approximately 0.1s to the present.^[1]: 605

Cosmic expansion history

The expansion of the universe is parameterized by a

${\displaystyle a=a(t)}$ (with time ${\displaystyle t}$ counted from the birth of the universe), defined relative to the present time, so ${\displaystyle a_{0}=a(t_{0})=1}$; the usual convention in cosmology is that subscript 0 denotes present-day values, so ${\displaystyle t_{0}}$ denotes the age of the universe. The scale factor is related to the observed

${\displaystyle z}$

${\displaystyle t_{\mathrm {em} }}$

${\displaystyle a(t_{\text{em}})={\frac {1}{1+z}}\,.}$

The expansion rate is described by the time-dependent

${\displaystyle H(t)}$

${\displaystyle H(t)\equiv {\frac {\dot {a}}{a}},}$

where ${\displaystyle {\dot {a}}}$ is the time-derivative of the scale factor. The first Friedmann equation gives the expansion rate in terms of the matter+radiation density ${\displaystyle \rho }$, the

${\displaystyle k}$,

${\displaystyle \Lambda }$,

${\displaystyle H^{2}=\left({\frac {\dot {a}}{a}}\right)^{2}={\frac {8\pi G}{3}}\rho -{\frac {kc^{2}}{a^{2}}}+{\frac {\Lambda c^{2}}{3}},}$

where as usual ${\displaystyle c}$ is the speed of light and ${\displaystyle G}$ is the gravitational constant. A critical density ${\displaystyle \rho _{\mathrm {crit} }}$ is the present-day density, which gives zero curvature ${\displaystyle k}$, assuming the cosmological constant ${\displaystyle \Lambda }$ is zero, regardless of its actual value. Substituting these conditions to the Friedmann equation gives

${\displaystyle \rho _{\mathrm {crit} }={\frac {3H_{0}^{2}}{8\pi G}}=1.878\;47(23)\times 10^{-26}\;h^{2}\;{\text{kg}}\;{\text{m}}^{-3},}$^[14]

where ${\displaystyle h\equiv H_{0}/(100\;\mathrm {km\;s^{-1}\;Mpc^{-1}} )}$ 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.^{[citation needed]}

It is standard to define the present-day density parameter ${\displaystyle \Omega _{x}}$ for various species as the dimensionless ratio

${\displaystyle \Omega _{x}\equiv {\frac {\rho _{x}(t=t_{0})}{\rho _{\mathrm {crit} }}}={\frac {8\pi G\rho _{x}(t=t_{0})}{3H_{0}^{2}}}}$

where the subscript ${\displaystyle x}$ is one of ${\displaystyle \mathrm {b} }$ for baryons, ${\displaystyle \mathrm {c} }$ for cold dark matter, ${\displaystyle \mathrm {rad} }$ for radiation (photons plus relativistic neutrinos), and ${\displaystyle \Lambda }$ for dark energy.^{[citation needed]}

Since the densities of various species scale as different powers of ${\displaystyle a}$, e.g. ${\displaystyle a^{-3}}$ for matter etc., the

${\displaystyle H(a)\equiv {\frac {\dot {a}}{a}}=H_{0}{\sqrt {(\Omega _{\rm {c}}+\Omega _{\rm {b}})a^{-3}+\Omega _{\mathrm {rad} }a^{-4}+\Omega _{k}a^{-2}+\Omega _{\Lambda }a^{-3(1+w)}}}}$

where ${\displaystyle w}$ is the equation of state parameter of dark energy, and assuming negligible neutrino mass (significant neutrino mass requires a more complex equation). The various ${\displaystyle \Omega }$ parameters add up to ${\displaystyle 1}$ by construction. In the general case this is integrated by computer to give the expansion history ${\displaystyle a(t)}$ and also observable distance–redshift relations for any chosen values of the cosmological parameters, which can then be compared with observations such as

Observations show that the radiation density is very small today, ${\displaystyle \Omega _{\text{rad}}\sim 10^{-4}}$; if this term is neglected the above has an analytic solution[15]

${\displaystyle a(t)=(\Omega _{\rm {m}}/\Omega _{\Lambda })^{1/3}\,\sinh ^{2/3}(t/t_{\Lambda })}$

where ${\displaystyle t_{\Lambda }\equiv 2/(3H_{0}{\sqrt {\Omega _{\Lambda }}})\ ;}$ this is fairly accurate for ${\displaystyle a>0.01}$ or ${\displaystyle t>10}$ million years. Solving for ${\displaystyle a(t)=1}$ gives the present age of the universe ${\displaystyle t_{0}}$ in terms of the other parameters.^{[citation needed]}

It follows that the transition from decelerating to accelerating expansion (the second derivative ${\displaystyle {\ddot {a}}}$ crossing zero) occurred when

${\displaystyle a=(\Omega _{\rm {m}}/2\Omega _{\Lambda })^{1/3}}$

which evaluates to ${\displaystyle a\sim 0.6}$ or ${\displaystyle z\sim 0.66}$ for the best-fit parameters estimated from the Planck spacecraft.^{[citation needed]}

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.^{[citation needed]}

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

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.^[citation needed]

These difficulties sharpened with the discovery of CMB anisotropy by the

Over the years, numerous simulations of ΛCDM and observations of our universe have been made that challenge the validity of the ΛCDM model, to the point where some cosmologists believe that the ΛCDM model may be superseded by a different, as yet unknown cosmological model.[16]^[17][20]

Lack of detection

Extensive searches for dark matter particles have so far shown no well-agreed detection, while dark energy may be almost impossible to detect in a laboratory, and its value is unnaturally small compared to vacuum energy theoretical predictions.^{[citation needed]}

Violations of the cosmological principle

The ΛCDM model has been shown to satisfy the cosmological principle, which states that, on a large-enough scale, the universe looks the same in all directions (isotropy) and from every location (homogeneity); "the universe looks the same whoever and wherever you are."^[21] The cosmological principle exists because when the predecessors of the ΛCDM model were being developed, there was not sufficient data available to distinguish between more complex anisotropic or inhomogeneous models, so homogeneity and isotropy were assumed to simplify the models,^[22] and the assumptions were carried over into the ΛCDM model.^[23] However, recent findings have suggested that violations of the cosmological principle, especially of isotropy, exist. These violations have called the ΛCDM model into question, with some authors suggesting that the cosmological principle is obsolete or that the Friedmann–Lemaître–Robertson–Walker metric breaks down in the late universe.^[16][24][25] This has additional implications for the validity of the cosmological constant in the ΛCDM model, as dark energy is implied by observations only if the cosmological principle is true.^[26][23]

Violations of isotropy

Evidence from galaxy clusters,^[27][28] quasars,^[29] and type Ia supernovae^[30] suggest that isotropy is violated on large scales.^{[citation needed]}

Data from the

Nevertheless, some authors have stated that the universe around Earth is isotropic at high significance by studies of the cosmic microwave background temperature maps.[46]

Violations of homogeneity

Based on N-body simulations in ΛCDM, Yadav and his colleagues showed that the spatial distribution of galaxies is statistically homogeneous if averaged over scales 260/h Mpc or more.^[47] However, many large-scale structures have been discovered, and some authors have reported some of the structures to be in conflict with the predicted scale of homogeneity for ΛCDM, including

• The Clowes–Campusano LQG, discovered in 1991, which has a length of 580 Mpc
• The Sloan Great Wall, discovered in 2003, which has a length of 423 Mpc,^[48]
• U1.11, a large quasar group discovered in 2011, which has a length of 780 Mpc
• The Huge-LQG, discovered in 2012, which is three times longer than and twice as wide as is predicted possible according to ΛCDM
• The Hercules–Corona Borealis Great Wall, discovered in November 2013, which has a length of 2000–3000 Mpc (more than seven times that of the SGW)^[49]
• The
  Giant Arc, discovered in June 2021, which has a length of 1000 Mpc^[50]

Other authors claim that the existence of structures larger than the scale of homogeneity in the ΛCDM model does not necessarily violate the cosmological principle in the ΛCDM model.[51]^[16]

El Gordo galaxy cluster collision

El Gordo is a massive interacting galaxy cluster in the early Universe (${\displaystyle z=0.87}$). The extreme properties of El Gordo in terms of its redshift, mass, and the collision velocity leads to strong (${\displaystyle 6.16\sigma }$) tension with the ΛCDM model.

Some authors have said the existence of the KBC void violates the assumption that the CMB reflects baryonic density fluctuations at ${\displaystyle z=1100}$ or Einstein's theory of

Hubble tension

Statistically significant differences remain in measurements of the Hubble constant based on the cosmic background radiation compared to astronomical distance measurements. This difference has been called the

The ${\displaystyle S_{8}}$ tension in cosmology is another major problem for the ΛCDM model.[16] The ${\displaystyle S_{8}}$ parameter in the ΛCDM model quantifies the amplitude of matter fluctuations in the late universe and is defined as

${\displaystyle S_{8}\equiv \sigma _{8}{\sqrt {\Omega _{\rm {m}}/0.3}}}$

Early- (e.g. from CMB data collected using the Planck observatory) and late-time (e.g. measuring weak gravitational lensing events) facilitate increasingly precise values of ${\displaystyle S_{8}}$. However, these two categories of measurement differ by more standard deviations than their uncertainties. This discrepancy is called the ${\displaystyle S_{8}}$ tension. The name "tension" reflects that the disagreement is not merely between two data sets: the many sets of early- and late-time measurements agree well within their own categories, but there is an unexplained difference between values obtained from different points in the evolution of the universe. Such a tension indicates that the ΛCDM model may be incomplete or in need of correction.^[16]

Some values for ${\displaystyle S_{8}}$ are 0.832±0.013 (2020 Planck),^[65] 0.766+0.020
−0.014 (2021 KIDS),^[66][67] 0.776±0.017 (2022 DES),^[68] 0.790+0.018
−0.014 (2023 DES+KIDS),^[69] 0.769+0.031
−0.034 -0.776+0.032
−0.033^{[70][71][72][73]} (2023 HSC-SSP), 0.86±0.01 (2024 EROSITA).^[74][75] Values have also obtained using peculiar velocities, 0.637±0.054 (2020)^[76] and 0.776±0.033 (2020),^[77] among other methods.

Axis of evil

The ΛCDM model assumes that the data of the

The actual observable amount of lithium in the universe is less than the calculated amount from the ΛCDM model by a factor of 3–4.^[83][16] If every calculation is correct, then solutions beyond the existing ΛCDM model might be needed.^[83]

Shape of the universe

The ΛCDM model assumes that the shape of the universe is flat (zero curvature). However, recent Planck data have hinted that the shape of the universe might in fact be closed (positive curvature), which would contradict the ΛCDM model.^[84][16] Some authors have suggested that the Planck data detecting a positive curvature could be evidence of a local inhomogeneity in the curvature of the universe rather than the universe actually being closed.^[85][16]

Violations of the strong equivalence principle

The ΛCDM model assumes that the

Several discrepancies between the predictions of cold dark matter in the ΛCDM model and observations of galaxies and their clustering have arisen. Some of these problems have proposed solutions, but it remains unclear whether they can be solved without abandoning the ΛCDM model.^[89]

Cuspy halo problem

The density distributions of dark matter halos in cold dark matter simulations (at least those that do not include the impact of baryonic feedback) are much more peaked than what is observed in galaxies by investigating their rotation curves.^[90]

Dwarf galaxy problem

Cold dark matter simulations predict large numbers of small dark matter halos, more numerous than the number of small dwarf galaxies that are observed around galaxies like the Milky Way.^[91]

Satellite disk problem

Dwarf galaxies around the Milky Way and Andromeda galaxies are observed to be orbiting in thin, planar structures whereas the simulations predict that they should be distributed randomly about their parent galaxies.^[92] However, latest research suggests this seemingly bizarre alignment is just a quirk which will dissolve over time.^[93]

High-velocity galaxy problem

Galaxies in the NGC 3109 association are moving away too rapidly to be consistent with expectations in the ΛCDM model.^[94] In this framework, NGC 3109 is too massive and distant from the Local Group for it to have been flung out in a three-body interaction involving the Milky Way or Andromeda Galaxy.^[95]

Galaxy morphology problem

If galaxies grew hierarchically, then massive galaxies required many mergers.

If galaxies were embedded within massive halos of cold dark matter, then the bars that often develop in their central regions would be slowed down by dynamical friction with the halo. This is in serious tension with the fact that observed galaxy bars are typically fast.^[99]

Small scale crisis

Comparison of the model with observations 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".^[100] 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.

High redshift galaxies

Observations from the

Massimo Persic and Paolo Salucci^[108] first 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 ${\textstyle M_{b}/L}$), weighted with the luminosity function ${\textstyle \phi (L)}$ over the previously mentioned classes of astrophysical objects:

${\displaystyle \rho _{\rm {b}}=\sum \int L\phi (L){\frac {M_{\rm {b}}}{L}}\,dL.}$

The result was:

${\displaystyle \Omega _{\rm {b}}=\Omega _{*}+\Omega _{\text{gas}}=2.2\times 10^{-3}+1.5\times 10^{-3}h^{-1.3}\simeq 0.003}$

where ${\displaystyle h\simeq 0.72}$.

Note that this value is much lower than the prediction of standard cosmic nucleosynthesis ${\displaystyle \Omega _{\rm {b}}\simeq 0.0486}$, 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".

The missing baryon problem is claimed to be resolved. Using observations of the kinematic Sunyaev–Zel'dovich effect spanning more than 90 % of the lifetime of the Universe, in 2021 astrophysicists found that approximately 50 % of all baryonic matter is outside dark matter haloes, filling the space between galaxies.^[109] Together with the amount of baryons inside galaxies and surrounding them, the total amount of baryons in the late time Universe is compatible with early Universe measurements.

Unfalsifiability

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.^[110]

Parameters

Planck Collaboration Cosmological parameters^[112]

	Description	Symbol	Value-2015[113]	Value-2018[114]
Indepen- dent para- meters	Physical baryon density parameter[a]	Ωb h2	0.02230±0.00014	0.0224±0.0001
	Physical dark matter density parameter[a]	Ωc h2	0.1188±0.0010	0.120±0.001
	Age of the universe	t0	(13.799±0.021)×109 years	(13.787±0.020)×109 years[117]
	Scalar spectral index	ns	0.9667±0.0040	0.965±0.004
	Curvature fluctuation amplitude, k0 = 0.002 Mpc−1	${\displaystyle \Delta _{R}^{2}}$	2.441+0.088 −0.092×10−9[118]	?
	Reionization optical depth	τ	0.066±0.012	0.054±0.007
Fixed para- meters	Total density parameter[b]	Ωtot	1	?
	Equation of state of dark energy	w	−1	w0 = −1.03 ± 0.03
	Tensor/scalar ratio	r	0	r0.002 <  0.06
	Running of spectral index	${\displaystyle dn_{\text{s}}/d\ln k}$	0	?
	Sum of three neutrino masses	${\displaystyle \sum m_{\nu }}$	0.06 eV/c2[c][111] : 40	0.12 eV/c2
	Effective number of relativistic degrees of freedom	Neff	3.046[d][111]: 47	2.99±0.17
Calcu- lated values	Hubble constant	H0	67.74±0.46 km s−1 Mpc−1	67.4±0.5 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	0.315±0.007
	Dark energy density parameter[b]	ΩΛ	0.6911±0.0062	0.6847±0.0073
	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 8h–1 Mpc	σ8	0.8159±0.0086	0.811±0.006
	Redshift at decoupling	z∗	1089.90±0.23	1089.80±0.21
	Age at decoupling	t∗	377700±3200 years[118]	?
	Redshift of reionization (with uniform prior)	zre	8.5+1.0 −1.1[119]	7.68±0.79

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.^[120] In accordance with Occam's razor, six is the smallest number of parameters needed to give an acceptable fit to the 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 theory (though, ideally, they may be related by a future "

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).[111] See also Planck (spacecraft).

Extended models

Extended model parameters^[118]

Description	Symbol	Value
Total density parameter	${\displaystyle \Omega _{\text{tot}}}$	0.9993±0.0019[121]
Equation of state of dark energy	${\displaystyle w}$	−0.980±0.053
Tensor-to-scalar ratio	${\displaystyle r}$	< 0.11, k0 = 0.002 Mpc−1 (${\displaystyle 2\sigma }$)
Running of the spectral index	${\displaystyle dn_{s}/d\ln k}$	−0.022±0.020, k0 = 0.002 Mpc−1
Sum of three neutrino masses	${\displaystyle \sum m_{\nu }}$	< 0.58 eV/c2 (${\displaystyle 2\sigma }$)
Physical neutrino density parameter	${\displaystyle \Omega _{\nu }h^{2}}$	< 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 (${\displaystyle \Omega _{tot}}$ 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 ${\displaystyle r}$), 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 ${\displaystyle r}$ should be between 0 and 0.3, and the latest results are within those limits.

See also

• Bolshoi Cosmological Simulation
• Galaxy formation and evolution
• Illustris project
• List of cosmological computation software
• Millennium Run
• Weakly interacting massive particles (WIMPs)
• The ΛCDM model is also known as the standard model of cosmology, but is not related to the Standard Model of particle physics.

References