# 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 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 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, modified gravity and theories of large-scale variations in the matter density of the universe.

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).

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.

Planck Collaboration Cosmological parameters
Description Symbol Value
Independent
parameters
Physical baryon density parameter Ωbh2 0.02230±0.00014
Physical dark matter density parameter Ωch2 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 Δ2
R
2.441+0.088
−0.092
×10−9
Reionization optical depth τ 0.066±0.012
Fixed
parameters
Total density parameter Ωtot 1
Equation of state of dark energy w −1
Sum of three neutrino masses mν 0.06 eV/c2
Effective number of relativistic degrees of freedom Neff 3.046
Tensor/scalar ratio r 0
Running of spectral index dns / d ln k 0
Calculated
values
Hubble constant H0 67.74±0.46 km s−1Mpc−1
Baryon density parameter Ωb 0.0486±0.0010
Dark matter density parameter Ωc 0.2589±0.0057
Matter density parameter Ωm 0.3089±0.0062
Dark energy density parameter ΩΛ 0.6911±0.0062
Critical density ρcrit (8.62±0.12)×10−27 kg/m3
Fluctuation amplitude at 8h−1 Mpc σ8 0.8159±0.0086
Redshift at decoupling z 1089.90±0.23
Age at decoupling t 377700±3200 years
Redshift of reionization (with uniform prior) zre 8.5+1.0
−1.1
Extended model parameters
Description Symbol Value
Total density parameter Ωtot 1.0023+0.0056
−0.0054
Equation of state of dark energy w −0.980±0.053
Tensor-to-scalar ratio r < 0.11, k0 = 0.002 Mpc−1 (2σ)
Running of the spectral index dns / d ln k −0.022±0.020, k0 = 0.002 Mpc−1
Physical neutrino density parameter Ωνh2 < 0.0062
Sum of three neutrino masses mν < 0.58 eV/c2 (2σ)

${\displaystyle {1 \over a(t_{\mathrm {em} })}=1+z.}$
${\displaystyle H(t)\equiv {\frac {\dot {a}}{a}},}$
${\displaystyle H^{2}=\left({\frac {\dot {a}}{a}}\right)^{2}={\frac {8\pi G}{3}}\rho -{\frac {kc^{2}}{a^{2}}}+{\frac {\Lambda c^{2}}{3}}.}$
${\displaystyle \rho _{\mathrm {crit} }={\frac {3H_{0}^{2}}{8\pi G}}=1.878\;47(23)\times 10^{-26}\;h^{2}\;{\text{kg}}\;{\text{m}}^{-3},}$
${\displaystyle \Omega _{x}\equiv {\frac {\rho _{x}(t=t_{0})}{\rho _{\mathrm {crit} }}}={\frac {8\pi G\rho _{x}(t=t_{0})}{3H_{0}^{2}}}}$
${\displaystyle H(a)\equiv {\frac {\dot {a}}{a}}=H_{0}{\sqrt {(\Omega _{c}+\Omega _{b})a^{-3}+\Omega _{rad}a^{-4}+\Omega _{k}a^{-2}+\Omega _{DE}a^{-3(1+w)}}}}$
${\displaystyle H(a)=H_{0}{\sqrt {\Omega _{m}a^{-3}+\Omega _{rad}a^{-4}+\Omega _{\Lambda }}}}$
${\displaystyle a(t)=(\Omega _{m}/\Omega _{\Lambda })^{1/3}\,\sinh ^{2/3}(t/t_{\Lambda })}$
${\displaystyle a=(\Omega _{m}/2\Omega _{\Lambda })^{1/3}}$
...
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