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'ΛCDM' or 'Lambda-CDM' is an abbreviation for 'Lambda-Cold Dark Matter'. It is frequently referred to as the 'concordance model' of big bang cosmology, since it attempts to explain cosmic microwave background observations, as well as large scale structure observations and supernovae observations of the accelerating expansion of the universe. It is the simplest known model that is in general agreement with observed phenomena.

★ Λ (Lambda) stands for the cosmological constant which is a dark energy term that allows for the current accelerating expansion of the universe. The cosmological constant is often described in terms of $Omega\_\{Lambda\}$, the fraction of the energy density of a flat Universe in the form of the cosmological constant. Currently, $Omega\_\{Lambda\}simeq$ 0.74, implying 74% of the energy density of the present universe is in this form.

★ Cold dark matter is the model where the dark matter is explained as being cold (i.e. its velocity is non-relativistic (v<non-baryonic, dissipationless (can not cool by radiating photons) and collisionless (e.i. the dark matter particles interact with each other and other particles only through gravity). This component makes up 22% of the energy density of the present universe. The remaining 4% is all of the matter (and energy) that makes up the atoms (and photons) that are the building blocks of planets, stars, and gas clouds in the universe.

★ The model assumes a nearly scale-invariant spectrum of primordial perturbations and a universe without spatial curvature. It also assumes that it has no observable topology, so that the universe is much larger than the observable particle horizon. These are predictions of cosmic inflation.
These are the simplest assumptions for a consistent, physical model of cosmology. However, ΛCDM is a ''model''. Cosmologists anticipate that all of these assumptions will not be borne out exactly, after more is learned about the applicable fundamental physics. In particular, cosmic inflation predicts spatial curvature at the level of 10⁻⁴ to 10⁻⁵. It would also be surprising if the temperature of dark matter were absolute zero. Moreover, ΛCDM says nothing about the fundamental physical origin of dark matter, dark energy and the nearly scale-invariant spectrum of primordial curvature perturbations: in that sense, it is merely a useful parameterization of ignorance.

Contents

Parameters

Extended models

References

Parameters

The model has six parameters. The Hubble constant determines the rate of expansion of the universe, as well as the critical density for closure of the universe, ρ₀. Densities for baryons, dark matter and dark energy are given as Ωs, which are the ratio of the true density to the critical density: ''e.g.'' $Omega\_b=$
ho_b/
ho_0. Since the ΛCDM model assumes a flat universe, these densities sum to one, and the density of dark energy is not a free parameter. The optical depth to reionization determines the red shift of reionization. Information about the density fluctuations is determined by the amplitude of the primordial fluctuations (from cosmic inflation) and the spectral index, which measures how the fluctuations change with scale ($n\_s=1$ corresponds to a scale-invariant spectrum).
The errors quoted are 1σ: that is, there is statistically a 68% likelihood that the true value falls within the upper and lower error bounds. The errors are not Gaussian, and they have been derived using a Markov chain Monte Carlo analysis by the Wilkinson Microwave Anisotropy Probe collaboration (Spergel ''et al.'' 2006) which also uses Sloan Digital Sky Survey and Type Ia supernova data.
{| class="wikitable"
! Parameter
! Value
! Description
|-
! colspan="3" align="center" | ''Basic parameters''
|-
| H₀
| $70.9^\{+2.4\}\_\{-3.2\}$ km s^-1 Mpc^-1
| Hubble parameter
|-
| Ω_b
| $0.0444^\{+0.0042\}\_\{-0.0035\}$
| Baryon density
|-
| Ω_m
| $0.266^\{+0.025\}\_\{-0.040\}$
| Total matter density (baryons + dark matter)
|-
| τ
| $0.079^\{+0.029\}\_\{-0.032\}$
| Optical depth to reionization
|-
| A_s
| $0.813^\{+0.042\}\_\{-0.052\}$
| Scalar fluctuation amplitude
|-
| n_s
| $0.948^\{+0.015\}\_\{-0.018\}$
| Scalar spectral index
|-
! colspan="3" align="center" | ''Derived parameters''
|-
| ρ₀
| $0.94^\{+0.06\}\_\{-0.09\}\; imes10^\{-26\}$ kg/m³
| Critical density
|-
| Ω_Λ
| $0.732^\{+0.040\}\_\{-0.025\}$
| Dark energy density
|-
| z_ion
| $10.5^\{+2.6\}\_\{-2.9\}$
| Reionization red-shift
|-
| σ₈
| $0.772^\{+0.036\}\_\{-0.048\}$
| Galaxy fluctuation amplitude
|-
| t₀
| $13.73^\{+0.13\}\_\{-0.17\}\; imes10^9$ years
| Age of the universe
|}

Extended models

Possible extensions of the simplest ΛCDM model are to allow quintessence rather than a cosmological constant. In this case, 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, which is determined by the energy scale of inflation. Other modifications allow for spatial curvature or a running spectral index, which are generally viewed as inconsistent with cosmic inflation.
Allowing these parameters will generally ''increase'' the errors in the parameters quoted above, and may also shift the observed values somewhat.
{| class="wikitable"
! Parameter
! Value
! Description
|-
| w
| $-0.926^\{+0.051\}\_\{-0.075\}$
| Equation of state
|-
| r
| (2σ)
| Tensor-to-scalar ratio
|-
| Ω_k
| $-0.010^\{+0.014\}\_\{-0.012\}$
| Spatial curvature
|-
| α
| $-0.102^\{+0.050\}\_\{-0.043\}$
| Running of the spectral index
|-
| $Sigma\; m\_$
u
| eV (2σ)
| Summed neutrino masses
|-
|}
These are consistent with a cosmological constant, $w=-1$, and no spatial curvature $Omega\_k=0$. Some 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 r should be between 0 and 0.3, and so should be tested in the near future.

References

★ Wilkinson Microwave Anisotropy Probe (WMAP) three year results: implications for cosmology, D. N. Spergel ''et al.'' (WMAP collaboration), , , , 2006

★ M. Tegmark ''et al.'' (SDSS collaboration), Cosmological Parameters from SDSS and WMAP, ''Phys. Rev.'' 'D69' 103501 (2004).

★ D. N. Spergel ''et al.'' (WMAP collaboration), First year Wilkinson Microwave Anisotropy Probe (WMAP) observations: determination of cosmological parameters, ''Astrophys. J. Suppl.'' '148' 175 (2003).

★ R. Rebolo ''et al.'' (VSA collaboration), Cosmological parameter estimation using Very Small Array data out to l=1500, Monthly Notices of the Royal Astronomical Society, Volume 353, Issue 3, pp. 747-759

★ J. P. Ostriker and P. J. Steinhardt, Cosmic Concordance, arXiv:astro-ph/9505066.