Lambda-CDM model

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The ΛCDM or Lambda-CDM model is a parametrization of the Big Bang cosmological model in which the universe contains a cosmological constant, denoted by Lambda, and cold dark matter. It is frequently referred to as the standard model of Big Bang cosmology, since it is the simplest model that provides a reasonably good match to the following observations:

the existence and structure of the cosmic microwave background
the large-scale structure in the distribution of galaxies
the 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 is extended by adding cosmological inflation, quintessence, and other elements that are current areas of research in cosmology. Some alternative models challenge the assumptions of the ΛCDM model, such as modified Newtonian dynamics, modified gravity, and large-scale variations in the matter density of the universe.^[1]

Overview

Lambda-CDM, Accelerated Expansion of the Universe. The time-line in this schematic diagram extends from the big bang/inflation era 13.7 Gyr ago to the present cosmological time.

Most modern cosmological models are based on the cosmological principle 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).^[2]

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. While 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.

Λ (Lambda) stands for the cosmological constant which is currently associated with a vacuum energy or dark energy inherent in empty space that explains the current accelerating expansion of space against the attractive (collapsing) effects of gravity from matter. A cosmological constant has negative pressure, $p=-\rho c^{{2}}$ ; this contributes to the stress-energy tensor in general relativity and therefore causes accelerating expansion. The cosmological constant is denoted as $\Omega _{{\Lambda }}$ , which is interpreted as the fraction of the total mass-energy density of a flat universe that is attributed to dark energy. Currently [2013], about 68.3% of the energy density of the present universe is estimated to be dark energy.

Cold dark matter is a form of matter necessary to account for gravitational effects observed in very large scale structures (anomalies in the rotation of galaxies, the gravitational lensing of light by galaxy clusters, enhanced clustering of galaxies) that cannot be accounted for by the quantity of observed matter. Dark matter is described as being cold (i.e. its velocity is non-relativistic [far below the speed of light] at the epoch of radiation-matter equality), non-baryonic (consisting of matter other than protons and neutrons), dissipationless (cannot cool by radiating photons) and collisionless (i.e., the dark matter particles interact with each other and other particles only through gravity and possibly the weak force). The dark matter component is currently [2013] estimated to constitute about 26.8% of the mass-energy density of the universe.

The remaining 4.9% [2013] comprises all ordinary matter observed as atoms, chemical elements, gas and plasma, the stuff of which visible planets, stars and galaxies are made.

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. While 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" or initial singularity, 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 is also necessary to address the "horizon problem" in the CMB. Indeed, it seems likely that the universe is larger than the observable particle horizon.

The model uses the FLRW 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.

History

The discovery of the Cosmic Microwave Background in 1965 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 early 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 realised 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 the model under-predicted observed large-scale galaxy clustering. These difficulties sharpened with the discovery of CMB anisotropy by COBE in 1992, and several alternatives including LambdaCDM and mixed cold+hot dark matter came under active consideration. The LambdaCDM model then became the standard 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, while in 2001 the 2dFGRS galaxy redshift survey measured the matter density to be near 25%; the large difference between these supports a positive Λ or dark energy. Much more precise measurements of the microwave background from WMAP in 2003 – 2010 have continued to support and refine the model.

There is currently active research into many aspects of the ΛCDM model, both to refine the parameters and possibly detect deviations. In addition, ΛCDM 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 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, brane cosmologies, the DGP model, and galileon theories.

Successes

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; the polarisation of the CMB; and the statistics of weak gravitational lensing.

Challenges

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 un-naturally 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. 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.

Parameters

The ΛCDM model is based on six parameters: physical baryon density, physical dark matter density, dark energy density, scalar spectral index, curvature fluctuation amplitude and reionization optical depth. 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 = 1.00, dark energy equation of state = -1, neutrino masses are small enough to be negligible. (See below for extended models which 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, including the Hubble constant and age of the universe, 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, globular cluster ages, are generally consistent with these, providing a check of the model, but are less accurately measured at present.

Parameter values listed below are from the Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP) temperature and polarization observations.^[3] These include estimates based on data from Baryon Acoustic Oscillations^[4] and Type Ia supernova luminosity/time dilation measurements.^[5] Implications of the data for cosmological models are discussed in Komatsu et al. ^[6] and Spergel et al.^[7] See also Planck (spacecraft).

Parameter	Value	Description
t₀	$13.75\pm 0.11\times 10^{9}$ years	Age of the universe
H₀	$70.4_{{-1.4}}^{{+1.3}}$ km s⁻¹ Mpc⁻¹	Hubble constant
Ω_bh²	$0.02260\pm 0.00053$	Physical baryon density
Ω_ch²	$0.1123\pm 0.0035$	Physical dark matter density
Ω_b	$0.0456\pm 0.0016$	Baryon density
Ω_c	$0.227\pm 0.014$	Dark matter density
Ω_Λ	$0.728_{{-0.016}}^{{+0.015}}$	Dark energy density
Δ_R²	$2.441_{{-0.092}}^{{+0.088}}\times 10^{{-9}}$ , k₀ = 0.002Mpc⁻¹	Curvature fluctuation amplitude
σ₈	$0.809\pm 0.024$	Fluctuation amplitude at 8h⁻¹Mpc
n_s	$0.963\pm 0.012$	Scalar spectral index
z_*	$1090.89_{{-0.69}}^{{+0.68}}$	Redshift at decoupling
t_*	$377730_{{-3200}}^{{+3205}}$ years	Age at decoupling
τ	$0.087\pm 0.014$	Reionization optical depth
z_reion	$10.4\pm 1.2$	Redshift of reionization

The "physical baryon density" Ω_bh² differs from the "baryon density" Ω_b in that the baryon density gives the fraction of the critical density made up of baryons (the critical density is the total density of matter/energy needed for the universe to be spatially flat, with measurements indicating that the actual total density Ω_tot is very close if not equal to this value, see below), while the physical baryon density is equal to the baryon density multiplied by the square of the reduced Hubble constant h,^[8] where h is related to the Hubble constant H₀ by the equation H₀ = 100 h (km/s)/Mpc.^[9] Likewise for the difference between "physical dark matter density" and "dark matter density".

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 (denoted r), which is determined by the energy scale of inflation. Other modifications allow for spatial curvature (Ω_tot may be different from 1), hot dark matter in the form of neutrinos, 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.

Parameter	Value	Description
Ω_tot	$1.0023_{{-0.0054}}^{{+0.0056}}$	Total density
w	$-0.980\pm 0.053$	Equation of state of dark energy
r	$<0.24$ , k₀ = 0.002Mpc⁻¹ (2σ)	Tensor-to-scalar ratio
d n_s / d ln k	$-0.022\pm 0.020$ , k₀ = 0.002Mpc⁻¹	Running of the spectral index
Ω_vh²	$<0.0062$	Physical neutrino density
Σm_ν	$<0.58$ eV (2σ)	Sum of three neutrino masses

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 r should be between 0 and 0.3, and the latest results are now within those limits.

References

↑ 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).
↑ Andrew Liddle. An Introduction to Modern Cosmology (2nd ed.). London: Wiley, 2003.
↑ Table 8 on p. 39 of Jarosik, N. et al. (WMAP Collaboration). "Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Sky Maps, Systematic Errors, and Basic Results" (PDF). nasa.gov. Retrieved 2010-12-04. (from NASA's WMAP Documents page)
↑ Percival, W. J. et al. (2010). "Baryon Acoustic Oscillations in the Sloan Digital Sky Survey Data Release 7 Galaxy Sample". Monthly Notices of the Royal Astronomical Society 401 (4): 2148–2168. arXiv:0907.1660. Bibcode:2010MNRAS.401.2148P. doi:10.1111/j.1365-2966.2009.15812.x.
↑ Riess, A. G. et al.. "A Redetermination of the Hubble Constant with the Hubble Space Telescope from a Differential Distance Ladder" (PDF). hubblesite.org. Retrieved 2010-12-04.
↑ E. Komatsu et al. 2010 (WMAP Collaboration). Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Interpretation (from NASA's WMAP Documents page)
↑ Spergel, D. N.; et al. (2003). "First‐YearWilkinson Microwave Anisotropy Probe(WMAP) Observations: Determination of Cosmological Parameters". The Astrophysical Journal Supplement Series 148: 175. arXiv:astro-ph/0302209. Bibcode:2003ApJS..148..175S. doi:10.1086/377226. Cite uses deprecated parameters (help)
↑ Appendix A of the LSST Science Book Version 2.0
↑ p. 7 of Findings of the Joint Dark Energy Mission Figure of Merit Science Working Group

Rebolo, R.; et al. (2004). "Cosmological parameter estimation using Very Small Array data out to ℓ= 1500". Monthly Notices of the Royal Astronomical Society 353 (3): 747. arXiv:astro-ph/0402466. Bibcode:2004MNRAS.353..747R. doi:10.1111/j.1365-2966.2004.08102.x. Cite uses deprecated parameters (help)
Ostriker, J. P.; Steinhardt, P. J. (1995). "Cosmic Concordance". arXiv:astro-ph/9505066 [astro-ph].

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