Lambda-CDM model

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

a cosmological constant, denoted by lambda, associated with dark energy;
the postulated cold dark matter, denoted by CDM;
ordinary matter.

It is the current standard model of Big Bang cosmology, as 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, 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 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 has been successful in modeling a broad collection of astronomical observations over decades. Remaining issues challenge the assumptions of the ΛCDM model and have led to many alternative models.

Overview

The ΛCDM model is based on three postulates on the structure of spacetime:

The cosmological principle, that the universe is the same everywhere and in all directions, and that it is expanding,
A postulate by Hermann Weyl that the lines of spacetime intersect at only one point, where time along each line can be synchronized; the behavior resembles an expanding perfect fluid,
General relativity that relates the geometry of spacetime to the distribution of matter and energy.

This combination greatly simplifies the equations of general relativity into a form called the Friedmann equations. These equations specify the evolution of the scale factor of the universe in terms of the pressure and density of a perfect fluid. The evolving density is composed of different kinds of energy and matter, each with its own role in affecting the scale factor. For example, a model might include baryons, photons, neutrinos, and dark matter. These component densities become parameters extracted when the model is constrained to match astrophysical observations. The model aims to describe the observable universe from approximately 0.1 s to the present.
The most accurate observations which are sensitive to the component densities are consequences of statistical inhomogeneity called "perturbations" in the early universe. Since the Friedmann equations assume homogeneity, additional theory must be added before comparison to experiments. Inflation is a simple model producing perturbations by postulating an extremely rapid expansion early in the universe that separates quantum fluctuations before they can equilibrate. The perturbations are characterized by additional parameters also determined by matching observations.
Finally, the light which will become astronomical observations must pass through the universe. The latter part of that journey will pass through ionized space, where the electrons can scatter the light, altering the anisotropies. This effect is characterized by one additional parameter.
The ΛCDM model includes an expansion of the spatial metric that is well documented, both as the redshift 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 in space. Also, since it originates from ordinary general relativity, it, like general relativity, 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 Λ 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,, 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 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 or based on the 2018 release of Planck satellite data, or more than 68.3% of the mass–energy density of the universe.
Dark matter is postulated in order to account for gravitational effects observed in very large-scale structures that cannot be accounted for by the quantity of observed matter.
The ΛCDM model proposes specifically cold dark matter, hypothesized as:

Non-baryonic: Consists of matter other than protons and neutrons
Cold: Its velocity is far less than the speed of light at the epoch of radiation–matter equality
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% of the mass–energy density of the universe. The remaining 4.9% 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.
The model includes a single originating event, the "Big Bang", which was not an explosion but the abrupt appearance of expanding spacetime containing radiation at temperatures of around 10¹⁵ K. This was immediately followed by an exponential expansion of space by a scale multiplier of 10²⁷ or more, known as cosmic inflation. The early universe remained hot 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 cosmological model consistent with the observed continuing expansion of space, the observed distribution of lighter elements in the universe, and the spatial texture of minute irregularities 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.

Cosmic expansion history

The expansion of the universe is parameterized by a dimensionless scale factor , defined relative to the present time, so ; the usual convention in cosmology is that subscript 0 denotes present-day values, so denotes the age of the universe. The scale factor is related to the observed redshift 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
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, 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.
The present-day density parameter for various species is defined as the dimensionless ratio
where the subscript is one of for baryons, for cold dark matter, for radiation, and 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 is the equation of state parameter of dark energy, and assuming negligible neutrino mass. 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
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 occurred when
which evaluates to or for the best-fit parameters estimated from the Planck spacecraft.

Parameters

Multiple variants of the ΛCDM model are used with some differences in parameters. One such set is outlined in the table below.

	Description	Symbol	Value-2018
rowspan="6"	Baryon density today	Ω_b ²
Cold dark matter density today	Ω_c ²		-
100 × approximation to r∗/DA	100		-
Reionization optical depth			-
Log power of the primordial curvature perturbations			-
Scalar spectrum power-law index	_s		-
rowspan="6"	Total matter density today	Ω_m ²	0.1428 ± 0.0011
Equation of state of dark energy		w₀ = −1	-
Tensor/scalar ratio		r_0.002 < 0.06	-
Running of spectral index		0	-
Sum of three neutrino masses		0.06 eV/	-
Effective number of relativistic degrees of freedom	N_eff		-
rowspan="10"	Hubble constant	₀
Age of the universe	₀	years	-
Dark energy density parameter	Ω_Λ		-
The present root-mean-square matter fluctuation, averaged over a sphere of radius 8h⁻¹ Mpc	₈		-
Redshift of reionization	_re		-

The Planck collaboration version of the ΛCDM model is based on six parameters: baryon density parameter; dark matter density parameter; scalar spectral index; two parameters related to curvature fluctuation amplitude; and the probability that photons from the early universe will be scattered once on route. 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.
The parameter values, and uncertainties, are estimated using 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 calculated.