\(\Lambda\)CDM Cosmological Model

Field equations

In order to describe an homogeneous and isotropic universe like in the framework of GR, one should solve the Einstein field equations using a homogeneous, isotropic, and expanding metric (the FLRW metric),

\[ ds^2 = dt^2 - a^2(t) \left(\frac{dr^2}{1-k r^2}+r^2 d\Omega\right) \,, \tag{1}\]

where \(k\) is the curvature and \(a(t)\) the scale factor of the universe, which is closely related to the redshift, \(a(t)=1/(1+z(t))\). We can use this relation to also connect the temporal coordinate to the redshift variable,

\[ \begin{aligned} a(t) &=\frac{1}{1+z(t)} \\ \frac{\dot a(t)}{a(t)} dt &= -\frac{dz}{1+z(t)} \\ dt &= -\frac{dz}{H(z)(1+z)} \,, \end{aligned} \tag{2}\]

where we have defined the Hubble parameter1 as \(H=\dot a/a\).

Friedman equations

Using the FLRW metric, Eq. (1), to solve the Einstein equations leads to the Friedmann equations, which are the basis of the standard cosmology. The first Friedmann equation can be written in terms of the cosmic density of radiation (\(\Omega_r\)), matter (\(\Omega_m\)), curvature (\(\Omega_k\)), and dark energy (\(\Omega_\Lambda\)),

\[ \begin{gathered} H(z) = H_0 \sqrt{\Omega_r (1+z)^4 + \Omega_m (1+z)^3 + \Omega_k (1+z)^2 + \Omega_\Lambda} \,. \end{gathered} \tag{3}\]

The model does not predict the values of these cosmic densities, so they should be adjusted using cosmological observations and measurements. For more information see [1].

List of Authors

  • Albert Einstein

  • Isaac Newton

References

[1]
Reyes Hung M A 2023 Exploration of possible signals beyond special relativity using high-energy astroparticle physics PhD thesis (U. Zaragoza (main))
[2]
Carmona J M, Cortés J L, Relancio J J and Reyes M A 2022 Time delays, choice of energy-momentum variables, and relative locality in doubly special relativity Phys. Rev. D 106 064045

Footnotes

  1. We will not discuss here the Hubble tension problem.↩︎