Standard Model of Cosmology

Prerequisites: Cosmology. Full path (17 pages)

Best explanations

Concrete The Standard Model of Cosmology: A Skeptic's Guide

Intuitive
Concrete
Abstract

Intuitive

The standard model of cosmology, also called the Lambda-CDM model, is the current best description of the universe as a whole. It says that the universe started in a hot, dense state and has been expanding and cooling ever since. Galaxies are carried apart by this expansion, like raisins in a rising dough, where every raisin sees all the others move away from it.

The model needs only a few ingredients. Ordinary matter, the atoms of stars, gas and planets, is about 5 percent of the energy content today. About 27 percent is cold dark matter (the "CDM"), which feels gravity but does not interact with light. The remaining 68 percent or so is dark energy, modelled as a cosmological constant (the "Lambda"), a constant energy density of empty space that makes the expansion speed up. Radiation and neutrinos were dominant early on but are negligible today.

On top of this smooth expanding background there were tiny density fluctuations in the very early universe. Gravity made the denser regions denser over billions of years, and they grew into the galaxies and clusters we see.

With about six adjustable numbers, the model fits the temperature pattern of the cosmic microwave background, the abundances of light elements made in the first minutes, and the large scale distribution of galaxies. The model does not say what dark matter and dark energy are, and some measurements disagree with it by up to about five standard deviations, most notably the value of the present expansion rate.

Concrete

Background. On large scales the universe is taken to be homogeneous and isotropic. The metric is then the Friedmann-Lemaître-Robertson-Walker (FLRW) metric

$ds^2 = -c^2 dt^2 + a(t)^2\left[\frac{dr^2}{1-kr^2} + r^2\left(d\theta^2 + \sin^2\theta\, d\varphi^2\right)\right],$

with scale factor $a(t)$ , normalised to $a(t_0) = 1$ today, and spatial curvature $k$ . Inserting it into the Einstein equation with a perfect fluid of density $\rho$ and pressure $p$ gives the Friedmann equations

$H^2 \equiv \left(\frac{\dot a}{a}\right)^2 = \frac{8\pi G}{3}\rho - \frac{kc^2}{a^2} + \frac{\Lambda c^2}{3}, \qquad \frac{\ddot a}{a} = -\frac{4\pi G}{3}\left(\rho + \frac{3p}{c^2}\right) + \frac{\Lambda c^2}{3}.$

Each component obeys $\dot\rho + 3H(\rho + p/c^2) = 0$ , so matter ( $p = 0$ ) dilutes as $a^{-3}$ and radiation ( $p = \rho c^2/3$ ) as $a^{-4}$ . Dividing by the critical density $\rho_c = 3H_0^2/(8\pi G)$ defines the density parameters, and the first Friedmann equation becomes

$H(a)^2 = H_0^2\left[\Omega_r a^{-4} + \Omega_m a^{-3} + \Omega_k a^{-2} + \Omega_\Lambda\right], \qquad \Omega_r + \Omega_m + \Omega_k + \Omega_\Lambda = 1 .$

Parameters. The base $\Lambda$ CDM model assumes $\Omega_k = 0$ and has six free parameters, commonly chosen as the physical baryon and cold dark matter densities $\Omega_b h^2$ and $\Omega_c h^2$ (with $H_0 = 100\,h$ km/s/Mpc), the angular size of the sound horizon at recombination $\theta_*$ , the optical depth to reionisation $\tau$ , and the amplitude $A_s$ and spectral index $n_s$ of the primordial fluctuations. The 2018 Planck fit gives approximately $\Omega_b h^2 \approx 0.0224$ , $\Omega_c h^2 \approx 0.120$ , $n_s \approx 0.965$ , $H_0 \approx 67.4$ km/s/Mpc, $\Omega_m \approx 0.315$ and $\Omega_\Lambda \approx 0.685$ . Distance ladder measurements of $H_0$ from supernovae calibrated by Cepheids give values around 73 km/s/Mpc; this disagreement is the Hubble tension.

Example: when did the expansion start to accelerate? Today radiation is negligible. With $\Omega_k = 0$ the acceleration equation reads $\ddot a/a = H_0^2\left(\Omega_\Lambda - \frac{1}{2}\Omega_m a^{-3}\right)$ , so $\ddot a > 0$ for $a^3 > \Omega_m/(2\Omega_\Lambda)$ . With the values above, $a > (0.315/1.37)^{1/3} \approx 0.61$ , which is redshift $z = 1/a - 1 \approx 0.63$ . Matter and $\Lambda$ had equal densities later, at $a = (\Omega_m/\Omega_\Lambda)^{1/3} \approx 0.77$ , or $z \approx 0.3$ . In the eras dominated by a single component, $a \propto t^{1/2}$ (radiation), $a\propto t^{2/3}$ (matter) and $a \propto e^{\sqrt{\Lambda/3}\,c t}$ ( $\Lambda$ ).

Structure. Small density contrasts $\delta = \delta\rho/\rho$ in cold dark matter grow during matter domination as $\delta \propto a$ on scales well inside the horizon. The primordial fluctuations are assumed to be adiabatic and Gaussian with a nearly scale-invariant power spectrum. Their imprint on the cosmic microwave background, through acoustic oscillations of the photon-baryon fluid before recombination, is the main source of the parameter values above.

For a particularly nice discussion see The Standard Model of Cosmology: A Skeptic's Guide by Douglas Scott.

Abstract

$\Lambda$ CDM is a model within general relativity: a spacetime $(M,g)$ solving the Einstein equation with cosmological constant, a specified matter content, and a statistical description of initial perturbations. It is built in two layers.

Background. The cosmological principle is the assumption that, on large scales, $M = \mathbb{R}\times\Sigma$ and the metric is $g = -c^2 dt^2 + a(t)^2 \gamma$ , where $(\Sigma,\gamma)$ is a maximally symmetric Riemannian 3-manifold. Up to quotients, $\Sigma$ is flat space, the 3-sphere or hyperbolic space, with isometry groups $E(3)$ , $O(4)$ or $O^+(3,1)$ for $k = 0, +1, -1$ . Homogeneity and isotropy force the energy-momentum tensor into perfect fluid form $T_{\mu\nu} = (\rho + p/c^2) u_\mu u_\nu + p\, g_{\mu\nu}$ with $u^\mu$ the comoving 4-velocity, and the Einstein equation reduces to the Friedmann equations for $a(t)$ . The components of base $\Lambda$ CDM are photons, three species of light neutrinos, baryons, pressureless cold dark matter that interacts only gravitationally, and $\Lambda$ , with $k = 0$ .

Perturbations. Structure is described by linear perturbation theory around FLRW. Perturbations of the metric and the matter split into scalar, vector and tensor parts under the isometries of $\Sigma$ , and these decouple at linear order. Coordinate changes act on the perturbations as gauge transformations, so physical statements are made with gauge invariant combinations or in a fixed gauge. In conformal Newtonian gauge with conformal time $\eta$ , scalar perturbations take the form

$ds^2 = a(\eta)^2\left[-(1+2\Psi)\,c^2 d\eta^2 + (1-2\Phi)\,\delta_{ij}\,dx^i dx^j\right].$

The linearised Einstein equations couple $\Phi$ and $\Psi$ to the perturbations of each component. Photons and neutrinos are described by phase space distributions obeying the Boltzmann equation, with Thomson scattering coupling photons to baryons; cold dark matter and baryons are treated as fluids. Without anisotropic stress $\Phi = \Psi$ ; free streaming neutrinos give a small difference.

Initial conditions. The initial state is a statistically homogeneous and isotropic Gaussian random field of adiabatic perturbations, specified by the power spectrum of the comoving curvature perturbation $\mathcal{R}$ ,

$\mathcal{P}_{\mathcal{R}}(k) = A_s\left(\frac{k}{k_*}\right)^{n_s - 1},$

with a pivot scale $k_*$ (Planck uses $0.05\ \text{Mpc}^{-1}$ ). On super-horizon scales $\mathcal{R}$ is conserved for adiabatic perturbations, which is why the initial conditions can be stated without knowing the physics that produced them. Inflation is the standard candidate for that physics, and it also predicts a spectrum of tensor modes that is not part of the six-parameter base model.

Observables. The model predicts the angular power spectra $C_\ell$ of temperature and polarisation anisotropies of the cosmic microwave background, the matter power spectrum and baryon acoustic oscillation scale, and distance-redshift relations. Because the initial field is Gaussian, these two-point functions contain all the information at linear order. Nonlinear structure formation on small scales goes beyond linear theory and is computed with N-body simulations.

Standard Model of Cosmology

Intuitive

Concrete

Abstract

Path to this page