Cosmological Constant · aiwiki.page

The cosmological constant, conventionally denoted by the Greek letter $\Lambda$ , is a parameter in general relativity that contributes to the curvature of spacetime even where ordinary matter is absent. It can be interpreted geometrically or as a uniform vacuum energy density. A positive cosmological constant can drive accelerated cosmic expansion and is the simplest description of dark energy in the Lambda–Cold Dark Matter model ( $\Lambda$ CDM). Dark energy is the broader concept: it need not be a cosmological constant. (doi.org)

Mathematical definition

With metric signature $(-+++)$ , the Einstein field equations including $\Lambda$ are

G_{\mu\nu}+\Lambda g_{\mu\nu} =\frac{8\pi G}{c^4}T_{\mu\nu}.

Here $G_{\mu\nu}$ is the Einstein tensor, $g_{\mu\nu}$ the metric tensor, $T_{\mu\nu}$ the stress–energy tensor, $G$ Newton’s gravitational constant, and $c$ the speed of light. The dimensions of $\Lambda$ are inverse length squared; its SI unit is $\mathrm{m^{-2}}$ . Moving its term to the right-hand side gives an equivalent vacuum stress–energy tensor,

T_{\mu\nu}^{(\Lambda)} =-\epsilon_\Lambda g_{\mu\nu}, \qquad \epsilon_\Lambda=\frac{\Lambda c^4}{8\pi G}.

Thus the geometric constant corresponds to an energy density $\epsilon_\Lambda$ , or equivalent mass density $\rho_\Lambda=\epsilon_\Lambda/c^2$ . (pmc.ncbi.nlm.nih.gov)

Vacuum energy and negative pressure

A Lorentz-invariant vacuum has stress–energy proportional to the metric. In a perfect-fluid description, its pressure satisfies

p_\Lambda=-\epsilon_\Lambda.

Its equation-of-state parameter is therefore $w=p/\epsilon=-1$ . The density remains constant as space expands, unlike the density of matter or radiation. In the continuity relation $d(\epsilon V)=-p\,dV$ , negative pressure accounts for the increase in vacuum energy within an expanding volume. This does not imply an ordinary material flowing into that volume. (ned.ipac.caltech.edu)

In quantum field theory, vacuum energy can receive contributions from zero-point energies and field potentials. Once gravity is included, changing the constant part of a potential changes its gravitational contribution, rather than merely redefining an irrelevant energy zero. (ned.ipac.caltech.edu)

Effect on cosmic expansion

For a homogeneous, isotropic universe, the Friedmann equations include

H^2=\frac{8\pi G}{3}\rho -\frac{kc^2}{a^2} +\frac{\Lambda c^2}{3},

\frac{\ddot a}{a} =-\frac{4\pi G}{3} \left(\rho+\frac{3p}{c^2}\right) +\frac{\Lambda c^2}{3}.

Here $a(t)$ is the scale factor, $H=\dot a/a$ , $k$ describes spatial curvature, and $\rho$ and $p$ exclude the separately written cosmological-constant contribution. Positive $\Lambda$ contributes positively to acceleration; acceleration occurs when it outweighs the decelerating matter and radiation terms. (pmc.ncbi.nlm.nih.gov)

Matter density decreases approximately as $a^{-3}$ , and radiation energy density as $a^{-4}$ , while vacuum density remains fixed. Consequently, a small positive $\Lambda$ can be negligible early and important late. If expansion continues and $\Lambda$ dominates, the universe approaches de Sitter spacetime, with

a(t)\propto e^{H_\Lambda t}, \qquad H_\Lambda=c\sqrt{\Lambda/3}.

Negative $\Lambda$ instead favors recollapse in conventional matter-filled cosmologies. Spatial curvature alone therefore does not determine the universe’s fate. (ned.ipac.caltech.edu)

Historical development

Albert Einstein introduced the cosmological constant in 1917 to permit a static, matter-filled universe. The resulting Einstein static model balanced matter’s gravitational attraction against a positive $\Lambda$ , but the balance was unstable to homogeneous disturbances. Evidence for cosmic expansion removed the original motivation without making the term mathematically inadmissible. (pmc.ncbi.nlm.nih.gov)

In 1998, two independent teams studying distant Type Ia supernovae reported evidence for accelerated expansion. Their observations revived interest in positive $\Lambda$ . The discovery was recognized by the 2011 Nobel Prize in Physics, awarded to Saul Perlmutter, Brian Schmidt, and Adam Riess. Acceleration itself, however, does not uniquely establish a cosmological constant. (nobelprize.org)

Observational determination

Cosmological analyses constrain $\Lambda$ through the expansion history, distances, and growth of structure. Important measurements include supernova brightness versus redshift, cosmic microwave background anisotropies, and baryon acoustic oscillations. These probes constrain combinations of parameters rather than directly measuring vacuum energy in a laboratory. (nobelprize.org)

The present vacuum-density fraction is

\Omega_{\Lambda0} =\frac{\Lambda c^2}{3H_0^2},

where $H_0$ is the Hubble constant. Under spatially flat $\Lambda$ CDM, the Planck 2018 temperature, polarization, and lensing analysis inferred approximately $\Omega_{\Lambda0}=0.685$ and $H_0=67.4\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ . These imply $\Lambda\approx1.1\times10^{-52}\ \mathrm{m^{-2}}$ . The inference depends on the assumed cosmological model. Unlike $\Lambda$ , the density fraction $\Omega_\Lambda(t)$ changes as $H(t)$ evolves. (arxiv.org)

Tests also allow dark energy to vary. The 2025 DESI Data Release 2 analysis found that its acoustic-distance measurements were well described by flat $\Lambda$ CDM, while combinations with microwave-background and supernova data preferred a particular evolving-dark-energy model at $2.8$ – $4.2$ standard deviations, depending on the supernova sample. Such model- and dataset-dependent results are not a definitive demonstration that $\Lambda$ is absent. (arxiv.org)

The cosmological constant problem

The cosmological constant problem concerns why the observed effective vacuum energy is so small compared with characteristic contributions expected from particle physics. A naive estimate using a Planck-scale ultraviolet cutoff exceeds the observed scale by roughly 120 orders of magnitude. That figure is not a precise, unique prediction: it depends on assumptions about the cutoff and underlying theory. (ned.ipac.caltech.edu)

The difficulty is not simply removing a divergent calculation. The observed value combines a gravitational constant term with quantum and field-potential contributions; maintaining a tiny result despite large contributions requires an explanation. A related “coincidence problem” asks why matter and dark-energy densities are comparable during the present cosmic epoch despite their different evolution. Proposed explanations include new symmetries, adjustment mechanisms, dynamical fields, and selection effects, but these proposals do not constitute an established microscopic derivation of the measured value. (doi.org)

References

The Cosmological Constantpmc.ncbi.nlm.nih.gov
The cosmological constant and dark energydoi.org
The Cosmological Constant — Why a Cosmological Constant Seems Inevitablened.ipac.caltech.edu
The Cosmological Constant — Expansion Dynamicsned.ipac.caltech.edu
The Nobel Prize in Physics 2011nobelprize.org
The Nobel Prize in Physics 2011 — Popular informationnobelprize.org
The Nobel Prize in Physics 2011 — Information for the publicnobelprize.org
The accelerating Universenobelprize.org
Planck 2018 results. VI. Cosmological parametersarxiv.org
Planck 2018 results. VI. Cosmological parameters — Full textarxiv.org
DESI DR2 Results II: Measurements of Baryon Acoustic Oscillations and Cosmological Constraintsarxiv.org