Cosmological constant ($\Lambda$)

The cosmological constant $\Lambda$ is the constant term proportional to the metric in the Einstein field equations. Interpreted as vacuum energy, it has $P_{\Lambda }=-{\rho }_{\Lambda }$, so its energy density is constant during expansion.

An expanding-universe regime in which a positive cosmological constant supplies most of the Friedmann equation. The approximately constant positive expansion rate gives exponential growth of the scale factor. Since matter, radiation, and curvature contributions decay with expansion, a positive cosmological constant eventually dominates a perpetually expanding model containing them; this does not guarantee that curvature dominates at an intermediate time.

The equality of a radiation density proportional to $(1+z{)}^4$ and a constant vacuum density. It is only a crossing of two components. Matter can dominate both at this epoch, so radiation-vacuum equality need not mark a change in the universe's dominant component.

For conserved pressureless matter and a true cosmological constant, equality follows by comparing the matter density proportional to $(1+z{)}^3$ with the constant vacuum density. This is not the onset of accelerated expansion: in a matter-plus-vacuum model acceleration starts at ${\rho }_m=2{\rho }_{\Lambda }$.

In a Friedmann equation written $H^2=8\pi G\rho /3+\Lambda /3-k{c}^2/a^2$, the symbol $\Lambda$ has units of inverse time squared. If ${\Lambda }_{\mathrm{geom}}$ is the geometrical constant measured in inverse length squared, $\Lambda =c^2{\Lambda }_{\mathrm{geom}}$. Natural units hide this distinction.

Nonrelativistic matter dilutes as $a^{-3}$, while a cosmological constant has constant energy density. The ratio of vacuum energy density to matter density consequently scales as $a^3$, so comparable contributions occur during a limited epoch. The cosmic coincidence problem asks why observers find these very different components comparable near their own epoch. This timing puzzle is distinct from the cosmological constant problem, which concerns the tiny absolute vacuum-energy scale.

The cosmological constant problem is the unexplained smallness of the gravitationally observable vacuum energy compared with the contributions expected from quantum field theory. In units with $\hslash =c=1$, an ultraviolet cutoff ${\Lambda }_{\mathrm{UV}}$ generically gives contributions of order ${\Lambda }_{\mathrm{UV}}^4$, whereas the observed dark-energy density is of order $10^{-47}{\mathrm{GeV}}^4$. The Einstein equation depends on the sum of a bare cosmological constant and these contributions; subtracting a constant from a nongravitational Hamiltonian does not remove its gravitational effect. Exact global supersymmetry gives zero energy to a supersymmetric vacuum, but supersymmetry breaking generally reintroduces a much larger positive scale.

A heavy field with mass $m$ introduces a natural vacuum-energy threshold scale of order $m^4$ up to couplings and loop factors. This is not a regulator-independent prediction: counterterms and renormalization are required. Comparing an electroweak mass to the observed vacuum density exposes a very large cancellation problem.

In $c=\hslash =1$ units, vacuum energy density has mass dimension four, Newton constant dimension minus two, and ${\Lambda }_{\mathrm{geom}}=8\pi G{\rho }_{\mathrm{vac}}$ dimension two. A value quoted in $\mathrm{Ge}{\mathrm{V}}^4$ is an energy density, not the geometric cosmological constant.