Is There a Relationship Between the Cosmological Constant and the Hubble Constant?

The cosmological constant Λ and the Hubble constant H₀ both appear in descriptions of cosmic expansion, so they are related—but they are not interchangeable. Λ is a parameter in the gravitational field equations, while H₀ is the measured expansion rate at the present cosmic time. To connect them quantitatively, you need a cosmological model that also accounts for matter, radiation, and spatial curvature.

What each quantity means

The cosmological constant Λ is usually interpreted as a constant energy density associated with empty space. In the usual convention, it has units of inverse length squared, such as m⁻². A positive Λ contributes to accelerated expansion; whether the universe’s total expansion is accelerating also depends on its other contents.

The Hubble parameter describes the expansion rate at a given time: $H(t)=\dot a(t)/a(t)$, where $a(t)$ is the universe’s scale factor. The symbol $H_0$ means its value today. Although called the Hubble constant, it is not constant throughout cosmic history; it changes as the universe expands and its contents evolve.

The relationship in the Friedmann equation

For a homogeneous, isotropic universe governed by general relativity, the expansion rate is described by the Friedmann equation. In one common form,

$$H^2(t)=\frac{8\pi G}{3}\rho(t)-\frac{k c^2}{a^2(t)}+\frac{\Lambda c^2}{3}.$$

Here $\rho$ is the mass-equivalent density of ordinary matter, dark matter, and radiation; $k$ describes spatial curvature under the chosen scale-factor convention; $G$ is Newton’s gravitational constant; and $c$ is the speed of light. Evaluating this equation today shows why Λ affects $H_0$: its contribution is $\Lambda c^2/3$. But the other terms contribute too.

Cosmologists often express the contributions as dimensionless density parameters. For the cosmological constant,

$$\Omega_\Lambda=\frac{\Lambda c^2}{3H_0^2}.$$

This equation is a useful direct connection between Λ and $H_0$. It does not mean that either quantity alone fixes the other: the value of $\Omega_\Lambda$ must also be known. In a spatially flat model, the present-day density parameters add to approximately one, with small contributions from radiation as well as matter and Λ.

Worked example using measured cosmological parameters

As an illustration, take representative values from the Planck 2018 analysis under the base flat ΛCDM model: $H_0\approx67.4$ km/s/Mpc and $\Omega_\Lambda\approx0.685$. Convert the Hubble parameter to SI units using $1\,\mathrm{Mpc}\approx3.086\times10^{22}$ m:

$$H_0\approx\frac{67.4\times10^3\ \mathrm{m/s}}{3.086\times10^{22}\ \mathrm{m}}\approx2.18\times10^{-18}\ \mathrm{s}^{-1}.$$

Rearrange the definition of $\Omega_\Lambda$ to estimate Λ:

$$\Lambda=\frac{3\Omega_\Lambda H_0^2}{c^2} \approx\frac{3(0.685)(2.18\times10^{-18}\ \mathrm{s}^{-1})^2}{(2.998\times10^8\ \mathrm{m/s})^2} \approx1.1\times10^{-52}\ \mathrm{m}^{-2}.$$

This estimate uses measured parameters interpreted within a particular cosmological model. It is not a universal conversion that lets you infer Λ from a Hubble measurement alone.

The special case: expansion driven only by Λ

If the universe contained only a positive cosmological constant and had no matter, radiation, or curvature contribution, the Friedmann equation would reduce to

$$H_\Lambda=c\sqrt{\frac{\Lambda}{3}}.$$

This is the expansion rate of a de Sitter universe. It is also the limiting expansion rate of a Λ-dominated universe in the far future, if Λ truly remains constant and other components become negligible. It is not generally equal to today’s $H_0$, because matter still contributes significantly to the present expansion rate.

Common mistakes

• Treating Λ and $H_0$ as the same kind of constant. Λ is a parameter in the gravitational equations; $H_0$ is the current value of a time-dependent expansion rate.
• Using $H_0=c\sqrt{\Lambda/3}$ for our universe today. That expression applies when Λ is the only relevant contribution. For a universe with matter, radiation, or curvature, use the full Friedmann equation.
• Assuming a measured $H_0$ uniquely determines Λ. A measurement constrains parameters within a model and alongside other data. Different assumptions about the universe’s contents or geometry can change the inferred values.
• Confusing Λ with all possible dark energy. A cosmological constant has constant density and a particular pressure-to-density relation. More general dark-energy models can have a density that changes over time, so they need not obey the same simple equation.

So the concise answer is that Λ contributes to the present expansion rate, and their connection is quantified by the Friedmann equation. The relationship becomes a simple formula only in the special case of a universe dominated entirely by Λ.