Cosmological Constant
The cosmological constant (Λ) is a constant term that can be added to Einstein's field equations of general relativity; it is mathematically equivalent to a uniform, unchanging energy density of empty space with negative pressure, which gravitates repulsively on cosmic scales and is the simplest candidate explanation for the observed accelerating expansion of the universe. Its measured value is extremely small, about 1.1×10−52 m−2.
Curious
When Einstein applied general relativity to the whole universe in 1917, his equations did not allow a universe that simply sat still: gravity pulled everything together, so the cosmos should either expand or collapse. At the time, astronomers believed the universe was static. To make the equations agree with that belief, Einstein added one extra term, which he called Λ (the Greek letter lambda). A positive Λ acts like a gentle repulsion spread evenly through space, balancing gravity's pull.
After Edwin Hubble's 1929 observations showed that distant galaxies are receding (see Edwin Hubble), the static universe was no longer needed, and Λ was widely set aside. It returned in 1998, when two teams studying distant supernovae found that the expansion is speeding up (see Dark Energy). A small positive Λ fits that data well, and it is now a core ingredient of the standard cosmological model, ΛCDM.
Exploring
| Symbol | Meaning | Unit |
|---|---|---|
| Λ | Cosmological constant | m−2 |
| ΩΛ | Fraction of critical density in the Λ component | dimensionless |
| H0 | Hubble constant (see Hubble Constant) | s−1 |
| c | Speed of light | m/s |
Deep Dive
Field equations with Λ. The modified equations read Gμν + Λgμν = (8πG/c4)Tμν. Moving Λ to the right-hand side shows it behaves like a fluid with energy density ρΛ = Λc2/(8πG) and pressure pΛ = −ρΛc2, that is, an equation-of-state parameter w = −1. In the Friedmann acceleration equation, ä/a = −(4πG/3)(ρ + 3p/c2) + Λc2/3, the pressure term makes a positive Λ push toward acceleration. Because its density does not dilute as space expands, Λ eventually dominates over matter, which is why the expansion is now accelerating.
Einstein's static universe was unstable. Einstein's 1917 model balanced Λ against matter density, but this balance is unstable: a small perturbation makes it expand or collapse. Independently, Alexander Friedmann (1922) and Georges Lemaître (1927) found expanding solutions of the original equations (see Georges Lemaître), so the original motivation for Λ was weaker than it first appeared. The widely repeated story that Einstein called Λ his “biggest blunder” comes from George Gamow's recollection in his 1970 autobiography My World Line; it is secondhand and cannot be independently checked, so this Codex records it as attributed to Gamow rather than as a verified quotation (see George Gamow).
The vacuum-energy problem (unresolved). Quantum field theory says empty space should have a zero-point energy, which behaves like a cosmological constant. Simple estimates of that energy, cut off at the Planck scale, exceed the observed value by roughly 120 orders of magnitude; estimates with lower cutoffs still miss by many tens of orders. No accepted mechanism explains why the observed Λ is so small but not zero. A second puzzle, the “coincidence problem”, asks why matter and Λ have comparable densities in the present epoch. Proposed responses include anthropic reasoning in a multiverse, dynamical dark energy, and modified gravity; none is established. Recent baryon acoustic oscillation measurements from the DESI survey (2024–2025) have reported hints, at roughly the 3-sigma level depending on the dataset combination, that dark energy might evolve rather than being a strict constant. This is a tentative result under active scrutiny and is not an established departure from Λ.
Concept Relationship Map
Requires: curved spacetime (Einstein field equations), Hubble constant
Enables: fate of the universe, age of the universe
Commonly confused with: dark energy in general (Λ is only the simplest model of it)
Sources
- Einstein, A. (1917). Sitzungsberichte der Preussischen Akademie der Wissenschaften, 142–152.
- Weinberg, S. (1989). “The cosmological constant problem.” Reviews of Modern Physics, 61(1), 1–23.
- Planck Collaboration (2020). “Planck 2018 results. VI. Cosmological parameters.” Astronomy & Astrophysics, 641, A6.
- Gamow, G. (1970). My World Line. Viking Press.