Cosmological constant
The cosmological constant, denoted by Lambda (Λ), is a mysterious force tied to the very fabric of spacetime, a concept Albert Einstein initially introduced then famously regretted. Today, it stands as the leading explanation for the accelerating expansion of our universe, hinting at a cosmos far stranger than we ever imagined. Yet, its true nature remains one of physics' deepest puzzles, potentially holding the key to a unified understanding of everything. Einstein first introduced the cosmological constant to maintain a static universe, later retracting it as his "biggest blunder" when cosmic expansion was discovered. Observations of distant supernovae in the late 1990s revealed the universe's expansion is accelerating, resurrecting the cosmological constant as the simplest explanation for a mysterious force called dark energy. Quantum mechanics predicts a vacuum energy vastly larger than observed, leading to the "cosmological constant problem," one of the most significant and perplexing discrepancies in modern physics.
AI Summary
The cosmological constant, denoted by Lambda (Λ), is a mysterious force tied to the very fabric of spacetime, a concept Albert Einstein initially introduced then famously regretted. Today, it stands as the leading explanation for the accelerating expansion of our universe, hinting at a cosmos far stranger than we ever imagined. Yet, its true nature remains one of physics' deepest puzzles, potentially holding the key to a unified understanding of everything.
- Einstein first introduced the cosmological constant to maintain a static universe, later retracting it as his "biggest blunder" when cosmic expansion was discovered.
- Observations of distant supernovae in the late 1990s revealed the universe's expansion is accelerating, resurrecting the cosmological constant as the simplest explanation for a mysterious force called dark energy.
- Quantum mechanics predicts a vacuum energy vastly larger than observed, leading to the "cosmological constant problem," one of the most significant and perplexing discrepancies in modern physics.
The Enigmatic Lambda
Imagine a fundamental number embedded in the universe itself — a value that dictates how space behaves. This is the cosmological constant, typically represented by the Greek letter Lambda (Λ). It's a concept with a fascinating history, evolving from a theoretical fix to one of the most profound mysteries in modern cosmology.
Initially, Albert Einstein added this constant to his equations of general relativity. His goal? To achieve a static universe, one that neither expanded nor contracted, aligning with the prevailing scientific view of his time. However, as we'll see, the universe had other plans.
A Universe Dominated by Dark Energy
Fast forward to today, and the cosmological constant has been dramatically revived. It's now the leading candidate for 'dark energy' — the invisible force believed to be driving the accelerated expansion of the universe. Current models suggest dark energy accounts for a staggering 68% of the universe's total mass-energy content.
This simple explanation for dark energy is a cornerstone of the standard model of cosmology, known as the ΛCDM model. Without it, our understanding of cosmic acceleration would require far more complex and speculative theories.
The Cosmological Constant Problem
Here's where things get truly perplexing. According to quantum field theory — the framework underlying all modern particle physics — empty space isn't truly empty. It's a vibrant, frothing sea of 'zero-point energy' fluctuations, constantly popping in and out of existence.
These quantum fluctuations should contribute to the cosmological constant, giving the vacuum a certain energy density. But when physicists calculate this theoretical vacuum energy, the numbers are unbelievably large. We're talking about a value that exceeds astronomical observations by a mind-boggling 120 orders of magnitude!
This enormous discrepancy has been dubbed 'the worst theoretical prediction in the history of physics.' It's known as the cosmological constant problem, and many believe that solving it could unlock deeper secrets about the fundamental nature of reality.
Einstein's 'Biggest Blunder'
Let's rewind to 1917, when Einstein published his general theory of relativity. His equations described a dynamic universe, one where gravity would naturally cause matter to pull together. This implied an expanding or contracting cosmos, not the static one Einstein (and others) believed in.
To force his theoretical universe to remain stationary, Einstein introduced the cosmological constant into his equations, essentially as a repulsive force to counteract gravity. He famously later expressed regret, calling it his 'biggest blunder.'
Indeed, just a few years later, in 1929, Edwin Hubble's observations confirmed that the universe was, in fact, expanding. This cosmic expansion had already been mathematically predicted by Alexander Friedmann's solutions to Einstein's original equations — without the constant!
Even with the constant, Einstein's static universe would have been inherently unstable. Any slight perturbation, like a tiny expansion, would release more vacuum energy, causing even more expansion, leading to a runaway effect. The universe couldn't stay still.
The Comeback: A Universe Speeding Up
For decades after Hubble's discovery, most physicists assumed the cosmological constant was zero, deeming it an unnecessary complication. The universe was thought to be expanding, but gradually slowing down due to the gravitational pull of its own matter.
Then, in the late 1990s, a startling discovery turned cosmology on its head. Two independent teams, led by Saul Perlmutter, Brian Schmidt, and Adam Riess, were observing distant Type Ia supernovae — stellar explosions used as 'standard candles' to measure cosmic distances.
The researchers expected to see these supernovae decelerating, but instead, they found the opposite: the expansion of the universe was accelerating! This profound revelation, announced in 1998, meant a mysterious repulsive force was at play, pushing galaxies apart ever faster.
The simplest explanation for this accelerating expansion was the reintroduction of Einstein's cosmological constant. It was no longer a blunder, but a necessary component of our cosmic story. This groundbreaking work earned Perlmutter, Schmidt, and Riess the Nobel Prize in Physics in 2011.
The Equation of Our Universe
The cosmological constant Λ finds its place directly in the Einstein field equations, the mathematical heart of general relativity. These equations describe how matter and energy warp spacetime, and how that warped spacetime dictates the motion of matter and energy.
Here, Rμν and R describe the curvature of spacetime, gμν is the metric tensor defining its geometry, and Tμν represents the energy and momentum content. The term Λgμν is where the cosmological constant makes its appearance, acting like an intrinsic energy density of the vacuum itself.
R_{\mu \nu }-{\tfrac {1}{2}}R\,g_{\mu \nu }+\Lambda g_{\mu \nu }=\kappa T_{\mu \nu }A positive cosmological constant implies a negative pressure within the vacuum. This 'negative pressure' behaves like an anti-gravitational force, pushing everything apart and driving the accelerated expansion we observe in the universe today.
Cosmic Composition: The Density Parameters
To understand the universe's makeup, cosmologists use 'density parameters,' denoted by Ω. These dimensionless values represent the ratio of a component's actual density to the 'critical density' — the tipping point that determines the universe's ultimate fate.
For a flat universe, which observations from missions like WMAP suggest ours is, the total density parameter is one. This total is composed of matter (Ωm), dark energy (ΩΛ), and curvature (Ωk, which is zero for a flat universe).
Recent data from the Planck Collaboration estimates ΩΛ to be around 0.68 to 0.7, meaning dark energy truly dominates the cosmic landscape.
• Ωm: The matter density parameter, encompassing both ordinary (baryonic) and dark matter. • ΩΛ: The dark energy density parameter, representing the cosmological constant's contribution. • Ωk: The curvature density parameter, zero for a flat universe.
Crucially, while the critical density of the universe changes over time as it expands, the energy density associated with the cosmological constant remains constant. This means the proportion of dark energy grows relative to matter, continually accelerating the expansion.
Unanswered Questions and Future Frontiers
Despite its success in explaining cosmic acceleration, the cosmological constant problem remains a gaping wound in our understanding. Why is the observed value so incredibly small compared to quantum predictions? And what exactly is dark energy?
Quantum Field Theory's Dilemma
Theoretical arguments, based on principles like dimensional analysis, consistently suggest that the vacuum energy should be huge, on the order of the Planck scale. Yet, our observations indicate a value roughly 10^120 times smaller.
Some advanced theories, like supersymmetry, even predict a cosmological constant of precisely zero, which only deepens the mystery. String theory, our best candidate for a 'theory of everything,' also struggles to naturally produce a small, positive cosmological constant.
The Anthropic Principle: A Controversial Explanation
One provocative idea, championed by physicists like Steven Weinberg, is the 'anthropic principle.' It suggests that if the cosmological constant were much larger or smaller, the universe wouldn't be conducive to the formation of life, galaxies, or even stable atoms.
In this view, we observe the specific value of the cosmological constant simply because it's the only one that allows for our existence. This often invokes the idea of a 'multiverse,' where countless universes exist with varying physical constants, and we simply reside in one compatible with life.
While intriguing, the anthropic principle is controversial among scientists, with critics pointing out that it doesn't explain the value, but rather sidesteps the question by appealing to selection effects. It remains an active area of debate.
The Search Continues
Physicists are constantly devising new experiments and theoretical approaches to probe the nature of dark energy. While direct detection in laboratories remains elusive, future cosmic surveys aim to map the universe's expansion with even greater precision, potentially revealing more about this cosmic enigma.
The cosmological constant has taken us on a remarkable journey — from Einstein's reluctant fix to the heart of the greatest unsolved problem in physics. Its story is far from over, and its ultimate resolution may redefine our understanding of spacetime, quantum mechanics, and the very existence of our universe.
Article
Cosmological constant
Sketch of the timeline of the Universe in the ΛCDM model. The accelerated expansion in the last third of the timeline represents the dark-energy dominated era.
In physical cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: Λ), alternatively called Einstein's cosmological constant, is a coefficient that Albert Einstein initially added to his field equations of general relativity. He later removed it; however, much later it was revived to express the energy density of space, or vacuum energy, that arises in quantum mechanics. It is closely associated with the concept of dark energy.
Einstein introduced the constant in 1917 to counterbalance the effect of gravity and achieve a static universe, which was then assumed. Einstein's cosmological constant was abandoned after Edwin Hubble confirmed that the universe was expanding. From the 1930s until the late 1990s, most physicists thought the cosmological constant to be zero. That changed with the discovery in 1998 that the expansion of the universe is accelerating, implying that the cosmological constant may have a positive value after all.
Since the 1990s, studies have shown that, assuming the cosmological principle, around 68% of the mass–energy density of the universe can be attributed to dark energy. The cosmological constant Λ is the simplest possible explanation for dark energy and is used in the standard model of cosmology known as the ΛCDM model.
According to quantum field theory, which underlies modern particle physics, empty space is defined by the vacuum state which is composed of a collection of quantum fields. All these quantum fields exhibit fluctuations in their ground state (lowest energy density) arising from the zero-point energy existing everywhere in space. These zero-point fluctuations should contribute to the cosmological constant Λ, but actual calculations give rise to an enormous vacuum energy. The discrepancy between theorized vacuum energy from quantum field theory and observed vacuum energy from cosmology is a source of major contention, with the values predicted exceeding observation by some 120 orders of magnitude, a discrepancy that has been called "the worst theoretical prediction in the history of physics!". This issue is called the cosmological constant problem, and it is one of the greatest mysteries in science with many physicists believing that "the vacuum holds the key to a full understanding of nature".
History
Cosmological constant
In the Principia (1687), Isaac Newton identifies a central force proportional to distance as one of the "two principal cases of attraction" alongside the inverse-square law of gravity; in modern physics, this linear force law is mathematically equivalent to the force associated with a cosmological constant.
The cosmological constant was originally introduced in Albert Einstein's 1917 paper entitled Cosmological considerations in the General Theory of Relativity. Einstein included the cosmological constant as a term in his field equations for general relativity because he was dissatisfied that otherwise his equations did not allow for a static universe: gravity would cause a universe that was initially non-expanding to contract. To counteract this possibility, Einstein added the cosmological constant. However, he was not happy about adding this cosmological term. He later stated "Since I introduced this term, I had always a bad conscience. ... I am unable to believe that such an ugly thing is actually realized in nature". Einstein's static universe is unstable against matter density perturbations.
In 1929, not long after Einstein developed his static theory, observations by Edwin Hubble indicated that the universe appears to be expanding; this was consistent with a cosmological solution to the original general relativity equations that had been found by the mathematician Alexander Friedmann, working on the Einstein equations of general relativity. Einstein reportedly referred to his failure to accept the validation of his equations—when they had predicted the expansion of the universe in theory, before it was demonstrated in observation of the cosmological redshift—as his "biggest blunder" (according to George Gamow).
It transpired that adding the cosmological constant to Einstein's equations does not lead to a static universe at equilibrium because the equilibrium is unstable: if the universe expands slightly, then the expansion releases vacuum energy, which causes yet more expansion. Likewise, a universe that contracts slightly will continue contracting.
However, the cosmological constant remained a subject of theoretical and empirical interest. Empirically, the cosmological data of recent decades strongly suggest that our universe has a positive cosmological constant. The explanation of this small but positive value is a remaining theoretical challenge, the so-called cosmological constant problem.
Some early generalizations of Einstein's gravitational theory, known as classical unified field theories, either introduced a cosmological constant on theoretical grounds or found that it arose naturally from the mathematics. For example, Arthur Eddington claimed that the cosmological constant version of the vacuum field equation expressed the "epistemological" property that the universe is "self-gauging", and Erwin Schrödinger's pure-affine theory using a simple variational principle produced the field equation with a cosmological term.
In 1990s, Saul Perlmutter at Lawrence Berkeley National Laboratory, Brian Schmidt of the Australian National University and Adam Riess of the Space Telescope Science Institute were searching for type Ia supernovae. At that time, they expected to observe the deceleration of the supernovae caused by gravitational attraction of mass according to Einstein's gravitational theory. The first reports published in July 1997 from the Supernova Cosmology Project used the supernova observation to support such deceleration hypothesis. But soon they found that supernovae were accelerating away. Both teams announced this surprising result in 1998. It implied the universe is undergoing accelerating expansion. The cosmological constant is needed to explain such acceleration. Following this discovery, the cosmological constant was reinserted in the general relativity equations.
Sequence of events 1915–1998
• In 1915, Einstein publishes his equations of general relativity, without a cosmological constant Λ. • In 1917, Einstein adds the parameter Λ to his equations when he realizes that his theory implies a dynamic universe for which space is a function of time. He then gives this constant a value that makes his Universe model remain static and eternal (Einstein static universe). • In 1922, the Russian physicist Alexander Friedmann mathematically shows that Einstein's equations (whatever Λ) remain valid in a dynamic universe. • In 1927, the Belgian astrophysicist Georges Lemaître shows that the Universe is expanding by combining general relativity with astronomical observations, those of Hubble in particular. • In 1931, Einstein accepts the theory of an expanding universe and proposes, in 1931, a model of a continuously expanding universe with zero cosmological constant (the Friedmann–Einstein universe). This is followed by another model, in 1932, with the Dutch physicist and astronomer Willem de Sitter in which both the cosmological constant and spatial curvature are set to zero (the Einstein–de Sitter universe). • In 1998, two teams of astrophysicists, the Supernova Cosmology Project and the High-Z Supernova Search Team, carried out measurements on distant supernovae which showed that the speed of galaxies' recession in relation to the Milky Way increases over time. The universe is in accelerated expansion, which requires having a strictly positive Λ. The universe would contain a mysterious dark energy producing a repulsive force that counterbalances the gravitational braking produced by the matter contained in the universe (see Standard cosmological model). For this work, Perlmutter, Schmidt, and Riess jointly received the Nobel Prize in Physics in 2011.
Equation
Cosmological constant
Estimated ratios of dark matter and dark energy (which may be the cosmological constant) in the universe. This image is made by NASA using the 9 year WMAP data. This is the final WMAP release.
The cosmological constant Λ appears in the Einstein field equations in the form ${\displaystyle R{\mu \nu }-{\tfrac {1}{2}}R\,g{\mu \nu }+\Lambda g{\mu \nu }=\kappa T{\mu \nu },}$ where the Ricci tensor Rμν, Ricci scalar R and the metric tensor gμν describe the structure of spacetime, the stress–energy tensor Tμν describes the energy density, momentum density and stress at that point in spacetime, and κ = 8πG/c4. The gravitational constant G and the speed of light c are universal constants. When Λ is zero, this reduces to the field equation of general relativity usually used in the 20th century. When Tμν is zero, the field equation describes empty space (a vacuum).
The cosmological constant has the same effect as an intrinsic energy density of the vacuum, ρvac (and an associated pressure). In this context, it is commonly moved to the right-hand side of the equation using Λ = κρvac. It is common to quote values of energy density directly, though still using the name "cosmological constant". The dimension of Λ is generally understood as length−2.
Using the Planck units, and the value evaluated in 2025 for the Hubble constant H0 = 76.5±2.2 (km/s)/Mpc = (2.48±0.07)×10−18 s−1, Λ has the value of ${\displaystyle {\begin{aligned}\Lambda =3\,\left({\frac {\,H{0}\,}{c}}\right)^{2}\Omega {\Lambda }&=1.4657\times 10^{-52}\ {\text{m}}^{-2}\\&=3.827\times 10^{-123}\,l{\text{P}}^{-2}\end{aligned}}}$ where ${\textstyle l{\text{P}}}$ is the Planck length. A positive vacuum energy density resulting from a cosmological constant implies a negative pressure, and vice versa. If the energy density is positive, the associated negative pressure will drive an accelerated expansion of the universe, as observed. (See Dark energy and Cosmic inflation for details.)
Density parameter Ω
The dimensionless density parameter Ω represents the ratio of the actual density of a component of the universe to the critical density. The total density parameter for a flat universe (concluded by WMAP), can be expressed as:
${\displaystyle \Omega {\text{total}}=\Omega {m}+\Omega {\Lambda }+\Omega {k}=1}$
• Ωm is the matter density parameter (including both baryonic and dark matter); • ΩΛ is the density parameter for dark energy (cosmological constant); and • Ωk describes the curvature of the universe which is 0 in a flat universe.
Instead of the cosmological constant itself, cosmologists often refer to the ratio between the energy density due to the cosmological constant and the critical density of the universe, the tipping point for a sufficient density to stop the universe from expanding forever (the dark energy density parameter). This ratio is estimated to be 0.714, according to results published by the Planck Collaboration in 2018 and clearly mentioned on WMAP. More intuitively, this parameter could be described as the fraction of the universe that is made up of dark energy. Note that this value changes over time: The critical density changes with cosmological time but the energy density due to the cosmological constant remains unchanged throughout the history of the universe, because the amount of dark energy increases as the universe grows but the amount of matter does not.
Equation of state
Another ratio that is used by scientists is the equation of state, usually denoted w, which is the ratio of pressure that dark energy puts on the universe to the energy per unit volume. This ratio is w = −1 for the cosmological constant used in the Einstein equations; alternative time-varying forms of vacuum energy such as quintessence generally use a different value. The value w = −1.028±0.032, measured by the Planck Collaboration (2018) is consistent with −1, assuming w does not change over cosmic time.
Value
Cosmological constant
Lambda-CDM, accelerated expansion of the universe. The time-line in this schematic diagram extends from the Big Bang/inflation era 13.7 Byr ago to the present cosmological time.
Observations announced in 1998 of distance–redshift relation for Type Ia supernovae indicated that the expansion of the universe is accelerating, if one assumes the cosmological principle. When combined with measurements of the cosmic microwave background radiation these implied a value of ΩΛ ≈ 0.7, a result which has been supported and refined by more recent measurements (as well as previous works). If one assumes the cosmological principle, as in the case for all models that use the Friedmann–Lemaître–Robertson–Walker metric, while there are other possible causes of an accelerating universe, such as quintessence, the cosmological constant is in most respects the simplest solution. Thus, the Lambda-CDM model, the current standard model of cosmology which uses the FLRW metric, includes the cosmological constant, which is measured to be on the order of 10−52 m−2. It may be expressed as 10−35 s−2 (multiplying by c2 ≈ 1017 m2⋅s−2) or as 10−122 ℓP−2 (where ℓP is the Planck length). The value is based on recent measurements of vacuum energy density, ρvac = 5.96×10−27 kg/m3 ≘ 5.3566×10−10 J/m3 = 3.35 GeV/m3. However, due to the Hubble tension and the CMB dipole, recently it has been proposed that the cosmological principle is no longer true in the late universe and that the FLRW metric breaks down, so it is possible that observations usually attributed to an accelerating universe are simply a result of the cosmological principle not applying in the late universe.
As was seen, by works of 't Hooft, Susskind and others, a positive cosmological constant has surprising consequences, such as a finite maximum entropy of the observable universe (see Holographic principle).
Predictions
Quantum field theory
A major outstanding problem is that most quantum field theories predict a huge value for the quantum vacuum. A common assumption is that the quantum vacuum is equivalent to the cosmological constant. Although no theory exists that supports this assumption, arguments can be made in its favor.
Such arguments are usually based on dimensional analysis and effective field theory. If the universe is described by an effective local quantum field theory down to the Planck scale, then we would expect a cosmological constant of the order of ${\textstyle M{\rm {pl}}^{2}}$ (${\textstyle 1}$ in reduced Planck units). As noted above, the measured cosmological constant is smaller than this by a factor of ~10120. This discrepancy has been called "the worst theoretical prediction in the history of physics".
Some supersymmetric theories require a cosmological constant that is exactly zero, which further complicates things. This is the cosmological constant problem, the worst problem of fine-tuning in physics: there is no known natural way to derive the tiny cosmological constant used in cosmology from particle physics.
No vacuum in the string theory landscape is known to support a metastable, positive cosmological constant, and in 2018 a group of four physicists advanced a controversial conjecture which would imply that no such universe exists.
Anthropic principle
One possible explanation for the small but non-zero value was noted by Steven Weinberg in 1987 following the anthropic principle. Weinberg explains that if the vacuum energy took different values in different domains of the universe, then observers would necessarily measure values similar to that which is observed: the formation of life-supporting structures would be suppressed in domains where the vacuum energy is much larger. Specifically, if the vacuum energy is negative and its absolute value is substantially larger than it appears to be in the observed universe (say, a factor of 10 larger), holding all other variables (e.g. matter density) constant, that would mean that the universe is closed; furthermore, its lifetime would be shorter than the age of our universe, possibly too short for intelligent life to form. On the other hand, a universe with a large positive cosmological constant would expand too fast, preventing galaxy formation. According to Weinberg, domains where the vacuum energy is compatible with life would be comparatively rare. Using this argument, Weinberg predicted that the cosmological constant would have a value of less than a hundred times the currently accepted value. In 1992, Weinberg refined this prediction of the cosmological constant to 5 to 10 times the matter density.
This argument depends on the vacuum energy density being constant throughout spacetime, as would be expected if dark energy were the cosmological constant. There is no evidence that the vacuum energy does vary, but it may be the case if, for example, the vacuum energy is (even in part) the potential of a scalar field such as the residual inflaton (also see Quintessence). Another theoretical approach that deals with the issue is that of multiverse theories, which predict a large number of "parallel" universes with different laws of physics and/or values of fundamental constants. Again, the anthropic principle states that we can only live in one of the universes that is compatible with some form of intelligent life. Critics claim that these theories, when used as an explanation for fine-tuning, commit the inverse gambler's fallacy.
In 1995, Weinberg's argument was refined by Alexander Vilenkin to predict a value for the cosmological constant that was only ten times the matter density, i.e. about three times the current value since determined.
Failure to detect dark energy
An attempt to directly observe and relate quanta or fields like the chameleon particle or the symmetron theory to dark energy, in a laboratory setting, failed to detect a new force. Inferring the presence of dark energy through its interaction with baryons in the cosmic microwave background has also led to a negative result, although the current analyses have been derived only at the linear perturbation regime. It is also possible that the difficulty in detecting dark energy is due to the fact that the cosmological constant describes an existing, known interaction (e.g. electromagnetic field).