Cosmological Constant

The cosmological constant went from being called Einstein’s biggest blunder to a fundamental field of research in contemporary physics. Nowadays, we think of it as a form of energy that opposes the pull of gravity and makes the expansion of the universe accelerate. However, the story of how it came to be like this is very interesting and also slightly mathematically unsettling. It is sure to be connected to the next great breakthrough in astrophysics as new observations are targeted at confirming its value and existence. It’s a rare comeback story in physics, which makes it even more exciting.

Measurements of supernova redshifts from 1997 that were used to show the accelerating expansion of the universe. Credit: ESO.

First, let’s go back to Einstein’s general relativity to explain how the constant originated. When Einstein first published the equations in 1915, they did not have the cosmological constant. These field equations are differential equations, so they are derived and solved using integrals. As with any indefinite integral, a constant C is added (or often forgotten by students), as the derivative of the constant is just 0. Einstein initially ignored a constant, as the predictions for the rate of change of the distribution of matter and energy in space remain valid no matter the constant. Nevertheless, in 1917 Einstein realized that his theory predicted an expanding and dynamic universe, where space changes with time. The scientific and philosophical consensus at the time was an eternal, unchanging universe, so Einstein introduced the cosmological constant Λ to make his equations static.

Einstein field equations with the cosmological constant Λ included.

However, very quickly the idea of a static universe collapsed, as I have talked about in this post on the expanding universe. In 1922, Alexander Friedmann showed that field equations were valid in a dynamic universe whatever Λ was. Furthermore, in the 1920s Hubble’s observations of redshift clearly showed that stars were moving away from us. In 1931, Einstein finally accepted his mistake and proposed a model of an expanding universe called Einstein-de Sitter space-time. Nobody expected that half a century later the cosmological constant was about to make a comeback.

In 1998, two independent teams lead by Saul Perimutter, Adam Riess and Brian Schmidt, showed that the expansion of the universe was accelerating. They measured the redshifts of supernova and found that their redshifts from one another was increasing at an accelerated rate. This implied a mysterious dark energy, which counteracts the pull of gravity, and a positive cosmological constant. Suddenly, we realised that Einstein was right after all, except in a strange reverse way. A cosmological constant exists, but it does not keep the universe static. It does just the opposite, pushing it apart at an increasing rate.

However, there is a big problem with the cosmological constant. We don’t know where it physically comes from. A theoretical idea is that a cosmological constant is equivalent to the vacuum energy of empty space, which arises from the creation of virtual particles. It is a constant quantum hum that persists in the emptiest regions of the universe. However, the force resulting from vacuum decay is 120 orders of magnitude greater than the measured cosmological constant! This is one of the most significant unsolved problems in physics and perhaps connecting the cosmological constant to quantum physics will require a unified theory. Without any theoretical explanation for the cosmological constant, it just appears like a neat mathematical trick for now. Or perhaps it is just an intristic property of the fabric of spacetime, but that would go against modern understanding of string and quantum loop theories.

The story doesn’t end here. The precise value of the cosmological constant is of tremendous importance for predicting the future and fate of the universe. When discussing dark energy, many scientist use the equation of state w. It is the ratio of pressure that dark energy puts on per unit volume. The equation is quite simple, with w being equal to the ideal gas constant R multiplied by temperature and divided by speed of light squared. If w = -1, the acceleration of the universe is constant and dark energy is the cosmological constant. If w < -1, which is called phantom dark energy, then the expansion is going to accelerate at such a rate that it will result in the big rip. For w > -1, the big crunch will occur as the dark energy will not be strong enough to counteract the pull of gravity.

The different scenarios for the end of the universe are based on the behavior of dark energy, If the strength of dark energy increases enough with time, we get a big rip caused by an ever-accelerating expansion. If the strength decreases, the gravitational attraction of objects within the universe eventually leads to a collapse. If dark energy is a cosmological constant, the universe continues expanding at a ‘mild ‘ constant accelerating rate and eventually ends in heat death. Credit: NASA.

The most precise measurement of the equation of state to date made by the Planck collaboration in 2018 put it at −1.028±0.032. This is close to the expected -1, however some physicists propose that dark energy is a scalar field so its value may differ in various regions of space. Ethan Siegel writes that the next decade, with advancements such the Vera C. Rubin observatory and the Euclid observatory we will be able to confirm if the cosmological constant is equivalent to dark energy, by increasing the accuracy of our measurement. Or perhaps we will find something totally surprising, like in 1998, which will shift our perspective on the cosmos.

As a sidenote, not all ends of the universe are directly connected with the value of w and dark energy. A very interesting apocalyptic scenario that I learned about in Katie Mack’s The End of Everything: (Astrophysically Speaking) is false vacuum decay. All physical systems tend to the most energetically stable equilibrium. The theory proposes that our universe is in a false equilibrium, like a valley in really tall mountains, but a more energetically stable value is still possible to reach, like a sea a few kilometers to the north. If by chance, in a process called bubble nucleation, a part of the universe would reach this more stable state, it would spread to the whole cosmos causing massive havok. And we wouldn’t see it coming because it would spread at the speed of light. It might even be happening somewhere far away right now…

The energy of the scalar field with a false and true vacuum. A transition to a true vacuum would be catastrophic for the universe, as the fundamental particles and constants might be changed. It would occur through quantum tunnelling. The probability of vacuum decay is a lot higher in the vicinity of black holes, yet this research is still recent and developing. Credit: Gary Scott Watson.


Astrokatie. (2018, September 14). The Universe, in Theory: Extra dimensions, black holes, and vacuum decay, oh my. The Universe, in Theory.

Cooper, K. (2018, June 19). The dark-energy deniers – Physics World. Physics World.

Dark Energy | COSMOS. (n.d.).

February 2021, A. M.-L. S. C. 16. (2021, February 16). What is the cosmological constant?

Siegel, E. (2020, December 25). Ask Ethan: Is Einstein’s Cosmological Constant The Same As Dark Energy? Forbes.

Wikipedia Contributors. (2019, May 9). Cosmological constant. Wikipedia; Wikimedia Foundation.

Published by Mateusz Ratman

High school student from Warsaw, Poland. JHU Class of 2026.

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