Decoding Einstein's Cosmological Constant

Continuation of the previous post - "Expansion of the Universe: Fifth Force as Dark Energy"

How did Einstein's Famous "Blunder" Unlock the True Nature of the Universe?

No seriously, Einstein was so smart that even his so-called 'blunder' turned out to be the key element that explained the accelerated expansion of the universe. The Cosmological Constant, since its introduction by Einstein in 1917, has to be the most chaotic yet elegant irony in the field of cosmology, and science itself! 

But currently, this constant is being speculated for its connection with the accelerated expansion of the universe, and now it is popularly known as the "Dark Energy's Number." That being said, Einstein's Cosmological Constant is doing much more than just explaining the unprecedented accelerated expansion of the universe. In this post, we'll be looking into the history of the Cosmological Constant, its ties with dark energy, and some unsolved problems relating to it that still remain in cosmology. 

History of the Cosmological Constant 

The Cosmological Constant, denoted by the Greek letter 'lambda' (Λ), was first introduced by Albert Einstein through his General Relativity equations. Einstein's General Relativity was tremendously successful in illustrating the complete picture of Newtonian gravity and in giving explanations to certain phenomena that were unexplainable by Newton's theory of gravity (like the peculiar orbit of the planet Mercury). 

But the Cosmological Constant first appeared in GR equations in 1917, when Einstein tried to explain the entire universe using his field equations (which are a set of ten partial differential equations). However, when he solved these equations, he found that the final solutions to the equations show that the universe is not static rather dynamic. It simply means that instead of an unchanging universe, he obtained a universe that either continually expands or continually contracts over time. 

Still, why did this result about the universe seem to be a huge flaw with his theory? It was not until 1929, the American astronomer Edwin Hubble showed that the universe is expanding with observational evidence drawn from his telescope. Now, Einstein's General Theory of Relativity was published in 1915, and its application to the universe as a whole was not studied until 1917, which means that Einstein, just like every other scientist believed that the universe was static and unchanging. 

For this very reason, Einstein introduced a new constant, the Cosmological Constant (Λ), to his field equations to get an unchanging universe. But he did so without violating any fundamental principles on which his theory was formulated, and the constant did not essentially demean its ability in explaining gravity or Mercury's orbit. However, in 1929, after Hubble showed that the universe was indeed expanding, Einstein realized that his equations did predict the dynamic nature of the universe way before Hubble's observation and that he wouldn't have to add the constant to make the universe static! Einstein called this addition of the Cosmological Constant into the field equations as his biggest blunder, and since 1929, he abandoned the constant from his field equations. 

Yet, that's not the end of the story. In the late 1990s, cosmologists discovered something very unexpected: the universe is expanding exponentially at an accelerated rate! And just like that, to explain this accelerated expansion of the universe, cosmologists revived the Cosmological Constant in Einstein's GR equations. Even now, cosmologists speculate that dark energy, the mysterious energy that drives the universal expansion, is represented by this constant. 

But how did the Cosmological Constant go from stabilizing Einstein's equation for a static universe to being a crucial factor that causes the accelerated expansion of the universe? 

For that, we'll need to delve into Einstein's GR equation and Friedmann's equations that were applied to the universe as a whole.

Friedmann Equations 

The Friedmann equations are actually derived from Einstein's field equations applied to the universe. These equations were developed and studied in 1922 upon by a Russian physicist, Alexander Friedmann. There are two main equations in the set of Friedmann equations that we'll need to understand the role of the Cosmological Constant - 

The velocity equation (first Friedmann equation)

The acceleration equation (second Friedmann equation)

In the Friedmann equations, a is the scale factor, which represents the evolution of space between two points in the universe. It can also be thought of as a function of time, a(t), which tells us about the rate of change of space between the points. If we differentiate it with respect to time t, we end up with the velocity, which is denoted by ȧ. Similarly, if we differentiate ȧ with respect to t, it tells us about the acceleration, denoted by:

࣫The term p is the pressure exerted by matter in the universe and ρ (rho) is the energy density of matter. Also, in the first Friedmann equation, H is called the Hubble parameter, and it is equal to ȧ/a. 

In the acceleration equation, if we analyze the equation without including the term with the Cosmological Constant Λ, the acceleration will come out to be negative. Since p and ρ are the only variables in the equation, p can be neglected (pressure is the same in the universe in large scales, so p=0) and the term ρ (rho), which represents the energy density of the universe, will always be positive. This is because as long as there is any matter in the universe, the energy density caused by the matter will always be positive. So technically, rho is the only contributing factor in the acceleration equation and always ρ>0, which means that the acceleration will always be negative, or the contraction of the universe is inevitable in any scenario. 

This is where Λ comes into action. Including the constant into the equation, with a condition that Λ>0, will counteract the inward pull of the universe caused by gravity, by providing an antigravity force that will expand the universe! In other words, Λ is the mathematical term responsible for the expansion of the universe. 

We saw that the second Friedmann equation shows how the Λ causes the expansion of the universe by counteracting its negative acceleration. Yet, that's not the only contribution of the Cosmological Constant in GR equations. It is also the main reason why our universe is flat. 

A flat universe essentially means that the net energy of the universe is zero, unlike a positively curved universe or a negatively curved universe. 

In the first Friedmann equation, the term k represents the curvature of space, and for a positively curved universe would have k > 0, and for a negatively curved universe, k < 0. For a flat universe, k = 0. But without the Λ term, the first equation gives a negative value for k, which shows a negatively curved (or hyperbolic) universe. However, by incorporating the Λ term into the equation, we achieve zero on both sides of the equation. This means that Λ adds matter (or density, energy) to the seemingly hyperbolic universe so that we now get a flat universe. 

So, not only Λ contribute to the expansion of the universe but it also gives an explanation for the missing energy that is required for a flat, balanced universe. This energy, termed the dark energy, makes up about 70% of the universe, with the rest being regular matter.

The Cosmological Constant is also being referred to as vacuum energy, which is thought to be an intrinsic property of space itself. But the close connection between the Cosmological Constant and vacuum energy invokes a problem that still remains one of the biggest unanswered questions in cosmology and theoretical physics - the Cosmological Constant Problem. For now, let's just save it for another post ;)

 

Click here to check the previous post on Universe Expansion and Dark Energy.