The cosmological constant puzzle

The cosmological constant puzzle

A bout a century ago, Albert Einstein introduced the famous lambda term, otherwise known as the cosmological constant, into his equations of the general theory of relativity. The introduction of such a term holds significance considering that dark energy is currently a hot topic of discussion in cosmology, and the recent observations being consistent with the fact that this cosmological constant may well be this energy driving the cosmos.

On February 4, 1917, Einstein wrote to his friend and fellow physicist Paul Ehrenfest, “I have…again perpetrated something about gravitational theory which somewhat exposes me to the danger of being confined to a madhouse.” He revealed these maddening thoughts four days later to the Royal Prussian Academy of Science, and it was published a week later by the academy. Thus, began the era of modern relativistic cosmology.

Why and what exactly did Einstein
introduce? Spurred by the success of his general theory of relativity, Einstein wanted to apply his equations to the whole universe and find a cosmological model for the same. For this, he assumed (erroneously as it turned out later) that the universe should be static. As a whole, it should not evolve in time. He was probably guided by the philosophical prejudice that the universe has no beginning or end. However, he found that his equations do not admit to a static solution.

Now, gravity as described by general relativity gives a large scale attraction, with all bodies are pulling each other and everything else together rather than remaining static. To get a static solution, he had to add a new term to his field equations which he denoted with the Greek letter, lambda. As it was a constant term having the dimension of curvature, it was also called as the cosmological constant. What the lambda term essentially introduces is a large scale repulsive force to balance all the attractive gravitational forces to give an ‘equilibrium’ size or the radius of a static universe and this is roughly the inverse square root of lambda.

Of course, we know that we have to balance forces in a physical system to get a static equilibrium configuration. This model became known as Einstein’s static universe. He estimated its radius as a few billion light years. Again, this repulsive force introduced by lambda was universal like gravity but increased linearly with distance like an elastic force on a spring.

Differing views
However a few years later, Edwin Hubble (using the then world’s largest telescope, the Mount Wilson telescope) observed that all distant galaxies show a red shift, implying that they are all receding or moving away from each other at large velocities and this recession was proportional to the distance of the galaxy. These observations definitely suggested a dynamic, expanding (with time) universe rather than a static one assumed by Einstein. With the weight of evidence increasing for expansion, Einstein realised that he could have predicted a dynamic model for the universe if he had not introduced the cosmological constant into his equations.

He reportedly remarked that it was his biggest blunder. But it turns out that his introduction of the lambda term was not his real blunder. The biggest blunder was in not realising that a static universe results only when there is no matter at all and moreover, he did not bother to check whether his static equilibrium configuration is stable or unstable. It is a common practice in physics to check whether the equilibrium is stable or unstable by making small perturbations about the equilibrium coordinate. An elementary calculation (first pointed out by Arthur Eddington, an astronomer, physicist and mathematician) shows that the Einstein’s universe is unstable to perturbations. It would either blow apart or collapse to zero radius. So, Einstein’s static solution was, in any case, untenable.

It is surprising that Einstein did not realise this. What is perhaps worse, he got the wrong Newtonian limit to his extended field equations which implied that the cosmic repulsive force falls off exponentially with distance, rather than increase with distance as it should. This was another real blunder where generations of physicists, even relativity specialists like Stephan Hawking and Wilhelm Eduard Weber, repeatedly parroted without realising that it was wrong. For instance in a 1983 Royal Society discussion, Hawking begins by saying that Einstein’s lambda term leads to an exponentially decreasing force.

The correct picture is that of a force increasing with distance as shown by Eddington. Instead of getting a shielded gravity field, we have now on large cosmic scales (distances) almost naked repulsion quite different from Einstein’s expected bargain. This large-scale cosmic repulsion resulting from the lambda term of Einstein was the first examples of dark energy, making the universe expand faster.

Interestingly, many of the recent cosmological observations are consistent with just a lambda term, i.e, Einstein’s cosmological constant dominating the universe to the extent of 70%. His ‘biggest blunder’ dominates the universe now. How do we know that dark energy dominates in the cosmos? Over the past two decades, astronomers have deduced that the expansion of the universe is faster now than it was before, say eight billion years ago. In other words, the universe is undergoing an accelerated expansion. Observations have been made of distant stellar explosions, of a particular type, i.e. Type Ia supernova.

Standard candles
This type of supernova explosion results when a white dwarf is pushed over the Chandrasekhar limit by accreting matter from a companion star. This limiting mass is about 1.5 solar masses. Above this, the white dwarf collapses under its gravity and the resulting billion degree temperatures incinerate the Carbon and Oxygen to iron in rapid nuclear fusion reactions. This is literally a thermonuclear celestial bomb releasing a very definite amount of energy (equal to what the sun would emit in its entire ten billion years life time).

So, these types of supernovae serve as ‘standard candles’. These distant explosions occurred nine billion years ago, long before Earth (or our solar system) was in existence.

They appear fainter substantially than they would if the universe had slowed down its expansion. This fainter appearance suggests that their distance from us is increasing at a faster rate, i.e. the expansion of the universe now is faster than it was then. The lambda term introduces a negative pressure and this gives rise to large scale repulsion making the universe accelerate faster and faster.

Einstein’s lambda term may also explain why the universe started expanding in the first place. He neither liked the cosmological constant nor the then formulation of quantum theory. Ironically, it is now believed that quantum vacuum energy should generate a lambda term and perhaps even Einstein gravity. If, in future, dark energy is indeed confirmed to be just the cosmological constant, Einstein should get a posthumous Nobel prize.

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