Dark energy, the mysterious force hovering over the Universe and the internet of clever scientific minds. I usually watch multiple videos mentioning dark energy covers 70% of our Universe, and we don’t know what it is. Some claim it’s the physical property of the space-time continuum. Some say it’s the unknown force that accelerates the expansion of the Universe. Let’s explore to the depth of our imaginary horizon, with some mathematics. Let’s see how Newton’s mathematics predicted the dynamics of the Universe and how Einstein’s did it. And how dark energy changed the dynamics of our thoughts.

First, I heard dark energy described as a mysterious force whose origin we didn’t know. At that time, I was like, I will solve it when I grow up. Now instead of solving it, I write stupid articles here at Formulon. I love doing this. Even when I started writing this, I didn’t know much about the basics and backstory of dark energy findings and mathematics. I just knew some taglines from articles, like it causes the acceleration in the expansion of the Universe. No more yapping. Let’s dive into the trench of exploration.

The Newtonian Universe

“Gravitatem in corpora univerfa fieri, eamaue proportionalem ejfe quantitati materu in Jingulis.”
“Planetas omnes in ie mutuo graves eiTe jam ante probavimus, ut & gravitatem in unumquemque feorfim fpectatum eiTe reciproce ut quadratum diftanti*e locorum a centro Planetae”[2]

English translation[1]

“That there is a power of gravity tending to all bodies, proportional to the several quantities of matter which they contain.”
“That all the planets mutually gravitate one towards another, we have proved before; as well as that the force of gravity towards every one of them, considered apart, is reciprocally as the square of the distance of places from the centre of the planet.”

These are the original wordings of Newton, which provide the basic structure of gravity and made the same rules for Heaven and Earth. We know the refined version of Newton’s gravitational law; some claim it as Newton’s fourth law. And the equation we learned in school is:

This Newtonian theory says the force is instantaneous. If I change the gravitational force of an object, the effect of it will be found at every point in the Universe through gravity. By this idea and the static Universe theory, the objects in the Universe will come closer to one point at the end. The idea is similar to the Big Bang theory: the Universe began with an explosion, the effect of the explosion stops, and the Universe comes back to one point or object. These are not discussing space; it’s the idea of mass and energy distributed in 3D Euclidean space. Again, after the energy burst, a new Universe forms — that’s my imagination of the Universe when I learn more than academics about it.

Einstein’s Universe

“Biggest Blunder”

Einstein said the term — the constant value which he included in his theory — the cosmological constant($\Lambda$), which he put in his field equation to balance the claim of a static Universe. This is what Einstein’s field equation represents.

In 1922, the Soviet mathematician Alexander Friedmann found the problem in Einstein’s field equation, which was published in 1917 as a part of the great General Relativity. He analyzed the equation and found two possibilities for the Universe using Einstein’s mathematics. They are: the Universe is expanding or contracting. First, Einstein refused Friedmann’s mathematics and said it was not correct, but later he agreed it was literally correct.

On the other side, Belgian astronomer Georges Lemaître showed Einstein’s equations predicted the expansion of the Universe. Later, American astronomer Edwin Hubble confirmed the expansion of the Universe using observations made by his associate, astronomer Milton Humason. They analyzed the redshift of galaxies and the Cepheid stars in those galaxies and found the speed of the galaxies. They found the galaxies are moving faster and faster when they are farther away from us. Even some galaxies are moving thousands of miles per second away from us. This observation is known as the Hubble–Lemaître Law, or Hubble’s law.

We also discussed symmetry and how the expansion of the Universe affects symmetry and energy conservation in another post. How Emmy Noether changed our understanding on Conservation laws?

The Dark Energy

Till the expansion, Einstein’s mathematics predicted well. But in 1998, two groups of astronomers found an anomaly in the Universe: the far-off supernova explosions were dimmer than the equations and previous calculations predicted. They specifically searched for Type Ia supernovae, which are known to have a certain level of luminosity. So they found something is making the supernovae dimmer. Using the brightness of the objects, they found the distance, and those objects are farther away than we expected. These data suggest the expansion of the Universe is getting accelerated. But we don’t know what it is. It’s not energy just because it contains the word “energy” in it. Actually, we humans have a lot of theories suggesting what it might be.

Theories around Dark Energy

Some scientists predict dark energy is some form of the energy of space-time itself, which is similar to the cosmological constant that accelerates the expansion of the Universe, which is a property of 4D space. And some people try to do mathematics this way. And some people suggest the quintessence theory, which says dark energy is a fluid or field filling the entire Universe, behaving opposite to normal matter. For information, quintessence is the theoretical fifth element of matter discussed by the Greeks. Some suggest it as wrinkles in space-time.

And the main and revolutionary theory is that dark energy is a great flaw in General Relativity. From one aspect, it’s good to think about multiple new theories. I support it. We need to think about new mathematics that draws a bigger circle than Einstein’s General Relativity. We see new discoveries are not explained by the standard model of our Universe. We also wrote an article about it. Just read it, if you have time. How James Webb and New Data Are Changing Our Ideas About the Universe?

From the Perspective of String Theory[5]

Recently, I found the Quanta Magazine article, “String Theory Can Now Describe a Universe with Dark Energy.” We have string theory, which was developed by mathematicians and physicists. When I asked one of my professors about it, she said it works more mathematically than physically. In some sense, we don’t have enough data and proofs that support string theory. And also, string theory struggled to give an explanation of dark energy. Older string theory models suggested zero or negative vacuum energy, but we observe a positive kind of vacuum energy.

Now two theorists, Bruno Bento and Miguel Montero, built a new string theory model which yields positive dark energy — technically a de Sitter space — which is extremely hard to achieve. They used quantum effects in extra dimensions to balance the force and result in positive dark energy. But there is a problem: this model predicts a Universe with 5 dimensions, not the 4 dimensions like ours. And the positive dark energy will get weaker over time and become unstable. The article says it doesn’t match reality, but it opens a new theoretical path for exploration.

Reference

• English version of Principia Mathematica
• Latin version of Principia Mathematica
• What is dark Energy? -NASA
• Relativity: special and General Theory, by A. Einstein
• String Theory Can Now Describe a Universe That Has Dark Energy