Dark energy, accelerated expansion and conservation of energy: how does it all fit together?

I have long had the opinion that those who manifest themselves in various forms of the forces of matter have one source. Or, in other words, they are so directly connected and mutually dependent on each other that they can turn into each other, and the energy of their actions is equivalent.
- Michael Faraday

For hundreds of years there was an idea according to which it is impossible to get something out of nothing, and it appeared long before we learned about the most important law of conservation in the Universe: the law of conservation of energy.



For example, a ball rolling down a hill, gaining speed and kinetic energy, does not get them out of nowhere! It turns into kinetic another form of energy - gravitational potential. And at any moment of the journey, the total amount of energy - its kinetic and potential - remains unchanged.

When considering a closed system, one that cannot receive energy from the outside or lose it, the law of conservation of energy works perfectly.



Earth, for example, is not a closed system. It receives light and energy from everywhere in the Universe, mainly from the Sun, and also radiates energy into space. The sun is also not a closed system, because it constantly radiates energy into space, but this energy is also not taken from nowhere. It is created by converting the energy of the rest mass into light and heat (and neutrinos) through nuclear fusion and E = mc ² . If you include all sources of outgoing and incoming energy for any system in the Universe, the conservation law works.



And although there are certain strange quantum phenomena that at first glance violate this law, on a scale from subatomic to cosmic, but if you calculate all the incoming and outgoing energy, the law will still work.

How then do we answer the question of one of the astrophysicists:

What is the source of energy for an accelerating universe?

Stunning in depth question. Let's start with what it means to live in an accelerating universe.



One can imagine the early Universe - at least its observable part - as a hot, dense and rapidly expanding sphere of matter and energy. But it is also known that if your universe is filled with matter and energy, then the force of gravity will do everything possible to pull this expanding Universe back. Therefore, simply put, we get the following: an initially rapidly expanding Universe, which gravity is trying to slow down, reverse and again lead to collapse.

In the picture above, the left diagram represents the expanding Universe in which gravity won. The initial expansion is not enough for the available matter and energy, and the Universe as a result reverses the expansion, and collapses in Great compression. The second image shows the Universe, in which only one subatomic particle was missing for re-compression. In this case, the expansion rate will decrease, asymptotically tending to zero, but not reaching it. The third case, the low-density universe, continues to expand forever, since the gravitational attraction of all matter and radiation is unable to withstand the initial expansion.



There remains the fourth, actual case of an accelerating Universe. This Universe is filled with dark energy, or the finite positive energy of a vacuum inherent in space itself. And in this case, the expansion of the Universe is not just “not slowing down enough” for a collapse to occur - it is so large that the distant galaxies are moving away from us faster and faster! This means that if we take the final sphere of matter and energy and let it expand, then its radius will increase as follows:



Our Universe is different from the case of repeated collapse (orange). It is neither a critical (green) case, nor a case of low density (blue). Our Universe is growing at an exponential rate, as the red curve shows. According to the best measurements, the energy density of the cosmos (the density of dark energy) does not increase or decrease, but remains constant.

But it can lead to confusion. If the energy density is constant and space expands, does this violate the law of conservation of energy? In other words, are we not producing more and more energy over time, which should actually be conserved?



Technically, you can get out of this complexity, noting that my picture is not entirely true. The large-scale Universe is governed by the General Theory of Relativity, which is the theory of gravity. Strictly speaking, energy is not defined in GR, so why worry about conserving it? But there is a tricky way to consider "energy", allowing us to prove its conservation even in such a paradoxical situation.

Remember that in addition to chemical, electrical, thermal, kinetic and potential energy, there is also work.


Vasily Ivanovich Alekseev - Soviet weightlifter, two-time Olympic champion and eight-time world champion

In physics, work is the application of force to an object directed to the same place where the object is moving. This means that if you apply upward force to a load moving upward, you are doing a positive job (adding energy to the system). But if you apply a force directed upward to a load moving downward, you do a negative job (remove energy from the system).

It is easier to imagine if you put a book on your hand and raise or lower it.



When you push in the direction of the book, you are doing a positive job and adding energy to the system. When you push in the opposite direction, you are doing negative work.

Now back to the universe. You have an expanding spherical object, and the sphere is filled with energy of constant density (dark energy). You are positive energy. Using gravity, pull this sphere inward. But what does the sphere do?



It is expanding. In other words, you pull in the opposite direction of expansion and do the negative work. Then why is expansion accelerating? Of course, the positive density of dark energy means that it is attracted to all matter in the Universe, but the rate of expansion depends on the magnitude of its negative pressure. And, answering the question, where does the energy for "dark energy" come from, we can say that it comes from the negative work performed on the expanding Universe!

In more technical terms, like Carroll, Press, and Turner in 1992:

This site does negative work on its surroundings because its pressure is negative. Assuming that it expands adiabatically, we can equate this negative work to an increase in the mass / energy of the site. Thus, you can get the correct equation of state of dark energy:

$$

So with mathematics, everything is in order. That's how energy can be stored even in a universe filled with dark energy!