DARK ENERGY AND THE END OF THE UNIVERSE

One of the most amazing things in this world is that human beings (a species of large-brained hominids) are capable of studying phenomena so infinitely removed from their everyday reality, such as the creation of the Universe or its possible final destiny. Achieving such a feat is possible thanks to the combination of modern technology, modern physics, and, above all, the immense power of Mathematics. In this article, we will see this power in all its splendor: with just three simple mathematical expressions, which we will explain step by step, we will be able to predict nothing less than the future of our Universe! As if it were a performance of a Shakespearean dramatic opera, the future of everything that exists around us will revolve around three possible scenarios: one of long and slow agony; another of sudden destruction; and yet another of regression and involution. In theatrical terms, the first scenario can be considered a "slow and boring play," the second involves a truly astonishing act, and the last contains a final act that is almost impossible to believe.

Welcome to the magnum opus of the end of our Universe!

Our current era: dark energy and accelerated expansion

In 1998, scientists discovered something surprising: the expansion of the Universe was not only not slowing down, as expected due to the effect of gravity, but was actually accelerating. Moreover, this acceleration had only recently begun! There are basically three fundamental ways to explain this surprising discovery:

1) The equations of general relativity must include a new constant of nature, called the "cosmological constant." This constant would act as a repulsive force that would explain the accelerated expansion.

2) The current vacuum state of the Universe has a positive energy density. This small energy density produces a gravitational repulsion that would explain the observed acceleration.

3) The vacuum energy density is not constant but depends on a new scalar field similar to the inflaton field that drove cosmic inflation. The value of this scalar field controls the total vacuum energy density and therefore the value of cosmic acceleration.

These three possible solutions represent the three scenarios we talked about in the introduction: the first leads to the slow and agonizing heat death of the Universe, the second can lead to a "sudden" destruction of the Universe, and the third leads to an incredible possibility: the Universe would slow down its expansion, stop, and slowly begin to contract!

Before studying these three scenarios, we must briefly analyze the fundamental characteristics of the quantum fields that constitute everything that exists in our Universe.

The vacuum stability of quantum fields

A quantum field is a complex field represented by two fundamental values: the field value (similar to the kinetic energy of classical fields) and the potential value (similar to classical potential energy). These values can be represented on a graph in the complex plane:

The total energy of the quantum field is the sum of both components. The "x" and "y" axes represent the imaginary and real parts of the kinetic energy, and the "z" axis the value of the potential. If we look at the image above, we see that at the top of the hill the field has kinetic energy 0 and potential energy V. Fields always tend to "roll" toward the point of minimum potential, so the field at that position is unstable: at a given moment, due to a quantum fluctuation or the tunneling effect, the field will "jump" and roll down to the lower part of the minimum potential.

We will now briefly analyze scenarios 1 and 2. Scenario 3 deserves special attention as its consequences are absolutely extraordinary.

Scenario 1: Thermodynamic death

This scenario is the simplest of all: the cosmological constant will cause the Universe to expand with a constant acceleration, causing galaxies to increasingly separate and the Universe to increasingly cool. At some point, stars and all matter will disintegrate and "dissolve" into the vast Cosmos, producing a Universe practically devoid of matter and energy. The Universe will be headed toward "heat death."

Scenario 2: Sudden destruction

The vacuum state of our Universe is determined by the vacuum of the Standard Model of particle physics. When the energy of our Universe dropped below 246 GeV, the so-called Higgs field fixed the vacuum state of our Universe (see this article ). There are indications that the vacuum of the Higgs field is not stable: if the potential of this field has a minimum below the current vacuum, there is a possibility that, due to quantum fluctuations or tunneling, the current vacuum state could "jump" instantly to the new vacuum state. This phenomenon would have disastrous effects: the new vacuum state would begin to expand in the form of a bubble traveling almost at the speed of light, destroying everything in its path.

Point B represents the current unstable vacuum of our Universe. At a certain point, the vacuum "jumps" through quantum tunneling to a lower vacuum state.

The final act of this scenario is apocalyptic! Furthermore, this "tragic opera" will have no witnesses: there is no way to detect the relativistic bubble of the new vacuum, and nothing will survive its passage. In any case, there is no need to panic: the probability of a tunnel-like "jump" is very low, and the probability that the bubble will reach our position in the near future is also very low. Both probabilities involve times hundreds or even thousands of times the current age of the Universe.

Scenario 3: Variable dark energy or quintessence

This scenario occurs if dark energy is not constant but is controlled by a scalar field similar to the inflaton field that drove cosmic inflation. In this case, the fate of the Universe will be determined by the characteristics of this scalar field called "quintessence." If the potential of this field is continuously decreasing and has an appropriate slope, then our Universe will go through a series of truly extraordinary phases. It is now that we discover the most incredible and surprising phenomena thanks to the power of Physics and Mathematics.

We'll call the scalar field that controls the expansion "psi" and the potential of said scalar field "V." The total energy of this field is given by:

Where p tot is the pressure and Rho tot is the total energy density.

The total pressure and energy density of the quintessence field are given by:

Below we'll present three simple equations that will allow us to determine the fate of our Universe. These equations are derived from the so-called Friedmann equations based on General Relativity:

1st) Friedmann equation of the scale factor (a) as a function of the total energy-pressure:

2nd) Friedman equation for the Hubble constant (H):

3rd) Equation of motion of a scalar field V:

Where G is Newton's gravitational constant. The upper dots indicate the first and second derivatives with respect to time.

The current experimental value of the dark energy density is tiny, almost zero. Therefore, the current value of the quintessence field must be almost zero. Furthermore, as we have explained, the potential of this field decreases continuously.

Considering the expression for the total energy and the values of the pressure and energy density of the scalar field, we clearly see that as V decreases and approaches zero, p Q increases and rho Q decreases, therefore, the value of the total energy increases (the quotient p/rho becomes larger). At a given time The total energy density reaches 1 and continues to increase. If we look at the first expression, we see that at that point ä (the acceleration of the Universe's scale factor) vanishes and begins to become negative. This implies something incredible:

the acceleration of the Universe begins to slow down.

Next, we find that as V continues to move toward more negative values, the total energy density decreases. When V reaches a certain negative value, the total energy density reaches zero. If we look at the second expression, we see that at that point, the Hubble constant H is also zero, and therefore the expansion of the Universe stops.

Finally, the above expressions together with equation 3 indicate that as V continues to decrease, H becomes negative, which implies that the Universe begins to contract.

The following graph summarizes the three previous stages:

The current moment is the t point, which has a value very close to zero. The psi point b marks the beginning of the phase in which dark energy begins to dominate, that is, the instant when accelerated expansion begins. Later, at the psi point dec, the expansion begins to slow down. Finally, at the psi point c, the Universe begins to contract.

This is where we come across the strangest phenomenon of all. So far, the analysis has been purely classical, since quantum phenomena are negligible on cosmological scales. However, one of the most important equations of quantum gravity, called the Wheeler-DeWitt equation, clearly indicates a close link between the expansion of the Universe and the flow of time. In fact, it indicates an asymmetry of time evolving from low-entropy states to high-entropy states. This equation suggests something truly surprising: if the expansion of the Universe stops , time stops! And if the Universe contracts , time changes sign ! While it is not clear how to correctly interpret the Wheeler-DeWitt equation, it is evident that the mere image of a Universe with time flowing backward is almost impossible to imagine. On the other hand, certain considerations based on the gravitational path integral seem to indicate that at the very instant in which the expansion stops, the wave function of the Universe vanishes, which would cause classical space-time and all the matter it contains to disappear.

Finally, we will attempt to estimate when this surprising phenomenon of braking and contraction can occur. To estimate the time in which each of these phases would occur, we will use a generic potential of the form:

In the accelerated expansion phase, the first (positive) term dominates, while the second is negligible. The second (negative) term begins to dominate the first at some point in the future.

The shortest possible time in the previous equation will occur when we have the highest possible value of psi or for the lowest possible value of my M. Since m is negligible in the past and in the present epoch we cannot know its value experimentally, we will select a reasonable and "natural" value between 0.01 and 1, for example m = 0.1mpl. Vo and V1 are chosen so that they agree with the experimental cosmological values. To perform specific calculations complying with the experimental limits, we will take the following values:

Where the last two values are the energy densities of matter and dark energy (relative to the critical energy density), respectively. With these values, we obtain the following graph:

The orange line represents the time remaining until the end of the expansion (tdec-to) and the blue line the time remaining until the beginning of the contraction of the Universe (tcon-to). For mpl/m = 10 the first time is 0.1 Ho and the second is 0.27 Ho, that is approximately 1.4 billion years for the first. For mpl/m = 50 the first time is only a few million years! And for larger values of mpl/m we get even smaller times. Although these times are large on a human scale, they are actually very small on a cosmological scale.

What is the most likely scenario for our Universe?

A priori, scenario 1 is considered the most likely because the cosmological constant seems to be the most "natural" and simple solution. However, our theories of quantum gravity, especially string theory, are seriously questioning this solution: expanding spacetime is unstable and as such must decay to a more stable state. Furthermore, the so-called "Swampland Conjectures" are severely restricting the types of possible inflation, to the point that the standard inflation of our consensus cosmological model is almost ruled out if these conjectures are correct. String theory and our general knowledge of quantum gravity indicate that dark energy must be driven by a scalar field (quintessence models). Thus, the values of this field and its potential would control the expansion of the Universe. If the "Swampland" conjectures are correct the accelerating expanding Universe should decay in a time on the order of the Hubble Ho time (the current age of the Universe) so the recent detection of this acceleration could indicate that the process has already begun!

In the coming years, the Vera Rubin and Euclid satellites, along with the SKA antenna array, will be able to find experimental evidence that will demonstrate whether the acceleration of the Universe is caused by a cosmological constant or a quintessence field. The results of these experiments will determine the ultimate future of everything in our Universe, and we poor mortals will be able to predict this future. Not bad for a species of "clever monkeys," don't you think?

Sources:

Rapidly Descending Dark Energy and the End of Cosmic Expansion