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This article comes from the official account of Wechat: back to Park (ID:fanpu2019), author: Dong Weiyuan

Friedman equation, proposed by Soviet physicist Friedman in 1922, is still the core equation for understanding the movement and evolution of the universe, describing a uniform and isotropic expanding universe. Now, physicists have finally found the analytical solution of the equation, and according to the new theory, our universe is as flat as it has been observed, not special.

When we find a strange object in the solar system, as long as we record a short orbit, we can tell whether the object is an asteroid within the system or a passer-by outside the system. How do astronomers know the whole leopard at a glance? In fact, it depends on the eccentricity e that we learned in high school.

We know that under the action of gravity, the orbit of a celestial body is a conic curve. If e > 1, the orbit is hyperbolic; if eigen1, the orbit is parabola. In both cases, the orbit is open, and the celestial body is an out-of-system passer-by and will not come back after a stroll around the sun. If e < 1, it means that the orbit is a closed ellipse and the celestial bodies revolve around the sun periodically.

Similarly, we need some indicative parameters when describing the evolution of the universe as a whole. The most basic and important one is the scaling factor a, which represents the proportional relationship between the coordinate distance and the actual physical distance in the coordinate system with the synchronous expansion / contraction of the universe. The so-called expansion of the universe means that an is getting bigger.

The current rate of expansion of the universe is about 7% more than the physical distance between two points every 1 billion years. If we define today's axiom 100 light-years, that is, the unit coordinate distance 1 in the co-motion coordinate system, corresponding to the actual physical distance of 100 light-years, then as the universe expands, the co-motion coordinate system will be stretched 1 billion years later, and the unit coordinate distance 1 corresponds to the actual physical distance of 107 light-years, that is, await 107 light-years.

Out of disdain for egocentrism and astronomical observations, we believe that all directions of the universe are expanding uniformly and synchronously, but at different rates at different stages of time. So a has nothing to do with the position and direction of space, but only a function of time a (t). If any reader has heard of the old man who is "uniform and spherically symmetrical everywhere", the scaling factor in cosmology is definitely one of the typical examples.

Just as displacement, velocity and acceleration are used to describe the motion of objects in high school physics textbooks, cosmologists rely on a,

And

The motion of the universe can be described, which represent the first derivative and the second derivative of a to time, that is, the speed and acceleration of the expansion of the universe. However, the dynamic equation of the universe is slightly more complicated than Newton's second law, because there is more than one factor that affects the first and second derivatives of time.

Equations of motion in cosmology if you open any textbook on cosmology, two equations will be mentioned at the beginning.

This is Friedman's first equation and second equation. In 1922, Soviet physicist Friedman (Alexander Friedmann,1888-1925) derived these equations from the general relativity equation. Today, 100 years have passed, it is still the theoretical core of the study of the evolution of the universe.

In order to look more refreshing, we can use the Planck system of units and agree on cantilever 8 π Group1, so the system of equations becomes

ρ represents the energy density, including the energy density ρ m in the form of mass and the energy density ρ r in the form of radiation. As the universe expands, the mass density will be diluted. After the physical length expands by a times, the volume expands by a 3 times, and the mass remains the same, so

In addition to the wave number being diluted, the wavelength of the radiation energy is also lengthened, and the frequency is correspondingly reduced, so

Therefore, the energy density can be divided into two terms ρ = μ a-3+ra-4. If the energy conversion between the two forms is ignored, μ and r are fixed constants.

Represents the curvature of the entire three-dimensional space of the universe at some point. Kull1 means that space as a whole is a "three-dimensional hyperspherical surface", and if you go far enough in any direction, you will return to the starting point. Khammer1 means that space has negative curvature like the saddle surface, except that the saddle surface is two-dimensional and the universe is three-dimensional. Krono corresponds to the "straightness" of space, and the current astronomical data show that the universe we live in should be very close to the "straight" situation. When we only discuss our universe, we often take kryp0 directly and then deduce it.

An is originally a "cosmological constant" in the equation of general relativity, which can be any constant in mathematical sense and represents the energy of vacuum itself in physical sense, and its density will not attenuate with expansion. This was not included in the original form of the Friedman equation because physicists would not have accepted that there was energy in a vacuum at all.

However, when the accelerated expansion of the universe is confirmed, the modern cosmological model has to pick up the A term and put it on the patch to represent the vacuum energy that accelerates the expansion of the universe. It is clear from Friedman's second equation that if A ≦ 0, the equation cannot be described.

The situation. However, what this vacuum energy is, it bothers cosmologists.

In fact, quantum field theory has put forward several theories about the source of vacuum energy, some of which have even been verified by experiments. Unfortunately, the vacuum energy given by either theory is much larger than the actual astronomical observation data, the most disparity is 120 orders of magnitude, and the smallest gap is more than 40 orders of magnitude. In other words, there must be some kind of suppression mechanism in the universe that can exactly counteract the vacuum energy that has been discovered in those experiments.

So far, there is no self-consistent theory that can explain the current situation of vacuum energy. In this situation, cosmologists have to call it "dark energy" and express it slightly rudely as a constant A. According to the observed data, the density of radiant energy in the universe is now negligible, and the ratio of all mass density ρ m, including dark matter, to vacuum energy density ρ An is about ρ m: ρ A ≈ 3:7. In other words, known matter and dark matter account for about 30% of the total, and the remaining 70% of energy is "dark energy" hidden in a vacuum.

As a result, there is a widespread saying in general popular science articles that "we know nothing about 70% of the energy in the universe", which is obviously not rigorous. Because this p An is the difference between the known huge vacuum energy and the unknown "suppression mechanism", the order of magnitude of the two is exactly the same, and both are much larger than the p An itself.

In the picture, orange represents the cognitive blank, and blue represents the theoretical explanation and empirical part, both of which are almost equal, so the unknown part is actually only 50%.

After really clarifying these relationships, we will feel that the confusion behind this "50%" far exceeds the original interpretation of the "70%." Everything we can see, whether twinkling stars or dim dust, even dark matter, which can only show us its existence through gravitational effects, all this accounts for close to zero in the form of energy we know.

If the proportion is written as a percentage, it will be a small number of tens or even hundreds of zeros after the decimal point. Fortunately, there are cosmological and astronomical observation data to save our confusion, otherwise we will not hesitate to believe that the real control of the universe is the vacuum itself!

There is another confusion that is also very puzzling. According to Friedman equation, the evolution law of mass density with time is ρ m~t-2, while the vacuum energy density does not change with time, p A constant. At this moment, 13.8 billion years after the birth of the universe, it is too coincidental for us to look up and see that ρ m and p An are of the same order of magnitude.

These signs make some researchers begin to doubt the validity of general relativity on the cosmic scale, perhaps the Friedman equation can not be applied to the cosmic scale at all. However, this suspicion is not as reasonable as it seems. You know, the challenges we face are of a huge order of magnitude, and even if we are as far away as general relativity and Newtonian mechanics, it is almost impossible to find a theoretical prediction that can be a hundredfold different.

We certainly believe that there must be a better theory to replace general relativity in the future, but if that theory can give the same result as general relativity on a light-year scale, it is more than 100 million times different from general relativity on the scale of ten billion light-years, it is really unimaginable. So now hastily abandon the Friedman equation to start a new stove, may not find a better way out.

It is strange to solve the equation. In the past 100 years since the birth of the Friedman equation, cosmologists have repeatedly studied its special solutions, but the most general analytical solution has not been solved until October 2022. The authors of this paper [1] are Latham Boyle of the Institute of Circular theoretical Physics in Canada and Neil Turok of the Higgs Center for theoretical Physics at the University of Edinburgh in Scotland.

Their idea of solving the equation is very easy to understand. The first step is to write the equation in the most general form:

The four terms on the right of the equation correspond to radiation energy density, mass density, curvature contribution and vacuum energy density, respectively. Then the time scale is transformed from coordinate time t to conformal time τ. The relationship between them is as follows:

So the Friedman equation becomes

By factoring the polynomial on the right side of the equation, the equation can be further written as

Because the maximum is only 4 times, A1, a2, A3, a4 can be expressed analytically by λ, κ, μ, r, of course, it may be plural.

After a dazzling transformation, a (τ) is solved by means of the elliptic integral tool. The approximate appearance of the analytical solution is

Where A1, A2, A3, A4, ζ, m are all analytical expressions containing A1, a2, A3, A4, λ, and C is the integral constant.

The binary function sn (z, m) is the soul of the analytical solution. It is one of the 12 Jacobian elliptic functions. Its properties are similar to sinusoidal functions. It is a double periodic function with periodicity on both the real axis and the imaginary axis. So a (τ) has a double periodicity, like a function defined on the surface of the tire. Of course, the surface of the tire is a double plane.

This mathematical structure not only looks very interesting, but also provides very novel ideas and tools for studying the evolution of the universe. Plural time is rich in content. Through an operation called "Wick rotation", the virtual time of the cycle becomes temperature, entropy becomes action, and then the phase of quantum state evolution. Many problems in quantum field theory can be interchanged with thermodynamic problems. Using this set of mathematical techniques, we can even easily deduce the black hole temperature and entropy in just a few steps. (see the previous article "temperature and mysterious virtual time" by Huipu.)

Plural time-related tools have been common in quantum field theory and black hole thermodynamics, but they are still pioneering in cosmology. After the side door hidden in the analytical solution of the equation was discovered, the researchers immediately noticed a large area of drooping fruit waiting to be picked.

With the help of the tools of black hole thermodynamics, the two researchers soon came up with a valuable result: of all possible universes, straight universes with zero curvature account for the vast majority. That is to say, the straightness of our universe is not a coincidence, but a natural result of evolution, just as air molecules must be evenly dispersed in the room. This conclusion strongly refutes the "fine adjustment hypothesis" which is strongly colored by "anthropocentrism".

In addition, the mechanism of vacuum energy evolution is mentioned at the end of the paper. Preliminary analysis shows that in a universe with positive initial vacuum energy, no matter what the initial value is, it will always decrease gradually in the process of evolution, which actually corresponds to the direction of entropy increase in the evolution process in the virtual time dimension. As for a more complete and detailed analysis and demonstration, we can only wait for the follow-up papers of the two researchers.

reference

[1] "Thermodynamic solution of the homogeneity, isotropy and flatness puzzles

(and a clue to the cosmological constant) "arXiv:2210.01142v2 [gr-qc]

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