Neutrino oscillations

Even though neutrinos in the Sun are mostly created as the "electron flavor", νe, when they travel 150 million kilometers to Earth, roughly a half of those neutrinos here are observed to have the muon or tau flavor: νμ, ντ. This is the "solar neutrino problem" which catalyzed a lot of neutrino research in the past 60 years.

https://en.wikipedia.org/wiki/Neutrino_oscillation

The standard theoretical model is to assume that there are three different neutrinos with different rest masses in the range 0.05 eV ... 0.5 eV: ν₁, ν₂, ν₃.

Let us have a reaction which creates an electron neutrino νe. The neutrino can be any of mass states ν₁, ν₂, ν₃. We do not know which it is. Various rest masses have various probability amplitudes.

Let α = e. The probability of observing the neutrino as the flavor β at the distance L from its creation is:

where mj are the various neutrino rest masses, and Uαj are constant components from the Pontecorvo et al. 3 × 3 matrix.

The formula on the right sums the waves of different particles: neutrinos with different rest masses m₁, m₂, m₃? That is strange. Is that allowed in quantum mechanics?

In quantum electrodynamics, the electron Dirac field is classical, and can describe an electron or a positron. There we have another case in which we sum waves of different particles: the electron and the positron.

Suppose that a neutrino with a rest mass m flies in vacuum. Obviously, the rest mass of the neutrino cannot change. There cannot be an oscillation of neutrinos if we look at their rest mass.

Anca Tureanu (2025) says that we cannot use the interference of the wave functions of the neutrino wave functions when the neutrinos have different masses

https://helda.helsinki.fi/bitstreams/9d374eb1-c3b2-4637-a770-43534a4d6e8d/download

https://quantumzeitgeist.com/university-helsinki-tureanu-challenges-decades-old/

Professor Anca Tureanu from the University of Helsinki says that, in quantum field theory, one is not allowed to calculate interference of wave functions of neutrinos of different rest masses.

A principle in quantum field theory, and in quantum mechanics, is that we can only calculate an interference of two waves if we cannot know which of the two wave histories happened. In the double-slit experiment, we cannot know if the photon passed the left or the right slit. We are allowed to calculate the interference pattern of the two different paths of the photon. But if we place a detector which can determine the path of the photon through the slits, the interference pattern disappears.

In the Nuclear Physics, Section B 1018 (2025) paper, Tureanu writes:

She claims that the end state of a reaction in quantum field theory is a mixed state. A mixed state has classical probabilities. There is no interference of classical probabilities.

But this is a strange claim. In quantum mechanics a "pure" state, that is, a superposition state, stays pure until it is measured. When we measure the outgoing particles, we obtain classical probabilities. The state is no longer pure. The state "decoheres".

Note that the pure state includes ALL outgoing particles. We cannot speak separately of the neutrino.

If we know the location of the reaction well, we do not know the mass state of the created neutrino

Suppose that we prepare the incoming particles in such a way that their momentum (and energy) are known extremely precisely. We measure the energies and the momenta of outgoing particles, except of the neutrino. That way we will know the energy and momentum of the created neutrino, and can determine its rest mass m.

The uncertainty principle says that we then cannot know the location of the reaction precisely:

       Δp Δx  ≥  ħ / 2.

The flavor probability formula has the parameter L which tells the distance to the reaction.

The energy-momentum relation is

       E²  =  p²  +  m².

Let E and |p| be roughly 1, and m ≈ 10⁻⁶. To determine a rough value of m from the energy-momentum relation, we have to know p to the precision better than Δp = 1/2 * 10⁻¹². We use natural units, so that ħ = 1. Then

      Δx  >  10¹².

We see that the formula

      exp( -i mj² L / (2 E) )

in the flavor probability equation above then has a relatively large uncertainty because of L:

       (10⁻⁶)² * 10¹² / 2 = 0.5.

That is, we know the rest mass of the neutrino, but do not know the neutrino oscillation phase too well. The uncertainty, actually, is quite a lot larger than 0.5 because we must measure momenta of several particles. Also, we chose Δp too optimistically above. A better choice might be 1/3 of the value we chose. Then the uncertainty 0.5 easily becomes 3 or more.

That is, we cannot know the phase of the neutrino oscillation. The flavor probability formula in this case gives the average over a whole cycle of the oscillation. The average is a constant and does not depend on the value of L. This is reasonable.

If we know L relatively precisely, then we cannot know the rest mass state of the neutrino. It makes sense to calculate the sum (interference) of different rest mass waves.

What is the created neutrino? Which state does it have?

If we know the location of the reaction relatively well, then we cannot know the rest mass of the created neutrino.

The created neutrino can then be seen as a "malformed" classical neutrino field wave, which does not have a definite rest mass until we measure the rest mass in some way.

There is only one neutrino particle which has 3 different states? In the model above, there is a single neutrino field which has (stable?) states of three different rest masses. Should we say that the neutrino is just one particle, which appears in three stable states?

An analogous question: are the electron and the positron a single particle which has two states?

Can a neutrino decay to a lower rest mass state? Could it emit a photon?

A particle reaction emits waves, not particles?

In bremsstrahlung, an electron can emit photon(s) of various energies and momenta.

https://en.wikipedia.org/wiki/Bremsstrahlung

Classically, the electron emits an electromagnetic wave of a complicated form, as the electron e passes the nucleus marked with +. For high energies of the photon, the quantum mechanical calculation yields probability amplitudes which differ surprisingly much from the probabilities one could naively assume from the Fourier decomposition of the classical treatment.

A basic principle of quantum mechanics is that you must treat everything as waves until a measurement is made. You cannot assume that a wave "has collapsed" until it is measured. A typical example of this principle is the Bell inequality: if you assume that there are "hidden variables" which determine the state into which a quantum wave will collapse, then you will calculate results which differ from quantum mechanical results.

This basic principle of quantum mechanics suggests that in a reaction, we must treat outgoing particles as waves, until the particle is measured in a detector.

However, we have a problem here. What is the shape of the bremsstrahlung wave? It cannot be the classical shape. From Feynman diagrams, we can calculate the probability amplitudes of various photons with a momentum p. Could it be that we ger the wave in the momentum space simply by superimposing the waves with different p, multiplied by its probability amplitude? The phase of an individual wave we may get from the position and the momentum change of the electron as it passes the nucleus.

It looks like the model of neutrino oscillation in Wikipedia is compatible with quantum mechanics.

Are the electron, the muon, and the tau particle states of the same particle with different rest masses? What about antiparticles?

The Dirac equation governs these particles, as well as their antiparticles. 

Do we know anything about interference of electron and positron waves? What about interference of electron and muon waves?

An electron colliding with a nucleus

The bremsstrahlung reaction creates photons. Let us forget photons for a moment, and look at the wave function of the electron colliding with the nucleus.

We may assume that the nucleus is moving at a velocity v, so that, depending on its path, the electron may gain or lose some kinetic energy.

                             -------------

                           /    v            \

     e-       --------      <-- ● Z      ----------->  p'

     p        -------------------------------------->  p''

The Schrödinger equation describes the behavior of the electron e-. The initial momentum of the electron is p.

In the diagram above, we have two alternative paths for the electron. Depending on the path, the electron exits with a momentum p' or p''.

The momentum of the nucleus changes by an amount p - p', or p - p''.

In the Schrödinger equation, the waves for different paths of the electron do interfere, even though the electron in each path has a different kinetic energy and momentum.

The collision can be considered a "reaction". The particle e- exits the reaction. The electron in each path has a different "state". 

The collision is actually a "double-slit" experiment, in which the nucleus forms the obstacle between the two "slits".

We conclude that in quantum mechanics, the waves of a particle do interfere, even though the waves correspond to different kinetic states of the particle.

In the case of neutrino oscillations, the neutrino can be in different rest mass states. Could it be that having a different rest mass removes the interference? Is having a different rest mass a "kinetic state"?

A simplified model of an excited hydrogen atom

An excited hydrogen atom has a "rest mass" which is larger than of the atom in the lowest energy state. What do we know about the interference of hydrogen atoms?

https://en.wikipedia.org/wiki/Double-slit_experiment

People have been able to run the double-slit experiment with large molecules, like the "buckyball". This suggests that we should be able to see interference between a hydrogen atom in the ground state and the same atom in an excited state.

If we have no way of knowing if the atom is excited or not, why would interference disappear? We can measure the rest mass later, and we will know. But the same hold for the electron in the previous section: we can measure the momentum of the electron later – this does not prevent interference from occurring between electrons which have different momenta.

            y position of proton

            ^

            |

            |

            |

            |

             --------------------------> x position of e-

Let us consider the following simplified model of a hydrogen atom. We have one spatial dimension in the world. The y coordinate tells the position of the proton. The x coordinate is the position of the electron. That is, we model both particles with one particle whose spatial coordinates are (x, y).

The potential:

       V  =  1 / | x  -  y |.

If we fix y to some value, the wave function is somewhat like a particle in a box.

The inertia of our one particle is larger in the y direction than to the x direction. The proton is heavier.

In this simple model, there most probably is interference between different momentum states of the proton?

***  WORK IN PROGRESS  ***

Solving the Hubble tension: slow down the expansion of the universe during the first 500 million years

Supernovae of type Ia, and Cepheids suggest that the universe is right now expanding at a rate 73 km/s / Mpc, and, furthermore, the expansion rate is accelerating (dark energy).

But the mapping the cosmic microwave background, and combining it to a ΛCDM model says that the expansion rate is only 67 km/s / Mpc.

This is called the Hubble tension. In our previous blog post we suggested a simple solution to the Hubble tension: slow down the expansion rate of the universe by a factor about 1/2X, when the universe is 150 million ... 500 million years old.

We do not know why the expansion of the universe started to speed up some 5 billion years ago. We cannot take for granted that the universe has obeyed the ΛCDM formula during the previous 8.7 billion years, either.

Let us investigate how people derive the figure 67 km/s / Mpc.

What we know of the cosmic microwave background

Literature seems to be certain that the "last scattering" temperature of the CMB was 3,000 K, at a precision +-0.5%. Since the temperature today is 2.73 K, we know that the redshift since the last scattering is

       z  =  3,000 K / 2.73 K

           = 1,100.

The spatial size of the sound horizon in comoving coordinates

https://www.mv.helsinki.fi/home/hkurkisu/Cosm9.pdf

In the link there is a textbook by Dr. Hannu Kurki-Suonio, in which the spatial size of the sound horizon is calculated:

where cs(t) is the sound speed (about 60% of c), and a(t) is the scale factor. The integral formula assumes a standard ΛCDM model. The speed of sound before the recombination (last scattering, or decoupling) is:

where ωb is the baryon density and ωγ is the radiation density. We know their approximate values by observing the current universe, and from knowing that the current temperature of the CMB is 2.725 K.

The expansion speed of the universe before recombination follows the formula:

There Ωr is the share of the radiation density of the critical density and Ωm is the share of matter of the critical density. Since |a| is very small, we can ignore the a² and a⁴ terms.

The value of the integral is:

The sound horizon size r* depends on the density of baryons and matter in the current universe. Its dependence on the Hubble constant H₀ in the current universe is not significant: if we calculate time backwards, the final collapse speed of the universe does not depend much on the collapse speed right now.

Apparently, the baryon density and the matter density in the current universe are known quite well. The radiation density we know very well, too. The redshift at the decoupling, zdec, is known well. Thus, if we assume the ΛCDM, we know the sound horizon size r* quite precisely.

The Hubble tension

Our best knowledge of the expansion speed of the current universe come from type Ia supernovae and Cepheids. The data from supernovae and Cepheids agrees about the value 73 km/s / Mpc.

If we assume the ΛCDM model, and calculate backward from the current universe, the cosmic microwave backround rises to the temperature 3,000 K at the redshift 1,100, when the scale factor a of the universe is 1/1,100th of its current value.

The last scattering / recombination / decoupling / end of baryon drag almost certainly happens at that temperature.

The Hubble H₀ constant now has the value 73 km/s / Mpc. The matter density of the current universe is Ωm = 0.31 times the critical density. We can proceed to calculate backward ΛCDM, and can determine the distance D(1,100) in comoving coordinates to the last scattering. That is, a photon scattered then moves the distance D(1,100) in comoving coordinates, before hitting our eyes.

   

                            /|

                         /   |

                      /      |

                   /         |

                / θ*       | r*

          o  -------------    last scattering

          |    D(1,100)                       

         /\                       comoving coordinates

An observer on Earth should see the sound horizon length in the cosmic microwave background to subtend an angle

       θ*  =  r* / D(1,100).

Recall that r* is given in comoving coordinates, too.

We can analyze the CMB very precisely from the Planck data, and obtain a value for θ* which is significantly smaller than the value which we calculated above. This is the Hubble tension. "Features" in the CMB look smaller in the sky than they should be.

The Hubble tension. The tension is between the assumption:

- we have measured H₀ correctly in the current universe, as well as other relevant parameters, and ΛCDM is correct,

and the fact:

- features in the CMB look smaller in the sky than what we calculated.

It is an oversimplification to say that the tension is just in the value of H₀.

A simple way to remove the tension is to assume that the universe expanded slower between the last scattering and the current time than assumed by ΛCDM. If the universe expands slower, then a photon sent from the last scattering has more time to travel in comoving coordinates, and the distance D(1,100) is longer. Consequently, the calculated angle θ* is smaller. We can make the angle so small that it agrees with the angle which we see in the sky.

Dark energy already showed us that the universe may expand surprisingly fast. It is not a stretch to claim that it can also expand surprisingly slowly.

The Hubble tension really is a tension in the angle θ* calculated from the current universe versus what we see in the CMB in the sky?

Let us measure the mass density Ωm in the current universe, as well as the value of the Hubble constant H₀ from type Ia supernovae.

Assuming ΛCDM, we can then calculate how the uneven distribution of the current galaxy clusters should show up in the CMB.

Let us check if galaxy surveys, combined with H₀ = 73 km/s / Mpc, really conflict with the CMB we see in the sky.

https://arxiv.org/html/2505.01661v1

Zhiwei Yang et al. (2025) derive H₀ from DESI, H0LiCOW, and Pantheon supernovae. DESI has measured the redshifts of millions of galaxies and built a 3D model of the current universe (actually, it is the universe in the past 10 billion years, since the redshifts z are up to 2 in the survey).

https://shsuyu.github.io/H0LiCOW/site/

H0LiCOW measures gravitational lensing of quasars and derives a value for H₀ from them.

https://arxiv.org/abs/2112.03863

Pantheon+ contains the light curves of 1,550 type Ia supernovae. They can be used to determine the distances to galaxies, as well as Ωm.

Zhiwei Yang et al. calculated H₀ = 73 km/s / Mpc and r* = 138 Mpc. The size of the sound horizon r* is given in the current universe.

The sound horizon size derived from the first 380,000 years of the universe is 147 Mpc. Why the difference?

Let us check what values for r* have other authors derived from DESI and other galaxy surveys.

https://en.wikipedia.org/wiki/Baryon_acoustic_oscillations

The Sloan Digital Sky Survey team got a value r* = 150 Mpc. The value differs a lot from the Yang et al. value.

https://arxiv.org/abs/2510.19149

E. A. Zaborowski et al. (2025) derive the value H₀ = 70 km/s / Mpc from DESI.

https://arxiv.org/abs/2406.18298

Tonghua Liu et al. (2024) list values obtained for r* in various studies: 143 Mpc, 137 Mpc, 144 Mpc. Their own result is 140 Mpc.

There seems to be too much variation in the estimates of the sound horizon size r* in the current universe. We cannot draw any conclusions. The Hubble constant H₀ derived in these calculations varies a lot, too. It is not clear if a Hubble tension exists in the calculated values.

Thus, currently, the tension is between the Hubble value 73 km/s / Mpc derived from Cepheids and supernovae versus the Hubble value calculated from the first 380,000 years of the universe plus a ΛCDM model.

What would the sound horizon angular size θ* be in ΛCMD if we tune the universe expansion rate?

In our blog we have brought forward an idea that the universe on the average follows a FLRW model, but there are "bounces" which make it to expand slower or faster. The current epoch of dark energy is a bounce to a faster expansion.

Let us check if there would be a Hubble tension in a CDM model where Λ would be zero "on the average", when the universe expands.

How much has dark energy recently accelerated the expansion relative to CDM?

We can model the expansion with newtonian gravity.

https://astronomy.stackexchange.com/questions/2194/what-was-the-age-of-the-universe-when-the-average-density-was-one-atmosphere

If we assume that H₀ = 67.8 km/s / Mpc, then the critical density is now

       ρc  =  8.4 * 10⁻²⁷ kg/m³.

The ratio of matter and dark matter to the critical density is believed to be

       Ωm  =  0.31.

Note that the estimate 0.31 depends on the cosmological model that we use. It is based on how much we believe that the gravity of the mass has slowed down the expansion of the universe since the Big Bang.

Let us assume that the expansion of the universe has slightly accelerated in the past 5 billion years. We calculate the amount of energy involved.

Let us have a spherical volume of the current universe. Let the radius be R.

If the mass density of the universe would be 1/0.31 = 3.2-fold, then newtonian gravity would asymptotically stop the expansion. Let us have a test mass m on the surface of the sphere R. The negative potential energy of m is linearly proportional to the mass M of R, or the density of R.

                     ×                   • ---> v

                               R        m

We conclude that the negative potential energy of m is -0.31 times its kinetic energy E = 1/2 m v².

Suppose that the velocity v of m has stayed the same for the past 9 billion years. Then dark energy has done ~ 0.31 * E of work on m, against the gravity potential. Quite a lot of work!

Let us use the "epoch" scale from the previous blog post. The scale factor is a ~ 1 / 2ⁿ during epoch n.

The time lengths of various epochs:

   t₀ : 9 billion years

   t₁ : 3 billion years

   t₂ : 1 billion years

   t₃ : 350 million years

   t₄ : 100 million years

   ...

   t₁₀: 140,000 years.

The contribution to the "angular distance" D of each epoch – this measures how much a photon propagates, in comoving spatial coordinates, during each epoch:

  D₀ : 1

  D₁ : 0.7

  D₂ : 0.5

  D₃ : 0.35

  D₄ : 0.25

  ...

  D₁₀: 0.03.

The sum D = 3.3.

Let us calculate the kinetic energy of our test mass m during each epoch. Recall that the current gravity potential of m is -0.31 * E, where E is the current kinetic energy of m. When a < 1, the gravity potential energy has added to the kinetic energy:

  0 : E

  1 : 1.31 E

  2 : 2 E

  3 : 3 E

  4 : 6 E

  5 : 11 E

  ...

  10 : 300 E.

If we want to make the angular distance D 8% larger, so that the Hubble tension would be removed, we have to reduce the kinetic energy of m a lot during epochs 2, 3, and 4. The expansion rate should only be 75%. That is, the kinetic energy must be reduced by E during epoch 2, and by 3 E during epoch 4.

Let us think. What does this mean? Initially, during epoch 10, the test mass m has a lot of kinetic energy. As m climbs up from the gravity potential well, it loses kinetic energy. The energy goes to the energy of the gravity field. In earlier blog posts we have suggested that retardation of gravity stores extra energy into the gravity field. Could that extra energy be so large that it could be 3 E during epoch 4?

Note that we have a fine-tuning problem here. How did nature "know" that the kinetic energy of m during epoch 10 must be set almost exactly to the negative gravity potential -300 E? As if nature would be fine-tuning the process, so that it is almost exactly a collapse of a cloud run backwards.

Entropy of the universe as a fine-tuning problem

The universe had a low entropy in the Big Bang. A low entropy is unlikely for a random system. Here we have another fine-tuning problem.

The entropy of the universe grows after the Big Bang, as the universe expands. Suppose that the universe follows an FLRW model, and starts to contract again. Suppose that energy is conserved. Then the macroscopic state of the universe in a Big Crunch should be very similar to the corresponding state at the Big Bang. The state is very hot, and particles have random motion. That is, the entropy has to start decreasing at some point before the Big Crunch. This would break the rule that entropy has to increase. What is going on?

Let us imagine an explosion in almost-newtonian mechanics and newtonian gravity. Let the explosion start from a very small volume and be almost uniform. The entropy of the system starts growing. In particular, photons will carry away a large amount of entropy. Let us assume that gravity will pull back most of the mass which exploded. Gravity will compress it back into a small volume. The end result might be a crystal which has a very low entropy. All entropy created after the explosion is carried away by photons.

Photons cannot escape from an FLRW model because there is nothing outside the model. Thus, an FLRW model leads us to a paradox: entropy should magically start decreasing at some point.

Let us assume that black holes do not exist in the FLRW model. It is not clear how they would behave in a Big Crunch.

Entropy bans a Big Crunch? Maybe a Big Crunch is not possible at all? Since FLRW models do allow a Big Crunch, we here have an argument against the correctness of FLRW models: they break the laws of entropy.

Suppose that we have an isolated physical system whose volume first increases and then decreases. A big volume contains more "degrees of freedom". Entropy can then increase. But if some strange law of physics (like an FLRW model) makes the volume to decrease, then there will be fewer degrees of freedom, and entropy may be forced to decrease.

How do we avoid the paradox of decreasing entropy? Either the system is:

1. not isolated, or

2. the system will expand forever.

Bounce-back (cyclic) models of the universe require that entropy has to decrease at some point. If one believes that entropy cannot decrease, then bounce-back models are ruled out.

Conclusions

We presented various thoughts about the Hubble tension and dark energy, but were unable to reach any definitive results. We will continue our investigation later.

https://en.wikipedia.org/wiki/Neutrino_oscillation

Next we will look at neutrino physics.

Dark energy: a balloon model

In December 2025 we had problems finding a reasonable mathematical model for retarded gravitation in an asymptotic Minkowski space. Let us try it again, this time for a spherical balloon model. The spacetime is then topologically different from a plane.

Picture publicdomainpictures.net

Can the equation of spacetime be local for a balloon model?

Let us think of the Schwarzschild solution. The solution extends through the entire asymptotically Minkowski space. The central mass M affects locations very far away.

For a spherical rubber balloon, the equation governing its stretching is local, if we assume that the pressure inside is uniform.

Friedmann equations in general relativity look local, but that is only because we assume a uniform mass distribution. Any retardation effects are masked.

A rubber string and a weight

                 rubber string

           ●----------------------------●

      point                            point

Let us draw points uniformly around a sphere, and connect them with tense rubber strings which run on the surface of the sphere. The rubber strings simulate the gravity between the mass points. If the balloon is static, or inflates or deflates at a constant rate, this model looks like an ordinary balloon: the tension in the strings is uniform along the string.

A rubber string and a weight which the string slows down

Let us consider the simplest possible model of a tense rubber string plus weight:

          

                                              force

                                         F  <------ 

         ==| ~~~~~~~~~~~~~~ ● ---> v

    wall         rubber string            M weight

We are interested in what kind of longitudinal waves form in the string when the weight moves to the right while the tension in the string pulls the weight to the left.

Free waves in the string are sine waves. Since the speed of the waves is finite, there is "retardation".

As the weight slows down, it creates new waves. The weight perturbs the string. Calculating the precise form of the wave probably requires a computer. But we are only interested in very crude estimates.

Do binary pulsars prove that there are no longitudinal gravitational waves? Are longitudinal waves confined between masses?

Binary pulsars have confirmed that transverse waves and linearized Einstein field equations explain the energy loss of a binary pulsar up to a precision of 0.1%.

In an earlier blog post we suggested that longitudinal waves must be "absorbed into" matter quickly, because they cannot propagate in empty space. This hypothesis would explain why binary pulsars match the transverse wave model so well.

Another hypothesis: longitudinal wave effects can only exist in the field between two masses M₁ and M₂ and can never escape to empty space.

https://en.wikipedia.org/wiki/Plasma_oscillation

Longitudinal electromagnetic waves exist in plasma, but they do not exist in empty space. Could this be analogous to our model of retarded gravity? Langmuir waves exist in a plasma which is, on the average, neutral. Since gravity charges always are positive and gravity always pulls, Langmuir waves cannot exist in a gravitating system.

A model with solid rubber inside

Let us have two masses M₁ and M₂ on the surface of the balloon. Visualizing a rubber string between the masses, and running on the surface of the balloon, is somewhat hard. Let us imagine that the rubber string runs inside the balloon.

The simplest case is when M₁ and M₂ are on the opposite sides of the balloon. A rubber string runs through the center of the balloon and connects them.

The balloon expands. The rubber string is not immediately aware of the expansion slowing down. This will cause oscillation in the expansion rate.

Photo publicdomainpictures.net

Let us have a solid rubber ball. The rubber will resist the expansion in most cases. But if there is a shock wave moving from the center toward the surface, the expansion of the rubber ball may actually accelerate at a certain time. "Dark energy" in this case is the energy of the shock wave.

This model predicts that the expansion rate of the universe will oscillate, while it, on the average, is quite similar to an FLRW model.

But we should find a mathematical formula, so that we could compare it to the observed dark energy.

The strength of dark energy seems to be weakening right now

https://en.wikipedia.org/wiki/Dark_energy

https://en.wikipedia.org/wiki/Dark_Energy_Spectroscopic_Instrument

The DESI project measures the 3-dimensional mass density variations of the universe right now, and compares it to the baryon acoustic oscillations after the Big Bang. The results of DESI, when combined to other observations, suggest that dark energy is weakening right now. Wikipedia says that dark energy density right now might be 10% less than 4.5 billion years ago.

This is consistent with our own rubber ball model: on the average, the expansion should be like an FLRW model, but there should be "bounces" from the retardation of gravity.

https://www.emergentmind.com/topics/chevallier-polarski-linder-cpl

In the link we have the definition for the CPL parametrization of the dark energy equation of state, w(a), where a is the universe scale factor. The current value of a is set to 1.

https://arxiv.org/abs/2503.14738

Abdul-Karim et al. (2025) write that the energy density of dark energy may have increased in the past. This is called a "phantom crossing". In our rubber ball model, a phantom crossing is expected: at some points the bouncing slows down the expansion, at other points speeds up the expansion.

The Hubble tension

Let us measure the baryon sound horizon size in the cosmic microwave background, and match it to the measured density fluctuations in the current universe. Assuming, for example, the standard ΛCDM model, we end up with a Hubble "constant" current value of

       67 km/s / Mpc.

But a direct measurement, based on Cepheids and type Ia supernovae yields a value

       73 km/s / Mpc.

The values differ too much to be just a statistical fluctuation.

https://arxiv.org/abs/2309.06795

Marco Raveri (2023) tries to use dark energy to make these values compatible. Let us check what he suggests. His conclusion is that dark energy modifications at "late" times, 0.01 < z < 3 cannot solve the Hubble tension.

Early dark energy (EDE)

https://academic.oup.com/mnras/article/533/4/3923/7750120

https://en.wikipedia.org/wiki/Recombination_(cosmology)

Xuejian Shen et al. (2024) write about an early dark energy (EDE) model, where an unknown field boosts the expansion of the universe just before the recombination (last scattering) which happens when the universe is 380,000 years old in a standard ΛCDM model.

Shen et al. say that this model can solve the Hubble tension. Is it so?

Let us start from the observed uneven matter distribution in the recent universe, and match it to the the uneven matter distribution in the cosmic microwave background (CMB). The baryon acoustic oscillations (BAO) make bumps into the uneven distribution. Let us match the bumps in recent matter and in the CMB.

Let us calculate time backward from the present time, and model matter distribution at earlier times. We use a standard ΛCMB model. When the universe has shrunk to one 1,100th of its size, the CMB has a temperature of 3,000 K, and the last scattering occurs.

The Hubble tension is that the BAO bumps in this calculation have a larger angular scale than in the measured and mapped CMB in the sky. That is, the "features" in the actual CMB map are smaller than our calculation backward in time predicts.

The early dark energy (EDE) hypothesis talks about what happened before the last scattering. The hypothesis does not help in any way in solving the Hubble tension. We have to check the literature. Can the EDE hypothesis be so much wrong?

https://www.researchsquare.com/article/rs-100387/latest.pdf

Karsten Jedamzik et al. (2020) write that EDE cannot resolve the Hubble tension.

 

Above, θ* is the angular size of the sound horizon, that is, the angular size of "features" in the CMB. We use coordinates comoving with matter. Then r* is the spatial size of the sound horizon (in those coordinates), and D(z*) is the spatial distance to the last scattering.

Jedamzik et al. say that only the quantity

       Ωm h²

affects the spatial distance to the last scattering, in a flat ΛCDM model. There, Ωm is the matter fraction of the critical density, and h is the value of the Hubble constant now.

The authors state that the Dark Energy Survey and the Kilo-Degree Survey have set strict limits on Ωm h², so that any attempt to solve the Hubble tension through manipulating Ωm h² will fail.

Let us check what they say about manipulating r*. Suppose that we calculate the development of a ΛCDM model backward in time. The matter density variations of galaxy clusters eventually turn into variations of the CMB as the temperature rises.

Solving the Hubble tension with a very slow expansion during the first billion years of the universe: also the James Webb paradox of too mature galaxies is solved?

The James Webb telescope has observed galaxies which in a standard ΛCDM model are only 500 million years old, but look much more mature, as if they were 1.5 billion years old.

Let us assume that the expansion during the first billion years of the universe was much slower than the ΛCDM says. We add an "extra" 1 billion years to the age of the universe, to the early phases of the universe. Adding that extra means that the universe expands much slower than in the ΛCDM model, during the first 1 billion years or so.

Then galaxies have much more time to mature. This would solve the James Webb paradox.

During that slow expansion during the first 1 billion years, photons from the last scattering had time to move a "longer" path than in the ΛCDM model. Let us explain what "longer" here means. If we model the expanding universe with coordinates comoving with matter, then the ruler of the spatial metric (say, a ruler 1 meter long) shrinks as time progresses. The "length" of the path of the photons is measured in these comoving coordinates.

Since the photons moved a longer path before coming to our eyes, the features in CMB will have smaller angular diameters in our eyes. This resolves the Hubble tension.

A quantitative calculation of making the universe to expand slower during the first 1 billion years

Let us make a quantitative calculation. We assume a matter-dominated universe with a zero pressure. Then we can use newtonian mechanics to calculate the development of the universe backwards from the present time.

              "feature" size

               <----->

              ●        ●        ●       ●

              ●        ●        ●       ●

                             o ~~~~~  last scattering

                             |     D

                            /\ 

                      observer       c ~ 1 / a

We set a = 1 as the current scale factor of the universe. In comoving coordinates, the speed of light

       c  ~  1 / a.

Let us look at eleven "epochs" where

       a  ≈  1 / 2ⁿ,

0 ≤ n ≤ 10.

The epoch 10 corresponds to the time of the last scattering, that is z ≈ 1,100.

In ΛCDM, the kinetic energy of a particle m in static (not co-moving coordinates) is very roughly

       1/2 m v²  ~  1 / a  =  2ⁿ.

That is, v ~ sqrt(2ⁿ).

This implies that the time the model spends in an epoch n is

       tₙ  ~  (1 / 2ⁿ)  /  sqrt(2ⁿ)

              = 1 / 2^(1.5 n)

              = 1 / 2.8ⁿ

              = 0.35ⁿ.

The sum of tₙ is t ~ 1 / (1 - 0.35) = 1.5.

The time lengths of various epochs:

   t₀ : 9 billion years

   t₁ : 3 billion years

   t₂ : 1 billion years

   t₃ : 350 million years

   t₄ : 100 million years

   ...

   t₁₀: 140,000 years.

The speed of light c ~ 1 / a = 2ⁿ. Light in that time propagates a distance in comoving coordinates:

       Dₙ  ~  2ⁿ  *  1 / 2^(1.5 n)

              = 1 / sqrt(2ⁿ)

              = 1 / 1.4ⁿ

              = 0.7ⁿ.

We are interested in the distance

               10

       D  =  ∑ Dₙ

              n = 0

(in comoving coordinates), which light travels from the last scattering, into the eye of the observer. If the contribution of epoch D₀ is 1, then the sum of the series is

       D  =  1 / (1 - 0.7) = 3.3.

The contributions of various epochs:

  D₀ : 1

  D₁ : 0.7

  D₂ : 0.5

  D₃ : 0.35

  D₄ : 0.25

  ...

  D₁₀: 0.03.

In particular, the contribution of epoch D₁₀ is only 0.03. We see that we have to modify the physics drastically during epoch 10, if we want to increase D by 8%. This shows that solving the Hubble tension by tampering only the first 400,000 years of the universe is an implausible solution.

The first one billion years of the universe corresponds to epochs 2, ..., 10. The sum of D₂, ..., D₁₀ is 1.6.

We see that if we make the universe to expand 16% slower during epochs 2, ..., 10, we can increase D by 8%. Then the angular size of "features" will be properly smaller to match how Planck and other probes saw the CMB.

But now we realize that we only have to add 160 million years to the early history of the universe. That would not explain James Webb observations.

If we double the time length of epoch 3 to 700 million years, we increase D by 11%, and add 350 million years to the early history of the universe. Could this be enough to solve the James Webb paradox?

Dark energy means that the universe has been expanding surprisingly fast during the past 7 billion years. Maybe the expansion rate fluctuates, and was surprisingly slow during the first 500 million years of the universe?

Conclusions

We still do not have a mathematical model which we could use to calculate the effects of retarded gravity. Intuitively, retardation should cause the universe to expand surprisingly slowly at certain times, and surprisingly fast at other times. That is, like a solid ball whose elastic interior can contract and expand.

We will write a new blog post about the Hubble tension. Literature strangely ignores solutions where the expansion of the universe is slowed down during the first 350 million years. Literature seems only to talk about modifying physics at z < 1 (the last 7 billion years), or at z ~ 3,000 (the first 100,000 years).

Why does literature ignore the most obvious solution? Slowing down the first 350 million years could solve also the James Webb paradox of "too" mature galaxies at z ~ 14.

de Broglie and Bohm: photon creation from a radio transmitter – the problem of new particles

Let us consider a simple semiclassical process which creates photons. Let us have two charged particles (charge Q) whose mass M is macroscopic. We let the particles pass close to each other, so that they send a radio wave with many photons.

                                                                    screen

                                                                          |

                                                                          |

                                                                          |

                              )   )   )   )   )  radio wave

           ● ---->              

          M, Q                 <---- ●

                                         M, Q

The radio wave will hit a screen far away, and we can determine the positions of individual photons.

We can calculate the shape of the radio wave classically, but we cannot calculate where the individual photons will hit the screen.

The process creates new particles. Is there a way to extend the de Broglie-Bohm model to this case?

Could this work: the photons "pre-exist" in the electric fields of the charges Q, and are "freed" in the scattering process? Then all the particles could have a hidden variable already at the outset of the experiment, and would sail in a deterministic way to the screen – just as in the double-slit experiment.

The outgoing photons have an uncountable number of possible momenta p. Even if we would have pre-existing photons in the field of Q, how would these obtain the required p?

Time asymmetry in the process

The initial configuration of two charges Q flying at each other is quite different from the final configuration where we have the wave function for the two charges Q plus many alternative configurations of photons. The entropy of the system grows, and there are more degrees of freedom in the final configuration.

What if there would be no growth of entropy? Then the initial and the final configurations should look similar. Maybe the number of the degrees of freedom does not grow any more?

What about inverting the direction of time? Then the degrees of freedom will shrink as time progresses.

Absorption of a photon by a hydrogen atom

An excited hydrogen atom can decay and send an individual photon. The atom can absorb the required momentum from the reaction as it launches the photon.

What about the reverse process? A wave function of the photon contains many cases in which the photon does not hit the atom at all. The photon passes the atom.

Can we somehow add more "worlds" in such a way that the the graph of possible worlds would look more time-symmetric?

In the case of a hydrogen atom absorbing a single photon, the degrees of freedom only shrink in the case of the absorption. In other cases, there is no change. A "wave function collapse" only happens in one branch of many-worlds.

Ignore photons and only consider (charged) particles which existed at the outset?

The de Broglie-Bohm approach might work if we only had undestructible particles. Let us have N such particles. Let us have a foliation of spacetime such that spacelike folios define "simultaneousness". Then we could replace the N particles with one particle moving in a space with 3 N spatial dimensions.

But we have a problem already with electrons: pair production and pair annihilation. There are no undestructible particles.

Give up determinism?

Albert Einstein said that God does not play dice. That is, the "next", measured, state of a physical system should follow deterministically from the "previous" state.

Since simultaneousness is not well defined in relativity, we maybe should say this: what happens at some point in time and space is determined by the light cone which precedes that point.

Why not require determinism also in the "initial" states of the particles? Why can the initial states be random? Does God play dice, after all?

Why is the time dimension different from spatial dimensions?

Defining a "branch" in many-worlds

Let us have a human living in spacetime. Most of the massive elementary particles in his brain survive for a long time. They are not annihilated. However, quite a few particles are replaced with other similar particles over his lifetime. How could we define the "branch" in many-worlds, in which he is living?

The brain consists mainly of electrons and of nuclei. These particles are, in most cases, conserved over 100 years.

Then we can apply the de Broglie-Bohm model to these particles, and obtain well defined "branches" of their configurations.

We do not try to define a branch in a more general sense: we do not define, e.g., which photons on Mars belong to his branch.

What if two photons collide and form an electron and a positron, and the electron subsequently is used as a new building block in his brain?

Could we simply accept a model in which the branch of this brain chooses "by random" the initial state of the electron, as it is integrated into the brain?

A new problem: which paths of separate particles are in the "same" branch of many-worlds?

Suppose that two electrons are created in pair productions, and these two electrons are integrated to the brain of the observer.

We can use the de Broglie-Bohm model to describe the paths of the two electrons. But what paths are in the "same" branch of many-worlds?

This might be the solution: we simply choose one de Broglie-Bohm path for each electron. Then the combination of those paths is taken to be in the "same" branch as the brain of the observer.

When a new particle is included into the brain, the brain performs a "quantum jump" to include a certain path of that particle.

Do we need precisely defined branches at all?

Definition of a branch does not change anything in the wave function. In the de Broglie-Bohm model, the path of a particle does not affect the wave function in any way.

However, picking a certain path may define the "real world".

Let us have an observer. Maybe he lives in the "real world", and the copies of him in other branches are just "empty" wave functions?

This does not sound natural. Why should a conscious observer always exist in the "real world" and not in the "empty" branches?

In the de Broglie-Bohm model, the empty branches do physically exist, too. They are just different branches of the wave function.

Maybe it is necessary to drop the concept of the "real world" and treat all the branches equal. This means that Schrödinger's cat is both dead and alive at the same time.

Do we have a reasonable definition of a "branch" now? A robot

Instead of an observer we may consider a computer or a robot which possesses a camera and other sensors. It also has arms for manipulating external objects.

Above we suggested that we can use the de Broglie-Bohm model to define a "branch" of many-worlds, such that an instance of the robot lives in that branch.

The robot will encounter various random (or pseudorandom) quantum phenomena, and adjust the contents of its memory accordingly. Its camera may register random photons hitting its CCD sensor. 

The robot is an almost classical machine. It probably has relatively exactly definable states, and it in most cases functions deterministically.

Is the de Broglie-Bohm model an awkward way to define a branch in this case? Why not use a traditional Copenhagen model where the next state of the machine depends probabilistically on the previous state?

Assigning hidden variables is awkward?

The observing subject

Suppose that we have a reasonable way to define a branch for the robot. The computer in the robot, in that branch, will have a similar world-view as a human observer. It will remember its history. In almost every branch, the robot notices that physical processes in its history behaved according to the probabilistic rules of quantum mechanics.

According to Rene Descartes, cogito ergo sum. That is, I as an observing subject do exist.

My subject might be attached to one branch of a robot, my physical body, or it might "jump" from branch to branch. I would not notice anything if my subject jumps around the branches of many worlds.

It sounds strange if I alternate between different branches. I very much feel that I am in just one branch. This paradox is similar to another paradox about time. In what sense is now different from moments in the past or in the future?

Conclusions

It would be nice if we could define a single "branch" in the many-worlds interpretation. Then we could imagine that a human observer lives in that branch.

The de Broglie-Bohm model gives us a way to define a branch for a robot. Simply plot the paths of the nuclei and the electrons in the mechanism of the robot.

However, that model is not very natural. Why should the branch exactly follow the strange paths of de Broglie-Bohm model?

Maybe a more natural way to define a branch is more fuzzy: it is the collection of those states of the robot which are "similar". For example, we may require that the bits in the memory of the robot have identical values in the collection of the states. This would be a "fuzzy branch".

We still have the question if an "observing subject" is somehow attached to a certain branch. This is a metaphysical problem which is hard to clarify and solve. What makes a certain branch "real" for the subject? How would the subject know if it jumps from branch to branch?

A similar metaphysical problem is in what sense is the world "now" more real than the past or the future. We have no answer.

de Broglie-Bohm model and the many-worlds interpretation of quantum mechanics

In the Copenhagen interpretation, the wave function collapses. That destroys information and spoils the time symmetry of the laws of physics.

The many-worlds interpretation would be better. If my "subject", that is, I as an observer, is located in a certain "branch" of possible world histories, then no collapse is required. It just happens that I as a subject am in a certain branch. Then all the other laws of physics can be time-symmetric. If we further assume that the path of the subject through the tree is determined by a time-symmetric method, then we have a model which is completely time-symmetric, and does not destroy information. This would be desirable.

https://journals.aps.org/pr/abstract/10.1103/PhysRev.85.166

A freely readable .pdf file:

https://quantum.country/assets4/Bohm1952.pdf

The de Broglie-Bohm model might provide us with a deterministic path for the observing subject, if we can extend it to cover the creation of new particles. The classic Bohm model is limited in that that it only covers a fixed number of particles, and uses the non-relativistic Schrödinger equation.

What constitutes a branch in the many-worlds interpretation?

https://physics.stackexchange.com/questions/15842/are-everettian-branchings-global-or-local

https://physics.stackexchange.com/questions/721175/how-to-compute-the-exact-universe-branches-in-many-worlds-interpretation

The definition of a branch is a question which comes up regularly on the Physics Stack Exchange.

Suppose that a single photon hits a charge-coupled device (CCD), and starts a cascade which registers the location which the photon hit. A quantum mechanical object photon "decoheres" into a macroscopic object or into a macroscopic process.

Many people seem to believe that this explains what a branch means in many-worlds. It is a local process of decoherence. However, this definition of a branch is not exact, nor mathematical. If the photon happens to hit the CCD at a slightly (1 micrometer) different location, is that another branch? Do branches actually form a continuum?

The de Broglie-Bohm model might give us a mathematical definition of a branch: it simply is the time development for fixed initial hidden variable values.

https://en.wiktionary.org/wiki/God_does_not_play_dice_with_the_universe

We might define a branch with certain initial hidden variable values the "true state of the material world". Albert Einstein wrote that "God does not play dice". The branch would be deterministic, and Einstein would be happy. Does this approach lead to any contradictions?

de Broglie-Bohm as a flow of probability?

https://en.wikipedia.org/wiki/De_Broglie%E2%80%93Bohm_theory

        |      |                        |

        |      |                                )        )

        |      |                        |

        |      |                                )        )

        |      |                        |

           --->

      incoming             double   diffracted wave

         wave                    slit       and interference

Let us think about the double-slit experiment. The incoming wave can be understood as a probability distribution, where the probability density is the squarw of the wave function value.

We may imagine a flow of probability, a flux, which takes a "probability parcel" forward from the double slit. That would give us a deterministic "path" for an incoming particle? As is well known, a de Broglie-Bohm particle will move along a strange, curvy path.

In a Feynman path integral, also fully unrealistic paths of a particle are calculated and added. If a de Broglie-Bohm path can be designated as the "true world", what about an unrealistic Feynman path?

https://archive.org/details/physicsphilosoph0000heis_n9m9

Werner Heisenberg writes in his 1958 book Physics and philosophy that the de Broglie-Bohm model has problems handling stationary states. The flow of probability is zero in a typical stationary state, say, for a particle in a box. That suggests that a particle would remain in a fixed place.

A particle in a box, intuitively, has two flows of probability, to opposite directions. The sum of these flows is zero.

    

             wave packet

     <--    ~~~~~~~~~~

                       ~~~~~~~~~~    -->

                       wave packet

Let us have a very long wave packet which describes a single particle. We use a beam splitter to divide the packet into two wave packets and let the packets collide so that they travel to opposite directions.

The flow of probability in the overlapping part of the two packets is zero. But the natural many-worlds interpretation for the process is that a single particle is moving either to the left or to the right in the diagram? 

No. In the many-worlds interpretation, "branching" means a process where the waves for various alternatives will not interfere much. The branching has not happened yet when the wave packets overlap.

Do other branches "exist"?

If we designate one de Broglie-Bohm branch as the "true world", the other branches still exist, in the sense that they interfere with the true world, and affect the probabilities how the true world develops in time. This is quite different from, say, newtonian mechanics.

Certain people have criticized the fact that all branches exist, also those that are not the "true world". But we cannot help it: if we prune some branches, we lose information, and the model no longer corresponds to quantum mechanics.

Generalizing to relativistic wave functions and creation and destruction of particles: quantum field theory

The de Broglie-Bohm model is based on the Schrödinger equation, which is nonrelativistic. A generalization to relativistic particle physics requires that we develop a de Broglie-Bohm model for quantum field theory.

Quantum field theory, in its present form, contains dubious methods like regularization and renormalization. It is unlikely that we can develop a satisfactory deterministic model for quantum field theory unless we can remove regularization and renormalization. In fact, a de Broglie-Bohm model might help us to solve the problems.

https://arxiv.org/abs/0904.2287

Hrvoje Nikolić (2009) describes a model where particles can be created and destroyed.

Nikolić assumes that a particle can have a finite lifetime. In the diagram, x⁰ is the time coordinate. "Virtual particles" pop up and disappear in a very short time. They are denoted with dots.

For the model to be deterministic, the position of those dots, (x⁰, x¹) should depend on the initial conditions (data) at some earlier time. Does it?

We can, of course, formally define that the locations of the dots are decided in the initial data, but that looks ugly.

Flow of probability when new particles are created

                        ~~~~~~~~~~~  real photon

                      /

       e- ----------------------------------

                           |

                           | virtual photon

       e- ----------------------------------

Let us consider a collision of two electrons. Photons and pairs can be created. We can draw Feynman diagrams and calculate the probability of each outcome. Above is a diagram for bremsstrahlung.

In the de Broglie-Bohm model of the double-slit experiment, there is a beautiful way to define the "path" of the particle, along the flow of probability in the Schrödinger equation. The path is meandering, but it is defined in a beautiful way.

But for a Feynman diagram, there is no such beautiful way. The real photon which appears above "breaks" the system into more degrees of freedom. If we have a probability flow in 6 spatial dimensions, there is no obvious way to make it to branch into a flow in 9 dimensions. What would decide the path for an arbitrary initial configuration of two electrons? How does the path suddenly branch to more dimensions?

In our blog we have several times written about the problem of "breaking" into more degrees of freedom. This seems to be associated with regularization and renormalization problems for Feynman diagrams.

Let us assign an arbitrary (irrational) real number r for the initial two electrons. We could design an ad hoc way to reproduce the correct Feynman probability distribution from the infinite number of decimals in r, and obtain a deterministic model. But that would be a very ugly solution.

Schrödinger's cat

The de Broglie-Bohm model for the Schrödinger equation solves a paradox associated with Schrödinger's cat, for the treatment with the Schrödinger equation.

The cat always is either dead or alive, in the "true world" determined by the hidden variables. Opening the box does not change anything. The cat does not suddenly "collapse" into either state.

Let us assume that we find the cat alive.

What remains of the paradox is the "empty" branch in which the cat is dead. The branch does exist in the de Broglie-Bohm model, though that branch is not "occupied" by the values of the hidden variables.

Schrödinger's cat involves a radioactive decay process which decides if the poison is released inside the box. Such a decay must be treated in quantum field theory, since new particles are created. And the de Broglie-Bohm model has no beautiful extension there. Thus, the paradox remains.

Conclusions

The de Broglie-Bohm model is a beautiful way to introduce determinism into the Schrödinger equation. We can define in a beautiful way what is a branch in the many-worlds interpretation. We avoid the "measurement problem" and many paradoxes in quantum mechanics.

But extending the de Broglie-Bohm model into quantum field theory depends on if we can find a beautiful model quantum field theory. Presumably, divergences, regularization, and renormalization should be removed. The problem of breaking into more degrees freedom may be a mathematical problem. Can we solve it?

Dark energy from retardation of gravity?

In our previous blog post we were able to derive an estimate for retarded gravity when a supernova core collapses to form a neutron star. We introduced a natural hypothesis which yields a numerical value.

Energy of the retarded gravity field in a collapse of a mass shell: the energy is significant

In the case of a collapsing sphere, the retarded field steals energy from the kinetic energy of the collapse.

Suppose that retardation keeps time running at a fraction h / 2 faster at the center of the collapsing shell than it would run if the signal speed were infinite. Then most of the retardation energy is stored in the interaction of the mass near the center, M, and the retarded field. The energy is

       ~ h / 2  *  M c²,

that is, linear in h.

If there is no mass near the center, the energy of the retarded field might be similar to the analogous electric field. Another way to estimate the energy of the gravity field is to calculate the pseudotensor. Generally, the energy density is quadratic in h.

https://blogs.helsinki.fi/kosmokoulu/files/2018/05/Solvalla_2018_detail.pdf

Mark Hindmarsh (2018) gives the following approximate formula for the energy density of gravitational waves:

The dot means the derivative over time. The brackets ⟨ ... ⟩ denote an average over an entire wave. The components hij are differences from the flat Minkowski metric.

Let us calculate a very rough estimate for the collapse into a neutron star. We assume that the mass 3 * 10³⁰ kg forms a thin shell. What is the energy of the retarded gravity field inside the shell?

In the previous blog post we calculated that h₀₀ ≈ 0.13 at the center, that is, time runs 6.4% faster at the center than for a flat metric.

Let us assume that the metric becomes flat when the signal to the center reaches from the radius r = 10 km of the neutron star which just formed. Then

       dh₀₀ / dt  =  h₀₀ c / r.

The energy density of the retarded gravity field is something like

       egw  =  c² / (32 π G)  *  h₀₀² c² / r²

               =  c⁴ / (32 π G)  *  h₀₀² / r²,

and the total energy

       E  =  1/24  *  c⁴ / G  *  h₀₀²  *  r

            = 1/24  *  8 * 10³³  *  1.5 * 10¹⁰

                * 0.13²  *  10⁴ J

     

            = 8 * 10⁴⁴ J.

This is interesting. The energy is of the same order of magnitude as we calculated in the previous blog post (13 versus 8) from the interaction of the mass near the center and the retarded gravity field (gravity potential).

Maybe the radial metric g₁₁ is stretched, too? In that case the energy density ggw must be doubled.

The energy of the pure retarded gravity field, without any mass in it, is significant!

What is the energy of the retarded field if we calculate it from the analogous electric field?

In the force formula,

       G  ~  1 / (4 π ε₀).

The energy density of a gravity field is then

       e  =   1 / (8 π G)  *  Eg²,

where Eg is the strength of the gravity field (force / mass).

If the potential is h₀₀ / 2 * c² over a distance of r = 10 km, the field strength is

       Eg  =  h₀₀ / 2  *  c² / r.

The energy density of the retarded gravity field is then

       e  =  c⁴ / (32 π G)  *  h₀₀² / r²

           = egw.

We see that the energy density e agrees with the one which we calculated from the formula for gravitational waves!

Does the energy of the retarded gravity field approach the total released energy in a collapse into a black hole?

The gravitational energy released in a collapse into a neutron star of a mass 3 * 10³⁰ kg is roughly 300 * 10⁴⁴ J. The energy in the retarded gravity field is something like 13 * 10⁴⁴ J – not a very large portion of the total energy.

What if the collapse would go further, so that the system becomes a black hole? Could the energy of the retarded field approach the total energy released in the collapse?

As the shell approaches the Schwarzschild radius r = 4.5 km, its speed approaches c.

               signal

               ----------------------------------------------->

               --> 0.4 c               --> c

              |                           |                          ×

       r = 10 km              r = 4.5 km           center

  

       potential              potential

       -0.2 c²                    -c²

Let the shell at r = 10 km send a signal to the center. When the signal arrives, the shell is already close to the Schwarzschild radius r = 4.5 km. But the observer at the center thinks that the radius is 6 km.

https://en.wikipedia.org/wiki/Schwarzschild_metric

The potential difference at the center versus close to the shell is then ~ 0.5 c². Then 

       h₀₀ / 2  *  c²  =  0.5 c²,

or h₀₀ = 1. The energy of the retarded gravity field is

       E  =  1/24 * c⁴ / G * h₀₀² * r.

The energy is (1 / 0.13)² * 0.45 = 30 times that which we calculated in the previous section, or 240 * 10⁴⁴ J.

The potential energy released is 

      M c²  =  3 * 10³⁰  *  9 * 10¹⁶ J

                =  2700 * 10⁴⁴ J.

Our calculation is extremely crude. It shows that the energy of the retarded gravity field approaches a significant portion of the released potential energy when a shell collapses into a black hole.

This suggests that in the reverse process, an explosion, the retarded gravity field might boost the expansion speed greatly, which might explain "dark energy" in the universe.

An explosion of a mass shell

Can a retarded gravity field release energy enough, so that it could accelerate the expansion?

In a collapse, some of the energy goes to the retarded field, so that kinetic energy grows slower. But kinetic energy does not decrease at any point.

Let us then imagine an explosion of a shell. As the shell expands, gravity slows down its expansion. A clock at the center "thinks" that the shell has expanded more than it actually has. The clock ticks faster than it would if the speed of signals were infinite.

Let us assume that the speed of the expansion approaches some constant value v > 0 as the radius r becomes very large.

                                              signal

                                             -------------------->

 0.4 c <--                        <-- c

              |                            |                       ×

           r = 10 km           rs = 4.5 km       center

           potential           potential

           -0.2 c²                -c²

Above we have the start of an expansion from an almost-black-hole. When the center receives the signal, a clock there "deduces" that the shell is at r = 9 km. But the velocity of the shell has already fallen substantially. Let us guess that the shell is at r = 8 km.

In the Schwarzschild metric, the potential at a radius r is

       (-1  +  sqrt(1  -  rs / r))  *  c².

The potential at 9 km is -0.29 c², and at 8 km it is -0.34 c². Then h₀₀ = 0.1.

Let us calculate the energy of the retarded gravity field:

       E = 1/24 * c⁴ / G * h₀₀² * r

           = 1/24  *  81 * 10³²  *  1.5 * 10¹⁰

              * 1² / 10²  *  8 * 10³

           = 16 * 10⁴⁴ J.

The negative potential is -900 * 10⁴⁴ J at r = 8 km. The energy of the retarded field is not very large compared to this.

Is there any reason why we should assume that the potential at the center is very different from -0.29 c²? Maybe it should be zero?

A cosmological model is uniform: the retarded gravity field is visible only in "static" coordinates?

In the collapse to a neutron star, we claimed that the retarded gravity field holds a substantial energy. Our claim is based on the potential difference between the mass shell center and close to the shell.

But in the universe, no such potential differences seem to be present. However, the equipotential state happens in the comoving coordinates. In "static" coordinates, there is a potential difference. That difference slows down the expansion of the universe.

If dark energy is in the retarded gravity field, then that energy is visible in static coordinates.

Let us assume that the universe is a large uniform expanding ball, embedded in an asymptotically Minkowski space.

Let us assume that gravity behaves, for some reason, "uniformly" in comoving coordinates, so that the ball expands uniformly, just like in cosmological models.

We probably can study the center of the ball in such a way that we ignore other parts. We only look at the "observable universe" whose radius is something like 30 billion light-years.

Then our considerations above about an expanding mass shell are relevant. The main difference is that the shell expands at almost the speed of light, and its radius is huge.

       E = 1/24 * c⁴ / G * h₀₀² * r.

The energy of the retarded field is linear in r for an almost-black-hole. The mass of a black hole is linear in r.

The mass inside a radius r is proportional to r³ in a cosmological model.

Can the retarded gravity field energy be released in a cascade?

Is there a reason why almost all the energy in the retarded gravity field should be released just now when the mass density of the universe is 0.3 times the "critical density"?

Yes: if the expansion no longer slows down, then all the energy in the retarded field must be released. Recall that the energy is there only because the expansion was slowing down. If the expansion slows down enough, it may start a cascade in which all the energy is released from the retarded field.

This would mean that the expansion of the universe may accelerate a little now, and then start slowing down again, as predicted by cosmological models. In the long run, the expansion, on the average, obeys cosmological models.

Let us try to estimate if the cascade happens in the expansion.

                           ^  bulge

                           |

      ____                                ___

              \         _____         /

                 \ • /            \ • /

            v <--                    --> v

       weight                     weight

Let us first consider a rubber membrane model. A ring of weights expands, and the membrane between them bulges upward as the velocity of the weights, v, slows down.

Could it be that no cascade happens?

If the energy stored in the bulge is just a few percents of the negative potential energy, then probably no cascade occurs.

We need better models and ideas to explain dark energy.

The length contraction of the gravity field of a mass moving at almost c

            \     /

              \ /

         m  • -->  ≈ c

              / \

            /     \

       "squeezed"

        gravity field

Suppose that a mass shell expands almost at the speed c. If the gravity field behaves like the electric Coulomb field, then the gravity field is very much "squeezed" in the direction of the movement of m. The field of m is not much felt inside the shell. This would mean that the a clock at the center of the shell would not be much slowed down. 

Does this make sense? If the shell is initially static, and the shell consists of photons, what happens?

The potential initially corresponds to a static shell. Then there is a pressure shock, as the shell is shot outward at the speed of light.

The center "knows" that the shell has moved far away, and can "calculate" the potential and adjust the clock rate accordingly.

If we shoot a photon shell at the speed of light toward the center, then the center does not "know" that it is at a lower potential as the shell approaches. In this case, a clock at the center should run at almost the rate of outer space. In this case, the gravity field behaves as it would be "squeezed", and not affect the center.

In the contracting case, the retarded gravity field can hold a lot of energy.

If the shell contracts at a velocity slightly less than c, an observer will receive a warning of the approaching shell wall early on.

Could it be that the speed of light inside the shell remains faster than at the shell? In that case, an observer at the center might see the contraction slowing down as the shell wall approaches?

We deduce the expansion speed of the universe from the redshift. In the case of a contracting universe, we look at the blueshift. The blueshift can, indeed, decrease if the clock rate at the shell wall slows down.

The redshift inside an expanding mass shell

                            ^ bulge

                            |

                            o  observer

      ____                                 ___

              \          _____          /

                 \ • /              \ • /

            v <--                    --> v

            weight           weight

Above is the rubber membrane model. Let us have an observer at the center, on the "bulge".

He sees a redshift in the receding shell. The redshift depends both on the velocity v and the rate of the clock on the bulge. If the bulge grows taller, then he sees a larger redshift.

The energy of the distorted radial metric in an expansion of a mass shell: it is 6 times the energy of the metric if time?

Thus far, we in this blog have for the retarded gravity field calculated the presumed energy of the distorted metric of time inside the sphere.

However, most of the energy in a gravitational wave seems to be in the distorted spatial metric. Let us try to estimate that energy heuristically.

      shell wall

         <-- |                    ×                      | -->

                                   clock rate

                                   d faster

Suppose that time flows a fraction d faster at the center of an expanding shell than close to the shell.

If we imitate the Schwarzschild metric, then radial distances are stretched by that same fraction 

       d  = h₀₀ / 2.

Let us put an extremely rigid material inside the shell. When the retarded gravity field tries to stretch the material, a negative radial pressure forms.

The negative pressure tries to flatten the spatial metric.

https://en.wikipedia.org/wiki/Komar_mass

The formula of the Komar mass gives us a guess of how much pressure is needed to flatten the metric. In the Komar formula, the  pressure in each spatial direction x, y, z is counted as an "energy density".

A positive mass-energy M c² stretches the radial metric. It can be "compensated" with an "equal" negative pressure.

In our example in a preceding section, h₀₀ = 0.1 and d = 0.05.

How much positive mass density do we need to create the d = 5% potential difference? Let ρ be the hypothetical energy density inside the shell. The potential difference between the center and the shell is then

       d c²  =  1/2 G M / r

                = 1/2 G * 4/3 π * 1 / c² * ρ r³ / r

                = 2/3 π G / c²  *  ρ r²

  =>

       ρ  =  3/2 * 1 / π * c⁴ / G  * 1 / r²  *  d.

The energy which we could hypothetically extract from a pressure ρ and a stretched metric by d is

       W  =  1/2 ρ d * 4/3 π r³

             = c⁴ / G  *  d² r

             = 1/4 c⁴ / G  *  h₀₀²  *  r.

The energy is 6X the energy which we got for the metric of time, g₀₀. In the expansion case of a preceding section, the energy thus is

       96 * 10⁴⁴ J.

The negative potential energy is -900 * 10⁴⁴ J.

The energy of the retarded field is probably too small to start a cascade which would accelerate the expansion.

However, for the collapse into a neutron star, this is big news. Above calculated that the energy of the retarded g₀₀ is something like 8 * 10⁴⁴ J, which is a bit low to explain the supernova explosion. A 7-fold energy 56 * 10⁴⁴ J explains it better.

But is the radial metric stretched for a retarded gravity field? Probably not

Retardation causes an observer at the center to miscalculate the radius of the mass shell. That should his clock to tick at a "wrong" rate.

But the spatial metric does not change if we slowly lower the shell down. Thus, retardation should have no effect on the spatial metric.

It seems likely that retardation does not affect the spatial metric.

The energy of the retarded gravity field in a Big Bounce

The observable universe came out from an explosion or a Big Bounce. Could it be that a lot of energy at that time was uploaded to the gravity field, and that energy only is released now when the mass density falls below the critical density?

                        pit

       ____                          ____  membrane

               • ------___------ •

         <-- a                     a -->

               mass       mass

In the first phase of the expansion, the expansion is accelerated. The retarded gravity field lags behind. In the rubber membrane model of an expansion of a mass shell, there is a "pit".

As gravity slows down the expansion, the pit becomes a bulge. What happens to the energy of the pit in the transition? If it becomes kinetic energy of the masses, then they might be accelerated, for a little while.

Dark energy is the expected result from any large explosion?

Let us assume that an unknown law of nature produces a large explosion. Retardation is present. Then it is expected that the gravity field of the system will contain a lot of energy which comes from retardation. The gravity field does not have time to settle down to its "lowest energy state". We can claim that "dark energy" is the expected result of such an explosion.

Since the energy in the retarded gravity field cannot escape as longitudinal gravitational waves from a spherically symmetric explosion, the energy has to be absorbed by the matter. The absorption process must be spherically symmetric, and must increase the energy of the matter. The natural way is to increase the radial velocity of the matter.

When the mass density of the universe falls a lot below the "critical density", we can regard that as the end stage of the explosion. Then the energy in the retarded gravity field must be absorbed by the matter. This explains why the expansion of the universe is accelerating right now.

When the energy has been absorbed, we can expect the universe to expand at an almost fixed rate ever after. Gravity will slow down the expansion only a little.

Our hypothesis has a serious weakness, though: it does not give any numerical prediction of how much dark energy we should observe. The hypothesis is purely qualitative.

The energy in the retarded gravity field gravitates, too. How does that affect the expansion of the universe? The mass density of the universe is significantly larger than the amount of matter and dark matter.

Dark energy implies that the metric of time is "upwards convex" close to Earth

By "close" we mean distances less than 5 billion light-years. Dark energy means that "close" galaxies are accelerating away from us – even though gravity should decelerate them.

Matter tends to be accelerated toward a place where the gravity potential is lower, or where the metric of time, |g₀₀|, is smaller in absolute terms.

If the potential is "sphere-like", so that its curvature is the same to all directions, we will observe all nearby objects accelerating uniformly away from Earth.

                  bulge

       -------                 -------  rubber membrane

               \•__----__•/

        v <--              --> v

       weight      weight

Let us consider a rubber membrane model of gravity. Let the weights recede from the center. The weights slow down, because of gravity. Then the membrane at the center of the ring of weights will form a "bulge" upward. The potential will be convex upward. But why would the potential be spherical?

Another problem: if we have a uniform, exploding dust ball, why can we assume that only its edges have a lot of mass, so that it is like the expanding ring of weights in the rubber model? The edge moves at almost the speed of light and has a lot of kinetic energy. This suggests that almost all the mass would be at the edges?

If the interior of the dust ball has only a little mass, then the upward bulge in the rubber membrane model can form. The interior will feel "dark energy".

Another problem: if the edge of the ball moves at almost the speed of light, then gravity will not decelerate it a lot, since it keeps moving at almost the speed of light. Why would a bulge upward form in the rubber model in that case?

The matter density of the universe has recently grown "thin" relative to the "critical" density – squeezed gravity fields

One of the implications is that we can use almost newtonian gravity to describe the gravity field locally. Maybe the effects of gravity are linear and can be summed for each mass element dm?

                                   squeezed

                                 gravity field

                                       \   |   /

                                         \ | /   ≈ c

                    ×                     • --->  

                                         / | \

             observer                   mass element dm

Let us consider a shell of mass expanding rapidly, at almost the speed of light.

The gravity field of each element dm is "squeezed" in the direction of the movement of dm. Does this mean that a clock at the center of the shell will tick faster than a clock close to the shell? If yes, then an observer inside the shell will feel a force which pulls him toward the shell. This could be the "dark energy"?

If we compare this to an expanding shell of electric charge, the squeezing of the electric field does not matter because the field is zero inside the shell, and we do not have a clock which would be affected by the electric potential.

If we have an expanding dust ball, the kinetic energy of its edges causes most of the gravitating mass-energy to be at the edges. Observers at a "typical" position in the dust ball will see an "upward convex" gravity potential.

But why would dark energy only show up after the universe is 8 billion years old and its density has dropped significantly below the critical density?

Conclusions

Let us close this blog post. We were not able to find a mathematical model for the "dark energy" caused by retardation, yet. We will return to this later.