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 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.

Can we figure out new ways to probe r* or H₀?

***  WORK IN PROGRESS  ***

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.