The word explanation in the question is expansion.

You mean a range of measurements of the hubble constant H.

You can ask AI . for properly interpreted measurements through

v=Hr, the definition. Valid only for really cosmological distances, and not devoid

Of disagreements.

In a homogeneous and isotropic cosmological model the the definition v= Hr is not true in general.

The minimum estimated value at the moment is about 68.5 km/sec/Mpc (although the actual value may be as much as 71.5 km/sec/Mpc. When I estimate nearby galaxy distances I use 70 km/sec/Mpc, since that's the value I first estimated in the 1960's, and is now the average of the two values shown above. In the future this will gradually increase, probably asymptotically approaching a value between 80 and 100 km/sec/Mpc, but we need billions of years of observations to confirm this range, let alone pin down the actual final value, and since none of US will be around for that long, WE will never know the answer.

However, as you yourself point out, even given the above you cannot use r = v/H to find the correct distance, for two reasons:

1. Most galaxies have random velocities relative to their neighbors (so-called "peculiar velocities"), particularly in clusters of galaxies, where the mutual gravitational interactions of the galaxies in the cluster can cause deviations from the "average" motion of galaxies at their distance to differ from the observed value by anywhere from 200 to 1000 km/sec, so even if the equation was correct, v might not be, in which case the estimated "r" would be wrong.

2. Just using the simple equation does not take into account the expansion of the space between the galaxy and us during the light-travel time required for both the original distance, and the expansion of that distance while the emitted light is passing through gradually expanding space. That requires a relativistic equation which relies on the "shape" of the Universe (undoubtedly hyperbolic on scales far larger than the Observable Universe, but apparently "flat" on scales as small as the current size of the "Observable' universe). This is similar to the fact that at any given place on the Earth, although the Earth is round, in most places (particularly at sea) it appears to be essentially "flat". These equations also rely on the estimated amount of mass of the Universe, relative to the so-called "critical" mass (somewhere in the range of 27%, but most of that is so-called "dark matter", so the actual value is uncertain). In general, this makes the actual distance at the time a distant galaxy's light was emitted LESS than the value obtained from r=v/H (though for v less than about 6 or 7000 km/sec, the error in the distance is considerably smaller than the "peculiar velocities" noted in '1".

This

“…To construct a homogeneous and isotropic cosmological model it is necessary to know the maximum amd minimum values of the expansions.…”

- really is a fundamentally now vague assertion, since really to say about the cosmological expansion, which is defined in mainstream cosmology as “space expansion”, what is fundamentally wrong, as that is rigorously shown in the Shevchenko-Tokarevsky’s really philosophical 2007 “The Information as Absolute” conception, recent version of the basic paper see

https://www.researchgate.net/publication/363645560_The_Information_as_Absolute_-_2022_ed

Compact, but essentially clear and rather in detail introduction in the conception see in

https://www.researchgate.net/publication/387933457_A_response_to_question_What_is_the_The_Information_as_Absolute_conception,

- and more concretely in cosmology, see SS&VT Planck scale informational physical model, in this case section “Cosmology” in

https://www.researchgate.net/publication/367397025_The_Informational_Physical_Model_and_Fundamental_Problems_in_Physics ,

- Matter’s utmost universal spacetime is [why? – see the links above, first pages] fundamentally absolute, fundamentally flat, fundamentally continuous, and fundamentally “Cartesian”, [5]4D spacetime with metrics (cτ,X,Y,Z,ct), where cτ,X,Y,Z, are 4 space dimensions, ctis the time dimension, which fundamentally cannot be impacted by anything in Matter and impact on anything in Matter, including so cannot expand.

However, everything in Matter is/are some disturbances in the Matter’s ultimate base - (at least) [5]4D dense lattice of primary elementary logical structures – (at least) [5]4D binary reversible fundamental logical elements [FLE], which is placed in the spacetime above, FLE “size” and “FLE binary flip time” are Planck length, lP, and Planck time, tP. The lattice indeed really can expand, and, at that, matter – first of all cosmological objects/structures - can move apart each other, and apart – in this case radially - from the point in the spacetime, where the first FLE was made at the Matter’s Beginning.

But, again, that happened/happens in the [5]4D spacetime above, while in the cosmology it is postulated, and the observations data are interpreted [including the expansion] provided this, that Matter’s spacetime is 4D pseudo Riemannian space, what is fundamentally wrong.

So here let to reiterate a passage from reDzennn comment in/to https://phys.org/news/2025-05-dark-evolving-energy.html “…while now the standard lCDM is something like to 1500s years cosmology, which was based on the postulate that Sun rotates around Earth, was in complete accordance with the experimental observations, and was successfully applicable in practice; say, Magellan in 1520 and others near 100 years after, well correctly traveled on whole Earth sphere navigating by this model...”

So, again, any really scientifically reasonable interpretations in cosmology will appear only after the Matter’s topology in the 5D spacetime [as that is proposed in the section “Cosmology” above – on surface of a 4D hypersphere] above will be developed.

Cheers