# How does the hypothesis of the "inconstant Hubble constant" solve the current crisis in cosmology?

It was published in a paper more or less like two months ago. I'd like to know also if more accurate measurements are necessary to close the gap between the model of the universe and the data reported.

For the record, here's the link to the paper: https://news.umich.edu/an-inconstant-hubble-constant-u-m-research-suggests-fix-to-cosmological-cornerstone/

• Well no, more accurate measurements made the Hubble tension appear in the first place... Nov 2 at 22:10

In the expanding universe the luminosity distance is related to the redshift by, $$\begin{equation} d_L=c(1+z)\int_0^z \frac{dz'}{H(z')} \end{equation}$$ where $$H(z')$$ - is the rate of the universe expansion at redshift $$z$$. You can determine the rate from the Friedmann equations (basically Einstein equations for the cosmological spacetime) $$\begin{equation} H(z)=H_0\sqrt{\frac{\rho(z)}{\rho_0}} \end{equation}$$ where $$\rho(z)$$ and $$\rho_0$$ is energy density at redshift $$z$$ and now correspondingly and $$H_0$$ is basically the universe rate of expansion right now that determines the slope of the Hubble law line.

Now, if you know the composition of the universe you may say how the function $$\rho(z)$$ behaves. What the authors of the paper do is that they take this theoretical function $$\rho(z)$$ (for the standard cosmological model $$\Lambda$$-CDM as well as for the model with dynamical dark energy) and simply replace the constant $$H_0$$ with some function $$H_0(z)$$. If you fit this function according to the supernova data for various redshifts you can relax somewhat the tension with the data obtained from CMB.

Now, the question is how do you interprete this fit. We could simply wrongly understood the behaviour of the function $$\rho(z)$$ but the authors argue against this point of view pointing out that they have to make the $$H_0$$ dynamical even for the model with dynamical dark energy (though they choose some specific values of the parameters and I wonder how rigidly they are constrained by the CMB observations) We could neglect some effect of the nearby inhomogeneity, some astrophysics. Or we must somehow modify the gravitational dynamics. All this is mentioned but no more than that. Basically this paper made some modified fit and it worked somewhat better. The interpretation is open.

The "crisis in cosmology" is a tension between the values of $$H_0$$ calculated from high-$$z$$ and low-$$z$$ astronomical data.

This paper modifies ΛCDM cosmology by making $$H_0$$ a function of $$z$$. That makes no sense given how $$H_0$$ is defined. $$H$$ is already a function of $$z$$, and $$H_0=H(z{=}0)$$ by definition. The assumption that $$H_0$$ depends on $$z$$ amounts to a restatement of the problem, not a solution.

They find a low-$$z$$ trend in $$H_0$$ that matches the high-$$z$$ value of $$H_0$$ when extrapolated. There is some content in this, but it's something that one would expect to be true for some relatively simple extrapolating function regardless of the reason for the discrepancy. It says almost nothing about the reason.

In general, you can expect to get a better fit in a model with more parameters. Even if the model has a strong theoretical foundation, you can't conclude that it's closer to reality than ΛCDM just because it fits the data better, if it has more parameters. That's doubly true of a model that was obviously chosen solely to solve the one specific problem that they applied it to.