<p>In the expanding universe the luminosity distance is related to the redshift by,
<span class="math-container">\begin{equation}
d_L=c(1+z)\int_0^z \frac{dz'}{H(z')}
\end{equation}</span>
where <span class="math-container">$H(z')$</span> - is the rate of the universe expansion at redshift <span class="math-container">$z$</span>. You can determine the rate from the Friedmann equations (basically Einstein equations for the cosmological spacetime)
<span class="math-container">\begin{equation}
H(z)=H_0\sqrt{\frac{\rho(z)}{\rho_0}}
\end{equation}</span>
where <span class="math-container">$\rho(z)$</span> and <span class="math-container">$\rho_0$</span> is energy density at redshift <span class="math-container">$z$</span> and now correspondingly and <span class="math-container">$H_0$</span> is basically the universe rate of expansion right now that determines the slope of the Hubble law line.</p>
<p>Now, if you know the composition of the universe you may say how the function <span class="math-container">$\rho(z)$</span> behaves. What the authors of the paper do is that they take this theoretical function <span class="math-container">$\rho(z)$</span> (for the standard cosmological model <span class="math-container">$\Lambda$</span>-CDM as well as for the model with dynamical dark energy) and simply replace the constant <span class="math-container">$H_0$</span> with some function <span class="math-container">$H_0(z)$</span>. 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.</p>
<p>Now, the question is how do you interpret this fit. We could simply wrongly understood the behaviour of the function <span class="math-container">$\rho(z)$</span> but the authors argue against this point of view pointing out that they have to make the <span class="math-container">$H_0$</span> 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.</p>

CC BY-SA 4.0
2021-08-18T14:12:53.153Z
2022-08-06T19:59:23.517Z
2022-08-06T19:59:23.517Z
OON
<p>The &quot;crisis in cosmology&quot; is a tension between the values of <span class="math-container">$H_0$</span> calculated from high-<span class="math-container">$z$</span> and low-<span class="math-container">$z$</span> astronomical data.</p>
<p>This paper modifies ΛCDM cosmology by making <span class="math-container">$H_0$</span> a function of <span class="math-container">$z$</span>. That makes no sense given how <span class="math-container">$H_0$</span> is defined. <span class="math-container">$H$</span> is already a function of <span class="math-container">$z$</span>, and <span class="math-container">$H_0=H(z{=}0)$</span> by definition. The assumption that <span class="math-container">$H_0$</span> depends on <span class="math-container">$z$</span> amounts to a restatement of the problem, not a solution.</p>
<p>They find a low-<span class="math-container">$z$</span> trend in <span class="math-container">$H_0$</span> that matches the high-<span class="math-container">$z$</span> value of <span class="math-container">$H_0$</span> 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.</p>
<p>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.</p>

CC BY-SA 4.0
2021-11-02T21:44:07.300Z
2021-11-02T21:44:07.300Z
benrg
<p>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.</p>
<p>For the record, here's the link to the paper: <a href="https://news.umich.edu/an-inconstant-hubble-constant-u-m-research-suggests-fix-to-cosmological-cornerstone/" rel="noreferrer">https://news.umich.edu/an-inconstant-hubble-constant-u-m-research-suggests-fix-to-cosmological-cornerstone/</a></p>

CC BY-SA 4.0
2021-11-02T22:10:54.397Z
Well no, more accurate measurements made the Hubble tension appear in the first place...
AtmosphericPrisonEscape
<cosmology><expansion><redshift><observable-universe><hubble-constant>
How does the hypothesis of the "inconstant Hubble constant" solve the current crisis in cosmology?
bestofthebeast
# How does the hypothesis of the "inconstant Hubble constant" solve the current crisis in cosmology?

(Asked by: username_0 on 2021-08-18)

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/>

**Tags:** cosmology, expansion, redshift, observable-universe, hubble-constant

> **Comments:**
> **username_1**: Well no, more accurate measurements made the Hubble tension appear in the first place...  

---

## 2 answers

### Answer by username_2 on 2021-08-18. Score: 6

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 interpret 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.


---

### Answer by username_3 on 2021-11-02. Score: 4

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.


---

