Interpretation of an anticorrelation between 𝐻0 and log10(𝜔𝐵𝐷)

[fab13]

TL;DR

I get below the following contours of a MCMC run with the main cosmological parameters for Brans-Dickce's theory without introducing a cosmological constant ($\Lambda=0$) and considering only baryonic matter component.

Could you justify the anticorrelation that I get between $H_0$ and $\omega_{BD}$ (actually $\log10(\omega_{BD}$) ?

If we take the relation :

$\Omega_{B D}=\frac{\omega_{B D}}{6}\left(\frac{F_0}{H_0}\right)^2-\frac{F_0}{H_0}$, then I can express $H_0$ as a function of $\omega_{BD}$ :

$H_0=\frac{-F_0+\sqrt{F_0^2+\frac{2 \Omega_{B D \omega_{B D} F_0^2}}{3}}}{2 \Omega_{B D}} .$

From this relation, we are expected to have a correlation instead of an anti-correlation since if $\omega_{BD}$ increases, then, $H_0$ will increase.

If someone could help me to justify my result (if it is true), this would be great.

 

[fab13]

Solution : the derivation of $H_0$ is wrong and the valide one is :

$\begin{aligned} & \Omega=\frac{\omega}{6}\left(\frac{F}{H}\right)^2-\frac{F}{H} \\ & \frac{E}{H}=A \\ & \Omega=\frac{\omega}{6} A^2-A \\ & \frac{\omega}{6} A^2-A-\Omega=0 \\ & A_{1 / 2}=\frac{1 \pm \sqrt{1+\frac{4 \omega \Omega}{6}}}{2}, A>0 \\ & \frac{F}{H}=\frac{1+\sqrt{1+\frac{4 \omega \Omega}{6}}}{2}=\frac{1}{2}+\sqrt{4+\frac{\omega \Omega}{6}} \\ & H=\frac{F}{\frac{1}{2}+\sqrt{4+\frac{\omega \Omega}{6}}} \end{aligned}$

So there is an anticorrelation between $H_0$ and $\omega_{BD}$