A Interpretation of an anticorrelation between š»0 and log10(šœ”šµš·)

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The discussion centers on the observed anticorrelation between H0 and log10(ωBD), which initially suggests a correlation instead. A derived equation expresses H0 as a function of ωBD, leading to confusion about the expected relationship. The correct derivation indicates that as ωBD increases, H0 actually decreases, confirming the anticorrelation. This resolution clarifies the mathematical relationship and supports the validity of the anticorrelation observed. The conclusion emphasizes that the initial derivation was incorrect, and the corrected formula aligns with the observed data.
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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.
triplot_TCL_HDF5_REFERENCE.png


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.
 
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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}##
 
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