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How is Hubble constant measured by the method of observing BAO?By just reading some articles it seems to me a more complex method than the astronomic scale.Can this method be explained in few words to point the idea how data is fetched from observing oscillations in power of emissions and is there some way to figure out why we encountered the crisys in measuring the Hubble constant?

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BAO experiments essentially measure two quantities, one parallel to the line-of-sight \begin{equation} \beta_{\parallel} = H(z_{\rm eff}) r_s (z_{\ast}) \end{equation} and the other perpendicular to the line-of-sight \begin{equation} \beta_{\perp} = \frac{ r_s (z_{\ast})}{D_A(z_{\rm eff})} \end{equation} Here $H(z)$ is the Hubble parameter, $ r_s (z_{\ast})$ is the comoving sound horizon at recombination (i.e. the standard ruler) and $D_A(z_{\rm eff})$ is the comoving angular distance to the observation redshift, $z_{\rm eff}$. The latter is computed as \begin{equation} D_A(z_{\rm eff}) = \int_0^{z_{\rm eff}} \frac{dz}{H(z)} \end{equation} The standard ruler $ r_s (z_{\ast})$ is well constrained by CMB experiments. For the shape of $H(z)$, one needs to assume some model (such as $\Lambda$CDM). Thus, by fitting the theoretical predictions for $\beta_{\parallel} $ and $\beta_{\perp}$ to the BAO data, we get indirect constraints on the expansion history of the universe, $H(z)$, and thus on the Hubble constant $H_0 = H(z=0)$. In a similar way to other probes of the early universe (as the CMB), this method gives a value of $H_0$ that is in tension with the direct measurement in the local universe (using the cosmic distance ladder). Note that even if BAO observations are made in the late universe (by looking at the large-scale distribution of galaxies), it is considered as a early probe because It provides a constraint on $ r_s (z_{\ast})$, that gives information about the primordial plasma.

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