I was reading an article and saw these 2 different BAO definitions.
anisotropic BAO;
$$\alpha_{\perp} \equiv \frac{d_A(z)}{r_d}\frac{r^{fid}_d}{d_A(z)^{fid}}$$
angular BAO;
$$\theta(z) = \frac{r_d}{d_A(1+z)}\frac{180}{\pi}$$
Of course there's difference in the equations but in terms of, how they obtained or what they actually measure (differently or the same)?
References:
Let's start off with a non-expanding Universe. The goal is to measure the angular size of the BAO sound horizon, $r_d$, which is a length. Let's say our distance to the galaxies we are using to measure the sound horizon is $d$. Since $r_d \ll d$, we can say the angular size (in radians) is approximately \begin{equation} \theta_{\rm no\ expansion} = \frac{r_d}{d} \end{equation} Now we need to account for the expansion. What we will directly measure is the angular diameter distance $d_A(z)$ at redshift $z$, but what we need to measure the angle above is the comoving distance $d_c(z)$. These are related by (https://arxiv.org/abs/astro-ph/9905116) \begin{equation} d_c(z) = (1+z) d_A(z) \end{equation} Therefore the angular size including expansion (in radians) is \begin{equation} \theta = \frac{r_d}{d_c(z)} = \frac{r_d}{(1+z) d_A(z)} \end{equation}
So far, so good. Now we get into the messiness of how BAO measurements are actually done, according to the paper you linked to (https://arxiv.org/abs/1910.14125).
Anisotropic BAO
This analysis assumes a fiducial cosmology. Since the point of BAO measurements is to measure cosmological parameters, the analysis needs to be able to allow for deviations from this fiducial cosmology after the fact.
In the fiducial cosmology, the angular size is \begin{equation} \theta^{fid} = \frac{r_d^{fid}}{(1+z) d^{fid}_A(z)} \end{equation} where $r_d^{fid}$ is the sound horizon and $d^{fid}_A(z)$ is the angular diameter distance at redshift $z$ in the fiducial cosmology. They then define the $\alpha_{\perp}$ parameter as the ratio of the angular size in a generic cosmology, to the fiducial cosmology \begin{equation} \alpha_\perp = \frac{\theta}{\theta^{fid}} = \frac{r_d}{d_A(z)} \frac{d_A^{fid}(z)}{r_d^{fid}} \end{equation}
They also define an $\alpha_{||}$ parameter for measurements of BAO along the line of size, which will also depend on the expansion rate $H(z)$, although you didn't explicitly ask about this.
Angular BAO
This is another analysis that does not assume a fiducial cosmology (the paper calls this analysis "model independent"). Here, $\alpha$ is defined as $\theta$, but just converted from radians to degrees \begin{equation} \alpha = \theta \frac{180}{\pi} = \frac{r_d}{(1+z)d_A(z)}\frac{180}{\pi} \end{equation}
