# The Difference between angular and anisotropic BAO

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:

1. https://arxiv.org/abs/1910.14125 equations 13 and 14

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 $$$$\theta_{\rm no\ expansion} = \frac{r_d}{d}$$$$ Now we need to account for the expansion. What we will directly measure is the angular diameter distance $$d_A(z)$$ at redshift $$z$$, but what we need to measure the angle above is the comoving distance $$d_c(z)$$. These are related by (https://arxiv.org/abs/astro-ph/9905116) $$$$d_c(z) = (1+z) d_A(z)$$$$ Therefore the angular size including expansion (in radians) is $$$$\theta = \frac{r_d}{d_c(z)} = \frac{r_d}{(1+z) d_A(z)}$$$$

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 $$$$\theta^{fid} = \frac{r_d^{fid}}{(1+z) d^{fid}_A(z)}$$$$ where $$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}$$ parameter as the ratio of the angular size in a generic cosmology, to the fiducial cosmology $$$$\alpha_\perp = \frac{\theta}{\theta^{fid}} = \frac{r_d}{d_A(z)} \frac{d_A^{fid}(z)}{r_d^{fid}}$$$$

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 $$$$\alpha = \theta \frac{180}{\pi} = \frac{r_d}{(1+z)d_A(z)}\frac{180}{\pi}$$$$

• So they measure the same thing, but one of them uses a fiducial cosmology and the other one does not ? Is that the only difference ? One measures the actual angle and the other one the angle w.r.t other cosmological model...but it's still the same I guess Commented Nov 21, 2021 at 16:22
• @SeVenVo1d Right, the fundamental observable is the angular size of the sound horizon. They have different ways of measuring it, which is a good thing because they can check that the different analyses (which make different assumptions and presumably have different sources of uncertainty) give consistent results. Commented Nov 21, 2021 at 18:20
• I see...The strange thing is that at the end of the article it says there's strong inconsistency between the angular and anisotropic BAO. Commented Nov 21, 2021 at 18:28
• @SeVenVo1d I see, I wasn't aware of that. That sounds very interesting! Commented Nov 21, 2021 at 18:29
• I am also not sure about that. I am also trying to understand the article and the meaning behind it. But yes, they are proposing a new way to measure the $H_0$. I know that there's also some tension between galaxy BAO and Ly$\alpha$ BAO. Yes..these are very interesting discrepancies. I am no expert either, but this method (using $M_B$ prior seems accepted in the literature). This method also looks promising. There are some things wrong with the $LCDM$. Let us hope someone finds it soon :p Commented Nov 21, 2021 at 19:53