Accelerating BAO Scale Fitting Using Taylor Series

The Universe is currently undergoing accelerated expansion driven by dark energy. Dark energy's essential nature remains mysterious: one means of revealing it is by measuring the Universe's size at different redshifts. This may be done using the Baryon Acoustic Oscillation (BAO) feature, a standard ruler in the galaxy 2-Point Correlation Function (2PCF). In order to measure the distance scale, one dilates and contracts a template for the 2PCF in a fiducial cosmology, using a scaling factor $\alpha$. The standard method for finding the best-fit $\alpha$ is to compute the likelihood over a grid of roughly 100 values of it. This approach is slow; in this work, we propose a significantly faster way. Our method writes the 2PCF as a polynomial in $\alpha$ by Taylor-expanding it about $\alpha = 1$, exploiting that we know the fiducial cosmology sufficiently well that $\alpha$ is within a few percent of unity. The likelihood resulting from this expansion may then be analytically solved for the best-fit $\alpha$. Our method is 48-85$\times$ faster than a directly comparable approach in which we numerically minimize $\alpha$, and $\sim$$12,000 \times$ faster than the standard iterative method. Our work will be highly enabling for upcoming large-scale structure redshift surveys such as that by Dark Energy Spectroscopic Instrument (DESI).