where $P_{\ell}$ are the Legendre polynomials, $\bar{n}$ is the number of quasars per steradian, and $f_{\mbox{\footnotesize sky}}$ is the fraction of the sky observed in the survey. In figure \[fig2\] we show the analytical covariance matrices for the linear and non-linear cases. For the sake of completeness, in the figure \[fig6\] of the Appendix section we compare the error estimates from this theoretical approach, for the linear and non linear cases, to other error estimates, namely, from the jackknife and bootstrap resampling methods.

![image](figure2a){width="61.00000%"} ![image](figure2b){width="61.00000%"}

We calculated the statistical significance of the BAO peak detection with the linear and non-linear analytical covariance matrices by following ref. [@Shanks], that is, we compute the $\chi^2$ statistics as a function of the scale dilation parameter $\alpha$, given by $$\label{chi}
\chi^2(\alpha) = [\omega(\theta) - \omega^{\mbox{\tiny FIT}}(\alpha \theta)]^T \mbox{\sc Cov}^{-1} [\omega(\theta) - \omega^{\mbox{\tiny FIT}}(\alpha \theta)] \, ,$$ where $\mbox{\sc Cov}^{-1}$ is the inverse of the covariance matrix; the brackets $[\,]$ and $[\,]^T$ represent column and row vectors, respectively; $\omega(\theta)$ is the 2PACF data points and $\omega^{\mbox{\tiny FIT}}(\alpha \theta)$ is one of two possible best-fit curves to the data: the one given by Eq. \[ajuste\], and the one given only by the power law and constant 