Revised LOFAR upper limits on the 21-cm signal power spectrum at $\mathbf{z}\approx \mathbf{9.1}$ using Machine Learning and Gaussian Process Regression

The use of Gaussian Process Regression (GPR) for foregrounds mitigation in data collected by the LOw-Frequency ARray (LOFAR) to measure the high-redshift 21-cm signal power spectrum has been shown to have issues of signal loss when the 21-cm signal covariance is misestimated. To address this problem, we have recently introduced covariance kernels obtained by using a Machine Learning based Variational Auto-Encoder (VAE) algorithm in combination with simulations of the 21-cm signal. In this work, we apply this framework to 141 hours ($\approx 10$ nights) of LOFAR data at $z\approx 9.1$, and report revised upper limits of the 21-cm signal power spectrum. Overall, we agree with past results reporting a 2-$\sigma$ upper limit of ${\mathrm{\Delta }}_{21}^2<(80{)}^2\mathrm{m}{\mathrm{K}}^2$ at $k=0.075h\mathrm{M}\mathrm{p}{\mathrm{c}}^{-1}$. Further, the VAE-based kernel has a smaller correlation with the systematic excess noise, and the overall GPR-based approach is shown to be a good model for the data. Assuming an accurate bias correction for the excess noise, we report a 2-$\sigma$ upper limit of ${\mathrm{\Delta }}_{21}^2<(25{)}^2\mathrm{m}{\mathrm{K}}^2$ at $k=0.075h\mathrm{M}\mathrm{p}{\mathrm{c}}^{-1}$. However, we still caution to take the more conservative approach to jointly report the upper limits of the excess noise and the 21-cm signal components.