Forecasting the Ly-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:math> forest <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mo stretch

We forecast the sensitivity of future correlations between cosmic microwave background (CMB) lensing and the Lyman-<a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>α</a:mi></a:math> forest power spectrum. This squeezed-limit bispectrum probes the connection between the Lyman-<c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>α</c:mi></c:math> transmission and the underlying large-scale matter density field. We recover the measured signal-to-noise (SNR) ratio obtained with <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mrow><e:mi>Planck</e:mi><e:mo stretchy="false">×</e:mo><e:mi>BOSS</e:mi></e:mrow></e:math>, and forecast a SNR of 10.3,14.8 and 20.2 for DESI combined with ACT, SO, and CMB-S4, respectively. For DESI and SO/CMB-S4, the correlation should be detectable at <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"><h:mi>SNR</h:mi><h:mo>≥</h:mo><h:mn>5</h:mn></h:math> in 5 redshift bins, useful for distinguishing our signal from several contaminants. Indeed, our forecast is affected by large theoretical uncertainties in the current modeling of the signal and its contaminants, such as continuum misestimation bias or damped-Lyman-<j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:mi>α</j:mi></j:math> absorbers. Quantifying these more accurately will be crucial to enable a reliable cosmological interpretation of this observable in future measurements. We attempt to enumerate the features required in future Lyman-<l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:mi>α</l:mi></l:math> forest simulations required to do so.