Probing new physics beyond $\Lambda$CDM with large-scale structure

These lecture notes expand a 90-minute lecture delivered at the Les Houches Summer School “The Dark Universe” (July 2025). The goal is to give an overview of how we probe imprints of new physics beyond \Lambda <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>Λ</mml:mi> </mml:math> CDM with observations of the large-scale structure (LSS). Two case studies are discussed: (i) primordial non-Gaussianity (PNG) as a window into the field content and interactions during inflation; (ii) properties of neutrinos and light relics, namely the mass of neutrinos and the effective number of relativistic species. We outline why LSS is powerful for these questions, what we can learn, current bounds, the main analysis challenges, and prospects with ongoing and upcoming galaxy surveys. We will also illustrate the role of higher-order statistics in extracting non-Gaussian information of LSS and challenges on observational and theoretical fronts in obtaining robust and high-precision constraints on each of these imprints.