  State-of-the-art 19th century spectroscopy led to the discovery of quantum
mechanics, and 20th century spectroscopy led to the confirmation of quantum
electrodynamics. State-of-the-art 21st century astrophysical spectrographs,
especially ANDES at ESO's ELT, have another opportunity to play a key role in
the search for, and characterization of, the new physics which is known to be
out there, waiting to be discovered. We rely on detailed simulations and
forecast techniques to discuss four important examples of this point: big bang
nucleosynthesis, the evolution of the cosmic microwave background temperature,
tests of the universality of physical laws, and a real-time model-independent
mapping of the expansion history of the universe (also known as the redshift
drift). The last two are among the flagship science drivers for the ELT. We
also highlight what is required for the ESO community to be able to play a
meaningful role in 2030s fundamental cosmology and show that, even if ANDES
only provides null results, such `minimum guaranteed science' will be in the
form of constraints on key cosmological paradigms: these are independent from,
and can be competitive with, those obtained from traditional cosmological
probes.
