  We present a comprehensive study shedding light on how thermal fluctuations
affect correlations in a Bose gas with contact repulsive interactions in one
spatial dimension. The pair correlation function, the static structure factor,
and the one-body density matrix are calculated as a function of the interaction
strength and temperature with the exact ab-initio Path Integral Monte Carlo
method. We explore all possible gas regimes from weak to strong interactions
and from low to high temperatures. We provide a detailed comparison with a
number of theories, such as perturbative (Bogoliubov and decoherent classical),
effective (Luttinger liquid) and exact (ground-state and thermal Bethe Ansatz)
ones. Our Monte Carlo results exhibit an excellent agreement with the tractable
limits and provide a fundamental benchmark for future observations which can be
achieved in atomic gases, cavity quantum-electrodynamic and
superconducting-circuit platforms.
