We study the role of bath-induced correlations in temperature estimation of cold Bosonic baths. Our protocol includes multiple probes, that are not interacting, nor are they initially correlated to each other. They interact with a Bosonic sample and reach a non-equilibrium steady state, which is measured to estimate the temperature of the sample. It is well-known that in the steady state such non-interacting probes may get correlated to each other and even entangled. Nonetheless, the impact of these correlations in metrology has not been deeply investigated yet. Here, we examine their role for thermometry of cold Bosonic gases and show that, although being classical, bath-induced correlations can indeed lead to sub-shot-noise precision for thermometry at low temperatures; e.g., for a probe of $30$ non-interacting impurities they can enhance the quantum Fisher information by two orders of magnitude. The proposed thermometry scheme here does not require precise dynamical control of the probes and tuning the parameters, as it is build upon the non-equilibrium steady state of a non-interacting system. Our results put forward new possibilities in thermometry at low temperatures, of relevance for instance in cold gases and Bose--Einstein condensates.
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