  We consider a scenario in which qubit-like probes are used to sense an
external field that linearly affects their energy splitting. Following the
frequency estimation approach in which one optimizes the state and sensing time
of the probes to maximize the sensitivity, we provide a systematic study of the
attainable precision under the impact of noise originating from independent
bosonic baths. We invoke an explicit microscopic derivation of the probe
dynamics using the spin-boson model with weak coupling of arbitrary geometry
and clarify how the secular approximation leads to a phase-covariant dynamics,
where the noise terms commute with the field Hamiltonian, while the inclusion
of non-secular terms breaks the phase-covariance. Moreover, unless one
restricts to a particular (i.e., Ohmic) spectral density of the bath modes, the
noise terms may contain relevant information about the frequency to be
estimated. Thus, by considering general evolutions of a single probe, we study
regimes in which these two effects have a non-negligible impact on the
achievable precision. We then consider baths of Ohmic spectral density yet
fully accounting for the lack of phase-covariance, in order to characterize the
ultimate attainable scaling of precision when $N$ probes are used in parallel.
Crucially, we show that beyond the semigroup (Lindbladian) regime the Zeno
limit imposing the $1/N^{3/2}$ scaling of the mean squared error, derived
assuming phase-covariance, generalises to any dynamics of the probes, unless
the latter are coupled to the baths in the direction perfectly transversal to
the encoding---when a novel scaling of $1/N^{7/4}$ arises. As our microscopic
approach covers all classes of dissipative dynamics, from semigroup to
non-Markovian ones (potentially non-phase-covariant), it provides an exhaustive
picture, in which all the different asymptotic scalings of precision naturally
emerge.
