  We consider the non-equilibrium zero frequency noise generated by a
temperature gradient applied on a device composed of two normal leads separated
by a quantum dot. We recall the derivation of the scattering theory for
non-equilibrium noise for a general situation where both a bias voltage and a
temperature gradient can coexist and put it in a historical perspective. We
provide a microscopic derivation of zero frequency noise through a quantum dot
based on a tight binding Hamiltonian, which constitutes a generalization of the
pioneering work of Caroli et al. for the current obtained in the context of the
Keldysh formalism. For a single level quantum dot, the obtained transmission
coefficient entering the scattering formula for the non-equilibrium noise
corresponds to a Breit-Wigner resonance. We compute the delta-$T$ noise as a
function of the dot level position, and of the dot level width, in the
Breit-Wigner case, for two relevant situations which were considered recently
in two separate experiments. In the regime where the two reservoir temperatures
are comparable, our gradient expansion shows that the delta-$T$ noise is
dominated by its quadratic contribution, and is minimal close to resonance. In
the opposite regime where one reservoir is much colder, the gradient expansion
fails and we find the noise to be typically linear in temperature before
saturating. In both situations, we conclude with a short discussion of the case
where both a voltage bias and a temperature gradient are present, in order to
address the potential competition with thermoelectric effects.
