  Correlation spectroscopy has been proposed as a spectroscopic technique for
measuring the coherence between the ground states in electromagnetically
induced transparency (EIT). While in time domain the steep dispersion in EIT
condition accounts for the robustness of the correlation linewidth against
power broadening, such physical insight was not directly established in the
frequency domain. We propose a perturbative approach to describe the
correlation spectroscopy of two noisy lasers coupled to a $\Lambda$-transition
in cold atoms, leading to EIT. Such approach leads to an analytical expression
that maps the intensity correlation directly in terms of the absorption and
dispersion of the light fields. Low and high perturbative regimes are
investigated and demonstrate that, for coherent light sources, the first oder
term in perturbation expansion represents a sufficient description for the
correlation. Sidebands resonances are also observed, showing the richness of
the frequency domain approach.
