“Overdense" and "underdense" often appear in various documents to refer to high density and low density respectively. And I know that "overdensity" is defined as
$$ \delta(x)=\frac{\rho(x)-\bar\rho}{\bar\rho} $$
Is there someone threshold, here denoted as $\delta_t$ to distinguish "overdense" and "underdense"?
Over- and underdensities are simply defined with respect to the mean density $\bar{\rho}$.
That is, if $\rho(x) > \bar{\rho}$, then $\delta(x)$ is positive and you call it an overdensity, whereas if $\rho(x) < \bar{\rho}$, then $\delta(x)$ is negative and you call it an underdensity.
In other words, the threshold you're requesting is $\bar{\rho}$.
In cosmology, one is more often interested in the overdensities, because these are the ones that will evolve into the "most interesting" structures, i.e. galaxies, groups, filaments, sheets, and clusters. The underdensities will evolve into voids, which are of course interesting too.
But not as beautiful.
