  Rest-frame mid- to far-infrared spectroscopy is a powerful tool to study how
galaxies formed and evolved, because a major part of their evolution occurs in
heavily dust enshrouded environments, especially at the so-called Cosmic Noon.
Using the calibrations of IR lines we predict the expected fluxes of lines and
features, with the aim to measure the star formation rate and the Black Hole
Accretion rate in intermediate to high redshift galaxies. The launch of the
James Webb Space Telescope will allow us a deep investigation of both the SF
and the BHA obscured processes as a function of cosmic time. We assess the
spectral lines and features that can be detected by JWST-MIRI in galaxies and
Active Galactic Nuclei up to redshift z= 3. We confirm the fine-structure lines
of [MgIV]4.49um and [ArVI]4.53um as good BHA rate tracers for the 1<z<3 range,
and we propose the [NeVI]7.65um line as the best tracer for redshifts of z<1.5.
We suggest the use of the [ArII]6.98um and [ArIII]8.99um lines to measure the
SF rate, for z<3 and z<2. At higher redshifts, the PAH features at 6.2um and
7.7um can be observed at z<3 and z<2.7 respectively. Rest-frame far-IR
spectroscopy is currently being collected in high redshift galaxies (z>3) with
the Atacama Large Millimeter Array. We confirm that the [CII]158um line is a
good tracer of the SF rate and can in most cases (0.9<z<2 and 3<z<9) be
observed, and we propose the use of the combination of [OIII]88um and [OI]145um
lines as an alternative SF rate tracer, that can be detected above z>3. We
conclude, however, that the current and foreseen facilities will not be able to
cover properly the peak of the obscured SF and BHA activities at the Cosmic
Noon of galaxy evolution and a new IR space telescope, actively cooled to
obtain very good sensitivities, covering the full IR spectral range from about
10um to 300um, will be needed.
