Rest-frame mid- to far-infrared (IR) 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 (
$1< z < 3$
). Using the calibrations of IR lines and features, recently updated with Herschel and Spitzer spectroscopy, we predict their expected fluxes with the aim to measure the Star Formation (SF) and the Black Hole Accretion (BHA) rates in intermediate to high redshift galaxies. On the one hand, the recent launch of the James Webb Space Telescope (JWST) offers new mid-IR spectroscopic capabilities that will enable for the first time a detailed investigation of both the SF and the BHA obscured processes as a function of cosmic time. We make an assessment of the spectral lines and features that can be detected by JWST-MIRI in galaxies and active galactic nuclei up to redshift
$z \sim 3$
. The fine structure lines of [MgIV]4.49
$\unicode{x03BC}\textrm{m}$
and [ArVI]4.53
$\unicode{x03BC}\textrm{m}$
can be used as BHA rate tracers for the
$1 \lesssim z \lesssim 3$
range, and we propose the [NeVI]7.65
$\unicode{x03BC}\textrm{m}$
line as the best tracer for
$z \lesssim 1.5$
. The [ArII]6.98
$\unicode{x03BC}\textrm{m}$
and [ArIII]8.99
$\unicode{x03BC}\textrm{m}$
lines can be used to measure the SF rate at
$z \lesssim 3$
and
$z \lesssim 2$
, respectively, while the stronger [NeII]12.8
$\unicode{x03BC}\textrm{m}$
line exits the JWST-MIRI spectral range above
$z \gtrsim 1.2$
. At higher redshifts, the PAH features at 6.2 and 7.7
$\unicode{x03BC}\textrm{m}$
can be observed at
$z \lesssim 3$
and
$z \lesssim 2.7$
, respectively. On the other hand, rest-frame far-IR spectroscopic observations of high redshift galaxies (
$z \gtrsim 3$
) have been collected with the Atacama Large Millimeter Array (ALMA) in the last few years. The observability of far-IR lines from ALMA depends on the observed frequency, due to the significant decrease of the atmospheric transmission at the highest frequencies (
$\gtrsim420\,\rm{Hz}$
). The [CII]158
$\unicode{x03BC}\textrm{m}$
line is a reliable tracer of the SF rate and can in most cases (
$0.9 \lesssim z \lesssim 2$
and
$2 \lesssim z \lesssim 9$
) be observed. Additionally, we propose the use of the combination of [OIII]88
$\,\unicode{x03BC}$
m and [OI]145
$\,\unicode{x03BC}$
m lines as an alternative SF rate tracer, that can be detected above
$z \gtrsim 3$
. Overall, we emphasize the importance of using multi-feature analysis to measure both BHA and SFR, since individual tracers can be strongly dependent on the local ISM conditions and vary from source to source. However, we conclude that the peak of the obscured SF and BHA activities at Cosmic Noon falls outside the wavelength coverage of facilities currently operating or under development. A new IR space telescope covering the full IR spectral range from about 10 to
$300\,\unicode{x03BC}\textrm{m}$
and actively cooled to achieve high sensitivity, will be needed.