  We report on the effects of cosmic rays (CRs) on the abundance of CO in $\rm
H_2$ clouds under conditions typical for star-forming galaxies in the Universe.
We discover that this most important molecule for tracing H$_2$ gas is very
effectively destroyed in ISM environments with CR energy densities $\rm
U_{CR}\sim(50-10^{3})\times U_{CR,Gal}$, a range expected in numerous
star-forming systems throughout the Universe. This density-dependent effect
operates volumetrically rather than only on molecular cloud surfaces (i.e.
unlike FUV radiation that also destroys CO), and is facilitated by: a) the
direct destruction of CO by CRs, and b) a reaction channel activated by
CR-produced He$^{+}$. The effect we uncover is strong enough to render
Milky-Way type Giant Molecular Clouds (GMCs) very CO-poor (and thus
CO-untraceable), even in ISM environments with rather modestly enhanced average
CR energy densities of $\rm U_{CR}\sim(10-50)\times\rm U_{CR,Gal}$. We conclude
that the CR-induced destruction of CO in molecular clouds, unhindered by dust
absorption, is perhaps the single most important factor controlling the
CO-visibility of molecular gas in vigorously star-forming galaxies. We
anticipate that a second order effect of this CO destruction mechanism will be
to make the H$_2$ distribution in the gas-rich disks of such galaxies appear
much clumpier in CO $J$=1--0, 2--1 line emission than it actually is. Finally
we give an analytical approximation of the CO/H$_2$ abundance ratio as a
function of gas density and CR energy density for use in galaxy-size or
cosmological hydrodynamical simulations, and propose some key observational
tests.
