  We study the production rate of ionizing photons of a sample of 588 H$\alpha$
emitters (HAEs) and 160 Lyman-$\alpha$ emitters (LAEs) at $z=2.2$ in the COSMOS
field in order to assess the implied emissivity from galaxies, based on their
UV luminosity. By exploring the rest-frame Lyman Continuum (LyC) with
GALEX/$NUV$ data, we find f$_{\rm esc} < 2.8\, (6.4)$% through median (mean)
stacking. By combining the H$\alpha$ luminosity density with IGM emissivity
measurements from absorption studies, we find a globally averaged
$\langle$f$_{\rm esc}\rangle$ of $5.9^{+14.5}_{-4.2}$ % at $z=2.2$ if we assume
HAEs are the only source of ionizing photons. We find similarly low values of
the global $\langle$f$_{\rm esc}\rangle$ at $z\approx3-5$, also ruling out a
high $\langle$f$_{\rm esc}\rangle$ at $z<5$. These low escape fractions allow
us to measure $\xi_{ion}$, the number of produced ionizing photons per unit UV
luminosity, and investigate how this depends on galaxy properties. We find a
typical $\xi_{ion} \approx 10^{24.77\pm0.04}$ Hz erg$^{-1}$ for HAEs and
$\xi_{ion} \approx 10^{25.14\pm0.09}$ Hz erg$^{-1}$ for LAEs. LAEs and low mass
HAEs at $z=2.2$ show similar values of $\xi_{ion}$ as typically assumed in the
reionization era, while the typical HAE is three times less ionizing. Due to an
increasing $\xi_{ion}$ with increasing EW(H$\alpha$), $\xi_{ion}$ likely
increases with redshift. This evolution alone is fully in line with the
observed evolution of $\xi_{ion}$ between $z\approx2-5$, indicating a typical
value of $\xi_{ion} \approx 10^{25.4}$ Hz erg$^{-1}$ in the reionization era.
