  We present the H{\alpha} luminosity function (LF) derived from a large sample
of Lyman break galaxies at z {\sim} 4.5 over the GOODS-South and North fields.
This study makes use of the new, full-depth Spitzer/IRAC [3.6] and [4.5]
imaging from the GOODS Re-ionization Era wide-Area Treasury from the Spitzer
program. The H{\alpha} flux is derived from the offset between the continuum
flux estimated from the best-fit spectral energy distribution, and the observed
photometry in IRAC [3.6]. From these measurements, we build the H{\alpha} LF
and study its evolution providing the best constraints of this property at high
redshift, where spectroscopy of H{\alpha} is not yet available. Schechter
parameterizations of the H{\alpha} LF show a decreasing evolution of
{\Phi^\star} with redshift, increasing evolution in L{^\star}, and no
significant evolution in the faint-end slope at high z. We find that star
formation rates (SFRs) derived from H{\alpha} are higher than those derived
from the rest-frame UV for low SFR galaxies but the opposite happens for the
highest SFRs. This can be explained by lower mass galaxies (also lower SFR)
having, on average, rising star formation histories (SFHs), while at the
highest masses the SFHs may be declining. The SFR function is steeper, and
because of the excess SFR(H{\alpha}) compared to SFR(UV) at low SFRs, the SFR
density estimated from H{\alpha} is higher than the previous estimates based on
UV luminosities.
