In the past years "quintessence" models have been considered which can prod
uce the accelerated expansion in the universe suggested by recent astronomi
cal observations. One of the key differences between quintessence and a cos
mological constant is that the energy density in quintessence, could be a s
ignificant fraction of the overall energy even in the early universe, while
the cosmological constant will be dynamically relevant only at late times.
We use standard big bang nucleosynthesis and the observed abundances of pr
imordial nuclides to put constraints on Omega (phi) at temperatures near T
similar to1 MeV. We point out that current experimental data do not support
the presence of such a field, providing the strong constraint Omega (phi)
(MeV)<0.045 at 2<sigma> C.L. and strengthening previous results. We also co
nsider the effect a scaling field has on cosmic microwave background (CMB)
anisotropies using the recent data from BOOMERANG and DASI combined with SN
Ia data, providing the CMB constraint Omega (phi)less than or equal to0.39
at 2 sigma during the radiation dominated epoch.