  Near-infrared observations of Lyman-break galaxies at redshifts z~3 are
beginning to provide constraints on ages, star-formation histories, dust
content, metallicities, and stellar masses. At present, uncertainties of more
than an order of magnitude are typical for many of these parameters. It is
nonetheless interesting to ask what the stellar-population models imply for the
existence and luminosities of Lyman-break galaxies at higher redshift. To this
end we examine the inferred star-formation rates in two well-studied samples of
galaxies as a function of redshift out to z = 10 for various best-fit and
limiting cases.
  Taken at face value, the generally young ages (typically 10^8 +- 0.5 yr) of
the z = 3 Lyman break galaxies imply that their stars were not present much
beyond z=4. By z = 6 the cosmic star-formation rate from the progenitors of
these galaxies is less than 10% of star-formation rate at z=3 +- 0.5, even for
maximally-old models, provided the derivative of the star-formation rate SFR(t)
is monotonic. The escaping Lyman-continuum radiation from such galaxies would
be insufficient to reionize the IGM. Thus other sources of ionizing photons
(e.g. very massive population III stars) may be needed, and the more normal
Lyman-break galaxies may be a phenomenon confined to redshifts z <~ 4. This
conclusion changes if SFR(t) was episodic, and we examine the parameters of
such bursty star-formation that might be consistent with both the z=2-4
luminosity functions and the z~3 spectral energy distributions.
