The impact of star formation sampling effects on the spectra of lensed
  $z>6$ galaxies detectable with $\textit{JWST}$
  The upcoming $\textit{James Webb Space Telescope}$ ($\textit{JWST}$) will
allow observations of high-redshift galaxies at fainter detection levels than
ever before, and $\textit{JWST}$ surveys targeting gravitationally lensed
fields are expected to bring $z\gtrsim 6$ objects with very low star formation
rate (SFR) within reach of spectroscopic studies. As galaxies at lower and
lower star formation activity are brought into view, many of the standard
methods used in the analysis of integrated galaxy spectra are at some point
bound to break down, due to violation of the assumptions of a well-sampled
stellar initial mass function (IMF) and a slowly varying SFR. We argue that
galaxies with SFR$\sim 0.1\ M_\odot$ yr$^{-1}$ are likely to turn up at the
spectroscopic detection limit of $\textit{JWST}$ in lensed fields, and
investigate to what extent star formation sampling may affect the spectral
analysis of such objects. We use the $\small{\text{SLUG}}$ spectral synthesis
code to demonstrate that such effects are likely to have significant impacts on
spectral diagnostics of, for example, the Balmer emission lines. These effects
are found to stem primarily from SFRs varying rapidly on short ($\sim$ Myr)
timescales due to star formation in finite units (star clusters), whereas the
effects of an undersampled IMF is deemed insignificant in comparison. In
contrast, the ratio between the HeII- and HI-ionizing flux is found to be
sensitive to IMF-sampling as well as ICMF-sampling (sampling of the initial
cluster mass function), which may affect interpretations of galaxies containing
Population III stars or other sources of hard ionizing radiation.
