Quantifying the effect of field variance on the H$alpha$ luminosity function with the New Numerical Galaxy Catalogue ($ u^2$GC)

We construct a model of H$alpha$ emitters (HAEs) based on a semi-analytic galaxy formation model, the New Numerical Galaxy Catalog ($ u^2$GC). In this paper, we report our estimate for the field variance of the HAE distribution. By calculating the H$alpha$ luminosity from the star-formation rate of galaxies, our model well reproduces the observed H$alpha$ luminosity function (LF) at $z=0.4$. The large volume of the $ u^2$GC makes it possible to examine the spatial distribution of HAEs over a region of (411.8 Mpc)$^3$ in the comoving scale. The surface number density of $z=0.4$ HAEs with $L_{rm Halpha} geq 10^{40}$ erg s$^{-1}$ is 308.9 deg$^{-2}$. We have confirmed that the HAE is a useful tracer for the large-scale structure of the Universe because of their significant overdensity ($>$ 5$sigma$) at clusters and the filamentary structures. The H$alpha$ LFs within a survey area of $sim$2 deg$^2$ (typical for previous observational studies) show a significant field variance up to $sim$1 dex. Based on our model, one can estimate the variance on the H$alpha$ LFs within given survey areas.