following subsections. A parallel line of reasoning applies to the detectability of normal galaxies in the 21cm line of neutral hydrogen, as discussed in Sect. 2.5.

The FIR luminosity function $\Phi(L_{60\mu})$
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Koranyi & Strauss (1997) have presented a conveniently formulated description of the far-infraed luminosity function, based on complete samples of galaxies from the IRAS 60 $\mu$m survey. Their paper analyzes the sample in the context of several cosmological models; here, parameters for the “$p=1$” case, which is the conventional cosmology, are adopted to describe the number density of luminous, FIR-selected galaxies.

The 60 $\mu$m differential luminosity function is given by

$$\Phi(L_{60}) = \left( \frac{\alpha}{L_{60}}+\frac{\beta}{L_{60}^*+L_{60}}\right) \Psi(L_{60})$$

with $\Psi(L_{60})$, the cumulative luminosity function, defined as $$\Psi(L_{60}) = C\left( \frac{L_{60}}{L_{60}^*}\right)^{-\alpha}
          \left(1 + \frac{L_{60}}{L_{60}^*}\right)^{-\beta}.$$ For the case of conventional cosmology, Koranyi & Strauss fit values of $\alpha= 0.49$, $\beta = 1.81$ and $L_{60}^*=10^{9.68}L_{\odot}$. The normalization constant, $C$, is related to their parameter $n_1 = 0.058$ Mpc$^{-3}$, through the relation $n_1 = \Psi[L_{min}(z_s)]$, where $L_{min}(z_s)$ is the minimum luminosity detectable in the sample, whose members are restricted to lie at redshifts greater than $z_s$. For Koranyi & Strauss’ analysis, $L_{min}(z_s)/L^* = 8.8{\times}10^{-3}$, which leads to $C=5.8{\times}10^{-3}$ Mpc$^{-3}$ ($H_o = 100$ km s$^{-1}$ Mpc$^{-1}$).

