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For an open matter-dominated Friedmann solution, put $u=H_0\sqrt{1-{\Omega }_0}\tau$, $a={\Omega }_0(\cosh u-1)/[2(1-{\Omega }_0)]$ and $\cosh u_0=2/{\Omega }_0-1$. The cosmological bolometric background from conserved sources has $J(u)=\sinh (2u)/4-2\sinh u+3u/2$, since $J^{\prime }=(\cosh u-1{)}^2$ and $dt=adu/[H_0\sqrt{1-{\Omega }_0}]$. At fixed $H_0$, the limit ${\Omega }_0\to 0$ is $cL{n}_0/(2H_0)$, whereas the Einstein-de Sitter universe gives $2cL{n}_0/(5H_0)$. Keeping the cosmic age fixed instead reverses this ordering: the two values are respectively $cL{n}_0{t}_0/2$ and $3cL{n}_0{t}_0/5$.

Use the same conserved-source interpretation, comoving number density $n_0$ and normalization $a_0=1$ as in the preceding part. DefineThe open matter-dominated Friedmann solution then has $a=C(\cosh u-1)$, $t=(C/\kappa )(\sinh u-u)$ and $dt=(C/\kappa )(\cosh u-1)du$. The present parameter satisfiesThe cosmological bolometric background from conserved sources is thereforeSince ${\cosh }^{2}u=(1+\cosh 2u)/2$, the open dust bolometric background integral isThis leaves the answer in the requested parameter form and makes no use of a radial-distance approximation. For arbitrary proper number density instead, the same calculation gives

For the conserved population, compare models with the same present $n_0,L,H_0$. As ${\Omega }_0\to 0$, $u_0\to \infty$ and $\cosh u_0\sim 2/{\Omega }_0$. The leading term in the square bracket is $\sinh (2u_0)/4\sim 2/{\Omega }_0^2$; the terms linear in $\sinh u_0$ and $u_0$ vanish after multiplication by ${\Omega }_0^2$. HenceThis is also immediate in the Milne model, for which $a=t/t_0$ and $t_0=H_0^{-1}$. If one compares at fixed cosmic age instead, the results are $L{n}_0{t}_0/2$ and $3L{n}_0{t}_0/5$, so the empty open model gives $5/6$ of the flat-model flux. Specifying what is held fixed is necessary because the two expansion histories have different ages at the same Hubble constant.

An isotropic source's bolometric luminosity is distributed over a wavefront of physical area $4\pi a_0^2{r}^2$. Each photon loses an energy factor $(1+z{)}^{-1}$ through cosmological redshift, and successive photons arrive with a further rate factor $(1+z{)}^{-1}$ through cosmological time dilation. Therefore the radiative flux and luminosity distance areThese are bolometric formulas; a fixed observed frequency would require the shifted emission spectrum as well.

Normalize the scale factor by $a_0=1$. Summing the radiative flux over the source counts on an FLRW past light cone makes the transverse shell-area factors cancel:Thus ${\mathcal{F}}_0$ is power per effective detector area, summed over the whole sky; an all-sky detector of effective area $A_{\det }$ receives $A_{\det }{\mathcal{F}}_0$. The corresponding intensity per solid angle is ${\mathcal{F}}_0/(4\pi )$. Restoring the speed of light multiplies these flux integrals by $c$.

Spatially uniform $n(t)$ by itself does not specify its time evolution. For a persistent population comoving with the expansion, with neither creation nor destruction, the cosmological continuity equation for source number is $\dot{n}+3Hn=0$. Its conserved comoving number density is $n_0=a^{3}n$, so the cosmological bolometric background from conserved sources becomes $L{n}_0\int adt$. For the Einstein-de Sitter universe,The last equality uses $H_0=2/(3t_0)$. The early-time improper integral converges despite the diverging physical source density. If instead the proper number density were maintained at a constant value $n_0$ by continuous source creation, the same geometry would give $3L{n}_0{t}_0/11$. For a completely unspecified source history the general integral above is the determined answer; constancy of individual luminosity alone does not imply number conservation.