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A K correction accounts for observing a redshifted spectrum through a fixed photometric passband, rather than the corresponding emitted band. In the same R-band astronomical magnitude convention,The luminosity distance already incorporates bolometric dimming; the additional correction depends on spectral shape and band response. At $z=3$, observed R-band light at $660\mathrm{nm}$ was emitted at $165\mathrm{nm}$, in the far ultraviolet. It is not a direct measurement of rest-frame R-band luminosity.

To fix the sign, let $L_{\nu }$ be emitted spectral luminosity and approximate R as narrow. Counting photon energy, arrival time and the transformed frequency interval gives $F_{{\nu }_o}=(1+z)L_{(1+z){\nu }_o}/(4\pi d_L^2)$. ThusFor an ordinary old elliptical galaxy, the far-ultraviolet spectrum is weak relative to the optical, so typically $L_{\nu }(165\mathrm{nm})/L_{\nu }(660\mathrm{nm})<1/4$. Then $K_R>0$: neglecting it overestimates the numerical absolute magnitude and underestimates the R-band luminosity. For a young, unobscured star-forming galaxy, hot stars make the far-ultraviolet strong, with an approximately flat $L_{\nu }$ spectrum often serving as a useful model. It gives $K_R=-2.5{\log }_{10}4\simeq -1.51$: neglecting it underestimates the numerical absolute magnitude and overestimates the R-band luminosity. Strong dust extinction or an unusual spectrum can reverse this latter sign; a galaxy label alone does not prove an exact correction. The band-integrated convention is given in Hogg and collaborators' definition of the K-correction.

The ionizing background is produced mainly by hot stars in star-forming galaxies and by accretion-powered quasars. The stellar contribution depends on star formation and the ionizing photon escape fraction; the quasar contribution depends on its evolving luminosity function. Radiative transfer through the intergalactic medium, including absorption, recombination emission, and cosmological redshifting, determines the radiation actually available for photoionization.

In the picture relevant to this 2009 paper, increasingly rare quasars cannot by themselves explain the high-redshift hydrogen ionizing field, so galaxies become especially important above $z\sim 3$. Harder quasar radiation is important for doubly ionizing helium around $z\sim 3$. Hydrogen reionization must be largely accomplished by $z\sim 6$. At lower redshift the evolution reflects declining source emissivity together with the increasing propagation distance of ionizing photons; near reionization, short absorption lengths and spatially uneven ionized regions prevent a simple uniform-background description. The background rate need not follow the quasar population alone: a roughly flat hydrogen photoionization rate over $2\lesssim z\lesssim 4$ is compatible with a changing mixture of galaxies and quasars. Faucher-Giguère and collaborators' 2009 background calculation and their preceding opacity analysis give this interpretation. These redshifts describe the broad historical picture, rather than precise universal transition times.