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The integrated halo star-formation efficiency is the fraction of a dark matter halo's reference baryonic mass converted into stars: ${ϵ}_{\ast }=M_{\ast }/(f_b{M}_h)$, with cosmic baryon fraction $f_b$. It is distinct from star-formation efficiency per free-fall time. Cooling, gas supply and retention, stellar feedback and active-galactic-nucleus feedback make it strongly dependent on dark matter halo mass. With stellar mass-to-light ratio ${{\rm Y}}_{\ast }$, a simple central-galaxy mapping is $L={ϵ}_{\ast }{f}_b{M}_h/{{\rm Y}}_{\ast }$. Changing this efficiency changes both the mapping and the logarithmic Jacobian between the dark matter halo mass function and the luminosity function.

The dark matter halo population follows gravitational hierarchical galaxy formation: many low-mass dark-matter haloes and an exponentially rare high-mass tail. The Press-Schechter halo mass function captures this broad form. A constant conversion from dark matter halo mass to luminosity would simply rescale that distribution and predict too much light from both very small and very large systems. The luminosity function instead has a shallower faint end and a sharp bright-end decline, often described by a Schechter function,Define the halo star-formation efficiency as ${ϵ}_{\ast }(M_h)=M_{\ast }/(f_b{M}_h)$, distinguishing this integrated baryon conversion from the star-formation efficiency per free-fall time. If the stellar mass-to-light ratio is ${{\rm Y}}_{\ast }$, a simple one-central-galaxy mapping is $L={ϵ}_{\ast }{f}_b{M}_h/{{\rm Y}}_{\ast }$. For a monotone mapping without scatter,Thus a mass-dependent conversion efficiency and its Jacobian change the shape, not just the horizontal scale, of the luminosity function. Scatter, satellites and the stellar mass-to-light ratio add further differences.

In shallow potential wells, photoheating can prevent gas accretion, while stellar feedback from radiation, stellar winds and supernovae heats or ejects gas. These effects lower halo star-formation efficiency toward low mass, helping flatten the faint end. The need for atomic or molecular line cooling also affects the smallest systems. Intermediate-mass dark-matter haloes retain gas and cool efficiently, so the integrated conversion reaches a maximum around galactic dark matter halo masses.

In massive dark-matter haloes, long optically thin gas cooling times and stable hot atmospheres reduce fresh cold-gas supply. Active-galactic-nucleus feedback can replenish the thermal energy lost in cooling and suppress a cooling flow, reducing halo star-formation efficiency and sharpening the bright cutoff. Massive group or cluster dark-matter haloes contain many distinct galaxies rather than one object with luminosity proportional to their entire mass. Galaxy mergers redistribute stars and can grow already formed bright galaxies, but do not turn all hot dark matter halo gas into new stars. The luminosity function is the gravitational dark matter halo population filtered through strongly mass-dependent baryonic physics. The plotted abundance and efficiency curves are qualitative examples, not observational fits or calibrated mass-to-light relations.