Mass and Environment as Drivers of Galaxy Evolution III: The constancy   of the faint-end slope and the merging of galaxies
  We explore using our continuity approach the underlying connections between the evolution of the faint-end slope of the stellar mass function of star-forming galaxies, the logarithmic slope of the sSFR-mass relation and the merging of galaxies. We derive analytically the consequences of the observed constancy of the faint-end slope since redshifts of at least z ~ 2. If the logarithmic slope of the sSFR-mass relation is negative, then the faint-end slope should quickly diverge due to the differential mass increase of galaxies on the star-forming main sequence, and this will also quickly destroy the Schechter form of the mass function. This problem can be solved by removing low mass galaxies by merging them into more massive galaxies. We quantify this process by introducing the specific merger mass rate (sMMR) as the specific rate of mass added to a given galaxy through mergers. For a modest negative value of the logarithmic slope of the sSFR-mass relation of beta ~ -0.1, an average sMMR ~ 0.1sSFR across the population is required to keep the faint-end slope constant with epoch, as observed. This in turn implies a merger rate of ~ 0.2sSFR for major mergers, which is consistent with the available observational estimates. More negative values of beta require higher sMMR and higher merger rates, and the steepening of the mass function becomes impossible to control for beta < ~ -0.6, for an observed value of the faint-end slope of ~ -1.4. The close link that is required between the in situ sSFR and the sMMR probably arises because both are closely linked to the buildup of dark matter haloes. These new findings further develop the formalism for the evolving galaxy population that we introduced earlier, and show how striking symmetries in the galaxy population can emerge as the result of deep links between the physical processes involved. 
Yingjie Peng, Simon J. Lilly, Alvio Renzini, C. Marcella Carollo
astro-ph.GA astro-ph.CO