We measure the merger fraction of massive galaxies using the UltraVISTA/COSMOS $Ks$-band selected catalog, complemented with the deeper, higher resolution 3DHST+CANDELS catalog selected in the HST/WFC3 $H$-band, presenting the largest mass-complete p
hotometric merger sample up to $zsim3$. We find that selecting mergers using the $H_{160}$-band flux ratio leads to an increasing merger fraction with redshift, while selecting mergers using the stellar mass ratio causes a diminishing redshift dependence. Defining major and minor mergers as having stellar mass ratios of 1:1 - 4:1 and 4:1 - 10:1 respectively, the results imply $sim$1 major and $lesssim$1 minor merger for an average massive (log$(M_{star}/M_{odot}) geqslant 10.8$) galaxy during $z=0.1-2.5$. There may be an additional $sim 0.5(0.3)$ major (minor) merger if we use the $H$-band flux ratio selection. The observed amount of major merging alone is sufficient to explain the observed number density evolution for the very massive (log$(M_{star}/M_{odot}) geqslant 11.1$) galaxies. We argue that these very massive galaxies can put on a maximum of $6%$ of stellar mass in addition to major and minor merging, so that their number density evolution remains consistent with observations. The observed number of major and minor mergers can increase the size of a massive quiescent galaxy by a factor of two at most. This amount of merging is enough to bring the compact quiescent galaxies formed at $z>2$ to lie at $1sigma$ below the mean of the stellar mass-size relation as measured in some works (e.g. Newman et al. 2012), but additional mechanisms are needed to fully explain the evolution, and to be consistent with works suggesting stronger evolution (e.g. van der Wel et al. 2014).
Using a mass-selected ($M_{star} ge 10^{11} M_{odot}$) sample of 198 galaxies at 0 < z < 3.0 with HST/NICMOS $H_{160}$-band images from the COSMOS survey, we find evidence for the evolution of the pair fraction above z ~ 2, an epoch in which massive
galaxies are believed to undergo significant structural and mass evolution. We observe that the pair fraction of massive galaxies is 0.15 pm 0.08 at 1.7 < z < 3.0, where galaxy pairs are defined as massive galaxies having a companion of flux ratio from 1:1 to 1:4 within a projected separation of 30 kpc. This is slightly lower, but still consistent with the pair fraction measured previously in other studies, and the merger fraction predicted in halo-occupation modelling. The redshift evolution of the pair fraction is described by a power law F(z) = (0.07 pm 0.04) * (1+z) ^ (0.6 pm 0.5). The merger rate is consistent with no redshift evolution, however it is difficult to constrain due to the limited sample size and the high uncertainties in the merging timescale. Based on the merger rate calculation, we estimate that a massive galaxy undergoes on average 1.1 pm 0.5 major merger from z = 3 to 0. The observed merger fraction is sufficient to explain the number density evolution of massive galaxies, but insufficient to explain the size evolution. This is a hint that mechanism(s) other than major merging may be required to increase the sizes of the massive, compact quiescent galaxies from z ~ 2 to 0.
