As galaxy formation and evolution over long cosmic time-scales depends to a large degree on the structure of the universe, the assembly history of galaxies is potentially a powerful approach for learning about the universe itself. In this paper we examine the merger history of dark matter halos based on the Extended Press-Schechter formalism as a function of cosmological parameters, redshift and halo mass. We calculate how major halo mergers are influenced by changes in the cosmological values of $Omega_{rm m}$, $Omega_{Lambda}$, $sigma_{8}$, the dark matter particle temperature (warm vs. cold dark matter), and the value of a constant and evolving equation of state parameter $w(z)$. We find that the merger fraction at a given halo mass varies by up to a factor of three for halos forming under the assumption of Cold Dark Matter, within different underling cosmological parameters. We find that the current measurements of the merger history, as measured through observed galaxy pairs as well as through structure, are in agreement with the concordance cosmology with the current best fit giving $1 - Omega_{rm m} = Omega_{rm Lambda} = 0.84^{+0.16}_{-0.17}$. To obtain a more accurate constraint competitive with recently measured cosmological parameters from Planck and WMAP requires a measured merger accuracy of $delta f_{rm m} sim 0.01$, implying surveys with an accurately measured merger history over 2 - 20 deg$^{2}$, which will be feasible with the next generation of imaging and spectroscopic surveys such as Euclid and LSST.
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