We aim to measure the major merger rate of star-forming galaxies at 0.9 < z <1.8, using close pairs identified from integral field spectroscopy (IFS). We use the velocity field maps obtained with SINFONI/VLT on the MASSIV sample, selected from the star-forming population in the VVDS. We identify physical pairs of galaxies from the measurement of the relative velocity and the projected separation (r_p) of the galaxies in the pair. Using the well constrained selection function of the MASSIV sample we derive the gas-rich major merger fraction (luminosity ratio mu = L_2/L_1 >= 1/4), and, using merger time scales from cosmological simulations, the gas-rich major merger rate at a mean redshift up to z = 1.54. We find a high gas-rich major merger fraction of 20.8+15.2-6.8 %, 20.1+8.0-5.1 % and 22.0+13.7-7.3 % for close pairs with r_p <= 20h^-1 kpc in redshift ranges z = [0.94, 1.06], [1.2, 1.5) and [1.5, 1.8), respectively. This translates into a gas-rich major merger rate of 0.116+0.084-0.038 Gyr^-1, 0.147+0.058-0.037 Gyr^-1 and 0.127+0.079-0.042 Gyr^-1 at z = 1.03, 1.32 and 1.54, respectively. Combining our results with previous studies at z < 1, the gas-rich major merger rate evolves as (1+z)^n, with n = 3.95 +- 0.12, up to z = 1.5. From these results we infer that ~35% of the star-forming galaxies with stellar masses M = 10^10 - 10^10.5 M_Sun have undergone a major merger since z ~ 1.5. We develop a simple model which shows that, assuming that all gas-rich major mergers lead to early-type galaxies, the combined effect of gas-rich and dry mergers is able to explain most of the evolution in the number density of massive early-type galaxies since z ~ 1.5, with our measured gas-rich merger rate accounting for about two-thirds of this evolution.
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