We examine the cosmological implications of the measurements of the linear growth rate of cosmological structure obtained in a companion paper from the power spectrum and bispectrum monopoles of the Sloan Digital Sky Survey III Baryon Oscillation Spectroscopic Survey Data Release 11, CMASS galaxies. This measurement was of f 0.43σ8, where σ8 is the amplitude of dark matter density fluctuations, and f is the linear growth rate, at the effective redshift of the survey, zeff = 0.57. In conjunction with cosmic microwave background (CMB) data, interesting constraints can be placed on models with non-standard neutrino properties and models where gravity deviates from General Relativity on cosmological scales. In particular, the sum of the masses of the three species of the neutrinos is constrained to mν < 0.49 eV (at 95 per cent confidence level) when the f 0.43σ8 measurement is combined with state-of-the-art CMB measurements. Allowing the effective number of neutrinos to vary as a free parameter does not significantly change these results. When we combine the measurement of f 0.43σ8 with the complementary measurement of fσ8 from the monopole and quadrupole of the two-point correlation function, we are able to obtain an independent measurements of f and σ8. We obtain f = 0.63 ± 0.16 and σ8 = 0.710 ± 0.086 (68 per cent confidence level). This is the first time when these parameters have been able to be measured independently using the redshift-space power spectrum and bispectrum measurements from galaxy clustering data only.