This work investigates the use of a well-known empirical correlation between the velocity dispersion , metallicity , and luminosity in H \beta of nearby H II galaxies to measure the distances to H II -like starburst galaxies at high redshifts .
This correlation is applied to a sample of 15 starburst galaxies with redshifts between z = 2.17 and z = 3.39 to constrain \Omega _ { m } , using data available from the literature .
A best-fit value of \Omega _ { m } = 0.21 ^ { +0.30 } _ { -0.12 } in a \Lambda -dominated universe and of \Omega _ { m } = 0.11 ^ { +0.37 } _ { -0.19 } in an open universe is obtained .
A detailed analysis of systematic errors , their causes , and their effects on the values derived for the distance moduli and \Omega _ { m } is carried out .
A discussion of how future work will improve constraints on \Omega _ { m } by reducing the errors is also presented .