  Most statistical inference from cosmic large-scale structure relies on
two-point statistics, i.e.\ on the galaxy-galaxy correlation function (2PCF) or
the power spectrum. These statistics capture the full information encoded in
the Fourier amplitudes of the galaxy density field but do not describe the
Fourier phases of the field. Here, we quantify the information contained in the
line correlation function (LCF), a three-point Fourier phase correlation
function. Using cosmological simulations, we estimate the Fisher information
(at redshift $z=0$) of the 2PCF, LCF and their combination, regarding the
cosmological parameters of the standard $\Lambda$CDM model, as well as a Warm
Dark Matter (WDM) model and the $f(R)$ and Symmetron modified gravity models.
The galaxy bias is accounted for at the level of a linear bias. The relative
information of the 2PCF and the LCF depends on the survey volume, sampling
density (shot noise) and the bias uncertainty. For a volume of $1h^{-3}\rm
Gpc^3$, sampled with points of mean density $\bar{n} = 2\times10^{-3} h^{3}\
\rm Mpc^{-3}$ and a bias uncertainty of 13\%, the LCF improves the parameter
constraints by about 20\% in the $\Lambda$CDM cosmology and potentially even
more in alternative models. Finally, since a linear bias only affects the
Fourier amplitudes (2PCF), but not the phases (LCF), the combination of the
2PCF and the LCF can be used to break the degeneracy between the linear bias
and $\sigma_8$, present in 2-point statistics.
