  We present an analytic expression for the axis ratio distribution of triaxial
dark matter halos driven from physical principles. Adopting the picture of
triaxial collapse based on the Zel'dovich approximation, we derive analytically
both the minor-to-major and the conditional intermediate-to-major axis ratio
distributions, and examine how they depend on the halo mass, redshift, and
cosmology. Our analytic model is tested against the simulation data given by
Jing & Suto in 2002, and found to reproduce the conditional
intermediate-to-major axis ratio distribution successfully and the
minor-to-major axis ratio distribution approximately. However, the trends of
our analytic axis-ratio distributions with mass and redshift are opposite to
what is found in N-body simulations. This failure of our analytic model puts a
limitation on analytic approaches based on the Lagrangian theory to the halo
ellipticity. Given the overall agreement with the simulation results, our model
provides a new theoretical step toward using the axis-ratio distribution of
dark halos as a cosmological probe. We also discuss several possibilities to
improve the model.
