The paper by Jenkins et al. (2000) combines data from multiple N-body simulations to predict the abundance of dark matter halos in Cold Dark Matter (CDM) universes over a wide range of masses, from $10^{11}$ to $10^{16} h^{-1} M_{\odot}$. The authors compare different simulations to assess systematic uncertainties, which are found to be less than 10–30% at all masses. They use "Hubble Volume" simulations of $\tau$CDM and $\Lambda$CDM cosmologies to predict the abundance of massive clusters with uncertainties below those expected in observational surveys. The study shows that the simulated mass function is nearly independent of epoch, cosmological parameters, and initial power spectrum when expressed in appropriate variables. This universality aligns with the predictions of the Press-Schechter model, although the model's mass function shape differs from the numerical results, overestimating the abundance of "typical" halos and underestimating that of massive systems. The authors also investigate the consistency of mass functions derived from different simulations and halo definitions, using the friends-of-friends (FOF) and spherical overdensity (SO) algorithms. They find that the mass functions are robustly determined with systematic uncertainties at or below 10% for FOF and somewhat higher for SO. The paper compares the simulated mass functions with analytic models, such as the Press-Schechter model and the Sheth-Tormen fitting formula. The Sheth-Tormen model is found to be a very good fit to the universal mass function, showing excellent agreement with the numerical data. Finally, the authors derive a general fitting formula that accurately describes the halo mass function over a wide range of cosmologies and redshifts. This formula is independent of the halo definition and is valid for a wide range of cosmic density values and power spectrum slopes. The formula is shown to be close to the Sheth-Tormen model and provides a significant improvement over the Press-Schechter model in terms of accuracy and coverage.The paper by Jenkins et al. (2000) combines data from multiple N-body simulations to predict the abundance of dark matter halos in Cold Dark Matter (CDM) universes over a wide range of masses, from $10^{11}$ to $10^{16} h^{-1} M_{\odot}$. The authors compare different simulations to assess systematic uncertainties, which are found to be less than 10–30% at all masses. They use "Hubble Volume" simulations of $\tau$CDM and $\Lambda$CDM cosmologies to predict the abundance of massive clusters with uncertainties below those expected in observational surveys. The study shows that the simulated mass function is nearly independent of epoch, cosmological parameters, and initial power spectrum when expressed in appropriate variables. This universality aligns with the predictions of the Press-Schechter model, although the model's mass function shape differs from the numerical results, overestimating the abundance of "typical" halos and underestimating that of massive systems. The authors also investigate the consistency of mass functions derived from different simulations and halo definitions, using the friends-of-friends (FOF) and spherical overdensity (SO) algorithms. They find that the mass functions are robustly determined with systematic uncertainties at or below 10% for FOF and somewhat higher for SO. The paper compares the simulated mass functions with analytic models, such as the Press-Schechter model and the Sheth-Tormen fitting formula. The Sheth-Tormen model is found to be a very good fit to the universal mass function, showing excellent agreement with the numerical data. Finally, the authors derive a general fitting formula that accurately describes the halo mass function over a wide range of cosmologies and redshifts. This formula is independent of the halo definition and is valid for a wide range of cosmic density values and power spectrum slopes. The formula is shown to be close to the Sheth-Tormen model and provides a significant improvement over the Press-Schechter model in terms of accuracy and coverage.