Accelerating computation of the density-field filtering scale σ(R) and non-linear mass by an order of magnitude

The non-linear mass is a characteristic scale in halo formation that has many applications across cosmology. Naively, computing it requires repeated numerical integration to calculate the variance of the power spectrum on different scales and determine which scales exceed the threshold for non-linear collapse. We accelerate calculation of both the non-linear mass and the rms amplitude of the power spectrum σ(R) by working in configuration space and approximating the correlation function as a polynomial at r ≤ 5 h^-1 Mpc. This enables an analytic rather than numerical solution for the non-linear mass, accurate across a variety of cosmologies to 0.1- $1{{\ \rm per\ cent}}$ in mass (depending on redshift) and 20-60× faster than the standard numerical method. We also present a further acceleration of the non-linear mass (400-1000× faster than the standard method) in which we determine the polynomial coefficients using a Taylor expansion in the cosmological parameters rather than re-fitting a polynomial to the correlation function. Our method is also 500× faster than the standard method for σ(R) for a typical case of N_R = 100 desired R values, with timing essentially independent of N_R. Our approach can be used for quick calculation of the halo mass function, halo mass-bias relation, and cosmological calculations involving the non-linear mass.