Intuitive usage: the interface should be as clear and simple as possible, allowing the user to evaluate complex quantities in one or a few lines of code. For this purpose, numerous fitting functions have been implemented

Performance: computationally intensive routines are, wherever possible, approximated using smart interpolation tables that rely on as few data points as possible given a desired accuracy. These tables are stored on disk between executions

Stand-alone: Colossus has no dependencies except for the standard numpy and scipy packages

Pure python: Colossus does not contain any non-python code or any code that needs to be compiled, and can thus be installed by simply cloning the repository or with an installer such as pip

Numpy compatibility: virtually all Colossus functions accept both numbers and numpy arrays as input, and return results in the corresponding dimensions

Large range of validity: Colossus tries to cover as wide a range of input parameters as possible. For example, the cosmology module works between redshifts of -0.995 and 200

Consistent units: Colossus follows a coherent set of physical units that are used for the input to and output from all functions

Reproducibility: Colossus contains a suite of about 90 unit tests that ensure that the code functions as expected on the user's machine and python distribution

the linear and logarithmic derivatives of density, d\rho /{dr} and d\mathrm{ln}(\rho )/d\mathrm{ln}(r)

the enclosed mass, M(\lt r)

\end{eqnarray} \tag{ 37 }

the circular velocity, {v}_{{\rm{c}}}(r)=\sqrt{{GM}(\lt r)/r}, and the maximum circular velocity, {v}_{\max }

and

SO radius and mass for a given overdensity definition and redshift

The three-parameter profile of Einasto (, 1965: #apjsaaee8cbib38) is described by a logarithmic slope that changes progressively with radius,\begin{eqnarray}&&{\rho }_{\mathrm{Einasto}}(r)={\rho }_{{\rm{s}}}\exp \left(-\displaystyle \frac{2}{\alpha }\left[{\left(\displaystyle \frac{r}{{r}_{{\rm{s}}}}\right)}^{\alpha }-1\right]\right).\end{eqnarray} \tag{ 38 } The user can either choose the shape parameter α or let it be determined by the fitting function of

The two-parameter Hernquist (,: #apjsaaee8cbib51) profile is given by the expression\begin{eqnarray}&&{\rho }_{\mathrm{Hernquist}}(r)=\displaystyle \frac{{\rho }_{{\rm{s}}}}{\left(\tfrac{r}{{r}_{{\rm{s}}}}\right){\left(1+\tfrac{r}{{r}_{{\rm{s}}}}\right)}^{3}}.\end{eqnarray} \tag{ 40 } This profile approaches power laws of slope -1 and -4 at small and large radii, respectively, turning over around the scale radius {r}_{{\rm{s}}}

The NFW profile

In order to account for the steepening at the splashback radius (Section, 4.4: #apjsaaee8cs4-4), Diemer & Kravtsov (,: #apjsaaee8cbib30,, DK14: #apjsaaee8cbib30) combined the Einasto profile at small radii with a steepening function in the outer profile,\begin{eqnarray}&&{\rho }_{\mathrm{DK}14}(r)={\rho }_{\mathrm{Einasto}}\times {\left[1+{\left(\displaystyle \frac{r}{{r}_{{\rm{t}}}}\right)}^{\beta }\right]}^{-\tfrac{\gamma }{\beta }},\end{eqnarray} \tag{ 42 } where β determines how rapidly this steepening happens and γ represents the limiting slope of the steepening term. Diemer & Kravtsov (,: #apjsaaee8cbib30) recommend (\beta,\gamma )=(4,8) or (6,4), depending on how the halo sample was selected (the profile shown in Figure, 4: #apjsaaee8cf4 uses (4,8)). The turnover radius {r}_{{\rm{t}}} depends on the location of the splashback radius and thus on the mass accretion rate. The, DK14: #apjsaaee8cbib30 profile makes sense only when combined with a prescription for the outer profile as described below

The mean density of the universe, \rho ={\rho }_{{\rm{m}}}(z). This term should always be included if the profile is evaluated at large radii

An estimate of the 2-halo term based on the linear matter-matter correlation function,\begin{eqnarray}&&{\rho }_{2{\rm{h}}}(r,z)={\rho }_{{\rm{m}}}(z)\xi (r,z)b(\nu ),\end{eqnarray} \tag{ 43 } where the bias b(\nu ) can be estimated based on a model of halo bias (Section, 3.3: #apjsaaee8cs3-3). The 2-halo term shown in the right panels of Figure, 4: #apjsaaee8cf4 corresponds to b = 6.1, appropriate for a very massive halo

A power-law outer profile (e.g., Diemer & Kravtsov,: #apjsaaee8cbib30) that can be used to approximate the profile of infalling matter (Bertschinger, 1985: #apjsaaee8cbib5) or mimic a 2-halo term. Mathematically the power-law outer term is described as\begin{eqnarray}&&{\rho }_{\mathrm{PL}}(r,z)={\rho }_{{\rm{m}}}(z)\displaystyle \frac{a}{\tfrac{1}{{\rho }_{\max }}+{\left(\tfrac{r}{{r}_{\mathrm{pivot}}}\right)}^{b}},\end{eqnarray} \tag{ 44 } where a is a normalization, b is the slope, and {r}_{\mathrm{pivot}} is an arbitrary pivot radius that can be set to either a fixed radius or an SO radius. The limiting density {\rho }_{\max } is introduced to avoid spurious contributions at small radii. The power-law outer profile shown in Figure, 4: #apjsaaee8cf4 has a = 1 and b = 1.5 (Diemer & Kravtsov,: #apjsaaee8cbib30)