Preprint typeset using LATEX style emulateapj v. 01/23/15

COLOSSUS: A PYTHON TOOLKIT FOR COSMOLOGY, LARGE-SCALE STRUCTURE, AND DARK MATTER HALOS

Benedikt DiemerInstitute for Theory and Computation, Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA 02138, USA;

benedikt.diemer@cfa.harvard.edu

ABSTRACTThis paper introduces Colossus, a public, open-source python package for calculations related to cosmology,the large-scale structure of matter in the universe, and the properties of dark matter halos. The code is designedto be fast and easy to use, with a coherent, well-documented user interface. The cosmology module implementsFLRW cosmologies including curvature, relativistic species, and different dark energy equations of state, andprovides fast computations of the linear matter power spectrum, variance, and correlation function. The large-scale structure module is concerned with the properties of peaks in Gaussian random fields and halos in astatistical sense, including their peak height, peak curvature, halo bias, and mass function. The halo moduledeals with spherical overdensity radii and masses, density profiles, concentration, and the splashback radius.To facilitate the rapid exploration of these quantities, Colossus implements about 40 different fitting functionsfrom the literature. I discuss the core routines in detail, with a particular emphasis on their accuracy. Colossusis available at bitbucket.org/bdiemer/colossus.Keywords: cosmology: theory - methods: numerical

1. INTRODUCTIONOver the past decade, python has emerged as the most popu-

lar programming language for data analysis in astronomy, par-ticularly for computations that do not demand large amountsof computing power. Many of those calculations are non-trivial but need to be implemented time and time again. Forexample, in the commonly accepted ΛCDM cosmology, dis-tances and times must be integrated numerically, increasingthe risk of programming errors and numerical inaccuracies.As a remedy, a number of cosmology calculators have beenpresented (e.g., Wright 2006; Astropy Collaboration et al.2013).

The sub-field of structure formation commonly relies onquantities that are even harder to compute accurately, for ex-ample the linear power spectrum of fluctuations, the varianceof the density field, or the matter-matter correlation function.The corresponding integrals tend to converge slowly, mak-ing their evaluation inefficient in an interpreted language likepython. When considering the structure of cold dark matterhalos on smaller scales, other calculations occur frequently,for example the conversion between halo mass definitions,halo density profiles, and their concentrations.

In this paper, I introduce Colossus, a python packagethat standardizes these calculations into a coherent, well-documented user interface (the name is an acronym for COs-mology, haLO, and large-Scale StrUcture toolS). Colossus isnot intended to be an all-encompassing library for structureformation but to provide a simple interface for basic calcu-lations. For example, Colossus does not replicate the func-tionality of galaxy-halo modeling codes such as HaloTools(Hearin et al. 2017) but it does compute many of the basicquantities that such codes rely on. In this vein, Colossus hasbeen developed with the following design goals in mind.

• Intuitive usage: The interface should be as clear andsimple as possible, allowing the user to evaluate com-plex quantities in a single or in a few lines of code. Forthis purpose, numerous fitting models have been pre-programmed.

• Performance: Computationally intensive routines are,wherever possible, approximated using smart interpo-lation tables with only as many data points as necessarygiven a desired accuracy. These tables are stored ondisk between executions.

• Stand-alone: Colossus has no dependencies except forthe standard numpy and scipy packages.

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

• Numpy integration: Virtually all Colossus functionsaccept both numbers and numpy arrays as input, andreturn results in the corresponding dimensions.

• Large range of validity: Colossus tries to cover as widea range of input parameters as possible. For example,the cosmology module works between redshifts of z =−0.995 (a = 200) and z = 200 (a = 0.05).

• Consistent units: Colossus follows a coherent set ofphysical units that are used for the input to and outputfrom all functions.

• Reproducibility: Colossus contains a suite of more than80 unit tests that ensure that the code functions as ex-pected on the user’s machine and python distribution.

Notably, accuracy is not listed among these principles. Al-though Colossus naturally strives to be as accurate as pos-sible, some functions trade accuracy for speed. Throughoutthe paper, the accuracy of functions and interpolation tables islisted carefully.

Sections 2–4 describe the main modules of Colossus in de-tail, Section 5 discusses future developments. The code repos-itory is hosted at bitbucket.org/bdiemer/colossus, but Colos-sus can also be automatically installed as a package by exe-cuting pip install colossus. An extensive online doc-umentation is available at bdiemer.bitbucket.io/colossus. The

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code repository includes tutorials that explain how to use eachmodule (in the form of Jupyter notebooks).

2. THE COSMOLOGY MODULEThe Colossus cosmology module implements the stan-

dard Friedman-Lemaitre-Robertson-Walker (FLRW) cosmol-ogy and includes the contributions from dark matter, baryons,dark energy, curvature, photons, and neutrinos. The underly-ing expressions can be found in a number of cosmology textbooks (e.g., Rich 2001; Ryden 2003; Dodelson 2003).

2.1. General DesignIn Colossus, the cosmology is set globally so that it does

not need to be passed to functions. If necessary, the user canstore multiple cosmology objects and activate them in turn.For convenience, the user can choose from an extensive list ofpre-defined sets of cosmological parameters (Table 1).

For many of its functions, the cosmology module relies onthe interpolation of lookup tables to speed up the evaluation.These tables are computed on demand, i.e., when a function isfirst evaluated for a given cosmology. The number of bins inthe tables was adjusted to obtain a particular accuracy. The in-terpolation uses cubic splines which are also used to invert thefunctions (e.g., give z(t) rather than t(z)) and evaluate deriva-tives (e.g. dz/dt or dt/dz). However, no guarantee is given onthe accuracy of the derivatives, particularly for higher-orderderivatives.

Table 2 gives an overview over the accuracy and rangeof validity of key functions in the cosmology module. Thequoted accuracy refers to the default parameters, but can bemodified by the user in a number of ways. First, interpo-lation can be turned off altogether, though at a significantperformance penalty. Second, the accuracy of many compu-tations (such as numerical integrations) can be altered fromthe default settings. Finally, the user can change the binningscheme of the lookup tables, though this is generally not rec-ommended.

Constructing all cosmological interpolation tables takesabout 1.7 seconds on a typical machine (2015 MacBook Pro,Python 3.5). This time is dominated by the correlation func-tion without which the calculation is reduced to about 0.15seconds. Importantly, interpolation tables are computed onlyonce: after the first execution, they are stored on disk andloaded on demand. To ensure that only matching tables areloaded for a given cosmology, all relevant parameters are ex-pressed as a string and converted to a unique hash identifierusing the md5 algorithm. All loaded tables are discardedwhen cosmological parameters are changed.

2.2. Initializing CosmologiesCosmology objects are initiated from a set of cosmolog-

ical parameters and settings, including the Hubble constantH0 (often denoted h ≡ H0/(100km/s/Mpc)), the primor-dial power spectrum index ns, the power spectrum normal-ization σ8, and the densities of certain species. The Colossuscosmology includes the densities of matter (dark matter andbaryons), baryons, dark energy, curvature, photons, neutrinos,and the sum of relativistic species (photos and neutrinos), de-noted ρm, ρb, ρde, ρk, ργ, ρν, and ρrel, respectively. Their frac-tion with respect to the critical density, ρc, are denoted Ωm,Ωb, Ωde, Ωk, Ωγ, Ων, and Ωrel, their values at z = 0 as ρm,0and Ωm,0 and so on. The user can specify whether a flat cos-mology is assumed and whether relativistic species should be

included. By default, the cosmology does contain relativisticspecies, namely radiation and neutrinos whose contributionsare computed as follows. The density of radiation today fol-lows from the Stefan-Boltzmann law for a blackbody,

ργ,0 = 4σSBT 4CMB,0 (1)

where σSB is the Stefan-Boltzmann constant. Dividing by thecritical density and converting to the appropriate units, we find

Ωγ,0 =4.4813 × 10−7

h2

(TCMB,0

K

)4

≈ 5.4 × 10−5 (2)

where we have assumed a default CMB temperature ofTCMB,0 = 2.7255K and the “planck15” cosmology (Table 1).The density of neutrinos is

Ων,0 =78

(411

)4/3

NeffΩγ,0 ≈ 0.69 Ωγ,0 ≈ 3.7 × 10−5 . (3)

where the effective number of neutrino species, Neff = 3.046,accounts for three neutrino species and subtle effects that in-fluence the ratio of the photon and neutrino temperatures. Theuser can, of course, set different values for TCMB,0 and Neff , butthe neutrino mass is currently fixed.

Finally, we need to compute the contributions from curva-ture and dark energy. If the cosmology is flat, Ωk,0 = 0 and

Ωde,0 = 1 −Ωm,0 −Ωγ,0 −Ων,0 . (4)

If the user has chosen a non-flat cosmology, Ωde,0 needs to beset and we compute

Ωk,0 = 1 −Ωde,0 −Ωm,0 −Ωγ,0 −Ων,0 . (5)

Dark energy is described by an equation of state parameterw(z) = P(z)/ρ(z) which can be −1 (a cosmological constantresulting in a ΛCDM cosmology), a constant other than −1(wCDM), or vary linearly with redshift according to Linder(2003),

w(z) = w0 + waz

z + 1. (6)

The user can also specify an arbitrary function for w(z).

2.3. Densities, Distances, TimesMany cosmological quantities rely on the Hubble constant

as a function of time, normalized to the z = 0 value,

E(a) =

√Ωm,0a−3 + Ωk,0a−2 + (Ωrel,0)a−4 + f (z)Ωde,0 (7)

where f (z) encapsulates any possible evolution of the darkenergy equation of state,

f (z) = exp(3∫ ln(1+z′)

0[1 + w(z′′)]d ln(1 + z′′)

). (8)

This expression evaluates to unity for a cosmological con-stant, to f (z) = a−3(1+w0) for w = w0, and to

f (z) = a−3(1+w0+wa)e−3wa(1−a) (9)

for w(z) = w0 + wa(1 − a) (Linder 2003). Colossus alwayscomputes E(z) exactly, i.e., without any interpolation. Thecritical density is evaluated as ρc(z) = ρc,0E2(z) whereas theother densities (ρm, ρb, ρde, ργ, ρν) are computed from theirz = 0 values and the redshift scalings given in Equation (7).A number of quantities depend on integrals of E(z), and these

Colossus 3

Table 1Pre-set cosmologies

ID rel H0 Ωm Ωb ns σ8 Reference Comment

planck15 yes 67.74 0.3089 0.0486 0.9667 0.8159 Planck Coll. 2016, Table 4 Best-fit, with external data (column 6)planck15-only yes 67.81 0.3080 0.0484 0.9677 0.8149 Planck Coll. 2016, Table 4 Best-fit, Planck only (column 2)planck13 yes 67.77 0.3071 0.0483 0.9611 0.8288 Planck Coll. 2014, Table 5 Best-fit, with external dataplanck13-only yes 67.11 0.3175 0.0490 0.9624 0.8344 Planck Coll. 2014, Table 2 Best-fit, Planck onlyWMAP9 yes 69.32 0.2865 0.0463 0.9608 0.8200 Hinshaw et al. 2013, Table 4 Best-fit, combined dataWMAP9-only yes 69.70 0.2814 0.0464 0.9710 0.8200 Hinshaw et al. 2013, Table 2 Max. likelihood, WMAP onlyWMAP9-ML yes 69.70 0.2821 0.0461 0.9646 0.8170 Hinshaw et al. 2013, Table 2 Max. likelihood, combined dataWMAP7 yes 70.20 0.2743 0.0458 0.9680 0.8160 Komatsu et al. 2011, Table 1 Best-fit, with BAO and H0WMAP7-only yes 70.30 0.2711 0.0451 0.9660 0.8090 Komatsu et al. 2011, Table 1 Max. likelihood, WMAP onlyWMAP7-ML yes 70.40 0.2715 0.0455 0.9670 0.8100 Komatsu et al. 2011, Table 1 Max. likelihood, with BAO and H0WMAP5 yes 70.50 0.2732 0.0456 0.9600 0.8120 Komatsu et al. 2009, Table 1 Best-fit, with BAO and SNeWMAP5-only yes 72.40 0.2495 0.0432 0.9610 0.7870 Komatsu et al. 2009, Table 1 Max. likelihood, WMAP onlyWMAP5-ML yes 70.20 0.2769 0.0459 0.9620 0.8170 Komatsu et al. 2009, Table 1 Max. likelihood, with BAO and SNeWMAP3 yes 73.50 0.2342 0.0413 0.9510 0.7420 Spergel et al. 2007, Table 5 Best fit, WMAP onlyWMAP3-ML yes 73.20 0.2370 0.0414 0.9540 0.7560 Spergel et al. 2007, Table 2 Max. likelihood, WMAP onlyWMAP1 yes 72.00 0.2700 0.0463 0.9900 0.9000 Spergel et al. 2003, Table 7/4 Best fit, WMAP onlyWMAP1-ML yes 68.00 0.3136 0.0497 0.9700 0.9000 Spergel et al. 2003, Table 1/4 Max. likelihood, WMAP onlyillustris no 70.40 0.2726 0.0456 0.9630 0.8090 Vogelsberger et al. 2014 Cosmology of the Illustris simulationbolshoi no 70.00 0.2700 0.0469 0.9500 0.8200 Klypin et al. 2011 Cosmology of the Bolshoi simulationmillennium no 73.00 0.2500 0.0450 1.0000 0.9000 Springel et al. 2005 Cosmology of the Millennium simulationEdS no 70.00 1.0000 0.0000 1.0000 0.8200 — Einstein-de Sitter cosmology

Note. — All cosmologies listed are flat ΛCDM cosmologies, i.e., have no curvature and w = −1 (or no dark energy in the case of an Einstein-de Sitter cosmology). The “rel” field indicates whether relativistic species (neutrinos and radiation) are included in the calculation which is thedefault. The exception are cosmologies that underlie simulations that do not explicitly follow relativistic species. The cosmology of the Illustris-TNGsimulation suite is equivalent to the “planck15” cosmology (Pillepich et al. 2018).

Table 2Accuracy and range of validity of key functions in the cosmology module

Function Dependent funcs. (incomplete list) § Range a Comput. acc. b Interp. acc. b Inverse Derivs. e

Hubble parameter E(z) Densities, times 2.3 −0.995 < z < 200 Mach. precision - no -Age of the universe - 2.3 −0.995 < z < 200 10−8 5 × 10−4 yes nComoving distance Angular diameter / luminosity dist. 2.3 −0.995 < z < 200 10−8 - yes nLinear growth factor Power spectrum, variance, corr. func. 2.4 −0.995 < z < 200 10−8 (low z) c 3 × 10−4 yes nLinear power spectrum Variance, correlation function 2.5 10−20 < k < 1020 5% (E&H98) 3 × 10−4 yes 1Variance and higher moments Peak height, non-linear mass 2.6 10−12 < R < 103 3 × 10−3 d 5 × 10−3 yes 1Correlation function 2-halo term 2.7 10−3 < R < 5 × 102 10−5 d 10−2 no 1

Note. — a) Many of the functions listed can be evaluated outside the given redshift range but have not been tested in those regimes. The ranges are given inthe native units of the cosmology module, i.e., Mpc/h for distances and radii, h/Mpc for wavenumbers. A number of functions (the power spectrum, variance,and correlation function) depend on redshift through the linear growth factor, which thus defines the redshift range over which they can be reliably computed.b) The table gives two different indicators of accuracy, namely the maximum error to which a quantity is computed (for example, in an integration), and theadditional error due to interpolation. The latter can be avoided by switching interpolation off, though at a (sometimes steep) performance penalty.c) The linear growth factor is computed to the standard integration accuracy of 10−8 at low redshift, but at high z relies on the fitting function of Gnedin et al.(2011) for which no estimate of the accuracy is given.d) The accuracy of the variance and correlation function refers to the accuracy of the integration and does not include the (generally much larger) inaccuraciesdue to the underlying approximation of the power spectrum (see Section 2.6 and 2.7 for estimates of those errors).e) Some functions can return their first derivative, some functions even higher orders. Note, however, that the accuracy of those derivatives is not guaranteedand likely to get worse at higher orders due to interpolation errors.

quantities are stored in interpolation tables to avoid repeatedevaluation of the integrals. For example, the age of the uni-verse is

t =1

H0

∫ ∞

z

1E(z) × (1 + z)

. (10)

All times are expressed in units of Gyr in Colossus. The line-of-sight comoving distance to a particular redshift is

dcom,los = c∫ z

0

1E(z)

(11)

where all distances are expressed in units of comovingh−1Mpc. For non-flat cosmologies, the transverse comoving

distance differs from this expression,

dcom,trans =

c√Ωk,0

sinh( √

Ωk,0

c dcom,los

)∀Ωk,0 > 0

dcom,los ∀Ωk,0 = 0c√−Ωk,0

sin( √−Ωk,0

c dcom,los

)∀Ωk,0 < 0

. (12)

Interpolation tables are used for the luminosity distancedlum = dcom,trans(1 + z) and the angular diameter distancedang = dcom,trans/(1 + z).

The Colossus results for densities, distances, and timesagree to a few times 10−4 or better with those computed byastropy, which was itself tested against several other calcula-tors (Astropy Collaboration et al. 2013). When interpolation

4 Diemer

Table 3Fitting functions implemented in Colossus

Model ID Reference Comments

Cosmology: Power spectrum

eisenstein98 Eisenstein & Hu 1998 Semi-analytical fit to the transfer function calibrated based on numerical calculationseisenstein98 zb Eisenstein & Hu 1998 The zero-baryon limit of the Eisenstein & Hu 1998 model (i.e., no baryon acoustic oscillations)

LSS: Mass function

press74 Press & Schechter 1974 Prediction based on the statistics of peaks in Gaussian random fields (FOF, uses δc(z))sheth99 Sheth & Tormen 1999 A calibration based on numerical results (FOF, uses δc(z)); see also Sheth et al. 2001jenkins01 Jenkins et al. 2001 A calibration based on numerical results (FOF, no z-dependence)reed03 Reed et al. 2003 High-mass correction to Sheth & Tormen 1999 model (FOF, uses δc(z))warren06 Warren et al. 2006 A calibration based on numerical results (FOF, no z-dependence)reed07 Reed et al. 2007 A model that takes the varying slope of the power spectrum into account (FOF, z-dependent)tinker08 Tinker et al. 2008 Calibration for SO halos with 200 ≤ ∆m ≤ 3200, explicit z-dependencecrocce10 Crocce et al. 2010 A calibration based on numerical results (FOF, uses δc(z))bhattacharya11 Bhattacharya et al. 2011 A calibration based on numerical results (FOF, explicit z-dependence)courtin11 Courtin et al. 2011 A calibration based on numerical results (FOF, uses fixed δc = 1.673)angulo12 Angulo et al. 2012 A calibration based on numerical results (FOF, no z-dependence)watson13 Watson et al. 2013 Both FOF and SO fits, explicit z-dependence in the latterbocquet16 Bocquet et al. 2016 A model for different mass definitions, redshifts, and both hydro and DM-only simulationsdespali16 Despali et al. 2016 A redshift and mass definition-dependent calibration for both ellipsoidal and SO halo finders

LSS: Bias

cole89 Cole & Kaiser 1989 Bias prediction based on the peak-background split model (see also Mo & White 1996)sheth01 Sheth et al. 2001 Bias model taking the ellipsoidal nature of halos into accounttinker10 Tinker et al. 2010 Fitting function that depends on the mass definition

Halo: Density profile

Einasto Einasto 1965 A three-parameter profile with smoothly varying slopeHernquist Hernquist 1990 A two-parameter profile with inner and outer logarithmic slopes of −1 and −4NFW Navarro et al. 1995, 1996, 1997 A two-parameter profile with inner and outer logarithmic slopes of −1 and −3DK14 Diemer & Kravtsov 2014 A profile function that models the steepening due to the splashback radius

Halo: Concentration

bullock01 Bullock et al. 2001 Universal model, c200c for any mass, redshift, and cosmologyduffy08 Duffy et al. 2008 Power-law fit, c200c, cvir, and c200m for 1 × 1011 < M < 1015 h−1 M, 0 < z < 2, WMAP5klypin11 Klypin et al. 2011 Power-law fit, cvir for 3 × 1010 < M < 5 × 1014 h−1 M, z = 0, WMAP7 cosmologyprada12 Prada et al. 2012 Fit based on peak height, c200c for any mass, redshift, and cosmologybhattacharya13 Bhattacharya et al. 2013 Power-law fit in ν, c200c, cvir, and c200m for 2 × 1012 < M < 2 × 1015 h−1 M, 0 < z < 2, WMAP7dutton14 Dutton & Maccio 2014 Power-law fit, c200c and cvir for M > 1010 h−1 M, 0 < z < 5, planck13 cosmologydiemer15 Diemer & Kravtsov 2015 Universal model, c200c for any mass, redshift, or cosmologyklypin16 m Klypin et al. 2016 Power-law fit, c200c and cvir for M > 1010 h−1 M, 0 < z < 5, planck13 or WMAP7 (function of M)klypin16 nu Klypin et al. 2016 Power-law fit, c200c and cvir for M > 1010 h−1 M, 0 < z < 5, planck13 (function of ν)

Halo: Splashback radius

adhikari14 Adhikari et al. 2014 Semi-analytical model, Rsp and Msp as a function of (Γ, z)more15 More et al. 2015 Numerical calibration, Rsp and Msp as a function of (Γ, z) or (M, z)shi16 Shi 2016 Semi-analytical model, Rsp and Msp as a function of (Γ, z)mansfield17 Mansfield et al. 2017 Numerical calibration, Rsp, Msp, and scatter as a function of (Γ, M, z)diemer17 Diemer et al. 2017 Numerical calibration, Rsp, Msp, and scatter as a function of (Γ, M, z) or (M, z)

Note. — Most Colossus modules (e.g., concentration and mass function) can automatically convert spherical overdensity mass definitions. There is,however, no simple conversion between friends-of-friends (FOF) and spherical overdensity (SO) definitions (More et al. 2011).

is used to speed up the evaluations (the default in Colossus),interpolation errors are well below 10−3 in all quantities andat all redshifts except for the luminosity distance at z < 0.1where errors can reach 7 × 10−3.

2.4. The Linear Growth FactorThe final quantity that relies on an integral of E(z) is the

linear growth factor, D+(z), the time-dependent normalizationof the linear fluctuations in the density field. The value ofD+ is influenced by relativistic species at high redshift and bydark energy at low redshift. Thus, the redshift range is splitinto multiple segments. At z > 10, D+ is approximated using

Equation 5 in Gnedin et al. (2011),

D+(a) =a +23

amr +amr

2 ln(2) − 3

×

2√1 + x +

(23

+ x)

ln

√1 + x − 1√

1 + x + 1

(13)

where x ≡ a/amr and amr ≡ Ωrel/Ωm is the epoch of matter-radiation equality. If relativistic species are not included inthe cosmology, D+(a) = a in this redshift regime. The rela-tivistic corrections become very small at low redshift and canbe neglected, but dark energy needs to be taken into accountinstead such that

D+(z) =52

Ωm,0E(z)∫ ∞

z

1 + z′

E(z′)3 dz′ . (14)

Colossus 5

10−9

10−5

10−1

103

P(k

)[h−

3M

pc3

]

Power spectrumeisenstein98 zb

eisenstein98

CAMB

−2

0

dln

(P)/d

ln(k

)

Slope

−0.05

0.00

0.05

P/P

CA

MB−

1

Model error

10−4 10−2 100 102 104

k [hMpc−1]

−0.0002

0.0000

0.0002

Pin

terp/P

exact−

1

Interpolation error

10−2

10−1

100

101

σ(R

)

Variance

−1

0

dln

(σ)/d

ln(R

)

−0.05

0.00

0.05

σ/σ

CA

MB−

1

10−2 100 102

R [h−1Mpc]

−0.01

0.00

0.01

σin

terp/σ

exact−

1

10−4

10−2

100

102

|ξ(R

)|

Correlation function

10−9

10−4

101

106

|dξ/dR|

−0.2

0.0

0.2

ξ/ξ C

AM

B−

1

10−3 10−1 101

R [h−1Mpc]

−0.01

0.00

0.01

ξ inte

rp/ξ

exact−

1

Figure 1. The linear matter power spectrum, variance, and correlation function computed numerically and from fitting functions. Left column: the top panelshows the power spectrum computed numerically using the Camb code (Lewis et al. 2000, dashed dark blue line) and the model of Eisenstein & Hu (1998, lightblue) as well as their formula for the zero-baryon limit (purple). The zero-baryon case is not distinguishable by eye, but its slope is visibly different (second row),particularly around the wiggles due to the baryon acoustic oscillations (BAO). The Eisenstein & Hu (1998) function does not take the baryonic pressure intoaccount, leading to excessive power at scales greater than k ≈ 100hMpc−1. Elsewhere, the model reproduces the numerical calculation to better than 5% (thirdrow). This error contains a small interpolation error (shown in the bottom row as the ratio of the interpolated to the exact prediction of the Eisenstein & Hu (1998)model). This interpolation error is a function of the binning scheme used for the interpolation table and was designed to be insignificant. The k-range shownis the range over which the Camb calculation is defined, but the Colossus power spectrum can be evaluated at any 10−20 < k < 1020hMpc−1. Center column:the variance, calculated using numerical integration (Equation 16). The Eisenstein & Hu (1998) approximation results in a variance that matches the numericalcomputation to better than 2% except at very small radii where the additional power at small scales begins to matter. The numerically computed power spectrumwould give increasingly poor results at small radii because of its limited k-range. The interpolation of σ results in errors of at most 0.5%. Right column: likecenter column, but for the correlation function. The relative errors due to both the approximate power spectrum and interpolation grow around the zero-crossing,but the difference is small in absolute units.

Here, E(z) does not include contributions from relativisticspecies, even if they are included in the cosmology. Differ-ent normalizations of the growth factor have been suggestedin the literature (e.g., Eisenstein & Hu 1999; Percival 2005),the Colossus normalization is chosen such that D+(a) = a inthe matter-dominated regime to match the high-redshift for-mula in Equation (13). In order to smooth a slight mis-matchof 10−3 between these expressions, they are interpolated inlog(a) space between z = 20 and z = 5, leading to inaccura-cies smaller than 3 × 10−4 at any redshift.

The growth factor calculation was tested against the iCos-mos calculator (icosmos.co.uk) and found to agree to muchbetter than a percent at low redshift. At high redshift, it is notclear that the two computations are directly comparable.

2.5. The Linear Matter Power SpectrumNumerous important quantities in structure formation are

based on the linear matter power spectrum, P(k), the ampli-tude of density fluctuations as a function of scale. The powerspectrum can be parameterized in terms of the primordial den-

sity field whose power spectrum is assumed to be a power law,

P(k) = T 2(k) × kns (15)

where T (k) is called the transfer function and ns is a cosmo-logical parameter, slightly less than 1 (Table 1). The transferfunction encapsulates the physics of growing and decayingperturbations, starting with the primordial power law and in-cluding effects such as the stagnation of growth during theradiation-dominated era, baryon acoustic oscillations (BAO),and various damping terms. After recombination, the evolu-tion simplifies because both baryons and dark matter behavelike pressureless fluids on large scales.

The left top panel of Figure 1 shows the power spec-trum calculated using the Boltzmann code Camb (Lewis et al.2000). At small k, T (k) = 1, meaning that the power spec-trum is equal to the primordial power law. The transfer func-tion starts to decrease around the horizon scale at the epochof matter-radiation equality. By definition, the linear powerspectrum captures only the linear contribution to the growth

6 Diemer

of perturbations but not their non-linear collapse. The timeevolution of the linear component is described by the lineargrowth factor, P(k, z) = D2

+(z)P(k, 0). The power spectrum isnormalized to give a particular variance σ(8 h−1Mpc, z = 0) =σ8 (Section 2.6).

While Colossus allows the user to supply a tabulated powerspectrum, for example from numerical calculations usingCamb or Cmbfast (Seljak & Zaldarriaga 1996), the default isto compute T (k) using the approximation of Eisenstein & Hu(1998). This semi-analytical fitting function is accurate to bet-ter than 5% if the effects of baryons are included (Figure 1).Colossus computes an interpolation table for the power spec-trum when it is first evaluated for a given cosmology. This ta-ble covers wavenumbers between 10−20 and 1020hMpc−1 anduses a non-uniform binning scheme with an increased densityof bins near the BAO features. The interpolation accuracy isbetter than 2 × 10−4 across the entire range of wave numbers(bottom left panel of Figure 1).

2.6. VarianceGiven the linear power spectrum, we can compute the vari-

ance of the density field as well as its higher-order moments,

σ2j (R, z) =

12π2

∫ ∞

0k2+2 jP(k, z)|W(kR)|2dk (16)

where W is a filter. Colossus offers multiple options for thisfilter, namely the most commonly used top-hat in real space,

Wtophat =3

(kR)3 [sin(kR) − kR × cos(kR)] . (17)

When this filter is used, σ0 quantifies the variance in spheresof radius R. Alternatively filters include a Gaussian,

Wgaussian = e−(kR)2, (18)

or a sharp k-space filter,

Wsharp−k = Θ(1 − kR) (19)

where Θ is the Heaviside step function. The variance andhigher-order moments grow with time according to the lineargrowth factor, σj(R, z) = D+(z)σj(R, 0).

The center column of Figure 1 compares σ ≡ σ0 calculatedfrom the numerically computed power spectrum and from theEisenstein & Hu (1998) approximation. The agreement is bet-ter than 2% at all relevant radii, the larger disagreement atvery small R is caused by two effects: first, the Eisenstein &Hu (1998) approximation overestimates the power at large kbecause it ignores the pressure from baryons, and second, theCamb power spectrum can only be computed to a finite wavenumber, meaning that σ is underestimated at small R.

The interpolation table for the variance covers a radial rangefrom 10−12 to 103 h−1Mpc. The integration accuracy is set to3 × 10−3 or better, the interpolation is accurate to 5 × 10−3 orbetter (center bottom panel of Figure 1).

2.7. Correlation FunctionThe linear matter-matter correlation function is given by yet

another integral over the power spectrum,

ξmm(R, z) =1

2π

∫ ∞

0k2P(k, z)

sin(kR)kR

dk . (20)

This integral converges slowly because of the fast frequencyand slow fall-off of the sine term at high kR, making effi-cient interpolation particularly important. Colossus reduces

the computation time by using two numerical tricks. First, theintegrand is exponentially suppressed at kR > 1000 becausethose fast oscillations contribute a negligible amount to theoverall integral. Second, Clenshaw-Curtis integration speedsup the integration of sinusoidal terms.

The integration accuracy is set to an error of at most 10−5.The inaccuracy due to the power spectrum approximation ismuch larger, up to about 4% between 10−2 h−1Mpc and thezero-crossing of the correlation function (Figure 1). Aroundthe zero crossing, the relative error grows because the ab-solute value of the function becomes small. The interpola-tion for the correlation function spans radii between 10−3 and500 h−1Mpc, and is accurate to better than 1% (bottom rightpanel of Figure 1).

3. THE LARGE-SCALE STRUCTURE MODULEThe large-scale structure (LSS) module covers the linear

and non-linear collapse of Gaussian random fields, includingdensity peaks, their peak height and curvature, as well as theabundance of collapsed peaks (the halo mass function) andbias. Functions that deal with matter in general (as opposedto collapsed peaks) are based in the cosmology module, andfunctions that deal with the mass, radius, or structure of col-lapsed peaks are based in the halo module. All fitting func-tions implemented in the LSS module are listed in Table 3.

3.1. Peak Height, Peak Curvature, and Non-Linear MassGiven the variance on a scale R, we can compute the sta-

tistical significance of the peaks corresponding to a particularhalo mass, or “peak height”. Here, some measure of halomass (e.g, Mvir) is translated into its Lagrangian radius, thecomoving radius of a sphere that encompasses halo’s mass atthe mean density of the universe,

ML = (4π/3)ρm(z = 0)R3L . (21)

The peak height is derived by comparing the inverse varianceon this radial scale with the overdensity above which densityfluctuations are expected to collapse into halos,

ν ≡δc

σ(ML, z)=

δc

σ(ML, z = 0) × D+(z). (22)

The critical overdensity for collapse,

δc,EdS =35

(3π2

)2/3

= 1.68647 (23)

is derived from the spherical top hat collapse model in anEinstein-de Sitter universe (Gunn & Gott 1972). In other cos-mologies, small corrections apply, namely

δc = δc,EdSΩm(z)0.0185 (24)

in non-flat cosmologies without dark energy and

δc = δc,EdSΩm(z)0.0055 (25)

in flat cosmologies with dark energy (Mo et al. 2010). Thesecorrections change δc by less than one percent for realisticcosmologies, and Colossus applies them only if requested bythe user. Finally, the non–linear mass, M∗, is defined as themass where σ(M∗) = δc, and thus ν(M∗) = 1.

The next higher-order property of peaks is their curvature(Bardeen et al. 1986),

〈x〉 =∇2δ

σ2. (26)

Colossus 7

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1dn/d

ln(M

)FOF

sheth99press74jenkins01reed03warren06reed07crocce10bhattacharya11courtin11angulo12watson13

109 1010 1011 1012 1013 1014 1015

MFOF (h−1M)

0.6

0.8

1.0

1.2

1.4

f/f

shet

h99

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

dn/d

ln(M

)

SO

tinker08watson13bocquet16 (DM)bocquet16 (hydro)despali16 (SO)despali16 (ellipsoidal)

109 1010 1011 1012 1013 1014 1015

M200m (h−1M)

0.6

0.8

1.0

1.2

1.4

f/f

tin

ker

08

Figure 2. Comparison of the halo mass function models implemented in Colossus, evaluated at z = 0 and for the planck15 cosmology. The left panel showsfriends-of-friends mass functions, the right panel SO mass functions. The top panels demonstrate that the differences between the models are subtle, at least whenviewed over a large range of halo mass. The bottom panels show the residual of the models with respect to the model of Sheth & Tormen 1999 (FOF) and Tinkeret al. 2008 (SO). For the SO mass functions, the differences between the models tend to increase toward higher overdensities. The Despali et al. (2016) modelwas additionally fit to mass functions found by an ellipsoidal halo finder, meaning that some disagreement with the conventional SO mass functions is expected.The difference between the Bocquet et al. (2016) calibrations for their dark matter-only and hydrodynamical simulations gives a hint as to the effect of baryonson the mass function.

The higher order moments of the variance (such as σ2) mustbe computed using a Gaussian filter rather than a top-hat be-cause the latter integral does not converge. One important is-sue with peak curvature is the cloud-in-cloud problem: while〈x〉 gives the average curvature of peaks of a certain size, notall of those peaks end up forming halos because they may getintegrated into other, larger peaks. Thus, 〈x〉 does not neces-sarily correspond to the average curvature of the peaks thatcreate halos of a particular mass (Bardeen et al. 1986).

3.2. Halo Mass FunctionThe mass function quantifies how many halos of a given

mass have formed at a given redshift and cosmology. In theexcursion set picture (Press & Schechter 1974; Bond et al.1991), the mass function is expected to be universal (i.e., in-dependent of redshift and cosmology) when expressed as themultiplicity function, f (σ). This function translates to thenumber of halos per logarithmic mass interval as

dNd ln(M)

= f (σ)ρm,0

Md ln(σ−1)d ln(M)

(27)

where σ(M) is the variance on the lagrangian scale of a haloas defined in Equation 16. The multiplicity function can beinterpreted as the fraction of mass that has collapsed to formhalos in a unit interval of ln(σ−1). Colossus can return themass function in units of either f (σ) or dN/d ln(M). Press &Schechter (1974) derived the generic prediction that

fPS(σ) =

√2π

δc

σexp

(−δ2

c

2σ2

)(28)

but this form was found to be in disagreement with numericalsimulations, leading to numerous improved fitting functions

for f (σ). The models implemented in Colossus are listed inTable 3, Figure 2 shows a comparison for halos defined via thefriends-of-friends algorithm (FOF, Davis et al. 1985), and forhalos defined via the spherical overdensity (SO) definition.

While the universality of f (σ) is still debated (e.g., Tinkeret al. 2008; Bhattacharya et al. 2011), its redshift evolution isagreed to be relatively mild. Some models encode an evolu-tion explicitly, some exhibit a slightly changing f (σ) due tothe weak redshift evolution of δc (Equations 24 and 25). Theuser can choose whether this evolution is taken into accountor not (unless the model explicitly specifies that δc should bea particular constant, e.g. Courtin et al. 2011). Finally, mod-els for the spherical overdensity (SO) mass function that canrescale between different mass definitions introduce a redshiftdependence simply because of the dependence of the over-density threshold on redshift (e.g., Tinker et al. 2008; Watsonet al. 2013; Despali et al. 2016).

The mass functions from Colossus were compared to thehmf package (Murray et al. 2013) and exhibit excellent agree-ment. However, some choices (such as the definition of thecollapse overdensity) differ in the default versions of the twocodes.

3.3. Halo BiasHalo bias quantifies the excess clustering of collapsed halos

over that of dark matter. Thus, bias can be defined as the ratioof the halo and linear matter power spectra,

b ≡

√Phalo(k)

Plin,matter(k). (29)

While this function is, in principle, expected to depend onboth halo mass and scale, many bias models ignore any scale

8 Diemer

0 1 2 3 4 5

ν200m

0

2

4

6

8

10

12b

cole89

sheth01

tinker10

Figure 3. Comparison of the halo bias models implemented in Colossus.

dependence and quantify the bias on large scales. Colossusimplements a few such models. A simple prediction for thebias can be derived from the peak-background split ansatz(Cole & Kaiser 1989; Mo & White 1996). Sheth et al. (2001)improved upon this prescription by taking the ellipsoidal na-ture of the collapse into account. Tinker et al. (2010) under-took a careful numerical calibration and included a prescrip-tion for the dependence of bias on the halo mass definition.As a result, their model exhibits a slight redshift dependencefor most mass definitions. Figure 3 compares those models asa function of peak height.

4. THE HALO MODULEThe halo module is concerned with the spherically aver-

aged structure of dark matter halos and their boundaries. Ta-ble 3 gives an overview of the fitting functions implementedfor halo density profiles, concentration, and the splashbackradius.

4.1. Spherical OverdensityThe most commonly used definition of a halo’s boundary

and mass is the spherical overdensity (SO) definition wherethe radius is defined to enclose a particular overdensity ∆ suchthat

M∆ =4π3

∆ρrefR3∆ (30)

where ρref is either the critical or mean matter density of theuniverse (e.g. Cole & Lacey 1996),

M∆m = M(< R∆m) =4π3

∆ρm(z)R3∆m , (31)

for example R200m and M200m, or

M∆c = M(< R∆c) =4π3

∆ρc(z)R3∆c , (32)

for example R200c and M200c. The labels Mvir and Rvir indicatea varying overdensity ∆vir(z) which Colossus computes usingthe approximation of Bryan & Norman (1998). Colossus of-fers a number of basic routines related to SO masses and radii,beginning with the computation of the density threshold ρref .Based on this threshold, we define a typical velocity

v∆ ≡

√GM∆

R∆

(33)

which we use to define the dynamical time of the halo as thetime it takes to cross 2R∆,

tdyn(z) ≡ tcross(z) =2R∆

v∆

= 23/2tH(z)(ρ∆(z)ρc(z)

)−1/2

. (34)

Notably, this time does not depend on the distribution of mat-ter inside the halo or even the halo radius, but only on ρ∆ andthe Hubble time,

tH(z) ≡1

H(z)=

√3

8πGρc(z). (35)

Alternatively, Colossus offers the time to pericenter (crossingR∆) or the orbital time (crossing 2πR∆) as definitions of thedynamical time.

If we wish to convert between R∆ and M∆ for different over-density definitions, the results do depend on the density pro-file. In particular, we need to solve Equation (30) numericallygiven a particular ρref and M(r). By default, Colossus uses aNavarro-Frenk-White profile (see Section 4.2) to convert be-tween definitions, but the user can also choose other profilemodels. The concentration can either be given by the user orautomatically computed using a model of the concentration-mass relation (Section 4.3).

The conversion between different overdensity definitionsconstitutes a special case of a more general conversion wherenot only the overdensity is varied but also the redshift. Here,the assumption is that the halo density profile is static in time,but that the spherical overdensity “pseudo-evolves” due to thechange in the critical or mean density (see, e.g., Diemand et al.2005; Cuesta et al. 2008; Diemer et al. 2013; Zemp 2014;More et al. 2015). This more general routine is also imple-mented in Colossus. All conversion routines are based on theBrent (1973) root finding algorithm with the default accuracyof 10−12.

4.2. Density ProfilesThe density profiles of dark matter halos have been stud-

ied extensively in the literature, generally based on the resultsof numerical simulations. All profile models implementedin Colossus describe the spherically averaged density profileρ(r). The implementation of each model relies on a base classwhich numerically computes numerous quantities, including:

• the linear and logarithmic derivatives of density, dρ/drand d ln(ρ)/d ln(r)

• the enclosed mass and cumulative pdf

• the surface density Σ and differential surface density ∆Σ

• the circular velocity vc =√

GM/R, as well as the max-imum circular velocity vmax

• spherical overdensity radius and mass for a given over-density definition and redshift.

A profile model can be fully specified by only the densityρ(r). The quantities listed above are, by default, computednumerically from the density unless a model implementationoverwrites them, for example when analytical solutions areavailable. In Colossus, the density profile is modeled as an“inner” profile (or 1-halo term) as well as an arbitrary numberof “outer” profiles. The left panels of Figure 4 show a com-parison of the inner profile models implemented in Colossus.

Colossus 9

10−1

100

101

102

103

104

105

106

ρ/ρ

m

Inner profiles

Einasto

Hernquist

NFW

DK14

10−3 10−2 10−1 100 101

r/Rvir

−4

−3

−2

−1

0

dlo

g(ρ

)/d

log(r

)

10−1

100

101

102

103

104

105

106

ρ/ρ

m

Outer profiles

EinastoEin.+ meanEin.+ power lawEin.+ mean + 2haloPower law2halo

10−2 10−1 100 101

r/Rvir

−4

−3

−2

−1

0

dlo

g(ρ

)/d

log(r

)

Figure 4. Halo density profiles (top panels) and their logarithmic slopes (bottom panels). All profiles correspond to a halo with Mvir = 1015 h−1 M and cvir = 5at z = 0. Left: a comparison of the Einasto, Hernquist, NFW, and DK14 profile forms. For realism and comparability, a power-law outer profiles as well as themean density of the universe were added to all profiles at large radii. Right: a comparison of the outer profile terms available in Colossus. The gray lines showthe density contributions due to the power law and correlation-function outer terms. The power-law term is cut off at a particular maximum density in order toavoid spurious contributions at very small radii.

• The three-parameter profile of Einasto (1965) is de-scribed by a logarithmic slope that changes progres-sively with radius,

ρEinasto = ρs exp(−

2α

[(rrs

)α− 1

]). (36)

The user can either choose the shape parameter α orlet it be determined by the fitting function of Gao et al.(2008),

α(ν) = 0.155 + 0.0095ν2 (37)

which gives α = 0.25 for the massive halo shown inFigure 4.

• The two-parameter Hernquist (1990) profile is given bythe expression

ρHernquist =ρs(

rrs

) (1 + r

rs

)3 . (38)

This profile approaches power laws of slope −1 and−4 at small and large radii, respectively, turning overaround the scale radius rs.

• The Navarro-Frenk-White (NFW) profile (Navarroet al. 1995, 1996, 1997) changes the outer slope of theHernquist profile to −3,

ρNFW =ρs(

rrs

) (1 + r

rs

)2 . (39)

• In order to account for the steepening at the splashbackradius (Section 4.4), Diemer & Kravtsov (2014, DK14)

combined the Einasto profile at small radii with a steep-ening function in the outer profile,

ρDK14 = ρEinasto ×

1 +

(rrt

)β−γβ

(40)

where β determines how rapidly this steepening hap-pens, and γ represents the limiting slope of the steep-ening term. Diemer & Kravtsov (2014) recommend(β, γ) = (4, 8) or (6, 4), depending on how the halosample was selected (the profile shown in Figure 4 uses(4, 8)). The turnover radius rt depends on the locationof the splashback radius and thus on the mass accre-tion rate. The DK14 profile makes sense only whencombined with a prescription for the outer profile asdescribed below.

Regardless of the parameterization of the given profile, theuser can initialize a profile from a mass and concentrationwhich are automatically converted to the native parameters(e.g., ρs and rs). Furthermore, Colossus provides an arbitraryspline-interpolated profile based on a table of either ρ(r) orM(r). Care needs to be taken when integrating over such pro-files, for example if the radial extent of the table is not suffi-cient to compute the surface density.

The left panels of Figure 4 show a comparison of these pro-file forms and their logarithmic derivatives. All models of theinner profile become somewhat unrealistic at r >

∼ Rvir wherethe outer profile begins to contribute significantly. Physically,the excess density at large radii is due to a combination of thenon-linear infall of matter into the halo and the statistical con-tribution from neighboring halos (the 2-halo term, e.g. Smithet al. 2003; Hayashi & White 2008). In Colossus, these con-tributions are modeled as the sum of an arbitrary combination

10 Diemer

of the following terms:

• The mean density of the universe, ρ = ρm(z). This termshould always be included if the profile is evaluated atlarge radii.

• An estimate of the 2-halo term based on the linearmatter-matter correlation function,

ρ2h = ρmξ(r)b(ν) (41)

where the bias b(ν) can be estimated based on a modelof halo bias (Section 3.3). The 2-halo term shown inthe right panels of Figure 4 corresponds to b = 6.1,appropriate for a very massive halo.

• A power-law outer profile (e.g., Diemer & Kravtsov2014) which can be used to approximate the profile ofinfalling matter (e.g., Bertschinger 1985) or mimic a 2-halo term. Mathematically the power-law outer term isdescribed as

ρPL = ρma

1ρmax

+

(r

rpivot

)b (42)

where a is a normalization, b is the slope, rpivot is anarbitrary pivot radius which can be set to either a fixedradius or an SO radius. The limiting density ρmax is in-troduced to avoid spurious contributions at small radii.The power-law outer profile shown in Figure 4 hasa = 1 and b = 1.5 (Diemer & Kravtsov 2014).

Finally, the density profile object provides powerful function-ality to fit profile models to data. The data can be density, en-closed mass, or (differential) surface density, error estimatesand covariances are taken into account if desired, and the fitcan be performed using either a least-squares or MCMC algo-rithm.

4.3. ConcentrationAs illustrated in the previous section, the most popular ex-

pressions for halo density profiles are characterized by a scaleradius, rs, in relation to which the profile steepens. Such pro-files are more naturally described by an SO mass and a con-centration, the ratio between the SO radius and the scale ra-dius, c∆ ≡ R∆/rs. If concentration can be quantified as a func-tion of mass and other parameters, a full description of thehalo profile can be derived from only an input mass, makingthe concentration-mass relation a paramount tool in halo mod-eling. However, the c-M relation turns out to depend redshiftand cosmology in a non-trivial fashion, and to exhibit signifi-cant halo-to-halo scatter. Thus, numerous models for the c-Mrelation have been proposed in the literature. Most works relyon NFW-based concentrations, i.e. fitting halo profiles withthe NFW form to derive rs, but other parameterizations ex-ist (e.g., Klypin et al. 2011, Prada et al. 2012; see Dutton &Maccio 2014 for a comparison with Einasto-based concentra-tions).

The Colossus concentration module implements severalmodels for the c-M relation (Table 3), Figure 5 shows a com-parison of their predictions at z = 0. The models broadlyfall into three categories. First, the c-M relation is reasonablywell described by a power law for a given redshift and cos-mology, and within a certain range of halo mass (Duffy et al.2008; Klypin et al. 2011; Dutton & Maccio 2014; Klypin et al.

109 1010 1011 1012 1013 1014 1015

M200c (h−1M)

2

4

6

8

10

12

14

c 200c

z = 0

WMAP7 Cosmology

bullock01

duffy08

klypin11

prada12

bhattacharya13

dutton14

diemer15

klypin16 m

klypin16 nu

Figure 5. Halo concentration at z = 0 for the bolshoi (WMAP7) cosmology(Table 1), as predicted by the various concentration models implemented inColossus.

2016). Second, concentration is strongly correlated with haloage, giving rise to models that base concentration on a de-scription of halo formation times (e.g., Bullock et al. 2001;Wechsler et al. 2002; Zhao et al. 2003, 2009). Finally, thec-M relation is almost universal with redshift when mass isexpressed as peak height (Section 3.1), leading to models thatparameterize the dependence either empirically or based onphysical arguments (Prada et al. 2012; Bhattacharya et al.2013; Diemer & Kravtsov 2015). Some noteworthy modelsare not implemented in Colossus, most commonly becausethey are not based on simple analytical expressions and thusdemand significant computation (e.g. Eke et al. 2001; Zhaoet al. 2009; Correa et al. 2015; Ludlow et al. 2016).

A further complication arises because the models quantifydifferent definitions of concentration, including c200m, c200c,and cvir. The conversion to the desired mass definition is per-formed automatically by Colossus and can be based on eitherNFW or DK14 profiles. This process is iterative because aninput mass in one definition may lead to an output concen-tration in another definition, which in turn changes the con-version between the masses. The conversion can introduceslight inaccuracies into the predictions for concentration be-cause the profile models do not describe real halos perfectly(Diemer & Kravtsov 2015). Moreover, the c-M relations mea-sured in simulations depend on technical details such as thefitting procedure and binning (Dooley et al. 2014; Meneghettiet al. 2014). Taking these effects into account would add sig-nificant complication and is beyond the scope of Colossus.

4.4. The Splashback RadiusThe splashback radius, Rsp, has recently been proposed as a

physically motivated definition of the halo boundary (Diemer& Kravtsov 2014; Adhikari et al. 2014; More et al. 2015)and has since been observed in stacked cluster density pro-files (More et al. 2016; Baxter et al. 2017; Chang et al. 2017).Unlike spherical overdensity radii, the splashback depends onthe dynamical state of a halo, namely on its mass accretionrate. A number of models for this dependence, and additionaldependencies on halo mass and redshift, have been proposed

Colossus 11

0 1 2 3 4 5

Mass accretion rate s or Γ

0.6

0.8

1.0

1.2

1.4

Rsp/R

200m

M200m = 1014, z = 0

WMAP7 Cosmology

adhikari14

more15

shi16

mansfield17

diemer17

Figure 6. Model predictions for the splashback radius as a function of massaccretion rate, evaluated for a halo of mass M200m = 1014 h−1 M at z = 0 inthe bolshoi (WMAP7) cosmology (Table 1).

(Table 3). Colossus provides a general function to evaluatethe model predictions for the splashback radius, splashbackmass, enclosed overdensity, and the scatter in those quanti-ties. Figure 6 compares the model predictions for the splash-back radius of halo with M200m = 1014 h−1M at z = 0.

The semi-analytical models of Adhikari et al. (2014) andShi (2016) are based on spherically collapsing shells whoseradial trajectories are integrated numerically. While the Ad-hikari et al. (2014) model predicts that Rsp/R200m should de-pend only on the mass accretion rate s ≡ d ln(M)/d ln(a), theShi (2016) model also predicts an evolution with redshift. Nu-merical calibrations have shown that Rsp/R200m does dependon redshift, regardless of the way Rsp was measured: fromstacked density profiles (More et al. 2015), non-sphericalsplashback shells (Mansfield et al. 2017), or particle orbits(Diemer 2017; Diemer et al. 2017). In these calibrations, themass accretion rate is defined as

Γ ≡∆ ln(M)∆ ln(a)

(43)

because the instantaneous rate is not a well-defined quantity insimulations. The time interval over which the mass accretionrate is measured has also varied slightly between the differ-ent models but is generally close to a dynamical time. Thedifferent definitions of the mass accretion rate complicate theinterpretation of the comparison in Figure 6.

5. FUTURE DEVELOPMENTI have presented Colossus, an open-source python toolkit

that is available at bitbucket.org/bdiemer/colossus. I antici-pate that the code will expand significantly over the comingyears, and that this development will be driven by the needsof its users. Additions could take the form of new functional-ity (e.g., new fitting functions), new physics (e.g., warm darkmatter), or entirely new modules (e.g., calculations related togalaxy formation).

Thus, all Colossus users are encouraged to suggestchanges, and to use the issue tracking system on BitBucket

to report bugs, unclear documentation, and feature requests.Most importantly, however, I invite collaborators! WhileColossus has hitherto been essentially a single-developerproject, I would like this situation to change in the future.

Colossus was born while I was working on my PhD the-sis, and I am grateful to Andrey Kravtsov for his mentoringand support during that time, as well as for contributing hisMCMC routine. I am also grateful to Matt Becker whose Cos-moCalc code was a great inspiration during the early develop-ment of Colossus. I thank all those who have used, tested,and discussed Colossus, namely Douglas Applegate, NealDalal, Daniel Eisenstein, Andrew Hearin, Wayne Hu, AndreyKravtsov, Alexie Leauthaud, Philip Mansfield, and SurhudMore. I am indebted to Savvas Koushiappas for suggestingthis paper, and to Sownak Bose for comments on a draft.Colossus makes extensive use of the numpy (numpy.org) andscipy (scipy.org) libraries. I gratefully acknowledge the fi-nancial support of an Institute for Theory and ComputationFellowship.

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