# Spline profile¶

This module implements a general density profile using spline interpolation. Please see Halo Density Profiles for a general introduction to the Colossus density profile module.

## Basics¶

This general profile is initialized with an arbitrary array of radii and densities or enclosed masses as input, and interpolates them using a splines (in log space). Note that there are three different ways of specifying the density profile:

• density and mass: Both density and mass are interpolated using splines.

• density only: In order for the enclosed mass to be defined, the density must be specified all the way to r = 0. In that case, the mass is computed numerically, stored, and interpolated.

• mass only: The density is computed as the derivative of the mass, stored, and interpolated.

In the following example, we create an NFW profile and use its density to initialize a spline profile:

from colossus.cosmology import cosmology
from colossus.halo import mass_so
from colossus.halo import profile_spline
from colossus.halo import profile_nfw

Mvir = 1E12
c = 10.0
mdef = 'vir'
z = 0.0
cosmology.setCosmology('planck15')
Rvir = mass_so.M_to_R(Mvir, z, mdef)

p_nfw = profile_nfw.NFWProfile(M = Mvir, c = c, mdef = mdef, z = z)
r = 10**np.arange(-2.0, 1.0, 0.02) * Rvir
rho = p_nfw.density(r)

p_spline = profile_spline.SplineProfile(r, rho = rho)


Of course, in this case it would be better to work with the NFW profile directly. Please see the Tutorials for more code examples.

## Module reference¶

class halo.profile_spline.SplineProfile(r, rho=None, M=None, spline_order=3)

An arbitrary density profile using spline interpolation.

Parameters
r: numpy array

rho: array_like

Density at radii r in physical $$M_{\odot} h^2 / {\rm kpc}^3$$. Does not have to be passed as long as M is passed.

M: array_like

Enclosed mass within radii r in $$M_{\odot} / h$$. Does not have to be passed as long as rho is passed.

spline_order: int

The order of the spline used. By default, a cubic spline (order 3) is used, but such splines can lead to ringing, especially if there are oscillating elements in the input mass or density array. In such cases, it can be helpful to set spline_order = 1.

Warning

If both mass and density are supplied to the constructor, the consistency between the two is not checked!

Methods

 MDelta(self, z, mdef) The spherical overdensity mass of a given mass definition. RDelta(self, z, mdef) The spherical overdensity radius of a given mass definition. RMDelta(self, z, mdef) The spherical overdensity radius and mass of a given mass definition. Vmax(self) The maximum circular velocity, and the radius where it occurs. circularVelocity(self, r) The circular velocity, $$v_c \equiv \sqrt{GM( densityInner(self, r) Density of the inner profile as a function of radius. Parameters r: array_like Radius in physical kpc/h; can be a number or a numpy array. Returns density: array_like Density in physical \(M_{\odot} h^2 / {\rm kpc}^3$$; has the same dimensions as r.

MDelta(self, z, mdef)

The spherical overdensity mass of a given mass definition.

Parameters
z: float

Redshift

mdef: str

The mass definition for which the spherical overdensity mass is computed. See Halo Mass Definitions for details.

Returns
M: float

Spherical overdensity mass in $$M_{\odot} /h$$.

RDelta

The spherical overdensity radius of a given mass definition.

RMDelta

The spherical overdensity radius and mass of a given mass definition.

RDelta(self, z, mdef)

The spherical overdensity radius of a given mass definition.

Parameters
z: float

Redshift

mdef: str

The mass definition for which the spherical overdensity radius is computed. See Halo Mass Definitions for details.

Returns
R: float

Spherical overdensity radius in physical kpc/h.

MDelta

The spherical overdensity mass of a given mass definition.

RMDelta

The spherical overdensity radius and mass of a given mass definition.

RMDelta(self, z, mdef)

The spherical overdensity radius and mass of a given mass definition.

This is a wrapper for the RDelta() and MDelta() functions which returns both radius and mass.

Parameters
z: float

Redshift

mdef: str

The mass definition for which the spherical overdensity mass is computed. See Halo Mass Definitions for details.

Returns
R: float

Spherical overdensity radius in physical kpc/h.

M: float

Spherical overdensity mass in $$M_{\odot} /h$$.

RDelta

The spherical overdensity radius of a given mass definition.

MDelta

The spherical overdensity mass of a given mass definition.

Vmax(self)

The maximum circular velocity, and the radius where it occurs.

Returns
vmax: float

The maximum circular velocity in km / s.

rmax: float

The radius where fmax occurs, in physical kpc/h.

circularVelocity

The circular velocity, $$v_c \equiv \sqrt{GM(<r)/r}$$.

circularVelocity(self, r)

The circular velocity, $$v_c \equiv \sqrt{GM(<r)/r}$$.

Parameters
r: array_like

Radius in physical kpc/h; can be a number or a numpy array.

Returns
vc: float

The circular velocity in km / s; has the same dimensions as r.

Vmax

The maximum circular velocity, and the radius where it occurs.

cumulativePdf(self, r, Rmax=None, z=None, mdef=None)

The cumulative distribution function of the profile.

Some density profiles do not converge to a finite mass at large radius, and the distribution thus needs to be cut off. The user can specify either a radius (in physical kpc/h) where the profile is cut off, or a mass definition and redshift to compute this radius (e.g., the virial radius $$R_{vir}$$ at z = 0).

Parameters
r: array_like

Radius in physical kpc/h; can be a number or a numpy array.

Rmax: float

The radius where to cut off the profile in physical kpc/h.

z: float

Redshift

mdef: str

The radius definition for the cut-off radius. See Halo Mass Definitions for details.

Returns
pdf: array_like

The probability for mass to lie within radius r; has the same dimensions as r.

deltaSigma(self, r, interpolate=True, interpolate_surface_density=True, accuracy=0.0001, min_r_interpolate=1e-06, max_r_interpolate=100000000.0, max_r_integrate=1e+20)

The excess surface density at radius r.

This quantity is useful in weak lensing studies, and is defined as $$\Delta\Sigma(R) = \Sigma(<R)-\Sigma(R)$$ where $$\Sigma(<R)$$ is the averaged surface density within R weighted by area,

$\Delta\Sigma(R) = \frac{1}{\pi R^2} \int_0^{R} 2 \pi r \Sigma(r) dr - \Sigma(R)$
Parameters
r: array_like

Radius in physical kpc/h; can be a number or a numpy array.

interpolate: bool

Use an interpolation table for the surface density during the integration. This can speed up the evaluation significantly, as the surface density can be expensive to evaluate.

interpolate_surface_density: bool

Use an interpolation table for density during the computation of the surface density. This should make the evaluation somewhat faster, but can fail for some density terms which are negative at particular radii.

accuracy: float

The minimum accuracy of the integration (used both to compute the surface density and average it to get DeltaSigma).

min_r_interpolate: float

The minimum radius in physical kpc/h from which the surface density profile is averaged.

max_r_interpolate: float

The maximum radius in physical kpc/h to which the density profile is integrated when using interpolating density.

max_r_integrate: float

The maximum radius in physical kpc/h to which the density profile is integrated when using exact densities.

Returns
DeltaSigma: array_like

The excess surface density at radius r, in physical $$M_{\odot} h/{\rm kpc}^2$$; has the same dimensions as r.

deltaSigmaInner(self, r, interpolate=True, interpolate_surface_density=True, accuracy=0.0001, min_r_interpolate=1e-06, max_r_interpolate=100000000.0, max_r_integrate=1e+20)

The excess surface density at radius r due to the inner profile.

Parameters
r: array_like

Radius in physical kpc/h; can be a number or a numpy array.

interpolate: bool

Use an interpolation table for the surface density during the integration. This can speed up the evaluation significantly, as the surface density can be expensive to evaluate.

interpolate_surface_density: bool

Use an interpolation table for density during the computation of the surface density. This should make the evaluation somewhat faster, but can fail for some density terms which are negative at particular radii.

accuracy: float

The minimum accuracy of the integration (used both to compute the surface density and average it to get DeltaSigma).

min_r_interpolate: float

The minimum radius in physical kpc/h from which the surface density profile is averaged.

max_r_interpolate: float

The maximum radius in physical kpc/h to which the density profile is integrated when using interpolating density.

max_r_integrate: float

The maximum radius in physical kpc/h to which the density profile is integrated when using exact densities.

Returns
DeltaSigma: array_like

The excess surface density at radius r, in physical $$M_{\odot} h/{\rm kpc}^2$$; has the same dimensions as r.

deltaSigmaOuter(self, r, interpolate=True, interpolate_surface_density=True, accuracy=0.0001, min_r_interpolate=1e-06, max_r_interpolate=100000000.0, max_r_integrate=1e+20)

The excess surface density at radius r due to the outer profile.

Parameters
r: array_like

Radius in physical kpc/h; can be a number or a numpy array.

interpolate: bool

Use an interpolation table for the surface density during the integration. This can speed up the evaluation significantly, as the surface density can be expensive to evaluate.

interpolate_surface_density: bool

Use an interpolation table for density during the computation of the surface density. This should make the evaluation somewhat faster, but can fail for some density terms which are negative at particular radii.

accuracy: float

The minimum accuracy of the integration (used both to compute the surface density and average it to get DeltaSigma).

min_r_interpolate: float

The minimum radius in physical kpc/h from which the surface density profile is averaged.

max_r_interpolate: float

The maximum radius in physical kpc/h to which the density profile is integrated when using interpolating density.

max_r_integrate: float

The maximum radius in physical kpc/h to which the density profile is integrated when using exact densities.

Returns
DeltaSigma: array_like

The excess surface density at radius r, in physical $$M_{\odot} h/{\rm kpc}^2$$; has the same dimensions as r.

density(self, r)

Density as a function of radius.

Abstract function which must be overwritten by child classes.

Parameters
r: array_like

Radius in physical kpc/h; can be a number or a numpy array.

Returns
density: array_like

Density in physical $$M_{\odot} h^2 / {\rm kpc}^3$$; has the same dimensions as r.

densityDerivativeLin(self, r)

The linear derivative of density, $$d \rho / dr$$.

This function should generally not be overwritten by child classes since it handles the general case of adding up the contributions from the inner and outer terms.

Parameters
r: array_like

Radius in physical kpc/h; can be a number or a numpy array.

Returns
derivative: array_like

The linear derivative in physical $$M_{\odot} h / {\rm kpc}^2$$; has the same dimensions as r.

densityDerivativeLinInner(self, r)

The linear derivative of the inner density, $$d \rho_{\rm inner} / dr$$.

Parameters
r: array_like

Radius in physical kpc/h; can be a number or a numpy array.

Returns
derivative: array_like

The linear derivative in physical $$M_{\odot} h / {\rm kpc}^2$$; has the same dimensions as r.

densityDerivativeLinOuter(self, r)

The linear derivative of the outer density, $$d \rho_{\rm outer} / dr$$.

This function should generally not be overwritten by child classes since it handles the general case of adding up the contributions from all outer profile terms.

Parameters
r: array_like

Radius in physical kpc/h; can be a number or a numpy array.

Returns
derivative: array_like

The linear derivative in physical $$M_{\odot} h / {\rm kpc}^2$$; has the same dimensions as r.

densityDerivativeLog(self, r)

The logarithmic derivative of density, $$d \log(\rho) / d \log(r)$$.

This function should generally not be overwritten by child classes since it handles the general case of adding up the contributions from the inner and outer profile terms.

Parameters
r: array_like

Radius in physical kpc/h; can be a number or a numpy array.

Returns
derivative: array_like

The dimensionless logarithmic derivative; has the same dimensions as r.

densityDerivativeLogOuter(self, r)

The logarithmic derivative of the outer density, $$d \log(\rho_{\rm outer}) / d \log(r)$$.

This function should generally not be overwritten by child classes since it handles the general case of adding up the contributions from outer profile terms.

Parameters
r: array_like

Radius in physical kpc/h; can be a number or a numpy array.

Returns
derivative: array_like

The dimensionless logarithmic derivative; has the same dimensions as r.

densityOuter(self, r)

Density of the outer profile as a function of radius.

This function should generally not be overwritten by child classes since it handles the general case of adding up the contributions from all outer profile terms.

Parameters
r: array_like

Radius in physical kpc/h; can be a number or a numpy array.

Returns
density: array_like

Density in physical $$M_{\odot} h^2 / {\rm kpc}^3$$; has the same dimensions as r.

enclosedMassOuter(self, r, accuracy=1e-06)

The mass enclosed within radius r due to the outer profile term.

Parameters
r: array_like

Radius in physical kpc/h; can be a number or a numpy array.

accuracy: float

The minimum accuracy of the integration.

Returns
M: array_like

The mass enclosed within radius r, in $$M_{\odot}/h$$; has the same dimensions as r.

fit(self, r, q, quantity, q_err=None, q_cov=None, method='leastsq', mask=None, verbose=True, tolerance=1e-05, maxfev=0, initial_step=0.1, nwalkers=100, random_seed=None, convergence_step=100, converged_GR=0.01, best_fit='median', output_every_n=100)

Fit the density, mass, or surface density profile to a given set of data points.

This function represents a general interface for finding the best-fit parameters of a halo density profile given a set of data points. These points can represent a number of different physical quantities: quantity can either be density, enclosed mass, surface density, or Delta Sigma (rho, M, Sigma, or DeltaSigma).

The data points q at radii r can optionally have error bars, and the user can pass a full covariance matrix. Please note that not passing any estimate of the uncertainty, i.e. q_err = None and q_cov = None, can lead to very poor fit results: the fitter will minimize the absolute difference between points, strongly favoring the high densities at the center.

There are two fundamental methods for performing the fit, a least-squares minimization (method = 'leastsq') and a Markov-Chain Monte Carlo (method = 'mcmc'). The MCMC method has some specific options (see below). In either case, the current parameters of the profile instance serve as an initial guess. Finally, the user can choose to vary only a sub-set of the profile parameters through the mask parameter.

The function returns a dictionary with outputs that depend on which method is chosen. After this function has completed, the profile instance represents the best-fit profile to the data points (i.e., its parameters are the best-fit parameters). Note that all output parameters are bundled into one dictionary. The explanations below refer to the entries in this dictionary.

Parameters
r: array_like

The radii of the data points, in physical kpc/h.

q: array_like

The data to fit; can either be density in physical $$M_{\odot} h^2 / {\rm kpc}^3$$, enclosed mass in $$M_{\odot} /h$$, or surface density in physical $$M_{\odot} h/{\rm kpc}^2$$. Must have the same dimensions as r.

quantity: str

Indicates which quantity is given in as input in q, can be rho, M, Sigma, or DeltaSigma.

q_err: array_like

Optional; the uncertainty on the values in q in the same units. If method == 'mcmc', either q_err or q_cov must be passed. If method == 'leastsq' and neither q_err nor q_cov are passed, the absolute different between data points and fit is minimized. In this case, the returned chi2 is in units of absolute difference, meaning its value will depend on the units of q.

q_cov: array_like

Optional; the covariance matrix of the elements in q, as a 2-dimensional numpy array. This array must have dimensions of q**2 and be in units of the square of the units of q. If q_cov is passed, q_err is ignored since the diagonal elements of q_cov correspond to q_err**2.

method: str

The fitting method; can be leastsq for a least-squares minimization of mcmc for a Markov-Chain Monte Carlo.

Optional; a numpy array of booleans that has the same length as the variables vector of the density profile class. Only variables where mask == True are varied in the fit, all others are kept constant. Important: this has to be a numpy array rather than a list.

verbose: bool / int

If true, output information about the fitting process. The flag can also be set as a number, where 1 has the same effect as True, and 2 outputs large amounts of information such as the fit parameters at each iteration.

tolerance: float

Only active when method == 'leastsq'. The accuracy to which the best-fit parameters are found.

maxfev: int

Only active when method == 'leastsq'. The maximum number of function evaluations before the fit is aborted. If zero, the default value of the scipy leastsq function is used.

initial_step: array_like

Only active when method == 'mcmc'. The MCMC samples (“walkers”) are initially distributed in a Gaussian around the initial guess. The width of the Gaussian is given by initial_step, either as an array of length N_par (giving the width of each Gaussian) or as a float number, in which case the width is set to initial_step times the initial value of the parameter.

nwalkers: int

Only active when method == 'mcmc'. The number of MCMC samplers that are run in parallel.

random_seed: int

Only active when method == 'mcmc'. If random_seed is not None, it is used to initialize the random number generator. This can be useful for reproducing results.

convergence_step: int

Only active when method == 'mcmc'. The convergence criteria are computed every convergence_step steps (and output is printed if verbose == True).

converged_GR: float

Only active when method == 'mcmc'. The maximum difference between different chains, according to the Gelman-Rubin criterion. Once the GR indicator is lower than this number in all parameters, the chain is ended. Setting this number too low leads to very long runtimes, but setting it too high can lead to inaccurate results.

best_fit: str

Only active when method == 'mcmc'. This parameter determines whether the mean or median value of the likelihood distribution is used as the output parameter set.

output_every_n: int

Only active when method == 'mcmc'. This parameter determines how frequently the MCMC chain outputs information. Only effective if verbose == True.

Returns
results: dict

A dictionary bundling the various fit results. Regardless of the fitting method, the dictionary always contains the following entries:

x: array_like

The best-fit result vector. If mask is passed, this vector only contains those variables that were varied in the fit.

q_fit: array_like

The fitted profile at the radii of the data points; has the same units as q and the same dimensions as r.

chi2: float

The chi^2 of the best-fit profile. If a covariance matrix was passed, the covariances are taken into account. If no uncertainty was passed at all, chi2 is in units of absolute difference, meaning its value will depend on the units of q.

ndof: int

The number of degrees of freedom, i.e. the number of fitted data points minus the number of free parameters.

chi2_ndof: float

The chi^2 per degree of freedom.

If method == 'leastsq', the dictionary additionally contains the entries returned by scipy.optimize.leastsq as well as the following:

nfev: int

The number of function calls used in the fit.

x_err: array_like

An array of dimensions [2, nparams] which contains an estimate of the lower and upper uncertainties on the fitted parameters. These uncertainties are computed from the covariance matrix estimated by the fitter. Please note that this estimate does not exactly correspond to a 68% likelihood. In order to get more statistically meaningful uncertainties, please use the MCMC samples instead of least-squares. In some cases, the fitter fails to return a covariance matrix, in which case x_err is None.

If method == 'mcmc', the dictionary contains the following entries:

x_initial: array_like

The initial positions of the walkers, in an array of dimensions [nwalkers, nparams].

chain_full: array_like

A numpy array of dimensions [n_independent_samples, nparams] with the parameters at each step in the chain. In this thin chain, only every nth step is output, where n is the auto-correlation time, meaning that the samples in this chain are truly independent.

chain_thin: array_like

Like the thin chain, but including all steps. Thus, the samples in this chain are not indepedent from each other. However, the full chain often gives better plotting results.

R: array_like

A numpy array containing the GR indicator at each step when it was saved.

x_mean: array_like

The mean of the chain for each parameter; has length nparams.

x_median: array_like

The median of the chain for each parameter; has length nparams.

x_stddev: array_like

The standard deviation of the chain for each parameter; has length nparams.

x_percentiles: array_like

The lower and upper values of each parameter that contain a certain percentile of the probability; has dimensions [n_percentages, 2, nparams] where the second dimension contains the lower/upper values.

getParameterArray(self, mask=None)

Returns an array of the profile parameters.

The profile parameters are internally stored in an ordered dictionary. For some applications (e.g., fitting), a simply array is more appropriate.

Parameters

Optional; must be a numpy array (not a list) of booleans, with the same length as the parameter vector of the profile class (profile.N_par). Only those parameters that correspond to True values are returned.

Returns
par: array_like

A numpy array with the profile’s parameter values.

setParameterArray(self, pars, mask=None)

Set the profile parameters from an array.

The profile parameters are internally stored in an ordered dictionary. For some applications (e.g., fitting), setting them directly from an array might be necessary. If the profile contains values that depend on the parameters, the profile class must overwrite this function and update according to the new parameters.

Parameters
pars: array_like

The new parameter array.

Optional; must be a numpy array (not a list) of booleans, with the same length as the parameter vector of the profile class (profile.N_par). If passed, only those parameters that correspond to True values are set (meaning the pars parameter must be shorter than profile.N_par).

surfaceDensity(self, r, interpolate=True, accuracy=0.0001, max_r_interpolate=100000000.0, max_r_integrate=1e+20)

The projected surface density at radius r.

The surface density is computed by projecting the 3D density along the line of sight,

$\Sigma(R) = 2 \int_R^{\infty} \frac{r \rho(r)}{\sqrt{r^2-R^2}} dr$
Parameters
r: array_like

Radius in physical kpc/h; can be a number or a numpy array.

interpolate: bool

Use an interpolation table for density during the integration. This should make the evaluation somewhat faster, depending on how large the radius array is.

accuracy: float

The minimum accuracy of the integration.

max_r_interpolate: float

The maximum radius in physical kpc/h to which the density profile is integrated when using interpolating density.

max_r_integrate: float

The maximum radius in physical kpc/h to which the density profile is integrated when using exact densities.

Returns
Sigma: array_like

The surface density at radius r, in physical $$M_{\odot} h/{\rm kpc}^2$$; has the same dimensions as r.

surfaceDensityInner(self, r, interpolate=True, accuracy=0.0001, max_r_interpolate=100000000.0, max_r_integrate=1e+20)

The projected surface density at radius r due to the inner profile.

Parameters
r: array_like

Radius in physical kpc/h; can be a number or a numpy array.

interpolate: bool

Use an interpolation table for density during the integration. This should make the evaluation somewhat faster, depending on how large the radius array is.

accuracy: float

The minimum accuracy of the integration.

max_r_interpolate: float

The maximum radius in physical kpc/h to which the density profile is integrated when using interpolating density.

max_r_integrate: float

The maximum radius in physical kpc/h to which the density profile is integrated when using exact densities.

Returns
Sigma: array_like

The surface density at radius r, in physical $$M_{\odot} h/{\rm kpc}^2$$; has the same dimensions as r.

surfaceDensityOuter(self, r, interpolate=True, accuracy=0.0001, max_r_interpolate=100000000.0, max_r_integrate=1e+20)

The projected surface density at radius r due to the outer profile.

This function checks whether there are explicit expressions for the surface density of the outer profile terms available, and uses them if possible. Note that there are some outer terms whose surface density integrates to infinity, such as the mean density of the universe which is constant to infinitely large radii.

Parameters
r: array_like

Radius in physical kpc/h; can be a number or a numpy array.

interpolate: bool

Use an interpolation table for density during the integration. This should make the evaluation somewhat faster, depending on how large the radius array is.

accuracy: float

The minimum accuracy of the integration.

max_r_interpolate: float

The maximum radius in physical kpc/h to which the density profile is integrated when using interpolating density.

max_r_integrate: float

The maximum radius in physical kpc/h to which the density profile is integrated when using exact densities.

Returns
Sigma: array_like

The surface density at radius r, in physical $$M_{\odot} h/{\rm kpc}^2$$; has the same dimensions as r.

update(self)

Update the profile object after a change in parameters.

If the parameters dictionary has been changed (e.g. by the user or during fitting), this function must be called to ensure consistency within the profile object. This involves deleting any pre-computed quantities (e.g., tabulated enclosed masses) and re-computing profile properties that depend on the parameters.

densityDerivativeLogInner(self, r)

The logarithmic derivative of the inner density, $$d \log(\rho_{\rm inner}) / d \log(r)$$.

Parameters
r: array_like

Radius in physical kpc/h; can be a number or a numpy array.

Returns
derivative: array_like

The dimensionless logarithmic derivative; has the same dimensions as r.

enclosedMass(self, r, accuracy=1e-06)

The mass enclosed within radius r.

Parameters
r: array_like

Radius in physical kpc/h; can be a number or a numpy array.

accuracy: float

The minimum accuracy of the integration.

Returns
M: array_like

The mass enclosed within radius r, in $$M_{\odot}/h$$; has the same dimensions as r.

enclosedMassInner(self, r)

The mass enclosed within radius r due to the inner profile term.

Parameters
r: array_like

Radius in physical kpc/h; can be a number or a numpy array.

Returns
M: array_like

The mass enclosed within radius r, in $$M_{\odot}/h$$; has the same dimensions as r.