"""
Module containing the `~halotools.mock_observables.mean_delta_sigma` function
used to calculate galaxy-galaxy lensing.
"""
from __future__ import absolute_import, division, print_function, unicode_literals
import numpy as np
from functools import partial
import multiprocessing
from .engines import mean_delta_sigma_engine
from ..mock_observables_helpers import (get_num_threads, get_separation_bins_array,
get_period, enforce_sample_respects_pbcs, enforce_sample_has_correct_shape)
from ..pair_counters.rectangular_mesh_2d import RectangularDoubleMesh2D
from ..pair_counters.mesh_helpers import _set_approximate_2d_cell_sizes
from ..pair_counters.mesh_helpers import _cell1_parallelization_indices
from ..pair_counters.mesh_helpers import _enclose_in_square
from ...utils.array_utils import custom_len
__all__ = ('mean_delta_sigma', )
__author__ = ('Andrew Hearin', 'Johannes Ulf Lange')
[docs]def mean_delta_sigma(galaxies, particles, effective_particle_masses,
rp_bins, period=None, verbose=False, num_threads=1,
approx_cell1_size=None, approx_cell2_size=None,
per_object=False):
r"""
Calculate :math:`\Delta\Sigma(r_p)`, the galaxy-galaxy lensing signal
as a function of projected distance.
The `mean_delta_sigma` function calculates :math:`\Delta\Sigma(r_p)` by calculating
the excess surface density of particles in cylinders surrounding the input galaxies.
The input particles should be a random downsampling of particles in the
same simulation snapshot as the model galaxies.
By using the ``effective_particle_masses`` argument, the function works equally well
with DM-only simulations as with hydro simulations that include
multiple species of particles with different masses and/or different downsampling rates.
Example calls to this function appear in the documentation below.
See also :ref:`galaxy_catalog_analysis_tutorial3`.
Parameters
----------
galaxies : array_like
Numpy array of shape (num_gal, 3) containing 3-d positions of galaxies.
Length units are comoving and assumed to be in Mpc/h, here and throughout Halotools.
See the :ref:`mock_obs_pos_formatting` documentation page for
instructions on how to transform your coordinate position arrays into the
format accepted by the ``galaxies`` and ``particles`` arguments.
particles : array_like
Numpy array of shape (num_ptcl, 3) containing 3-d positions of particles.
Length units are comoving and assumed to be in Mpc/h, here and throughout Halotools.
effective_particle_masses : float or ndarray
Float or array storing the effective mass of each particle in units of Msun with h=1 units.
If passing in an ndarray, must be of shape (num_ptcl, ),
one array element for every particle.
If passing in a single float, it will be assumed that every particle
has the same mass (as is the case in a typical DM-only simulation).
The effective mass is simply the actual mass multiplied by the downsampling rate.
For example, if your simulation has a particle mass of 10**8 and you are using a
sample of particles that have been randomly downsampled at a 1% rate, then
your effective particle mass will be 10**10.
See the Examples section below for how this can be calculated
from Halotools-provided catalogs.
rp_bins : array_like
Numpy array of shape (num_rbins, ) of projected radial boundaries
defining the bins in which the result is calculated.
Length units are comoving and assumed to be in Mpc/h, here and throughout Halotools.
period : array_like
Length-3 sequence defining the periodic boundary conditions
in each dimension. If you instead provide a single scalar, Lbox,
period is assumed to be the same in all Cartesian directions.
Length units are comoving and assumed to be in Mpc/h, here and throughout Halotools.
per_object : bool, optional
Boolean flag specifying whether the function will return the per-object
lensing signal. Default is False, in which the returned array will be
an average over the entire sample. If True, the returned ndarray will
have shape (num_gal, num_rbins)
num_threads : int, optional
Number of threads to use in calculation, where parallelization is performed
using the python ``multiprocessing`` module. Default is 1 for a purely serial
calculation, in which case a multiprocessing Pool object will
never be instantiated. A string 'max' may be used to indicate that
the pair counters should use all available cores on the machine.
approx_cell1_size : array_like, optional
Length-3 array serving as a guess for the optimal manner by how points
will be apportioned into subvolumes of the simulation box.
The optimum choice unavoidably depends on the specs of your machine.
Default choice is to use Lbox/10 in each dimension,
which will return reasonable result performance for most use-cases.
Performance can vary sensitively with this parameter, so it is highly
recommended that you experiment with this parameter when carrying out
performance-critical calculations.
approx_cell2_size : array_like, optional
Analogous to ``approx_cell1_size``, but for sample2. See comments for
``approx_cell1_size`` for details.
Returns
-------
Delta_Sigma : array_like
Numpy array of shape (num_rbins-1, ) storing :math:`\Delta\Sigma(r_p)`
in comoving units of :math:`h M_{\odot} / {\rm Mpc}^2` assuming h=1.
If per_object is True, Delta_Sigma will instead have shape (num_gal, num_rbins)
Examples
--------
For demonstration purposes we will calculate `mean_delta_sigma` using a mock
catalog generated with the `~halotools.sim_manager.FakeSim`
that is generated on-the-fly.
>>> from halotools.sim_manager import FakeSim
>>> halocat = FakeSim()
Now let's populate this halo catalog with mock galaxies.
>>> from halotools.empirical_models import PrebuiltHodModelFactory
>>> model = PrebuiltHodModelFactory('leauthaud11', threshold = 11.)
>>> model.populate_mock(halocat)
Now we retrieve the positions of our mock galaxies and transform the arrays
into the shape of the ndarray expected by the `~halotools.mock_observables.mean_delta_sigma`
function. We transform our *x, y, z* points into the array shape used by the pair-counter by
taking the transpose of the result of `numpy.vstack`. This boilerplate transformation
is used throughout the `~halotools.mock_observables` sub-package:
>>> x = model.mock.galaxy_table['x']
>>> y = model.mock.galaxy_table['y']
>>> z = model.mock.galaxy_table['z']
>>> galaxies = np.vstack((x, y, z)).T
The `~halotools.mock_observables.return_xyz_formatted_array` function
also performs this same transformation, and can also be used to place mock
galaxies into redshift-space for additional observational realism.
Let's do the same thing for a set of particle data:
>>> px = model.mock.ptcl_table['x']
>>> py = model.mock.ptcl_table['y']
>>> pz = model.mock.ptcl_table['z']
>>> particles = np.vstack((px, py, pz)).T
The default Halotools catalogs come with ~1e6 particles.
Using this many particles may be overkill: in many typical use-cases,
the `mean_delta_sigma` function converges at the percent-level using
an order of magnitude fewer particles.
The code below shows how to (optionally) downsample these particles
using a Halotools convenience function.
>>> from halotools.utils import randomly_downsample_data
>>> num_ptcls_to_use = int(1e4)
>>> particles = randomly_downsample_data(particles, num_ptcls_to_use)
>>> particle_masses = np.zeros(num_ptcls_to_use) + halocat.particle_mass
>>> total_num_ptcl_in_snapshot = halocat.num_ptcl_per_dim**3
>>> downsampling_factor = total_num_ptcl_in_snapshot/float(len(particles))
>>> effective_particle_masses = downsampling_factor * particle_masses
>>> rp_bins = np.logspace(-1, 1, 10)
>>> period = model.mock.Lbox
>>> ds = mean_delta_sigma(galaxies, particles, effective_particle_masses, rp_bins, period)
Take care with the units. The values for :math:`\Delta\Sigma` returned by
the `mean_delta_sigma` functions are in *comoving* units of
:math:`h M_{\odot} / {\rm Mpc}^2` assuming h=1,
whereas the typical units used to plot :math:`\Delta\Sigma` are in
*physical* units of :math:`M_{\odot} / {\rm pc}^2` using the value of
little h appropriate for your assumed cosmology.
The code shown above demonstrates how to calculate :math:`\Delta\Sigma` via the excess
surface density of mass using the z-axis as the axis of projection. However, it may be useful
to project along the other Cartesian axes, for example to help beat down sample variance.
While the `mean_delta_sigma` function is written to always use the "third" dimension as the
projection axis, you can easily hack the code to project along, say, the y-axis by simply
transposing your y- and z-coordinates when you pack them into a 2-d array:
>>> particles = np.vstack((px, pz, py)).T
>>> galaxies = np.vstack((x, z, y)).T
Using the above ``particles`` and ``galaxies`` and otherwise calling the `mean_delta_sigma`
function as normal will instead calculate the surface mass density by projecting
along the y-axis.
See also
--------
:ref:`galaxy_catalog_analysis_tutorial3`
"""
# Process the inputs with the helper function
result = _mean_delta_sigma_process_args(
galaxies, particles, effective_particle_masses, rp_bins,
period, num_threads, approx_cell1_size, approx_cell2_size)
x1in, y1in, x2in, y2in, w2in = result[0:5]
rp_bins, period, num_threads, PBCs, approx_cell1_size, approx_cell2_size = result[5:]
xperiod, yperiod = period[:2]
rp_max = np.max(rp_bins)
search_xlength, search_ylength = rp_max, rp_max
# Compute the estimates for the cell sizes
approx_cell1_size, approx_cell2_size = (_set_approximate_2d_cell_sizes(
approx_cell1_size, approx_cell2_size, period))
approx_x1cell_size, approx_y1cell_size = approx_cell1_size
approx_x2cell_size, approx_y2cell_size = approx_cell2_size
# Build the rectangular mesh
double_mesh = RectangularDoubleMesh2D(
x1in, y1in, x2in, y2in,
approx_x1cell_size, approx_y1cell_size,
approx_x2cell_size, approx_y2cell_size,
search_xlength, search_ylength, xperiod, yperiod, PBCs)
# Create a function object that has a single argument, for parallelization
# purposes
counting_engine = partial(mean_delta_sigma_engine, double_mesh, x1in, y1in,
x2in, y2in, w2in, rp_bins)
# # Calculate the cell1 indices that will be looped over by the engine
num_threads, cell1_tuples = _cell1_parallelization_indices(
double_mesh.mesh1.ncells, num_threads)
if num_threads > 1:
pool = multiprocessing.Pool(num_threads)
result = pool.map(counting_engine, cell1_tuples)
delta_sigma = np.sum(np.array(result), axis=0)
pool.close()
else:
delta_sigma = counting_engine(cell1_tuples[0])
if per_object:
return delta_sigma
else:
return np.mean(delta_sigma, axis=0)
def _mean_delta_sigma_process_args(
galaxies, particles, effective_particle_masses, rp_bins,
period, num_threads, approx_cell1_size, approx_cell2_size):
period, PBCs = get_period(period)
if PBCs is False:
_result = _enclose_in_box(
galaxies[:, 0], galaxies[:, 1], galaxies[:, 2],
particles[:, 0], particles[:, 1], particles[:, 2])
_x1, _y1, _z1, _x2, _y2, _z2, period = _result
galaxies[:, 0] = _x1
galaxies[:, 1] = _y1
galaxies[:, 2] = _z1
particles[:, 0] = _x2
particles[:, 1] = _y2
particles[:, 2] = _z2
galaxies = enforce_sample_has_correct_shape(galaxies)
particles = enforce_sample_has_correct_shape(particles)
effective_particle_masses = np.atleast_1d(effective_particle_masses)
if len(effective_particle_masses) == 1:
effective_particle_masses = np.zeros(particles.shape[0]) + effective_particle_masses[0]
else:
msg = "Must have same number of ``particle_masses`` as particles"
assert effective_particle_masses.shape[0] == particles.shape[0], msg
enforce_sample_respects_pbcs(galaxies[:, 0], galaxies[:, 1],
galaxies[:, 2], period)
enforce_sample_respects_pbcs(particles[:, 0], particles[:, 1],
particles[:, 2], period)
x1 = galaxies[:, 0]
y1 = galaxies[:, 1]
x2 = particles[:, 0]
y2 = particles[:, 1]
rp_bins = get_separation_bins_array(rp_bins)
rp_max = np.max(rp_bins)
if period is None:
PBCs = False
x1, y1, x2, y2, period = (
_enclose_in_square(x1, y1, x2, y2,
min_size=[rp_max*3.0, rp_max*3.0]))
num_threads = get_num_threads(num_threads, enforce_max_cores=False)
if approx_cell1_size is None:
approx_cell1_size = [rp_max, rp_max]
elif custom_len(approx_cell1_size) == 1:
approx_cell1_size = [approx_cell1_size, approx_cell1_size]
if approx_cell2_size is None:
approx_cell2_size = [rp_max, rp_max]
elif custom_len(approx_cell2_size) == 1:
approx_cell2_size = [approx_cell2_size, approx_cell2_size]
return (x1, y1, x2, y2, effective_particle_masses, rp_bins,
period, num_threads, PBCs, approx_cell1_size, approx_cell2_size)
def _enclose_in_box(x1, y1, z1, x2, y2, z2, min_size=None):
"""
Build box which encloses all points, shifting the points so that
the "leftmost" point is (0,0,0).
Parameters
----------
x1,y1,z1 : array_like
cartesian positions of points
x2,y2,z2 : array_like
cartesian positions of points
min_size : array_like
minimum lengths of a side of the box. If the minimum box constructed around the
points has a side i less than ``min_size[i]``, then the box is padded in order to
obtain the minimum specified size.
Returns
-------
x1, y1, z1, x2, y2, z2, Lbox
shifted positions and box size.
"""
xmin = np.min([np.min(x1), np.min(x2)])
ymin = np.min([np.min(y1), np.min(y2)])
zmin = np.min([np.min(z1), np.min(z2)])
xmax = np.max([np.max(x1), np.max(x2)])
ymax = np.max([np.max(y1), np.max(y2)])
zmax = np.max([np.max(z1), np.max(z2)])
xyzmin = np.min([xmin, ymin, zmin])
xyzmax = np.max([xmax, ymax, zmax])-xyzmin
x1 = x1 - xyzmin
y1 = y1 - xyzmin
z1 = z1 - xyzmin
x2 = x2 - xyzmin
y2 = y2 - xyzmin
z2 = z2 - xyzmin
Lbox = np.array([xyzmax, xyzmax, xyzmax])
if min_size is not None:
min_size = np.atleast_1d(min_size)
if np.any(Lbox < min_size):
Lbox[(Lbox < min_size)] = min_size[(Lbox < min_size)]
return x1, y1, z1, x2, y2, z2, Lbox