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Package cosmolopy :: Module distance

Module distance

source code

Cosmological distance measures.

Mostly follows David Hogg's pedagogical paper arXiv:astro-ph/9905116v4 .

Distance units are Mpc, time units are seconds.

Functions
 
get_omega_k_0(**cosmo)
'Spatial curvature density' omega_k_0 for a cosmology (if needed).		 source code
 
set_omega_k_0(cosmo)
Returns the cosmo dictionary with omega_k_0 set. See get_omega_k_0.		 source code
 
e_z(z, **cosmo)
The unitless Hubble expansion rate at redshift z.		 source code
 
hubble_z(z, **cosmo)
The value of the Hubble constant at redshift z.		 source code
 
hubble_distance_z(z, **cosmo)
The value of the Hubble distance at redshift z.		 source code
 
comoving_integrand(z, **cosmo)
The derivative of the comoving distance with redshift: dd_c/dz.		 source code
 
comoving_distance(z, z0=0, **cosmo)
The line-of-sight comoving distance (in Mpc) to redshift z.		 source code
 
proper_motion_distance(z, **cosmo)
Returns comoving_distance_transverse.		 source code
 
comoving_distance_transverse(z, **cosmo)
The transverse comoving distance (in Mpc) to redshift z.		 source code
 
angular_diameter_distance(z, z0=0, **cosmo)
The angular-diameter distance (Mpc) to redshift z.		 source code
 
luminosity_distance(z, **cosmo)
The luminosity distance to redshift z.		 source code
 
diff_comoving_volume(z, **cosmo)
The differential comoving volume element dV_c/dz/dSolidAngle.		 source code
 
comoving_volume(z, **cosmo)
The comoving volume out to redshift z.		 source code
 
lookback_integrand(z, **cosmo)
The derivative of the lookback time with redshift: dt_L/dz.		 source code
 
lookback_time(z, z0=0.0, **cosmo)
The lookback time (in s) to redshift z.		 source code
 
age(z, use_flat=True, **cosmo)
The age of the universe as seen at redshift z.		 source code
 
age_flat(z, **cosmo)
The age of the universe assuming a flat cosmology.		 source code
 
quick_distance_function(function, zmax=20.0, zmin=0.0, zstep=0.001, return_inverse=False, k=3, **cosmo)
Return an interpolation function that will give distance as a funtion of z		 source code
 
quick_age_function(zmax=20.0, zmin=0.0, zstep=0.001, return_inverse=False, **cosmo)
Return an interpolation function that will give age as a funtion of z		 source code
 
quick_redshift_age_function(zmax=20.0, zmin=0.0, zstep=0.001, **cosmo)
Return an interpolation function giving z as a funtion of age of the universe.		 source code
 
light_travel_distance(z, z0=0, **cosmo)
The light travel distance to redshift z.		 source code
 
redshift_d_light(dl, z_guess=6.0, fmin_args={}, **cosmo)
The redshift corresponding to a given light travel distance.		 source code
Variables
  __package__ = 'cosmolopy'
Function Details

get_omega_k_0(**cosmo)

source code 

'Spatial curvature density' omega_k_0 for a cosmology (if needed).

If omega_k_0 is specified, return it. Otherwise return:

1.0 - omega_M_0 - omega_lambda_0

set_omega_k_0(cosmo)

source code 

Returns the cosmo dictionary with omega_k_0 set. See get_omega_k_0.

Note that cosmo is not passed as **cosmo for once. This function modifies the dictionary in place and returns the result.

e_z(z, **cosmo)

source code 

The unitless Hubble expansion rate at redshift z.

In David Hogg's (arXiv:astro-ph/9905116v4) formalism, this is equivalent to E(z), defined in his eq. 14.

Modified (JBJ, 29-Feb-2012) to include scalar w parameter

hubble_z(z, **cosmo)

source code 

The value of the Hubble constant at redshift z.

Units are s^-1

In David Hogg's (arXiv:astro-ph/9905116v4) formalism, this is equivalent to H_0 * E(z) (see his eq. 14).

hubble_distance_z(z, **cosmo)

source code 

The value of the Hubble distance at redshift z.

Units are Mpc.

In David Hogg's (arXiv:astro-ph/9905116v4) formalism, this is equivalent to D_H / E(z) = c / (H_0 E(z)) [see his eq. 14], which appears in the definitions of many other distance measures.

comoving_integrand(z, **cosmo)

source code 

The derivative of the comoving distance with redshift: dd_c/dz.

See equation 15 of David Hogg's arXiv:astro-ph/9905116v4

Units are Mpc.

comoving_distance(z, z0=0, **cosmo)

source code 

The line-of-sight comoving distance (in Mpc) to redshift z.

See equation 15 of David Hogg's arXiv:astro-ph/9905116v4

Units are Mpc.

Optionally calculate the integral from z0 to z.

Returns

d_co: ndarray
Comoving distance in Mpc.

Examples

>>> import cosmolopy.distance as cd
>>> cosmo = {'omega_M_0' : 0.3, 'omega_lambda_0' : 0.7, 'h' : 0.72}
>>> cosmo = cd.set_omega_k_0(cosmo)
>>> d_co = cd.comoving_distance(6., **cosmo)
>>> print "Comoving distance to z=6 is %.1f Mpc" % (d_co)
Comoving distance to z=6 is 8017.8 Mpc

proper_motion_distance(z, **cosmo)

source code 

Returns comoving_distance_transverse.

Units are Mpc.

Examples

>>> import cosmolopy.distance as cd
>>> cosmo = {'omega_M_0' : 0.3, 'omega_lambda_0' : 0.7, 'h' : 0.72}
>>> cosmo = cd.set_omega_k_0(cosmo)
>>> d_M = cd.proper_motion_distance(6., **cosmo)
>>> print "Transverse comoving distance to z=6 is %.1f Mpc" % (d_M)
Transverse comoving distance to z=6 is 8017.8 Mpc

comoving_distance_transverse(z, **cosmo)

source code 

The transverse comoving distance (in Mpc) to redshift z.

This is also called the proper motion distance, D_M.

See equation 16 of David Hogg's arXiv:astro-ph/9905116v4

Units are Mpc.

This is the distance d_m, such that the comoving distance between two events at the same redshift, but separated on the sky by some angle delta_theta is d_m * delta_theta.

Examples

>>> import cosmolopy.distance as cd
>>> cosmo = {'omega_M_0' : 0.3, 'omega_lambda_0' : 0.7, 'h' : 0.72}
>>> cosmo = cd.set_omega_k_0(cosmo)
>>> d_M = cd.comoving_distance_transverse(6., **cosmo)
>>> print "Transverse comoving distance to z=6 is %.1f Mpc" % (d_M)
Transverse comoving distance to z=6 is 8017.8 Mpc

angular_diameter_distance(z, z0=0, **cosmo)

source code 

The angular-diameter distance (Mpc) to redshift z.

Optionally find the angular diameter distance between objects at z0 and z (only implemented for omega_k_0 >= 0).

See equations 18-19 of David Hogg's arXiv:astro-ph/9905116v4

Units are Mpc.

Examples

>>> import cosmolopy.distance as cd
>>> cosmo = {'omega_M_0' : 0.3, 'omega_lambda_0' : 0.7, 'h' : 0.72}
>>> cosmo = cd.set_omega_k_0(cosmo)
>>> d_a = cd.angular_diameter_distance(6., **cosmo)
>>> print "Angular diameter distance = %.1f Mpc" % (d_a)
Angular diameter distance = 1145.4 Mpc

luminosity_distance(z, **cosmo)

source code 

The luminosity distance to redshift z.

Units are Mpc.

See, for example, David Hogg's arXiv:astro-ph/9905116v4

diff_comoving_volume(z, **cosmo)

source code 

The differential comoving volume element dV_c/dz/dSolidAngle.

Dimensions are volume per unit redshift per unit solid angle.

Units are Mpc**3 Steradians^-1.

See David Hogg's arXiv:astro-ph/9905116v4, equation 28.

Examples

>>> import cosmolopy.distance as cd
>>> cosmo = {'omega_M_0' : 0.3, 'omega_lambda_0' : 0.7, 'h' : 0.72}
>>> cosmo = cd.set_omega_k_0(cosmo)
>>> dVc = cd.diff_comoving_volume(6.0, **cosmo)
>>> print "dV/dz/dSolidAngle at z=6 is %.3g Mpc**3" % (dVc)
dV/dz/dSolidAngle at z=6 is 2.63e+10 Mpc**3

comoving_volume(z, **cosmo)

source code 

The comoving volume out to redshift z.

See David Hogg's arXiv:astro-ph/9905116v4, equation 29.

Examples

>>> import cosmolopy.distance as cd
>>> cosmo = {'omega_M_0' : 0.3, 'omega_lambda_0' : 0.7, 'h' : 0.72}
>>> cosmo = cd.set_omega_k_0(cosmo)
>>> Vc = cd.comoving_volume(6.0, **cosmo)
>>> print "Vc = %.3g Mpc**3" % (Vc)
Vc = 2.16e+12 Mpc**3
>>> import cosmolopy.distance as cd
>>> cosmo = {'omega_M_0' : 0.3, 'omega_lambda_0' : 0.0, 'h' : 0.72}
>>> cosmo = cd.set_omega_k_0(cosmo)
>>> Vc = cd.comoving_volume(6.0, **cosmo)
>>> print "Vc = %.3g Mpc**3" % (Vc)
Vc = 1.68e+12 Mpc**3

lookback_integrand(z, **cosmo)

source code 

The derivative of the lookback time with redshift: dt_L/dz.

See equation 30 of David Hogg's arXiv:astro-ph/9905116v4

Units are seconds.

lookback_time(z, z0=0.0, **cosmo)

source code 

The lookback time (in s) to redshift z.

See equation 30 of David Hogg's arXiv:astro-ph/9905116v4

Units are s.

Optionally calculate the integral from z0 to z.

Returns

t_look: ndarray
Lookback time in seconds.

age(z, use_flat=True, **cosmo)

source code 

The age of the universe as seen at redshift z.

Age at z is lookback time at z'->Infinity minus lookback time at z.

See also: lookback_time.

Units are s.

Examples

>>> import cosmolopy.distance as cd
>>> import cosmolopy.constants as cc
>>> cosmo = {'omega_M_0' : 0.3, 'omega_lambda_0' : 0.7, 'h' : 0.72}
>>> cosmo = cd.set_omega_k_0(cosmo)
>>> t = cd.age(6.0, **cosmo)
>>> print "age at z=6.0 = %.3g Gyr" % (t/cc.Gyr_s)
age at z=6.0 = 0.892 Gyr

age_flat(z, **cosmo)

source code 

The age of the universe assuming a flat cosmology.

Units are s.

Analytical formula from Peebles, p. 317, eq. 13.2.

Examples

>>> import cosmolopy.distance as cd
>>> import cosmolopy.constants as cc
>>> cosmo = {'omega_M_0' : 0.3, 'omega_lambda_0' : 0.7, 'h' : 0.72}
>>> cosmo = cd.set_omega_k_0(cosmo)
>>> t = cd.age_flat(6.0, **cosmo)
>>> print "age at z=6.0 is %.3g Gyr" % (t/cc.Gyr_s)
age at z=6.0 is 0.892 Gyr

quick_distance_function(function, zmax=20.0, zmin=0.0, zstep=0.001, return_inverse=False, k=3, **cosmo)

source code 

Return an interpolation function that will give distance as a funtion of z

If return_inverse is True, will also return a function giving z as a function of distance.

Inputs

function -- the distance function to interpolate (can be any callable that takes a redshift argument plus cosmology keywords).

k -- spline order (scipy.interpolate.InterpolatedUnivariateSpline)

Returns

distfunc

or

distfunc, zfunc

Examples

>>> import cosmolopy.distance as cd
>>> import cosmolopy.constants as cc
>>> cosmo = {'omega_M_0' : 0.3, 'omega_lambda_0' : 0.7, 'h' : 0.72}
>>> cosmo = cd.set_omega_k_0(cosmo)
>>> distfunc, redfunc = cd.quick_distance_function(cd.luminosity_distance, return_inverse=True, **cosmo)
>>> d = distfunc(6.3333)
>>> z = redfunc(d)
>>> "%.1g" % (distfunc(6.3333)/cd.luminosity_distance(6.3333, **cosmo) - 1.0)
'-2e-16'
>>> "%.1g" % (z/6.3333 - 1.0)
'0'

quick_age_function(zmax=20.0, zmin=0.0, zstep=0.001, return_inverse=False, **cosmo)

source code 

Return an interpolation function that will give age as a funtion of z

Units are s.

If return_inverse is True, will also return a function giving z as a function of age.

Returns

agefunc

or

agefunc, redfunc

Examples

>>> import cosmolopy.distance as cd
>>> import cosmolopy.constants as cc
>>> cosmo = {'omega_M_0' : 0.3, 'omega_lambda_0' : 0.7, 'h' : 0.72}
>>> cosmo = cd.set_omega_k_0(cosmo)
>>> agefunc = cd.quick_age_function(**cosmo)
>>> t = agefunc(6.0)
>>> print "age at z=6.0 is %.3g Gyr" % (t/cc.Gyr_s)
age at z=6.0 is 0.892 Gyr

quick_redshift_age_function(zmax=20.0, zmin=0.0, zstep=0.001, **cosmo)

source code 

Return an interpolation function giving z as a funtion of age of the universe.

Units of time are s.

Returns

redfunc

Examples

>>> import cosmolopy.distance as cd
>>> import cosmolopy.constants as cc
>>> cosmo = {'omega_M_0' : 0.3, 'omega_lambda_0' : 0.7, 'h' : 0.72}
>>> cosmo = cd.set_omega_k_0(cosmo)
>>> redfunc = cd.quick_redshift_age_function(**cosmo)
>>> z = redfunc(1.0 * cc.Gyr_s)
>>> print "When age=1.0Gyr z=%.2f" % (z)
When age=1.0Gyr z=5.49

light_travel_distance(z, z0=0, **cosmo)

source code 

The light travel distance to redshift z.

Units are Mpc.

Examples

>>> import cosmolopy.distance as cd
>>> cosmo = {'omega_M_0' : 0.3, 'omega_lambda_0' : 0.7, 'h' : 0.72}
>>> cosmo = cd.set_omega_k_0(cosmo)
>>> dlookback = cd.light_travel_distance(3.0, 2.0, **cosmo)
>>> print "Lookback distance from z=2 to 3 is %.2g Mpc" % (dlookback)
Lookback distance from z=2 to 3 is 3.3e+02 Mpc

redshift_d_light(dl, z_guess=6.0, fmin_args={}, **cosmo)

source code 

The redshift corresponding to a given light travel distance.

Units are the same as light_travel_distance (Mpc).

Examples

>>> import cosmolopy.distance as cd
>>> import cosmolopy.constants as cc
>>> cosmo = {'omega_M_0' : 0.3, 'omega_lambda_0' : 0.7, 'h' : 0.72}
>>> cosmo = cd.set_omega_k_0(cosmo)
>>> z = cd.redshift_d_light(10. * cc.c_light_Mpc_Gyr, **cosmo)
Optimization terminated successfully.
         Current function value: 0.000112
         Iterations: 26
         Function evaluations: 52
>>> print "Redshift for a lookback time of 10Gyr is z=%.3f" % (z)
Redshift for a lookback time of 10Gyr is z=2.025

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