Douglas Burke


Astro::Cosmology - calculate cosmological distances, volumes, and times


This module provides a set of routines to calculate a number of cosmological quantities based on distance and time. Some are a bit complex - e.g. the volume element at a given redshift - while some, such as the conversion between flux and luminosity, are more mundane.

To calculate results for a given cosmology you create an Astro::Cosmology object with the desired cosmological parameters, and then call the object's methods to perform the actual calculations. If you aren't used to objects, it may sound confusing; hopefully the SYNOPSIS section below will help (after all, a bit of code is worth a thousand words). The advantage of using an object-orientated interface is that the object can carry around the cosmological parameters, so you don't need to keep on specifying them whenever you want to calculate anything; it also means you can write routines which can just accept an Astro::Cosmology object rather than all the cosmological parameters.

This module requires that the PDL distribution is installed on your machine; PDL is available from CPAN or


Whilst I believe the results are accurate, I do not guarantee this. Caveat emptor, as the Romans used to say...


  use Astro::Cosmology qw( :constants );

  # what is the luminosity distance, in metres, for
  # a couple of cosmologies
  my $z   = sequence(10) / 10;
  my $eds = Astro::Cosmology->new;
  my $sn  = Astro::Cosmology->new( matter => 0.3, lambda => 0.7 );

  my $de  = 1.0e6 * PARSEC * $eds->lum_dist($z);
  my $ds  = 1.0e6 * PARSEC * $sn->lum_dist($z);

  # let's change the parameters of the $sn cosmology
  $sn->setvars( lambda=>0.6, matter=>0.2 );


If H0 is set to 0, then the units used are the Hubble distance, volume per steradian, or time. If greater than zero, distances are measured in Mpc, volumes in Mpc^3/steradian, and time in years.


  • The comoving volume routine gives a slightly smaller answer than Figure 6 of Carroll, Press & Turner for z ~ 100. It could be due to differences in the numerical methods, but I've not yet investigated it thoroughly.

  • A year is defined to be YEAR_TROPICAL seconds. Let me know if this is wrong.


The following calculations were cobbled together from a number of sources, including the following (note that errors in the documentation or code are mine, and are not due to these authors):

  Distance measures in cosmology, Hogg, D.W., astro-ph/9905116
  Perlmutter et al. 1997, ApJ, 483, 565
  Carroll, Press & Turner 1992, ARAA, 30, 499
  Weinberg, S., sections 14.6.8, 15.3.25
  Sandage, A. 1961, ApJ, 133, 355-392

In the following all values are in "natural" units: Hubble distance, volume, or time.

Symbols used in the following:

  om    is omega_matter
  ol    is omega_lambda
  ok    is 1 - om - ol
  kappa is sqrt( abs(ok) )


For cosmologies with no lambda term, the luminosity distances (dl) are calculated by the standard formulae:

  empty:     dl = 0.5 * z * (2+z)
  flat:      dl = 2 * ( 1+z - sqrt(1+z) )
  otherwise: dl = (2 / (om*om)) *
                  ( om*z + (om-2) * (sqrt(1+om*z)-1) )

For non-zero lambda cosmologies, the luminosity distance is calculated using:

  closed:  dl = (1+z) * sin( kappa * dc ) / kappa
  open:    dl = (1+z) * sinh( kappa * dc ) / kappa
  flat:    dl = (1+z) * dc

where dc is the comoving distance, calculated by numerical integration of the following from 0 to z:

  1.0 / sqrt( (1+z)^2 * (1+om*z) - z*(2+z)*ol )

The comoving distance is always calculated by numerical integration of the above formula. The angular diameter and proper motion distances are defined as dl/(1+z)^2 and dl/(1+z) respectively.


If dm is the proper motion distance, then the comoving volume vc is given by

 flat:   vc = dm^3 / 3
 open:   vc = dm * sqrt(1+ok*dm^2) - asinh(dm*kappa) /
              ( 2 * kappa * ok )
 closed: vc = dm * sqrt(1+ok*dm^2) - asin(dm*kappa) /
              ( 2 * kappa * ok )

The differential comoving volume, dvc, is calculated using the proper motion distance, dm, and the differential proper motion distance, ddm, by

  dvc = dm^2 * ddm / sqrt( 1 + ok*dm^2 )


  ddm = dc * sqrt( 1 + abs(ok) * dm^2 )


The lookback time is calculated by integration of the following formula from 0 to z:

 1.0 / ( (1+z) * sqrt( (1+z)^2 * (1+om*z) - z*(2+z)*ol ) )

Flux and Magnitudes

The conversion between absolute and apparent magnitudes is calculated using:

  $app_mag = $abs_mag + 25 + 5 * $cosmo->lum_dist($z)->log10();

The conversion between flux and luminosity is calculated using

  $lumin = FOURPI * $dl * $dl * $flux


  $dl = $cosmo->lum_dist($z) * 1.0e8 * PARSEC

Note that these equations do not include any pass-band or evolutionary corrections.

Integration Technique

All integrations are performed using Romberg's method, which is an iterative scheme using progressively higher-degree polynomial approximations. The method stops when the answer converges (ie the absolute difference in the values from the last two iterations is smaller than the ABSTOL parameter, which is described in the new method).

Typically, the romberg integration scheme produces greater accuracy for smooth functions when compared to simpler methods (e.g. Simpson's method) while having little extra overhead for badly-behaved functions.


Currently the following constants are available via use Astro::Cosmology qw( :constants ):

  • LIGHT - the speed of light in m/s.

  • PARSEC - one parsec in metres.

  • STERADIAN - one steradian in degrees^2.

  • YEAR_TROPICAL - one tropical year in seconds.

  • PI - defined as 4.0 * atan(1.0,1.0) [this is in uppercase, whatever this document may say]

  • FOURPI - 4.0 * PI [again PI should be in upper case here]

Please do not use this feature, as it will be removed when an 'Astronomy constants' is created - e.g. see the astroconst package at .


This document uses the $object->func(...) syntax throughout. If you prefer the func($object,...) style, then you need to import the functions:

  use Astro::Cosmology qw( :Func );

Most functions have two names; a short one and a (hopefully) more descriptive one, such as pmot_dist() and proper_motion_distance().

Most of the routines below include a sig: line in their documentation. This is an attempt to say how they `thread' (in the PDL sense of the word). So, for routines like lum_dist - which have a sig line of dl() = $cosmo->lum_dist( z() ) - the return value has the same format as the input $z value; supply a scalar, get a scalar back, send in a piddle and get a piddle of the same dimensions back. For routines like abs_mag - with a sig line of absmag() = $cosmo->abs_mag( appmag(), z() ) - you can thread over either of the two input values, in this case the apparent magnitude and redshift.


Utility routines

  • new

  • version

  • stringify

  • setvars

  • matter/omega_matter, lambda/omega_lambda, h0/hO

Distance measures

  • lum_dist/luminosity_distance

  • adiam_dist/angular_diameter_distance

  • pmot_dist/proper_motion_distance

  • comov_dist/comoving_distance

Volume measures

  • comov_vol/comoving_volume

  • dcomov_vol/differential_comoving_volume

Time measures

  • lookback_time



  my $cosmo = Astro::Cosmology->new(
                matter => 0.3, lambda => 0.7 );
  my $cosmo = Astro::Cosmology->new(
                { matter => 0.3, lambda => 0.7 } );

Create the object with the required cosmological parameters. Case does not matter and you can use the minimum number of letters which remain unique (the parsing is done by the PDL::Options module).

The options can be specified directly as a list - as shown in the first example above - or in a hash reference - as shown in the second example. You can not mix the two forms within a single call. The options are:

  OMEGA_MATTER or MATTER  - default 1.0
  OMEGA_LAMBDA or LAMBDA  - default 0.0
  H0           or HO      - default 50.0
  ABSTOL                  - default 1.0e-5

If H0 is set to 0, then answers are returned in units of the Hubble distance, volume, or time, otherwise in Mpc, Mpc^3/steradian, or years.

ABSTOL (absolute tolerance) is used as a convergence criteria when integrating functions as well as whether values are close enough to 0. You should not have to worry about it.


  print "Version is " . Astro::Cosmology->version . "\n";
  if ( $cosmo->version > 0.9 ) {

Returns the version number of the Astro::Cosmolgy module as a string. This method is not exported, so it has to be called using either of the two methods shown above.


  print $cosmo;

Returns a string representation of the object. The operator "" is overloaded by this function, so that print $cosmo gives a readable answer.


  $cosmo->setvars( matter => 0.3, lambda => 0.7 );

Change the cosmological parameters of the current object. The options are the same as for new.

omega_matter or matter

  $cosmo->omega_matter( 1.0 );
  my $omega = $cosmo->omega_matter;

If supplied with an argument, sets the value of Omega_matter. Returns the current value of the parameter.

omega_lambda or lambda

  $cosmo->omega_lambda( 0.8 );
  my $lambda = $cosmo->omega_lambda;

If supplied with an argument, sets the value of Omega_lambda. Returns the current value of the parameter.

h0 or hO

  $cosmo->h0( 75 );
  my $cosmo->$h0 = h0;

If supplied with an argument, sets the value of H0. Returns the current value of the parameter.

lum_dist or luminosity_distance

  sig: dl() = $cosmo->lum_dist( z() )

  my $dl = $cosmo->lum_dist( $z );

returns the luminosity distance, for a given redshift, $z, for the current cosmology.

adiam_dist or angular_diameter_distance

  sig: da() = $cosmo->adiam_dist( z() )

  my $da = $cosmo->adiam_dist( $z );

returns the angular diameter distance, for a given redshift, $z, for the current cosmology.

pmot_dist or proper_motion_distance

  sig: dm() = $cosmo->pmot_dist( z() )

  my $dm = $cosmo->pmot_dist( $z );

returns the proper motion distance, for a given redshift, $z, for the current cosmology.

comov_dist or comoving_distance

  sig: dc() = $cosmo->comov_dist( z() )

  my $dc = $cosmo->comov_dist( $z );

returns the line-of-sight comoving distance, for a given redshift, $z, for the current cosmology.

comov_vol or comoving_volume

  sig: dv() = $cosmo->comov_vol( z() )

  my $dv = $cosmo->comov_vol( $z );

returns the comoving volume out to a given redshift, $z, for the current cosmology. Does not work if omega_matter and omega_lambda are both 0.0.

dcomov_vol or differential_comoving_volume

  sig: ddv() = $cosmo->dcomov_vol( z() )

  my $ddv = $cosmo->dcomov_vol( $z );

returns the differential comoving volume at a given redshift, $z, for the current cosmology. Does not work if omega_matter and omega_lambda are both 0.0.


  sig: t() = $cosmo->lookback_time( zmax() )
  sig: t() = $cosmo->lookback_time( zmin(), zmax() )

  my $delta_t = $cosmo->lookback_time( [$zmin,] $zmax );

Returns the lookback time between $zmin and $zmax. If $zmin is not supplied it defaults to 0.0.

abs_mag or absolute_magnitude

  sig: absmag() = $cosmo->abs_mag( appmag(), z() )

  my $absolute_mag = $cosmo->abs_mag( $apparent_mag, $z );

Returns the absolute magnitude - excluding K and evolutionary corrections - for the given apparent magnitude.

app_mag or apparent_magnitude

  sig: appmag() = $cosmo->app_mag( absmag(), z() )

  my $apparent_mag = $cosmo->app_mag( $absolute_mag, $z );

Returns the apparent magnitude for a given absolute magnitude. As with abs_mag, the K- and evolutionary-corrections are left up to the user.


  sig: lumin() = $cosmo->luminosity( flux(), z() )

  my $lumin = $cosmo->luminosity( $flux, $z );

Returns the luminosity of a source of a given flux. As with abs_mag, the K- and evolutionary-corrections are left up to the user.

The spatial units of the flux must be cm^-2, so a flux in erg/cm^2/s will be converted into a luminosity in erg/s.


  sig: flux() = $cosmo->flux( lumin(), z() )

  my $flux = $cosmo->flux( $lumin, $z );

Returns the flux of a source of a given luminosity. As with abs_mag, the K- and evolutionary-corrections are left up to the user.

The spatial units of the flux is cm^-2, so a luminosity in erg/s will be converted into a flux in erg/cm^2/s.


Add ability to request a particular unit; for example have $cosmo->lum_dist() return cm rather than Mpc.

Add the ability to use Pen's approximations ("Analytical Fit to the Luminosity Distance for Flat Cosmologies with a Cosmological Constant", 1999, ApJS, 120, 49).

There is currently no method to calculate the age of the universe at a given redshift.


Thanks to Brad Holden for trying out early versions of this module and for providing some of the test code.

The cosmology routines make use of code based on routines from

  NUMERICAL METHODS: FORTRAN Programs, (c) John H. Mathews 1994
  NUMERICAL METHODS for Mathematics, Science and Engineering, 2nd Ed, 1992
  Prentice Hall, Englewood Cliffs, New Jersey, 07632, U.S.A.

The "Integration Technique" section of the documentation is based on that from from the Math::Integral::Romberg module by Eric Boesch (available on CPAN).


PDL, Math::Integral::Romberg.


Copyright (C) Douglas Burke <> 1999, 2000, 2001.

All rights reserved. There is no warranty. This program is free software; you can redistribute it and/or modify it under the same terms as Perl itself.