Annu. Rev. Astron. Astrophys. 1988. 26:
561-630
Copyright © 1988 by . All rights reserved |

**3.1. The Coordinate r as a Function of Redshift**

The relation between time and distance for a light ray is given by the null geodesic of the space-time interval whose metric is

(15) |

Orienting the axes so that
=
= 0 and putting *ds* = 0
gives the basic equation of the problem as

(16) |

Using Equation 12, we finally obtain

(17) |

which if *R*(*t*) is a known function of time will give
*r*(*t*). This is related to the redshift
*z*
/
_{0} by the
Lamaitre Equation

(18) |

where *R*_{0} and *R*_{1} are the scale
factors at the times of light reception and fight emission, respectively.

The time variation of *R* is given by the solution of the dynamical
Friedmann equation

(19) |

which is fundamental to the standard model. Integration of this equation
gives *R*(*t*), which when put in Equation 17 gives the
*r*(*z*) connection via
Equation 18. This in turn, when put in Equations 9-11 gives
*V*(*z*), *which solves the problem in closed form.*

Two special cases illustrate the method. Consider first the Euclidean
case of *k* = 0. The well-known solution of Equation 19 for
*R*(*t*_{i}) at time *t*_{i} is

(20) |

When this is put into the right side of Equation 17 and integrated,
using Equation 18 to relate *z* with the ratio of the scale factors, we
obtain

(21) |

which, with
*t*_{0} = 2/3*H*_{0}^{-1}, where
*H*_{0} =
/ *R*, gives

(22) |

This, put into Equation 11, gives

(23) |

for the volume enclosed in redshift distance *z* for the Euclidean case
(*k* = 0).

Consider next an empty universe (no mass). In this case,
= 0, *k* = - 1,
and Equation 19 integrates directly to

(24) |

Equation 24 put into the right side of Equation 17 gives

(25) |

Noting that
*t*_{0} = *H*_{0}^{-1} in this case,
we obtain, after reduction,

(26) |

Using Equation 1, and remembering that *q*_{0} = -
/
*R*_{0}*H*_{0}^{2} = 0
in this case, gives
*R*_{0} = *c*/*H*_{0}, hence

(27) |

Equations 26 and 27, substituted into Equation 10, give
*V*(*z*) for an empty universe explicitly.

We have now introduced the dimensionless deceleration parameter
*q*_{0},
which is convenient in expressing the general case. This parameter
first arose in the literature via series expansions of the relevant
observational equations
(Heckmann 1942,
Robertson 1955,
McVittie 1956,
Davidson 1959),
where no recourse to the solution of the Friedmann
equation was needed. Before
Mattig's (1958,
1959)
exact (closed)
solutions were known, Taylor expansions of *R*(*t*) were made
backward in
time starting with the time of observation *t*_{0}. This
required no
knowledge of the Friedmann solution but merely an assumption that
*R*(*t*)
is well enough behaved for a Taylor series to exist. These series
expressions for the *V*(*z*), *m*(*z*), and
(*z*) tests
sufficed for small
redshifts, but not for redshifts of arbitrarily large size. In
contrast, Equations 21, 22, 26, 27 are exact for all values of *z*. It
follows that their use in Equations 9 and 10 for the volumes also
apply to any value of the redshift. The reason is that we have used
the complete solution for *R*(*t*) from Equation 19 in
Equation 17 rather than a Taylor series.

The value of *q*_{0} determines the size of the space
curvature via
Equation 1. This, in turn, is related to the matter density
(Hoyle & Sandage
1956) by

(28) |

For any arbitrary
value (hence
*q*_{0} value) we seek general formulae
for *r*(*z*) and *Rr*(*z*). Equations 21, 22, 26,
and 27 are special cases of
these formulae. We need the general solution of the Friedmann equation
(Equation 19) for any arbitrary value of the curvature
*kc*^{2} / *R*^{2}.

Mattig (1958) shows that this solution is

(29) |

and

(30) |

for all values of *q*_{0}. A transparent derivation of
these equations in terms of the parametric cycloid and hypercycloid
development angle was given by
Sandage (1961b).