1. Introduction
One of the most interesting aspects of investigating cosmic models is determination of cosmological distances of the extragalactic objects in these models such as the proper distance, luminosity distance, angular diameter distance (the proper distance of the object when its light was emitted) and distance modulus.
In a previous study five general cosmic models were developed using comprehensive selection ratios of the universe components which are luminous matter, dark matter, photons, neutrinos, and dark energy [1]. Although some physical properties of the universe were discussed in these models, we concentrate in this article on precise determination of cosmic distances of the extragalactic objects in terms of their redshifts.
The importance of this study comes from the fact that the cosmic distances mentioned above are most commonly used in many cosmological researches. As a consequence of specifying cosmic distances, interesting estimations were computed in the general models at the present time. These estimations are horizon distance, total mass within the horizon distance, mass of matter (luminous and dark matter) within the horizon distance, and its equivalent numbers of both the Milky Way-like galaxies and the Coma-like clusters.
Description of methodology is given in Section 2, while results are presented in Section 3, accuracy and errors are illustrated in Section 4, and discussion and conclusion are shown in Section 5.
2. Methodology
The proper distance of an extragalactic object at present time 
is given by
(1)
where 
is the radial comoving coordinate of the object now, and 
is expressed as
(2)
where 
is the scale factor of the universe expansion, 
the speed of light, 
the emission time of the object photon and 
the curvature of space in the universe. Let 
then Equation (2) becomes
(3)
Substituting by Equation (3) in Equation (1) we get
(4)
From [1] we have seen that the expansion speed of the universe is
(5)
where 
and 
are the density parameters of the dark energy, matter and radiation at the present time respectively. 
is the Hubble parameter at the present time. Equation (5) can be written as
(6)
Substituting by Equation (6) in Equation (4) we find
(7)
where
, 
is the redshift of the object.
The luminosity distance 
angular diameter distance 
and distance modulus of the object at the present time are expressed respectively as
(8)
(9)
(10)
where 
are the apparent and absolute magnitudes of the object. The distances 
and 
are measured in Mpc.
It is obvious from Equation (7) that the horizon distance of the universe at the present time is given by
(11)
where 
The Robertson-Walker metric of spacetime which describes the expansion or contraction of homogeneous and isotropic universe is given by
(12)
where 
are the comoving coordinates of a point in space. The volume of three dimensional sphere can be written as
(13)
where 
is the determinant of the spatial part of the Robertson-Walker metric matrix and given by
(14)
Since the space of the universe is flat at the present time, hence 
and
(15)
Substituting by (15) in (13) we have
(16)
For flat space (1) and (2) give
(17)
Substituting by (17) in (16) we get at the present time
(18)
Similarly the volume of sphere of radius 
is expressed as
(19)
From the previous study [1] it is obvious that at any given cosmic time the total density of the universe is given by
or
(20)
where the parameters 
and 
are all defined exactly as in the previous study. At the present time Equation (20) gives
(21)
The total mass within 
is given by Equations (19), (21) as
(22)
The mass of matter within 
is
or
(23)
Since the masses of the Milky Way galaxy 
and the Coma cluster 
are 
, [2], where 
is the solar mass. Therefore the equivalent numbers of the Milky Way-like galaxies and the Coma-like clusters to 
are respectively
(24)
(25)
3. Results and Discussion
The proper distance-redshift relation (7) in the general models up to 
is plotted in Figure 1, where the distributions increase with 
and they coincide on each other for 
but they slightly diverge from each other afterwards such that the distributions of the general models 
and 
are at the top followed by those of the general models 
and 
respectively.
The distributions of the luminosity distance-redshift relation in the general models up to 
are shown in Figures 2(a)-(e). In each figure the upper curve represents Equation (8) and the lower curve represents the relation
(26)
where 
is the decelerating parameter of the universe at the present time [3,4]. It is noticeable that 
continuously raises with 
however, Equation (26) gives less estimation of 
than that of Equation (8) for
and the difference between the two estimations increases also with 
This is due to the repulsive effect of the dark energy. Initially, when the universe has small volume the effect of gravity is greater than the effect of dark energy, so the initial expansion rate of the universe is slow just as in Friedmaun models. However, there will come a point when the repulsive effect of the dark energy will equal and then exceeds the effect of gravity and the universe will expand in continuously increasing rate. In this case, distant galaxies will have greater distances from us than their distances in the case of the Friedmaun models [5].
Figure 3 illustrates the distributions of the angular diameter distance—redshift relation (9) in the general models up to 
These distributions coincide on each other for 
then they start diverging slightly from each other with the distributions of the general models 
and 
are at the top followed by those of the general models 
and 
respectively. All distributions raise up appreciably fast until the value 
given in each general model in Table 1, then the distributions decrease gradually with increasing 
The distance modulus-redshift relation is determined using Equations (8) and (10) and represented by the
Figure 1. The proper distance redshift relation in the general cosmic models up to z = 6.
Table 1. Maximum values of the angular diameter distances in the general cosmic models.
distributions in the general models up to 
exhibited in Figure 4. It is prominent that 
increases continuously with 
in all general models, and the distributions of these models coincide on each other.
Values of 
and 
were computed in the general models at the present time using Equations (11) and (22)-(25) respectively. These values and their possible ranges are shown in Table 2. More details about calculations of these estimations are given in the next section.
4. Accuracy and Errors
In computing the proper distance 
of an extragalactic object at the present time in terms of its redshift 
one can get an average value of 
to good degree of accuracy by applying Equation (7), where the parameters 
and 
have their average values in the general model 
obtained from [1].
Nevertheless, more accurate value of 
can be obtained by calculating 
in all the possible observed 163 models of the case A [1]. Assume the average of these 163 values of 
is 
and the maximum and minimum values of 
are
respectively. Let 
Then the accurate value of the proper distance of the extragalactic object is
(27)
This procedure was used to determine the average estimations of the parameters 
and 
with their possible ranges in the general models as illustrated in Table 2.
5. Conclusions
In this paper distributions of the proper distance, luminosity distance, angular diameter distance and distance modulus were investigated in terms of the redshift of an
Figure 3. The angular diameter distance redshift relation in the general cosmic models up to z = 6.
Figure 4. The distance modulus redshift relation in the general cosmic models up to z = 6.
Table 2. Estimation of the horizon distance , the total mass and the mass of matter within the horizon distance, the equivalent numbers of the Milky way-like galaxies and the Coma-like clusters to the mass of matter in the general cosmic models at the present time.
extragalactic object in the light of the five general cosmic models which were constructed in a previous paper. It is found that all of these cosmological distances increase continuously with redshift except the angular diameter distance which increases very rapidly with redshift to maximum value and then it decreases gradually with increasing 
Calculations of the horizon distance, the total mass and the mass of matter within the horizon distance, the equivalent numbers of the Milky Way-like galaxies and the Coma-like clusters of galaxies to the mass of matter were all obtained in the general cosmic models at the present time.
6. Acknowledgements
This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The author, therefore, acknowledges with thanks DSR technical and financial support.