Four Cosmological distances were determined in the light of the closed cosmic model which was presented in a previous study. Each of these distances was obtained in terms of the redshift of an extragalactic object. It is found that the luminosity distance of the extragalactic object in the closed cosmic model, the observed model and model are approximately the same up to z = 0.1535. However, the luminosity distance in the close cosmic model approximately agrees with its value in the observed model up to z = 0.6442. Estimations of the horizon distance of the universe, 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 computed in the closed cosmic model at the present time.
In [
The proper distance of an extragalactic object at the present time is given by
where c is the speed of the light and the emission time of the object photon [
where
The Hubble parameter in the expanding cosmic model at time t is
The density parameters in the expanding cosmic model at time t are
where
where is the cosmic time in Gyr.
Substituting by (2) in (1) we get
The density parameters in the closed cosmic model at time t are
where
and the Hubble parameter in the closed cosmic model at time t is
using (5), (14) we have
substituting by (12), (17) in (19) we get
Using (3)
Let, then Equation (20) can be written as
Similarly we can get
Substituting by (21)-(23) in (13) the proper distance of the extragalactic abject in the closed cosmic model at the present time becomes
and its luminosity distance, angular diameter distance and distance modulus are respectively
The horizon distance of the universe in the closed cosmic model at the present time is given by Equation (28).
It is obvious from
The total density of the universe in the closed cosmic model at time t is
Substituting by (14)-(16) in (30) we get
where. At the present time we have
Hence, the total mass and the mass of matter within in the closed cosmic model at the present time are respectively expressed as
The equivalent numbers of the Milky Way-like galaxies and the Coma-like clusters to are respectively written as
where are the masses of the Milky Way galaxy and the Coma cluster of galaxies respectively.
Results of the closed cosmic model would be compared with those of the general cosmic model A which best represents observations [2,3].
The proper distance—redshift relation (24) in the closed cosmic model and the corresponding relation in the general cosmic model A [
The luminosity distance—redshift relation (25) in the closed cosmic model and the corresponding relations in the general cosmic model A and the model [
The angular diameter distance—redshift relation (26) in the closed cosmic model and the corresponding relation in the general cosmic model and A [
The distance modulus-redshift relation (27) in the closed cosmic model and the corresponding relation in the general cosmic model A [
1) The horizon distance of the universe .
2) The total mass within is .
3) The mass of matter in within is .
4) Number of the Milky Way-like galaxies equivalent to the mass of matter within is .
5) Number of the Coma-like cluster of galaxies equivalent to the mass of matter within is
.
Errors in these estimations were obtained by similar way as in [
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 extragalactic object in the light of the closed cosmic model which was constructed in a previous paper. It is found that all of these cosmological distances increase continuously with redshift except the angular diameter distance which steeply increases with redshift towards maximum value at, then it decreases gradually with increasing z. It is also found that the luminosity distance of the extragalactic object in the closed cosmic model, observed model and model are approximately similar up to z = 0.1535. However, the luminosity distance in the closed cosmic model approximately agrees with its value in the observed model up to z = 0.6442. Evaluations of the horizon distance of the universe, 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 calculated in the closed cosmic model at the present time.
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.