Varshni,Y.P.: 1976, Astrophys.Space Sci., 43, 3.
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Abstract. It is shown that the cosmological interpretation of the red shift in the spectra of quasars leads to yet another paradoxical result: namely, that the Earth is the center of the Universe. Consequences of this result are examined.It appears, however, that the significance of this coincidence of four emission-line red shifts at 1.955 has remained unappreciated. Also, it is perhaps not recognized that there are many other similar groups where the red shifts of two or more quasars lie very close together. In this paper we discuss a very serious consequence of these coincidences assuming the red shift hypothesis for quasar spectra.
In 1968 there were 20 QSOs with emission-line red shifts greater than 1.9. Burbidge calculated the probability for a chance coincidence of these four red shifts from the equation
r ! 1 r-k
P = -------- --- (1-1/n) (1)
k k!(r-k)! n^k
where Pk is the probability of the chance coincidence of k red shifts, where the total
number of possible intervals is n=(total range in red shift measured)/(size of box
determined by measurement errors or other effects) and the total number of red shifts
is r. Burbidge took the size of box, dz, to be 0.002. For the origin of Equation (1),
reference may be made to Parratt (1961) and Piper (1968).It may be argued, however, that Burbidge considered only a small range of z : namely, between z=1.93 and 2.36; and the number of QSOs in the population considered was also small. In the intervening years spectral data on a large number of quasars has become available. We shall consider all quasars for which spectral data for one or more emission lines have been published, except those for which the red shift z < 0.2. In the region z < 0.2, there are a few objects which have occasionally been listed as quasars. Some of these (for example, B234, B264, and Ton 256) are N-type galaxies or related objects, while some others are quasars. The total number of quasars with z < 0.2 is quite small and will not have any significant effect on our numerical results. As of June 1975 there were 384 quasars in the category that we are considering. The range of z is 0.2 to 3.53. Using the same size of box as that used by Burbidge, we find that the probability of a chance coincidence of the red shifts of the same 4 QSOs is Pk=9.24x10-5. This value is about 50 times as large as the one calculated by Burbidge (1968). Although this is a small probability, the significance of this is possibly not appreciated.
Before proceeding further, we wish to make one point clear. The resulting low probability given above, in itself, does not confer any special status to the red shift of 1.955. We would have obtained the same result had the four QSOs been at any other red shift between z=0.2 to 3.53. A special status for 1.955 arises because of the assumed identifications of lines. We clarify this point further. The red shift 1.955 is obtained because the two strong lines observed in these QSOs are identified with C IV 1549 and Lyman alpha 1216. The ratio of these two wavelengths is 1.274. Another ratio close to this number is that of [O III] 4363 to [Ne V] 3426. If for some reason we identify the two observed lines with 4363 and 3426, we get z=0.049. The important point to note is that now the coincidence will occur at z=0.049, but the probability Pk will change only slightly. Similarly, if we were to identify the two observed lines with C III 977 and N IV 765.1, the corresponding red shift will be 3.69, but the probability Pk will be about the same.
Fig.1.Diagrammatic representation of the spectra of 5 quasars belonging to group 18 of
Table I.
The heights of the lines represent their strengths, except for 4C 05.46, for which the observers
have not given the strengths of lines. Two third height indicates medium strength, one third weak.
The spectrum of the quasar 1055+20 has not been investigated below 4000 Å, and that of 3C 204
has not been investigated above 4950 Å.
Table I represents the first classification of QSOs on the basis of their spectra. As a matter of fact, some QSOs with very different red shifts belong to the same group, but this will introduce complications in our discussion, hence we shall not consider this point further in this paper.
For each of these groups, we have calculated Pk and the results are shown boldface in the second column of Table I. It was assumed that the uncertainty in the reported z values is ±0.001 and thus the red shift width of a group was equal to z(highest)-z(lowest)+0.002. From the multiplicative law of probability, the probability of these 57 sets of coincidences occurring in this system of 384 QSOs is about 3x10-85. We hope this number will be convincing evidence that the coincidences are real and cannot be attributed to chance. As discussed earlier for z=1.955, this calculation does not imply any special status to the red shifts at which these coincidences occur. These coincidences are to be clearly distinguished from the `peaks' that some authors have claimed in the red shift distribution.
Next we must consider how these coincidences can be physically explained. First let us consider the conventional view that the red shifts of the QSOs are cosmological in origin. If we assume that the Universe is homogeneous and isotropic (the cosmological principle), it is well known that the most general expression for the line-element is the Robertson-Walker line element, which has the form
2
2 2 2 R (t) 2 2 2 2 2
ds = c dt - ---------------- { dr + r (dtheta + sin theta dphi ) }
(1 + 0.25 k r^2)
where R(t) is the scale factor (also called the expansion parameter), k is the curvature
index, and r, theta, and phi are co-moving coordinates.Consider the universe at a particular time to. Then we have dt=0, and the line element for three-dimensional space at the time to becomes
2
2 R (to) 2 2 2 2 2
ds = ---------------- { dr + r (dtheta + sin theta dphi ) } (3)
(1 + 0.25 k r^2)
At a later time t1 we would have exactly the same line element, except that every
interval ds would be multiplied by the factor R(t1)/R(to). Thus the Robertson-Walker
line element implies that the distance between two co-moving observers can be written
as
D(t) = Do R(t) (4)
where Do is a constant. It is also clear that although the distance between observers
changes as a function of world time, the relative distance does not change.
Also, it is known that the red shift z is connected with R(t) by the relationship
L R(t)
---- = 1 + z = ------ (5)
Lo R(to)
where Lo and L are the wavelengths of the emitted and received light signals,
respectively. From this discussion it is obvious that if two or more quasars have the same
value of z, they are at the same distance (though in different directions) from the
Earth. In other words, assuming the cosmological red-shift hypothesis, the quasars
in the 57 groups in Table I are arranged on 57 spherical shells with Earth as the center.
This is certainly an extraordinary result. Some of the possibilities that we shall
consider to accommodate this result may be disturbing, but we must consider these
possibilities dispassionately.
We are essentially left with only one possibility - No.3 in the cosmological red-shift interpretation. However, before we accept such an unaesthetic possibility, we must raise the question : Are the `red shifts' real ? We wish to point out that we have proposed an alternative explanation of the spectra of quasars (Varshni, 1973, 1974, 1975; Menzel, 1970; Varshni and Lam, 1974) which is based on sound physical principles, does not require any red shifts, and has no basic difficulty.
There is a second part to this paper.
Einstein
distinguishes between two main criteria [for a good
theory]: (a) the external confirmation of a theory, which informs
us in experimental checks of the correctness of the theory, and (b) the
inner perfection of a theory which judges its `logical simplicity' or
`naturalness'.
- G. Holton (1973)
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