Varshni,Y.P.: 1976, *Astrophys.Space Sci.*, **43**, 3.

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A few years ago it was pointed out by G. Burbidge (1968) that a number of quasars have an absorption-line red shift of 1.95. He also drew attention to the fact that there are four quasars : namely, 3C 191, PKS 0119-04, PHL 938, and BSO 6, whose emission-line red shifts are extremely close to 1.955. The absorption-line red shift at 1.95 has been subsequently discussed by the Burbidges and by others in a number of papers.

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^kwhere 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,

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

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

D(t) = Do R(t) (4)where

L R(t) ---- = 1 + z = ------ (5) Lo R(to)where

- Coincidence in distances could be possible if there were clustering. However, an examination of the coordinates of the various members of individual groups shows that in most cases there is no such correlation. Hence, this explanation has to be ruled out.
- Quasars may be arranged like atoms in a crystal lattice, with the Earth being either at an empty lattice site or at a suitable interstitial site. Should that be the case, one would expect some pattern or regularity in the directions of quasars belonging to a certain group. No such evidence is found and this possibility must also be abandoned.
- The Earth is indeed the center of the Universe. The arrangement of quasars on certain spherical shells is only with respect to the Earth. These shells would disappear if viewed from another galaxy or a quasar. This means that the cosmological principle will have to go. Also, it implies that a coordinate system fixed to the Earth will be a preferred frame of reference in the Universe. Consequently, both the Special and the General Theory of Relativity must be abandoned for cosmological purposes.

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.

- Burbidge,G.: 1968,
*Astrophys.L.Letters***154**, L41. - Holton,G. (revised by S.G.Brush): 1973,
*Introduction to Concepts and Theories in Physical Science*, Addison-Wesley, Reading. - Menzel,D.H.: 1970, in H.G. Groth and P.Wellmann (eds),
*Spectrum Formation in Stars with Steady-State Extended Atmospheres*, N.B.S. Special Publication**332**, p.134. - Parrat,L.G.: 1961,
*Probability and Experimental Errors in Science*, John Wiley and Sons, Inc. New York. - Piper,D.E.: 1968,
*Nature***219**, 846. -
Varshni,Y.P.: 1973,
*Bull.Amer.Phys.Soc.***18**, 1384. - Varshni,Y.P.: 1974,
*Bull.Amer.Phys.Soc.***6**, 213, 308. -
Varshni,Y.P.: 1975,
*Astrophys. Space Sci.***37**, L1. -
Varshni,Y.P., Lam,C.S.: 1974,
*J.Roy.Astron.Soc.Canada***68**, 264.

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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