THE RED SHIFT HYPOTHESIS FOR QUASARS :

IS THE EARTH THE CENTER OF THE UNIVERSE ? (Part II)

Varshni,Y.P.: 1977, Astrophys.Space Sci. 51, 121.

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Abstract. It is pointed out that Stephenson (1977) has used incorrect dz, and has also made an arithmetical error, which invalidates his claims. Tests for randomness of quasar red-shift clusters, using correct dz, have been carried out and it is shown that at least for clusters having three red shifts or more, the distribution is highly non-random. The model of the Universe proposed by Stephenson does not in any way explain these red-shift clusters; it merely substitutes one paradox by another.
In a recent paper ( Varshni, 1976; hereafter to be referred to as Paper I) the author has presented evidence for 57 coincidences in the apparent red shifts of quasars. Stephenson (1977) has raised the valid question whether some such clustering could arise due to random processes. He proposes two tests, both of which are reasonable, to check this point. However, his results and conclusions are erroneous because he has used incorrect dz in his calculations; and has also made an arithmetical error. We show in this note that, when the correct dz is used, the essential conclusions of Paper I are substantiated. All the necessary data required for calculations in the present paper are taken from Paper I.

Test 1

We start from the binomial distribution formula 
                  r !    1         r-k
          P  = -------- --- (1-1/n)                                  (1)
           k   k!(r-k)! n^k
In the present context, 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, dz) and the total number of red shifts is r. Then the number of random coincidences having k or more red shifts (represented here by Tk) is given by 
                   k-1
                   __
                   \
     Tk = n ( 1 -  /_  Pj )                                          (2)
                   j=0
The first test of Stephenson (1977) compares the calculated values of Tk with the reported ones (Table I of Varshni, 1976). Care must be exercised in the determination of dz to be used in Equation (2).

Fig.1.Dermination of the size of box, dz, when there are two red shifts in a cluster.

In Figure 1, we show the case of two red shifts, z1 and z2. Let 0.001 be the uncertainty in their values. We distinguish three cases :

when the errors are subtractive. (dz)min=0, if the right-hand side of Equation (c) is negative.

A reference to Figure 1 shows that case (b) is as probable as case (c), and the correct dz to use in the present situation is that given by case (a). When there are several red shifts, dz=z(highest)-z(lowest). Stephenson (1977) has used (dz)max, which invalidates his results. We may also point out that Stephenson is inconsistent in the value of (dz)max that he uses. For k>=3 he uses the correct value of (dz)max, but for k > 2 he uses a value 0.004, while the correct value is 0.0053.

In Table I we show, for k >=2, 3 and 4, the observed and calculated values of Tk for the three cases, (a), (b) and (c). A comparison of columns 2 and 4 shows that for k >=3 the clustering is highly non-random.

TABLE I

Observed and calculated values of the number of clusters of red shifts 
  k  Observed  Average Calculated  Average Calculated  Average Calculated
        Tk       dz    Tk for dz   (dz)max   Tk for    (dz)min   Tk for
                                            (dz)max             (dz)min
--------------------------------------------------------------------------
 >= 2   57     0.0033    56.8      0.0053    78.8      0.0016    31.3
 >= 3   25     0.0046    12.1      0.0066    21.1      0.0026     4.6
 >= 4    9     0.0051     2.0      0.0071     4.6      0.0032     0.6

Test 2

A comparison of the percentage of quasars which are in clusters with the percentage of the z space that these quasars occupy can be used as some sort of a measure of clustering.

Stephenson (1977) claims that the sum of (dz)max values is 16 percent of the whole range of z for the 384 quasars. The number of quasars in Table I of Paper I is 40 percent of the total. Actual calculations using the data given in Table I of Paper I give the following results for the three cases enumerated above.

It is obvious that Stephenson has made some arithmetical error. The results given above show that 40 percent of the quasars occupy 5.6 percent of the whole range of z. It is also of interest to apply the chi-square test to the present problem. In Table II, for an average dz = 0.0033, we show the observed and caleulated number of clusters for various values of k. We must note here that for k = 5 and 6, there is just one cluster for each, and the calculated value in each case is very small; thus it is not very meaningful to include such cases in a Chi-square type of test. If we exclude k = 5 and 6, we find that Chi^2=90.2 and the significance level, alpha about 10^-18 . Clearly , the clustering is highly non-random. (Purely as an academic curiosity we may note here that alpha about 10^-97, if k = 5 and 6 are included this is merely to indicate the direction of change in the value of alpha in the latter case.)

The foregoing tests clearly show that at least for k > = 3 the clustering of quasar red shifts (if there be one !) is highly non-random, thereby fully substantiating the paradox presented in Paper I. It is obvious that for the purpose of the arguments leading to the paradox, the exact number of clusters is not important.

In the last paragraph of his paper, Stephenson (1977) proposes an interesting but contrived model of the Universe to accommodate non-random clustering of quasar red shifts within the framework of the cosmological interpretation. However, this model merely replaces one unaesthetic possibility by another. Instead of having Earth at the center, now we have to assume that the Universe evolved in fits and starts of quasar production. The concept of preferred epochs for quasar production is hardly any more aesthetic than that of a preferred position for the Earth. There is no `logical simplicity', or `naturalness' about the proposed suggestion. Merely attributing the non-random clusters to a certain capricious property of the evolution of the Universe at certain arbitrary values of epochs does not explain anything. We are reminded of a well-known quotation due to Newton : `To tell us that every species of things is endowed with an occult specific quality by which it acts and produces manifest effects, is to tell us nothing.'

There is a first part to this paper.

REFERENCES

  1. Stephenson,C.B.: 1977, Astrophys.Space Sci., 51, 117.
  2. Varshni,Y.P.: 1976, Astrophys.Space Sci., 43, 3.
He who finds what he seeks makes, in general, a good school exercise; intent on what he wants, he often neglects the signs, sometimes minimal, which indicate something else than the object of his attention. The real researcher must pay attention to signs which will reveal the existence of an unexpected phenomenon.
- Louis Leprince-Ringuet, Des Atomes et des Hommes
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