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In this form, the distance modulus M has an intuitive interpretation. The original Hubble relation, based on the first order Doppler effect, takes the form
${H}_{0}D=cz$ where D is the proper distance to the galaxy with recessional speed cz. For larger values of redshift, and when acceleration is taken into account, the revised Hubble law takes the form
${H}_{0}D=czk\left(z,{q}_{0},{j}_{0}\right)$ , upon neglect of higher order terms, but including a period of deceleration represented by j_{0}. Hence it follows, upon taking the logarithm of the preceding equation and utilizing (8), that
$M=-\mathrm{log}D$ . However, the following argument does not make use of the value of M, since one is going to subtract from
$\mathrm{log}{H}_{0}$ for an accelerating universe,
$\mathrm{log}{H}_{0}$ for a decelerating universe. To simplify the notation, primes will be used to denote the latter, so that upon subtracting, and suitably rewriting the result, one has

$\mathrm{log}\left(\frac{{H}_{0}}{{{H}^{\prime}}_{0}}\right)=M-{M}^{\prime}+\mathrm{log}\left(\frac{k\left(z,{q}_{0},{j}_{0}\right)}{k\left(z,{{q}^{\prime}}_{0},{j}_{0}\right)}\right)$ (9)

After setting ${q}_{0}=-0.55$ for the accelerating universe as given in [7], and ${q}_{0}=0.5$ for the decelerating EdS universe, equation (9) takes the form

$\mathrm{log}\left(\frac{{H}_{0}\left(-0.55\right)}{{H}_{0}\left(0.5\right)}\right)=M-{M}^{\prime}+\mathrm{log}\left({\frac{1+0.775z-0.274z}{1+0.25z-0.125{z}^{2}}}^{2}\right)$ (10)

One can determine M from (8) as a function of redshift since the value of
${H}_{0}\left(-0.55\right)$ is known, or alternatively from its definition in (6). However, in order to obtain
${M}^{\prime}$ it will be necessary for the astronomers to re-interpret their data under the assumption that the universe is decelerating, so that one can check whether
${H}_{0}\left(0.5\right)$ is equal to the BAO value of H_{0} after allowing for the measurement uncertainties. In the present case, given the lack of such empirical information for
${M}^{\prime}$ , one can proceed as follows: Set the ratio of the two Hubble constants, which will be denoted by
$\eta $ , equal to
${H}_{0}\left(CDL\right)/{H}_{0}\left(BAO\right)$ ; in addition, for simplicity, take their mean values as fiducial, so that one has
$\eta =73.24/67.6=1.083$ . Next, find the behavior of
$M-{M}^{\prime}$ as function of redshift by using (10) with
$\eta =1.083$ . Since it is mentioned in [7]that the primary fit was based on the redshift range,
$0.0233<z<0.15$ , it is of interest to determine
$M-{M}^{\prime}$ for, say, z = 0.03 and z = 0.15. One finds for these two redshifts,
$M-{M}^{\prime}=0.0323$ , and
$M-{M}^{\prime}=0.0084$ , respectively. Thus
$M-{M}^{\prime}$ is approaching zero with increasing redshift, for this range of redshifts. The redshift
${z}^{*}$ for which
$M-{M}^{\prime}=0$ is of special interest, since under these circumstances log
$\eta =k\left(-0.55,{z}^{*},{j}_{0}\right)/k\left(0.5,{z}^{*},{j}_{0}\right)$ . Because of the lack of empirical values for
${M}^{\prime}$ one cannot predict
${H}_{0}\left(0.5\right)$ , so that a reasonable alternative is to predict the values of
${z}^{*}$ for which
$M-{M}^{\prime}=0$ . From (10), when
$M-{M}^{\prime}=0$ , one obtains for
${z}^{*}$ , after removing the logarithms, the quadratic equation

${\left({z}^{*}\right)}^{2}-\frac{0.775-\eta 0.250}{0.274-\eta 0.125}{z}^{*}+\frac{\eta -1}{0.274-\eta 0.125}=0$ (11)

For $\eta =1.083$ , one has the two roots ${z}^{*}=0.17$ , and ${z}^{*}=3.47$ . Because terms of $O\left({z}^{3}\right)$ have been omitted, the higher redshift value is possibly less reliable than the lower value, and in any case this higher value is unacceptable. Since, as indicated in [7], more than 600 SNe Ia were used in a Hubble plot, with redshifts ranging, $0.01<z<0.4$ , it follows that the higher value of z* lies well outside the range of redshifts considered in the CDL work, unlike the lower value. Also, if one takes the fiducial value of $\eta $ to be that for ${H}_{0}\left(CDL\right)/{H}_{0}\left(CMB\right)$ so that $\eta =73.24/67.90=1.079$ , one finds that the two roots are ${z}^{*}=0.16$ , and ${z}^{*}=3.47$ . Hence the smaller root is clearly more sensitive to the value of $\eta $ than the larger root, which, as indicated above, is to be excluded.

Although the above analysis makes it reasonable that the origin of the disagreement about the Hubble constant is due to the fact that the CMB and the BAO values are for a decelerating universe, in contrast with the CDL value, which is clearly for an accelerating universe, the analysis does not provide a rigorous proof that the universe is decelerating, since the above lower values of
${z}^{*}$ , when uncertainties are allowed for, have not yet been confirmed empirically, and consequently there is always the possibility, until it is shown otherwise, that the CMB and BAO values for H_{0} could be brought up to be in agreement with the CDL value. Therefore, other possible proofs that the universe is decelerating, and that the speed of light in IGS is ~2c/3, are required. In the next section, two further predictions of the model that support this proposed reduction of the speed of light in IGS are described. They have been mentioned earlier in [2][4][5], but for completeness, they are briefly presented here, together with some additional comments.

4. Additional Predictions of the Model

A prediction that fully supports the model, although it does not confirm it, was first discussed in [2], and in greater detail in [5]. It is based on attempts to find correlation of neutrinos with gamma ray bursts (GRBs). Significantly, to date, none have been found. The most impressive search, in terms of the number of null events, is that of the Boreximo collaboration, as given in Agostino et al. [20]. They searched from 2007-2015 for neutrinos and anti-neutrinos in the energy range, 1.5 MeV - 17 MeV, correlated with 2350 observations of GRBs, and found no statistically significant excess above background. Since in the model the speed of light through the dark energy of IGS is ~2c/3, and since for these energies neutrinos travel very close to c, if D is a typical distance to the cosmological sources of the GRBs, the time difference for the arrival of the neutrinos and the GRBs would be ~D/2c, after neglecting the time the GRBs spent traveling through the Milky Way with speed c. Since the GRBs come from cosmologically distant sources, D/2c is of the order of millions of years, so that even if there were a sufficient fluence of neutrinos to be detectable, the neutrinos would have arrived millions of years earlier than the GRBs. Hence the model predicts that there cannot be any correlation of neutrinos with GRBs, which is what is observed. This has also been shown to be true with numerous other searches as well [21]- [30]. To be sure, it could be the case that none of the GRB sources produce a sufficient fluence of neutrinos to be detectable above the background, which would therefore give rise to the same null result. However, there is a possible counter-example to this alternative: although Artsen et al. [31]reported that the IceCube collaboration had detected two PeV neutrinos without any correlation with GRBs, it is nevertheless possible that a GRB was produced along with these very high energy neutrinos, but the GRB will not arrive until millions of years in the future. Furthermore, as emphasized in [5], if one assumes that none of the GRB sources produced a measureable fluence of neutrinos, that amounts to thousands of assumptions: one assumption for each case of detected GRB unaccompanied by neutrinos. Hence, according to Occam’s Razor, this large number of assumptions makes this explanation less preferable than that of the proposed model that uses only one assumption about the reduced speed of light through IGS.

Although the above absence of correlations supports the model, as well as the proposed resolution of the disagreement over the Hubble constant, neither fully confirms it. On the other hand, the following tests based on alternative sources for discordant redshift galaxies would confirm the model, and can be carried out quite readily. It was shown by Bahcall [32]that of the 64 cases of discordant redshifts, compiled and edited by Arp [33], ~40 of them could be explained as due to accidental superposition, i.e., the higher redshift galaxy (HRG) being behind the lower redshift galaxy (LRG), at a cosmological distance, and shining through. He concluded that these ~40 cases were sufficiently close in number to the 64 observed cases that all the cases could be explained as due to accidental superposition. However, upon re-examining Bahcall’s estimate, I found that he had rounded-up two of the numbers he had used in the product that led to the ~40 cases, and that when I did not round up these numbers, the product yielded only 30 cases that could be explained as due to accidental superposition. This suggested that about half the remaining cases are not due to accidental superposition, but are due to something else. It turns out the proposed model can explain the remaining cases, because it predicts that the dark energy of IGS has an index of refraction of ~1.5, while galaxies have an index of refraction of unity. Consequently, a light ray from an HRG that is not behind the LRG, but off to the side, angle-wise, upon being incident upon a LRG, at a suitable angle relative to the local normal, could be refracted in such way that it would pass through the LRG, and after being further refracted upon exiting the LRG, it could travel along a path that would bring it to the astronomer who would see the HRG as a discordant redshift galaxy, while at the same time by suitably redirecting the telescope, depending on the location of the discordant redshift image, the astronomer would see the HRG directly. (Please see Figure 1 in [4].) Since galaxies do not have sharp edges, a light ray will encounter a variable behavior of n upon entering and leaving the LRG, as well as entering the Milky Way, so that the above scenario is highly simplified, but the qualitative picture should be valid. Hence it is a prediction of the model that the proposed HRGs will be found as a supplementary source of discordant redshifts. However, if they should not be found, this would not necessarily falsify the model, since it could turn out that the numbers Bahcall proposed did not accurately describe the situation. On the other hand, the model would definitely be falsified if it were found that an HRG was suitably located off to the side of a LRG in such a way that it should have given rise to a discordant redshift but did not, since that would mean the ray from the HRG had most likely gone through the LRG with negligible refraction, and would thereby demonstrate that IGS does not have an observably higher index of refraction than the LRG, in conflict with the model.

5. Conclusion

The above work has shown that one can resolve the current disagreement about the Hubble constant by assuming that the CMB and BAO determinations of H_{0} are not for the accelerating ΛCDM universe, but for a decelerating EdS universe, in which the density parameter for the dark energy in the EdS universe satisfies the relation
${\Omega}_{de}={\Omega}_{\Lambda}$ . However, unlike the cosmological term, the dark energy in this model does not have a negative pressure associated with it, but instead it has an index of refraction n. As a test of this proposal, it was shown that the CDL’s higher value for H_{0} could be lowered to agree with the CMB and BAO determinations by re-evaluating the CDL value of H_{0} for a decelerating universe, rather than for an accelerating universe. In order to obtain the additional distance in the EdS decelerating universe that is needed to explain the increased apparent magnitude of the SNe Ia that led astronomers [9][10][11]to infer that the universe is accelerating, it is necessary to assume the speed of light through the dark energy of IGS is reduced to c/n, where n ≈ 1.5. This assumption gives rise to a challenging problem: How can such a very low density substance as the dark energy have an index of refraction comparable to some types of glass, and moreover do it without introducing any dispersion? Furthermore, since it has been shown recently from a binary neutron star merger that the resulting gravitational waves (GWs) Abbott et al. [34][35]arrive at essentially the same time (~1.7s) as the GRBs Goldstein et al. [36]and Savchenko et al. [37], this index of refraction for electromagnetic radiation would have to hold for gravitational waves as well! Consistent with this, the Borexino collaboration, as reported in d’Agostino et al. [38], did not find any neutrinos above background correlated with the arrival of GWs. The only explanation that seems to embrace all these findings, as remarked in [5], is that one is possibly encountering consequences of that long-sought unified theory of electromagnetism and gravitation that might predict how dark energy could influence the propagation of EM radiation and GWs in this way. However, before one can conclude that dark energy does indeed have such extraordinary properties, it is essential to ascertain whether the above predictions concerning the Hubble constant and discordant redshifts actually hold. Hopefully this work, as well as those that preceded it, will encourage astronomers to undertake such investigations.

Conflicts of Interest

The authors declare no conflicts of interest.

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