_{1}

^{*}

This work revises and extends the author’s previous work (2015), Journal of Modern Physics, 6, 78- 87, by proposing that the index of refraction n of intergalactic space (IGS) is of electromagnetic origin. This leads to a theoretical expression for n that agrees very well with the least squares value obtained previously. A table comparing the fractional distance increase predicted by the two differently obtained indices is given. This revised view requires that the high energy charged particles found in cosmic rays originate from high energy neutral particles, presumably high energy gamma rays, that were able to travel through the IGS without energy loss due to Cherenkov radiation. An alternative explanation for the counter indication from the IceCube findings of Abassi, R., et al. (2012) Nature, 484. 351-353 is proposed, which might also explain the findings of Aartsen et al. (2013) Physical Review Letters, 111, 021103. Since the model predicts galaxies act as divergent lenses, a geometrical analysis and corresponding figure describing this effect is given, as well as a table for a range of angles to the image galaxy relative to the direction to a target galaxy that is divergently lensed. The reduction of the speed of light in the IGS leads to a revision of the Planck (2015) value of the Hubble constant of ~68 km·s
^{-1}·Mpc
^{-1} to ~47 km·s
^{-1}·Mpc
^{-1}, and hence an age for the Einstein-de Sitter universe greater than that of the oldest white dwarfs in the Galaxy, thereby resolving a long-standing problem with this model of the universe.

In a previous work [^{9} yr. In Section 6 there are concluding remarks.

As remarked above, it was assumed in I that the index of refraction n of the dark energy was not of electromagnetic origin so as to eliminate the possibility of energy loss by Cherenkov radiation by high energy charged particles traveling through the IGS, which would tend to negate their appearance in cosmic rays, in conflict with observation. However, in attempting to make a model in which n was not of electromagnetic origin, the following theoretical objection was found. Although, as is well-known, a free charged particle cannot emit a photon, since it would violate the conservation of energy and momentum, in contrast, as discussed in a quantum theory approach to the emission of a photon in the Cherenkov effect by Cox [

While alternative 1) cannot be ruled out at present, it turns out alternative 2) is to some extent supported by current research in cosmic rays, see, e.g., Milgrom and Usov [

Nevertheless, this conclusion has to be re-examined in view of the fact that although under any reasonable form of the interaction of light with the dark energy, one would expect the dispersion at the very high frequencies of the gamma rays would be such as to bring the index n down to a value very close to unity, one has to take into account that since one is dealing with an interaction for which one has no analytical knowledge, n could be so sufficiently greater than unity that neutrinos associated with the gamma rays could have arrived much earlier than the gamma rays themselves. In other words, the neutrinos could have arrived before IceCube was built! In addition, one could imagine detecting neutrinos now, and the associated gamma-ray bursts being detected some unknown number of years in the future. As a possible example, Aartsen et al. [

Now, if one assumes that n is indeed of electromagnetic origin, and one further assumes that the dark energy is a medium that is linear, isotropic, and non-dispersive (in the optical frequencies, as was found by the SNeIa investigators) so that

(It should be noted that, to avoid confusion, the subscript “μ” is used above rather than the customary “m”, since the latter subscript is used here to refer to matter rather than to magnetism.) Since it was found in I by a least squares analysis that n = 1.49, if one is going to employ (1) to obtain this number, it follows that one needs some dimensionless physical quantity related to the dark energy described by

This quantity should be greater than unity, and its square root should yield the above value for n. It was found by trial and error that setting,

yields a reasonably close fit to the desired value. Thus in I, the fiducial values were

so that

and for

As a means of attempting to determine whether the above fairly good agreement between the least squares determination of n and the square root determination is a coincidence, in this section n will be obtained in both ways for another set of values for

This value on n is to be compared with the least squares fit to the LCDM model that, with these new values of

Although these two values are within 0.7% of each other, since it is actually n − 1 that is involved in the fractional distance increase, the percentage disagreement is larger, ~2%, but it is still quite small. Also, as noted above in Section 2, the fit to the data for the LCDM model as given in [

and, for the above values of

This expression, as noted in I, has to be integrated numerically. For the Einstein-de Sitter universe, with

Also, as in I, one has that

The comparison is given in

z | |||||||
---|---|---|---|---|---|---|---|

0.1 | 0.0205 | 0.0190 | −0.0015 | −7.3 | 0.0186 | −0.0019 | −9.3 |

0.2 | 0.0379 | 0.0357 | −0.0022 | −5.8 | 0.0350 | −0.0029 | −7.7 |

0.3 | 0.0528 | 0.0505 | −0.0023 | −4.4 | 0.0495 | −0.0033 | −6.3 |

0.4 | 0.0657 | 0.0638 | −0.0019 | −2.9 | 0.0625 | −0.0032 | −4.9 |

0.5 | 0.0768 | 0.0758 | −0.0010 | −1.3 | 0.0743 | −0.0025 | −3.3 |

0.6 | 0.0858 | 0.0867 | 0.0009 | 1.0 | 0.0850 | −0.0008 | −0.9 |

0.7 | 0.0948 | 0.0966 | 0.0019 | 2.0 | 0.0949 | 0.0001 | 0.1 |

0.8 | 0.1022 | 0.1059 | 0.0037 | 3.6 | 0.1039 | 0.0017 | 1.7 |

0.9 | 0.1087 | 0.1145 | 0.0058 | 5.3 | 0.1124 | 0.0037 | 3.4 |

1.0 | 0.1145 | 0.1225 | 0.0080 | 6.9 | 0.1202 | 0.0057 | 5.0 |

that it is probable that the interaction between light and the dark energy is of an electromagnetic nature, even though at this stage one cannot give more details, such as to whether it is only the dielectric constant K that is involved, or whether it is only the relative permeability

As one goes to sufficiently higher redshifts, as discussed in I, one eventually reaches a value of z for which the thermodynamic phase change that according to the model transformed the dark matter into dark energy is no longer present, and as a consequence, the dark energy that resides in the IGS has been transformed back into dark matter. Under these circumstances, n = 1, and hence it follows that

where, _{m} includes baryonic matter, the resulting density of dark matter in the IGS should be greater than that associated with the galaxies in order that

Finally, as noted in Section 2, due to Cherenkov radiation loss, charged particles traveling through the IGS will be slowed down to speeds

As was pointed out near the end of Section 3 in I, light traveling through the galactic halos would transition from traveling at a lower speed in the IGS to higher speeds in the halos, and would finally achieve speed c in the bulk of the galaxies, and hence galaxies would act as diverging lenses. In

1 | 1.1 | 1.2 | 1.3 | 1.4 | 1.5 | 1.6 | |
---|---|---|---|---|---|---|---|

2.5 | 0.15 | 0.05 | - | - | - | - | - |

2.6 | 0.20 | 0.10 | - | - | - | - | - |

2.7 | 0.24 | 0.14 | 0.04 | - | - | - | - |

2.8 | 0.29 | 0.19 | 0.09 | - | - | - | - |

2.9 | 0.34 | 0.24 | 0.14 | 0.04 | - | - | - |

3.0 | 0.38 | 0.28 | 0.18 | 0.08 | - | - | - |

3.1 | 0.43 | 0.33 | 0.23 | 0.13 | 0.03 | - | - |

3.2 | 0.47 | 0.37 | 0.27 | 0.17 | 0.07 | - | - |

3.3 | 0.52 | 0.42 | 0.32 | 0.22 | 0.12 | 0.02 | - |

3.4 | 0.57 | 0.47 | 0.37 | 0.27 | 0.17 | 0.07 | - |

3.5 | 0.61 | 0.51 | 0.41 | 0.31 | 0.21 | 0.11 | 0.01 |

3.6 | 0.66 | 0.56 | 0.46 | 0.36 | 0.26 | 0.16 | 0.06 |

that depends upon n and

In

The main difficulty in making such observations would be in obtaining target galaxies located relative to foreground galaxies at angles such that the image galaxy does not lie along a line of sight that passes through the central part of the foreground galaxy, thereby making the image unobservable. Other difficulties include: the fact that the shape of the halo could depart significantly from the assumed spherical shape; the recognition that one indeed has the divergent lens image of the target galaxy, and not some similar-looking galaxy; the possibility that a galaxy somewhat behind the foreground galaxy that is being gravitationally lensed has its image taken to be the target galaxy, and the galaxy that was gravitationally lensed is taken to be the divergently lensed image; finally, there may be cases where it has been assumed that one has an example of gravitational lensing, whereas instead, the lensed galaxy is in fact the image galaxy of a galaxy that has been divergently lensed.

One of the major objections to the present model, which takes as its trial basis the Einstein-de Sitter universe, is that it gives rise to an age of the universe that is smaller than the maximum age of the white dwarfs in the globular clusters in the Galaxy. For reviews, see, e.g., Krauss and Chaboyer [

where ^{9} yr, based on ^{9} yr, a significant disagreement. In contrast, for the accelerating universe, since it is now expanding faster than the above^{9} yr [

A resolution of this conflict for the presently proposed model has been found that is based on a re-examina- tion of the methodology used in obtaining a value for the Hubble constant. Thus, whereas the determination of the ages of the white dwarfs rely solely on measurements made within the Galaxy, the determination of the Hubble constant relies on measurements that make use of light coming from distant galaxies, and hence light coming from outside the Galaxy. Consequently, such measurements necessarily are based on light that has gone through the IGS where, unlike the situation within the Galaxy, in which the speed of light is c, the speed of light is c/n, and as will be shown next, this modifies the value of the Hubble constant, as presently determined, to yield an age for the Einstein-de Sitter universe larger than the ages of the white dwarfs, thereby eliminating the above conflict.

Now the way the Hubble relation,

and hence_{0} is [T]^{−1}, the way one obtains a time is from the speed of light. Since in the present model, light traveling through the IGS does not travel with speed c, but with speed c/n, the above Doppler expression should be replaced by

where _{0} by

The corresponding age of the Einstein-de Sitter universe, instead of (13), now becomes

which is in very good agreement with the above age of the universe obtained by the accelerating model, and hence the conflict of the Einstein-de Sitter model with the maximum age of the white dwarfs is eliminated. Interestingly, the white dwarfs’ estimated maximum age can be used to set a lower bound on n using (17). Clearly, one must have that the age of the universe is greater than ~13 ´ 10^{9} yr, and hence n > ~1.35, which is obviously well satisfied. It should be pointed out that a value close to the Planck value [^{−}^{1}・Mpc^{−1}, hence, rounding off to 70 km・s^{−1}・Mpc^{−1}, one obtains ^{9} yr, which is still sufficient to eliminate the conflict with the maximum age of the white dwarfs.

One obtains the same result as (16) by using, instead of the Doppler expression, the modified FLRW line element, ^{2} = 0, expanding a(t) as

Since_{c} cancels, and (2) reduces to (10), which only involves ρ_{de}/ρ_{m}, and hence n determined in this way is independent of the Hubble constant. Likewise, in determining n by the least squares method, that corrects with the aid of the factor, _{Λ} given in (7) to X_{m} given in (8) is involved, in which, again, the density ρ_{c} cancels. To show this, one has that

_{Λ} denotes the integral in (7), and since the ratio Ω_{Λ}/Ω_{m} is independent of

ρ_{c}, only the factor _{m}, the integral f_{m} in (8) is independent of ρ_{c}, and although

the factor _{m} has the same value as ρ_{c}. Consequently, upon taking the ratio X_{Λ}/X_{m}, once again ρ_{c} cancels, and hence since_{c}, for _{c}, and hence it is independent of the Hubble constant.

Finally, although the proposed new value of the Hubble constant has been written as_{0}, that is based on the speed of light in the IGS as being c. If the proposed reduction of the speed of light through the IGS proves correct, the asterisk superscript could be eliminated, and the present value of _{0}.

The present work revises the view expressed in I that the interaction of light with the dark energy was not of an electromagnetic nature. As was shown above, the assumption of an electromagnetic interaction leads to a value of n in good agreement with that obtained from a least squares fit to the accelerating model for the fiducial values of density parameters used in I, and to an excellent agreement with that model when more recent values are used. Further, as was pointed out, the assumption that the interaction is electromagnetic means that one has to assume that the high energy charged particles found in cosmic rays could not have reached the Galaxy through the IGS, because of the energy they would have lost due to Cherenkov radiation. Instead, the model supports the widely discussed view that the high energy charged particles seen in cosmic rays are produced through collisions with the baryonic matter in the Galaxy by incoming high energy gamma rays that presumably originate from gamma-ray bursters, or collapsars, and/or some other high energy gamma-ray sources. The absence of predicted accompanying neutrinos from the gamma ray bursts [

Turning next to astronomical tests of the proposed model: as discussed in Section 4, according to the model, there should be divergent lensing of more distant galaxies by foreground galaxies. If one fails to find any examples of divergent lensing, this would rule out this proposed alternative to the accelerating universe. Also, as discussed in Section 5, the ages of the oldest white dwarfs in the Galaxy can be accommodated by the present model, because it predicts a smaller value for the Hubble constant than its present value.

Significant theoretical challenges remain. One has to show that despite the extremely low density of the dark energy, it is possible for it to give rise to such a large index of refraction, and to determine through dispersion relations, its frequency dependence,_{tot} from the Cosmic Microwave Background Radiation, neutrinos, and possible spatial curvature have been omitted; these too will have to be included in future more detailed models.

I am thankful to Professor Adam Riess for a helpful critical comment concerning the previous work. I would also like to thank Professor Thomas Murphy for another helpful critical comment as well.

Frank R.Tangherlini, (2015) A Possible Alternative to the Accelerating Universe II. Journal of Modern Physics,06,1360-1370. doi: 10.4236/jmp.2015.69141