_{1}

^{*}

This work continues the previous study (2018) Journal of Modern Physics. 9, 1827-1837, that proposes that the disagreement arises because the cosmic microwave background (CMB) value for the Hubble constant
*H*
_{0} is actually for a universe which is decelerating rather than accelerating. It is shown that when
*H*
_{0} of Freedman et al. (2019) Astrophysical Journal, 882: 34 (24 pp.) is re-determined for redshift
*z *= 0.07, by replacing
*q*
_{0 }=
−0.53 with
*q*
_{0 }
= −0.5, the new lower value is in excellent agreement (0.1%) with the CMB
*H*
_{0}. The model is modified to include the clustering of galaxies, and the recognition that there are clusters that do not experience the Hubble expansion, such as the Local Group, and hence, in accordance with the model, within the Local Group the speed of light is
*c*. The bearing of this result on the neutrino and light time delay from SN1987a is discussed. It is suggested that the possible emission of a neutrino from the blazar TXS-0506+56, that was flaring at the time, as well as possible neutrino emission earlier, may arise instead from a more distant source that happens to be, angle-wise, near the blazar, and hence the correlation is accidental. The model is further modified to allow for a variable index of refraction, and a comparison with the ΛCDM model is given. The age of the universe for different values of
* H*
_{0} is studied, and comparison with the ages of the oldest stars in the Milky Way is discussed. Also, gravitational wave determination of
*H*
_{0} is briefly discussed.

In the previous work [_{0}, it was proposed that the cosmic microwave background (CMB) [_{0}, although based on the flat ΛCDM model of an accelerating universe, with the dark energy possessing a negative pressure supplied by the cosmological term [_{0}, and hence they are distance determinations of H_{0}, as was shown explicitly in [_{0}. To resolve this ambiguity, it was noted that the SHoES local distance ladder (LDL) determination (Note: Formerly described as cosmic distance ladder (CDL) determination.) of H_{0} of Riess et al. [_{0}, with q 0 = − 0.55 , and hence this determination unambiguously describes an accelerating universe. It was further noted that the empirical inequality H 0 ( LDL ) > H 0 ( CMB ) can be readily understood if the Hubble constant H 0 ( CMB ) is actually for a decelerating universe, rather than an accelerating universe, as is customarily assumed. This follows from the fact that since H 0 = a ˙ 0 / a 0 , at the present epoch, where a ( t ) is the expansion parameter for the Friedmann, LeMaître, Robertson, Walker (FLRW) line-element, and since a 0 ( LDL ) = a 0 ( CMB ) , because the diminished brightness of the Type Ia supernovae (SNe Ia) has to be the same for both models, it follows a ˙ 0 ( LDL ) > a ˙ 0 ( CMB ) . This latter inequality is what one expects if H 0 ( CMB ) is for a decelerating universe, assuming that for a sufficiently large redshift, a ˙ ( t ) was essentially the same for both models. Then, because of the acceleration of the one model, and the deceleration of the other, at the present epoch, a ˙ 0 will be greater for the accelerating model than for the decelerating model, and hence the above inequality. Thus, in effect, the decelerating model predicts the sign of the inequality between the two competing values of the Hubble constant. It was also shown in [_{0} would agree with H 0 ( CMB ) . This suggestion was further described in a contributed talk [_{0} had been made by the Carnegie-Chicago collaboration of Friedman et al. [_{0}, corresponding to a greater distance than SHoES for a given redshift. This finding can be taken as support for the view expressed above that the redshift distances are greater for the proposed model than for SHoES; this will be discussed in greater detail in Section 2. In Section 3, the clustering of galaxies, that was not taken up previously, will be taken up, and its importance for the model described. This discussion will include a less ad hoc explanation for the relatively short arrival time difference between photons and neutrinos from SN1987a than what was given in [_{0} associated with this model will be examined, and compared with the age of the oldest stars in the Milky Way. In Section 6, there are concluding remarks.

As was pointed out in the Introduction, if the value found for H 0 ( CMB ) is interpreted as being for a decelerating universe, rather than for an accelerating universe, the proposed model predicts the sign of the inequality that has been found from observation to be H 0 ( LDL ) > H 0 ( CMB ) . In order to clarify this interpretation further, assume: 1) that the decelerating model gives the correct distances as a function of redshift, as will be justified below from the excellent agreement that is obtained with H 0 ( CMB ) , and 2) that the distances from the ΛCDM model provide a good approximation to the distances from the decelerating model that results in the ambiguity described above in the Introduction. Then, in obtaining the value of H_{0} from the distances associated with the ΛCDM model, it would be quite reasonable to assume, as the Planck collaboration does, that one is determining the Hubble constant for an accelerating universe. This would also be the case for the BAO determinations. Importantly, it is possible to verify the above re-interpretation of H 0 ( CMB ) by re-determining the value for H 0 ( LDL ) by replacing the negative deceleration parameter currently used, q 0 = − 0.55 , with the positive deceleration parameter, q 0 = 0.50 , appropriate to the Einstein de Sitter (EdS) decelerating universe, to see whether the subsequently revised value of the Hubble constant comes down reasonably close to the lower CMB value. In Equation (10) in [_{0} could be written as

log ( H 0 ( − 0.55 ) H 0 ( 0.50 ) ) = M − M ′ + log ( 1 + 0.775 z − 0.274 z 2 1 + 0.25 z − 0.125 z 2 ) , (1)

where M and M ′ are the redshift-dependent distance moduli for the accelerating and decelerating universes, respectively, and the next term, in which z is the redshift, involves the kinematic corrections to the lowest order Hubble-LeMaître relation

k ( z , q 0 , j 0 ) = { 1 + 1 2 [ 1 − q 0 ] z − 1 6 [ 1 − q 0 − 3 q 0 2 + j 0 ] z 2 } , (2)

where j_{0} is the jerk given by a ⃛ 0 a 0 2 / a ˙ 0 3 [_{0} is also unity, independently of epoch. If, as was done in [

H 0 ( 0.50 ) = H 0 ( − 0.55 ) D L D ′ L ( 1 + 0.25 z − 0.125 z 2 1 + 0.775 z − 0.274 z 2 ) . (3)

Since there is going to be a comparison with the work in [

H 0 = ( D L / D ′ L ) 71.51 ± 1.37 km ⋅ s − 1 ⋅ Mpc − 1 . (4)

One sees that the SHoES value of H 0 ( LDL ) does come down towards the 2018 Planck CMB value of H 0 = 67.36 ± 0.54 km ⋅ s − 1 ⋅ Mpc − 1 [_{0} used in [^{2} term. However, if one omits it in obtaining (4), but keeps the other terms the same, one finds the ratio of the kinematic factors is still 0.966, so the omission of the z^{2} term in the kinematic factor is not significant for this low value of redshift. Next, without changing the luminosity distances found in [

H 0 ( 0.50 ) = H 0 ( − 0.53 ) ( 1 + 0.25 z 1 + 0.765 z ) . (5)

Then, with H 0 ( − 0.53 ) = 69.8 ± 2.0 km ⋅ s − 1 ⋅ Mpc − 1 from [^{−1}∙Mpc^{−1}, and with z = 0.07, one has

H 0 ( 0.50 ) = 67.43 ± 1.93 km ⋅ s − 1 ⋅ Mpc − 1 ， (6)

in excellent agreement (0.1%) with the above CMB value [_{0} be recalculated for the entire range or redshifts they used, upon replacing q 0 = − 0.53 , with q 0 = 0.50 , in order to compare with the CMB value of H_{0}; but at this writing, this has not been done. The value in (6) is also in excellent agreement with the latest BAO results, as given in Addison et al. [_{0}, with the fractional weights w_{i} given by σ i − 2 / ∑ i σ i − 2 , i = 1, 2, 3, one obtains

〈 H 0 〉 = ∑ i w i H 0 ( i ) = 67.57 ± 0.52 km ⋅ s − 1 ⋅ Mpc − 1 . (7)

Hence the agreement of 〈 H 0 〉 with (6), apart from the uncertainties, is again excellent (0.3%). (Note: The uncertainties for H 0 = 67.78 − 0.87 + 0.91 have been rounded to H 0 = 67.78 ± 0.90 ). As remarked earlier, the CMB and BAO determinations are based on distance determinations that do not explicitly involve the deceleration parameter, unlike the LDL determinations that do, and hence they can be alternatively attributed to a decelerating model that gets the same distance as the ΛCDM model to within experimental error. However, this simple observation seems to fail, in view of the recent HOLiCOW XIII result of Wong et al. [

Finally, it should be mentioned that from the gravitational wave (GW) standard siren measurement of the distance of the binary neutron star merger [_{0}, they used v H = H 0 D , where v H is the Hubble flow velocity, D is the distance to the source, and v H = c z . Since for this case z ≈ 0.01 , they could ignore higher order corrections involving k ( z , q 0 , j 0 ) , that would have contributed of the order of one percent. More recently, a less uncertain determination of H_{0} has been made by Hotokezaka et al. [_{0} are too uncertain to decide between the CMB and BAO lower values, and the LDL higher values of H_{0}, although, as H.-Y. Chan et al. [_{0} to a precision of approximately two percent in five years, and approximately one percent within ten years, and this should enable a judgment between the two discordant values to be made. According to the decelerating model, since one is dealing with a distance measurement, the result should agree with the CMB and BAO values, or the model would be wrong. However, it should be noted that the low velocity relation v H = H 0 D , that they used to determine H_{0} depended on the fact that the redshift of the source galaxy was so small that the higher order corrections involving k ( z , q 0 , j 0 ) could be neglected. But if some of the future GWs are from binary neutron star mergers that are at higher redshifts, say, z ≈ 0.07 , so that it would be necessary to include the next order correction, k ( z , q 0 ) = ( 1 + 1 2 ( 1 − q 0 ) z ) , then the determination of H_{0} would have to include q_{0}, and if they set, q 0 = − 0.53 , in accordance with the current accelerating universe paradigm, the result should agree with the value of H_{0} found in [_{0}, but in determining whether a major paradigm shift from the current accelerating universe to a decelerating universe is needed. This view is in contrast with alternative explanations for the Hubble constant disagreement, such as in [

In the model that has been developed up until now galactic clusters were neglected for simplicity. An additional purpose of this work is to revise the model by taking this clustering into account, and to examine some of the resulting consequences. One such consequence will be that it explains why the difference of arrival times between neutrinos and photons from SN1987a [^{5} lyr from the MW, since it is well within the Local Group, is not within a sea of dark energy, but only within the tight cluster’s dark matter, for which n = 1 . Therefore, the photons that came to the earth from the explosion associated with SN1987a traveled with speed c, and hence the enormous time delay did not occur, that would have occurred if they had traveled through dark energy from the LMC to the MW at a speed of approximately 2c/3. Although this was the explanation for the absence of a lengthy time delay that was given previously in [

As was first noted in [

In the previous work it was assumed that for the redshift range 0 ≤ z ≤ 0.6 , the index of refraction of the dark energy n is a constant, and it was found that a least squares fit to the ΛCDM model for various values of Ω_{m} yielded the following range of values for n given by 1.47 ≤ n ≤ 1.54 . For the values 1.47 ≤ n ≤ 1.50 , the values of the distances predicted by the model for the redshift range 0.1 ≤ z ≤ 0.5 were less than that for the ΛCDM model, whereas, as discussed in Section 2, the model requires that the distances should be greater than that for the ΛCDM model, when the latter is combined with the LDL determinations of H_{0}. On the other hand, for n = 1.54 , as shown in

The above considerations lead to the conclusion that the revised model should introduce a variable index of refraction n ( z ) for 0 ≤ z ≤ 1.7 , and a constant index n ( z ) = 1 , for z ≥ 1.7 . The behavior of the function n ( z ) is determined by the conditions: n ( 1.7 ) = 1 , and n ( 0.5 ) should have the value required to fit the ΛCDM model at the redshift z = 0.5 , for the present density values Ω m = 0.315 , Ω Λ = 0.685 . To arrive at a possible expression for n ( z ) , it is convenient to review the original argument in [

( n − 1 ) a ˙ d t / a = ( n − 1 ) d ln a . (8)

This expression is to be integrated from the time of emission of the light to the present, in order to obtain the total fractional amount of extra expansion. However since it is redshift rather than time that is measured, it is appropriate to convert this relation to one that is a function or redshift using a = a 0 / ( 1 + z ) , which holds for arbitrary FLRW expanding universes. Under this assumption, and the assumption that n is now a function of redshift, n = n ( z ) , the integral of (8) takes the form

∫ 0 z ( n ( z ′ ) − 1 ) d ln ( 1 + z ′ ) . (9)

Now in [

n ( z ) = n 0 − b z , 0 ≤ z ≤ 1.7 , n ( z ) = 1 , z ≥ 1.7. (10)

Under this assumption, (9) takes the form

∫ 0 z ( n 0 − b z ′ − 1 ) d ln ( 1 + z ′ ) . (11)

The value of this integral is

( n 0 − 1 + b ) ln ( 1 + z ) − b z , (12)

so that the original luminosity distance D L is increased by the factor ( 1 + ( n 0 − 1 + b ) ln ( 1 + z ) − b z ) , and hence the resulting luminosity distance D ′ L is given by

D ′ L = ( 1 + ( n 0 − 1 + b ) ln ( 1 + z ) − b z ) D L . (13)

Then, following the argument on page 82 of [

δ m = 5 log ( 1 + ( n 0 − 1 + b ) ln ( 1 + z ) − b z ) ， (14)

d = log ( 1 + ( n 0 − 1 + b ) ln ( 1 + z ) − b z ) . (15)

Note that in the limit b → 0 , the above expressions reduce to the previous ones for a constant index of refraction, upon setting n 0 = n , and note also that δ m = 5 d . The parameters n 0 and b are next determined from the requirements described above. Thus from (10), one has

n 0 − b 1.7 = 1 ， (16)

and at z = 0.5 , one has

log ( Χ Λ ( 0.5 ) / Χ m ( 0.5 ) ) = d ( 0.5 ) = log ( 1 + ( n 0 − 1 + b ) ln ( 1.5 ) − b ( 0.5 ) ) ， (17)

where Χ Λ , Χ m are defined and worked out in Equations (17), (18) in [

Χ Λ = ( 0.315 ) − 1 / 2 ∫ 0 0.5 ( ( 1 + z ) 3 + 2.1746 ) − 1 / 2 d z = 0.4387 ， (18)

where the integral has been evaluated numerically, while by direct integration one has

Χ m = ∫ 0 0.5 ( 1 + z ) − 3 / 2 d z = 0.3670 ， (19)

hence log ( Χ Λ ( 0.5 ) / Χ m ( 0.5 ) ) = 0.0775 = d ( 0.5 ) . Since from (16) n 0 − 1 = b 1.7 , upon substitution in (17), the following relation for b results

log ( 1 + 2.7 b ln ( 1.5 ) − 0.5 b ) = 0.0775 ， (20)

from which one obtains, with the aid of (16), the following values for the two parameters

b = 0.3285 , n 0 = 1.558. (21)

Obviously, the above values are only approximate because of the uncertainties in Ω m and the value of redshift for which n = 1 . It will be noted that for z = 0.5 , from (10) and (21) one has that n ( 0.5 ) = 1.394 . This is significantly lower than the values for constant n that was found earlier, depending on the values of Ω m , to be in the range 1.47 - 1.5. It is interesting to compare the values of log ( Χ Λ / Χ m ) with the values of d, as was done in earlier papers, to see how well this model fits the ΛCDM model. This is done in

z | log ( Χ Λ / Χ m ) | d | Δ | Δ / R Λ % |
---|---|---|---|---|

0.005 | 0.00110 | 0.00121 | 0.00011 | 10.0 |

0.02 | 0.00433 | 0.00470 | 0.00037 | 8.5 |

0.04 | 0.00864 | 0.00930 | 0.00066 | 7.7 |

0.06 | 0.01273 | 0.01367 | 0.00094 | 7.4 |

0.08 | 0.01675 | 0.01786 | 0.00111 | 6.6 |

0.10 | 0.02061 | 0.02189 | 0.00128 | 6.2 |

0.30 | 0.05330 | 0.05467 | 0.00137 | 2.6 |

0.50 | 0.07752 | 0.07752 | 0 | 0 |

0.70 | 0.09580 | 0.09367 | −0.00213 | -2.2 |

0.90 | 0.10989 | 0.10505 | −0.00484 | -4.4 |

1.10 | 0.12095 | 0.11284 | −0.00811 | -6.7 |

1.30 | 0.12979 | 0.11783 | −0.01196 | -9.2 |

1.50 | 0.13698 | 0.12056 | −0.01642 | -12.0 |

1.70 | 0.14292 | 0.12140 | −0.02152 | -15.1 |

1.90 | 0.14790 | 0.12140 | −0.02650 | -17.9 |

2.10 | 0.15212 | 0.12140 | −0.03072 | -20.2 |

integral (11) from z = 1.7 to z = 2.1 vanishes.

In concluding this section, it should be emphasized that the above linear model for the index of refraction is only a preliminary attempt to go beyond the constant index of refraction of the previous publications. It is possible a deeper understanding of dark matter and its proposed phase transition to dark energy will emerge from the comparison of the above predictions with that of the ΛCDM model, as well as with other accelerating models, that will lead to an improved model for the behavior of the index of refraction.

The age of the universe, i.e. the time back to the Big Bang where the expansion parameter a(t) was arbitrarily small, was first taken up in [

The recent Planck CMB rounded value of the Hubble constant is H 0 = 67.4 ± 0.5 km ⋅ s − 1 ⋅ Mpc − 1 [

G 0 0 = − κ T 0 0 − Λ ， (22)

where κ ≡ 8 π G / c 4 . For T 0 0 = ρ c 2 , where ρ is the matter density consisting of the dark matter mass density and the baryonic mass density, while κ − 1 Λ is the density of dark energy in the ΛCDM model. This is the simplified two component energy density source model for ΛCDM that has been used throughout this work. Upon introducing the FLRW line element for a flat universe (analysis for a slightly closed universe is in [

d s 2 = c 2 d t 2 − a ( t ) 2 δ i j d x i d x j , i , j = 1 , 2 , 3. (23)

After the metric is substituted in G 0 0 , and suitably rewritten,(22) takes the form

a ˙ 2 2 − 4 π G ρ a 2 3 − Λ a 2 c 2 6 = 0. (24)

Since the matter source tensor T ν μ is that for cold dark matter T ν μ = d i a g ( T 0 0 , 0 , 0 , 0 ) , there is no pressure. It follows from the covariant conservation law T ν μ ; μ = 0 that ρ a 3 is a constant of the motion. Hence, upon introducing the mass M ≡ 4 π ρ a 3 / 3 , (24) may also be written as

a ˙ 2 2 − G M a − Λ a 2 c 2 6 = 0. (25)

In this form, one recognizes (25) as the Newtonian equation for a test particle moving radially outside a fixed spherical body of mass M with kinetic energy per unit mass T = a ˙ 2 / 2 , and potential energy per unit mass V = − ( G M / a ) − Λ a 2 c 2 / 6 , so that the total energy per unit mass is T + V = 0 . The fact that the mass of the test body cancels out completely from the equation can be seen as a manifestation of the Newtonian principle of equivalence, inertial mass equals gravitational mass, which is not only obeyed by the standard gravitational potential energy term, in accordance with Newtonian mechanics, but by the cosmological potential energy per unit mass term as well. One sometimes refers to the cancelled mass that would multiply M as the “passive” gravitational mass, while M, since it comes from the source tensor T ν μ , is described as the “active” gravitational mass. However, since the source energy-momentum tensor exists in special relativity where there is no gravitation, M is necessarily inertial mass. But since it also has been shown that inertial mass is passive gravitational mass, it follows that all three masses are the same in general relativity, as they are in Newtonian mechanics. Thus, at this level, general relativity has a deep connection with Newtonian mechanics, so that if general relativity were to break down, as has been occasionally proposed as an alternative explanation for the diminished brightness of the SNe Ia, Newtonian mechanics would also have to break down as well, at its most basic level, something that is frequently not discussed in such proposals.

Returning now to the derivation of F which, although standard, is given here for completeness. From (25), after solving for a ˙ , and taking the positive root, one has

a ˙ / a = ( ( 8 π G ρ / 3 ) + ( Λ c 2 / 3 ) ) 1 / 2 . (26)

The negative root, not indicated above, corresponds to the descending branch of the ΛCDM universe, since, as discussed in conjunction with the EdS universe in [

H / H 0 = Ω m 1 / 2 ( ( 1 + z ) 3 + ( Ω Λ / Ω m ) ) 1 / 2 . (27)

Since d t = H − 1 d a / a , it follows, after integrating, that the age of the universe is given by

T 0 ≡ ∫ 0 T 0 d t = H 0 − 1 Ω m − 1 / 2 ∫ 0 a 0 ( ( 1 + z ) 3 + ( Ω Λ / Ω m ) ) − 1 / 2 a − 1 d a . (28)

Then, upon using a − 1 d a = − a 0 d z / ( 1 + z ) , and T 0 = H 0 − 1 F , one has

F = Ω m − 1 / 2 ∫ 0 ∞ ( ( 1 + z ) 3 + ( Ω Λ / Ω m ) ) − 1 / 2 ( 1 + z ) − 1 d z . (29)

Since Ω Λ / Ω m = 2.1746 , as noted preceding (18), and Ω m − 1 / 2 = 1.7817 , (29) yields F = 0.951 . Since, more accurately, one has Ω m = 0.315 ± 0.017 [

F = 0.951 ± 0.006. (30)

However, in view of the uncertainty in H_{0}, and the simplification of a two component flat ΛCDM model, it is clearly appropriate to set F = 0.95 in what follows.

If one now determines the age of the universe for the SHoES LDL value of H_{0} given by Riess et al. [

T 0 = H 0 − 1 F = 12.6 ± 0.2 Gyr . (31)

This age is in possible conflict with the ages of the globular clusters in the MW, for which the cut-off as to lower age was given as 12.6 Gyr by Krauss and Chaboyer [

T 0 = 12.7 ± 0.4 Gyr . (32)

It is in good agreement with (31), and again in possible disagreement with the above stellar ages. However, there is another issue arising with HOLiCOW’s determination of H_{0}. Since their determination involves a distance determination of H_{0}, and according to the arguments presented earlier in Section 2, it should agree with the CMB and BAO determinations to within the uncertainties, which it obviously does not, and hence, either the decelerating model proposed here is wrong, or there is possibly some problem with HOLiCOW’s determination of H_{0}. To examine this, it will be noticed that their value for H_{0} is based on six independent determinations that can be divided into two groups, A and B. Group A has four members (units, kms^{−1}∙Mpc^{−1}, in what follows are omitted for brevity): 68.9 − 5.1 + 5.4 , 71.0 − 3.3 + 2.9 , 71.6 − 3.9 + 3.8 , 71.7 − 4.5 + 4.8 , which have been simplified to those given in column A below, since more accuracy is not needed for the analysis. The second group B has two members: 78.2 − 3.4 + 3.4 , 81.1 − 7.1 + 8.0 , which again have been simplified in column B, and so one has

A B 69.8 ± 5.3 78.2 ± 3.4 71.0 ± 3.1 81.1 ± 7.6 71.6 ± 3.9 71.7 ± 4.7 (33)

Before proceeding further, it is desirable to re-evaluate H_{0} using the data in the above columns, and the weighted average method used earlier, to see how it compares with their more precise determination of H_{0}. It was found that 〈 H 0 〉 = 73.3 ± 2.0 , which is in perfect agreement with the value of H_{0} obtained by HOLiCOW, given by 73.3 − 1.8 + 1.7 and is therefore in more than sufficient agreement with their value to justify the analysis that follows. It was found that the weighted average of the values in column A yielded 〈 H 0 ( A ) 〉 = 71.1 ± 2.2 , while that in column B yielded 〈 H 0 ( B ) 〉 = 78.7 ± 4.2 . One sees that the difference, ignoring the uncertainties in the mean values, 〈 H 0 ( B ) 〉 − 〈 H 0 ( A ) 〉 = 7.6 is significantly greater than the difference 〈 H 0 ( A ) 〉 − H 0 ( CMB ) = 3.7 , so that one may question combining the data in B with that in A. A further objection is the age of the universe implied by 〈 H 0 ( B ) 〉 , since one has ( 〈 H 0 ( B ) 〉 ) − 1 = 12.5 ± 0.7 Gyr , and, if this is to be associated with the flat ΛCDM universe, then upon multiplication by F = 0.95, one has T 0 ( B ) = 11.9 ± 0.7 Gyr , an age range that is hardly acceptable for the globular clusters, let alone the oldest stars in the MW. This suggests that in determining the value of the Hubble constant by HOLiCOW III, one should reject the data from column B, and base the determination solely on that from column A, and clearly the small value for the difference 〈 H 0 ( A ) 〉 − H 0 ( CMB ) at 1.6σ, does not disprove the proposed decelerating model.

It is now appropriate to examine the age of the decelerating universe predicted by the revised model, in which the index of refraction is no longer a constant n, but a variable n ( z ) . It will be assumed that the age takes the form

T 0 = ( 2 / 3 ) 〈 n 〉 H 0 − 1 ， (34)

where 〈 n 〉 is the average of the index of refraction over a suitable redshift distance ζ that is determined from the integral

〈 n 〉 = ζ − 1 ∫ 0 ζ n ( z ) d z . (35)

To obtain , it will be assumed (to be discussed further below) that just as the CMB determines a Hubble constant for a decelerating universe, the age that it predicts is also for the decelerating universe, hence

T 0 = ( 2 / 3 ) 〈 n 〉 H 0 − 1 = 0.95 H 0 − 1 ， (36)

so that 〈 n 〉 = 1.43 . In evaluating the above integral, it is convenient to break it up into two parts: The first part extends from the terrestrial observer to the edge of the local group, since n ( z ) = 1 over this range. For simplicity, this redshift distance will be taken to be the redshift l determined by the diameter of the local group, since it is going to prove to be negligible. To be sure, since the local group is a tight cluster, it is not expanding, but one can imagine a galaxy, with negligible peculiar velocity, located just outside the local group, that is experiencing the Hubble expansion at the redshift l . To obtain a value for l it is convenient to use the expression for the first order Doppler effect c z / 〈 n 〉 = H 0 D L , since the desired value of redshift is so small, higher order terms may be neglected. Now, with D L taken to be the diameter of the local group, that is estimated to be ~10^{7} lyr [

ζ 2 − 2 b − 1 ( n 0 − 〈 n 〉 ) ζ + 2 b − 1 l ( n 0 − 1 + ( b / 2 ) l ) = 0. (37)

After inserting the numerical values, n 0 = 1.56 , 〈 n 〉 = 1.43 , b = 0.3285 , to sufficient accuracy, (36) becomes

ζ 2 − 0.79 ζ + 3.75 × 10 − 4 = 0. (38)

The larger root ζ = 0.79 is clearly the root of physical interest. It is significant that ζ exceeds the redshift z = 0.5, where the model and ΛCDM are in perfect agreement (by selection) with the diminished brightness of the SNe Ia. On the other hand, as noted in the previous section, the function n ( z ) should emerge from a theory that goes beyond the present model. It is possible in such a development ζ will also emerge, and hence one will be able to use (34) to obtain 〈 n 〉 , and compare it with the value 〈 n 〉 = 1.43 found above. Also, a matter for further investigation is the assumption that the age of the decelerating universe is 0.95 H 0 − 1 , where H_{0} is the CMB Hubble constant that is claimed here to be that for a decelerating universe, since, as was shown above, F = 0.95 was obtained for the flat ΛCDM accelerating universe. Thus, rigorously, the age 13.8 ± 0.1 Gyr is actually the age of an accelerating universe that presumably has the same age as that of the decelerating universe proposed here. However, under the assumption the distances used in obtaining the Hubble constant found in [

0.95 H 0 − 1 = 0.95 ( 69.8 ± 2.0 km / s / Mpc ) − 1 = 13.2 ± 0.2 Gyr . (39)

As was shown in previous discussions, the age of the decelerating universe is greater than the age of the accelerating universe, consequently, while the age 13.8 ± 0.1 Gyr used here for the decelerating universe is qualitatively correct, since it is greater than the above age of the accelerating universe, it will require further theoretical study to determine whether the use of F = 0.95 in (36) is justified in obtaining the age of the decelerating universe.

Meanwhile, just as this work was nearing completion, it was reported that the Atacama Cosmology Telescope (ACT) group [

Undoubtedly, the major support for the model, so far, is the finding in Section 2 that when q 0 = − 0.53 for an accelerating universe, that was used by Freedman et al. [_{0}, is replaced by q 0 = 0.50 for a decelerating universe, and the distance they determined is assumed to be the same as that required for the model for the redshift z = 0.07 , then their value of H_{0} comes down to excellent agreement (0.1%) with the CMB value, and likewise to within 0.3% of the weighted average of the BAO values for H_{0}, despite the uncertainties of a few to several percent in these different measured values of H_{0}. This result partially validates the claim underlying this work that the disagreement about the Hubble constant stems mainly from the proposal that the CMB and BAO values for H_{0} are actually for a decelerating universe, rather than for an accelerating universe, as is currently believed. However, to fully validate the claim, it remains to be seen what value Freedman et al. [_{0} when its re-evaluation has been made for the full range of redshifts that were used in obtaining their present result.

Additional support for the model stems from the fact, as discussed in Section 3, that there are tight clusters, such as the local group, of which the Milky Way and the LMC are members, in which the gravitational binding of the dark matter content is so strong that these clusters do not undergo the Hubble expansion, and, as a consequence, in accordance with the model, their dark matter does not undergo a phase change into dark energy, so that the speed of light within these tight clusters remains c, as it does within galaxies themselves. This leads to a simple explanation as to why the light from SN1987a in the LMC was able to arrive 3 hours after the neutrinos that were detected. On the other hand, the reduction in the speed of light that is traveling through the dark energy of the space that is experiencing the Hubble expansion leads to the prediction that there should not be any correlation between neutrinos and gamma ray bursts (GRBs). As reported in [

In Section 4, an attempt was made to improve on the original assumption of a constant index of refraction by introducing a variable index of refraction n ( z ) that was taken to be linear for simplicity. However, this complicated the determination of the age of the universe, as discussed in Section 5, and clearly more study of this issue is needed. On the other hand, the analysis showed that the higher values of the Hubble constant found by different groups is problematic, since their values lead to ages of the universe in possible conflict with the ages of the oldest stars in the Milky Way.

In conclusion, as noted earlier in [

I would like to thank Prof. George Fuller for his long-term interest in the work, and helpful suggestions. I am further indebted to Prof. Thomas Murphy for critical comments.

The author declares no conflicts of interest regarding the publication of this paper.

Tangherlini, F.R. (2020) A Possible Solution to the Disagreement about the Hubble Constant II. Journal of Modern Physics, 11, 1215-1235. https://doi.org/10.4236/jmp.2020.118076