_{1}

^{*}

The current disagreement about the Hubble constant
*H*
_{0} was described as a “Crisis in Cosmology”, at the April (2018) Meeting of the American Physical Society, and hence its resolution is of utmost importance. This work proposes that the solution to the disagreement between the Planck Collaboration cosmic microwave background (CMB) value of
*H*
_{0}, together with the very close BOSS Collaboration baryon acoustic oscillation (BAO) value, and the significantly higher value of
*H*
_{0} found by the SHOES Collaboration cosmic distance ladder (CDL) work, is due to the fact that the CMB and BAO values of
*H*
_{0} are not for an accelerating universe, as generally believed, but are actually the values for a decelerating universe. In contrast, the CDL value of
*H*
_{0} is indeed that for an accelerating universe. It is shown that by replacing the negative deceleration parameter in the expression for log
*H*
_{0} in the CDL work by a positive deceleration parameter, the value of
*H*
_{0} can be brought down to agree with the CMB and BAO lower values. There is a brief review of the author’s decelerating model based on the Einstein de Sitter universe, augmented by a model of dark energy that does not have a negative pressure, but instead has a non-dispersive index of refraction
*n*, causing the speed of light through the dark energy of intergalactic space to be reduced to
*c/n*. As reported earlier, this assumption is sufficient to accommodate the increase in apparent magnitude of the Type Ia supernovae (SN
_{e} Ia). Additional support for the model is presented, together with a proposal for astronomical falsification.

The current disagreement about the Hubble constant H_{0} is widely recognized as a serious problem in cosmology, indeed, as indicated in the Abstract, it has been recently described as a “Crisis in Cosmology.” The purpose of this work is to show that there is a possible solution to the problem based on the author’s proposal in [^{−1}∙Mp∙c^{−1}, that utilizes the ΛCDM model, and the SHOES collaboration [^{−1}∙Mp∙c^{−1}, is due to the CMB value for H_{0} actually being that for a decelerating universe, whereas the CDL value for H_{0} is indeed, as presented, that for an accelerating universe. The BOSS collaboration [^{−1}∙Mp∙c^{−1}, is so close to the CMB value, and because it is simpler to deal with analytically, it will be used to justify the decelerating interpretation of the lower CMB and BAO values. It will also be shown that when the CDL expression for logH_{0}, given in [_{tot} sufficiently close to unity, a closed universe might alternatively be used. It is also pointed out there that the closed universe yields an upper bound on Λ that is too small to accommodate the accelerating universe, and if the EdS universe can be seen as the limit of a closed universe with infinite radius, then the upper bound on Λ goes to zero.

Since the above results do not necessarily prove that the universe is decelerating, but only make it reasonable, in Section 4, two tests of the decelerating model, that have been discussed earlier, particularly in [

In the decelerating model, it is assumed that the dark energy of intergalactic space (IGS), instead of having a negative pressure, that causes the expansion of the universe to be presently accelerating, has rather an index of refraction n, so that the speed of light through IGS is reduced to c/n. A least squares fit of the EdS universe to the accelerating ΛCDM universe, that is presently favored to explain the increased apparent magnitude of the Type Ia supernovae (SNe Ia), found by Riess et al. [

δ m = 5 log ( 1 + ( n − 1 ) ln ( 1 + z ) ) (1)

while the corresponding logarithmic fractional increase in distance d is given by

d = log ( 1 + ( n − 1 ) ln ( 1 + z ) ) (2)

The above value for d is used when comparing the logarithmic distance to the standard ruler of the BAO that is fitted by the ΛCDM model, with that fitted by the EdS model that has been augmented by d; see, e.g., Equation (16) in [

The resolution of the disagreement about the Hubble constant H_{0} that is proposed here begins with the observation that the CMB value not only disagrees with the CDL value, but is significantly lower than the CDL value. This would be the case if the CMB value of H_{0} were not for an accelerating universe, but were actually for a decelerating universe. In contrast, the CDL value of H_{0} is certainly for an accelerating universe, because it explicitly involves the deceleration parameter q_{0}, to which is assigned the value q 0 = − 0.55 [_{0} arises from the observation that since H 0 ≡ a ˙ 0 / a 0 , where a(t) is the expansion parameter for the flat Friedmann LeMaître Robertson Walker (FLRW) line element

d s 2 = c 2 d t 2 − a ( t ) 2 d r 2 + r 2 ( d θ 2 + sin 2 θ d ϕ 2 ) , and since a 0 is necessarily the same for both models, and also since a ˙ 0 ( a c c e l . ) > a ˙ 0 ( d e c e l . ) , then H 0 ( a c c e l . ) > H 0 ( d e c e l . ) . The inequality for the expansion parameter velocities arises in the following way: In the very early universe, before the accelerating term became significant, both models would be expanding at essentially the same rate, but as the effect of the accelerating term became larger, the ΛCDM universe’s deceleration became increasingly less than that of the EdS universe, and eventually the accelerating term became dominant, and caused the universe to accelerate, so that by the times (which are different, less for the ΛCDM universe) each universe has expanded to a 0 the above inequality for the Hubble constants resulted. To be sure, the above qualitative analysis does not guarantee that H 0 ( a c c e l . ) > H 0 ( d e c e l . ) by the observed amount, but it is suggestive, and it serves to motivate the quantitative analysis given further below. It will be shown there that by replacing in the expression for H 0 ( C D L ) a value for the deceleration parameter that is the value for a decelerating universe, so that q 0 > 0 instead of q 0 < 0 as was used in [_{0} that agrees with H 0 ( B A O ) which, as noted above, is essentially the same as H 0 ( C M B ) .

But before carrying out this analysis, it is necessary to show how it could be possible that the reported CMB and BAO values of H_{0} could be that for a decelerating universe, since they are presented as being the Hubble constants for the accelerating ΛCDM universe. The key is to note a well-known ambiguity in the term “dark energy.” On the one hand, such as in the author’s work, one of the interpretations leads to a density parameter for dark energy Ω d e ≈ 0.7 that is needed to arrive at Ω t o t ≈ 1 , since, as is well-known, dark matter and baryonic matter together only account for about 30% of Ω t o t . This was shown independently of the CMB determination, by Verde et al. [_{0} from their measurements is rather complicated, and well beyond the scope of this work, it will be shown for the comparable BAO determination of the Hubble constant, that H 0 ( B A O ) can be alternatively interpreted as being for a decelerating universe that has the same density of dark energy as the accelerating ΛCDM universe, but does not have the negative pressure associated with the cosmological term.

As was shown in [

c H 0 − 1 Χ Λ ( z ) = c H 0 − 1 Χ m ( z ) ( 1 + ( n − 1 ) ln ( 1 + z ) ) (3)

where Χ Λ ( z ) and Χ m ( z ) are dimensionless integrals derived from the field equations for energy densities for the two different universes, and in which each side of the above equation is equal to the effective distance to the standard ruler of the BAO, while z is its redshift. Actually there are two distances that are determined observationally: The angular distance D A ( z ) = a 0 r ( z ) / ( 1 + z ) , and the longitudinal distance D L ( z ) = a 0 r ( z ) ( 1 + z ) , so that the effective distance a 0 r ( z ) is given by a 0 r ( z ) = ( D A ( z ) D L ( z ) ) 1 / 2 . For the determination of these distances, that are presented as supportive of the flat ΛCDM universe, see Anderson et al. [_{0} from the left hand side of (3) by setting a 0 r ( z ) = c H 0 − 1 Χ Λ ( z ) would seem to yield the Hubble constant for an accelerating universe given by H 0 = c Χ Λ / a 0 r ( z ) , which is the current interpretation. However, one can alternatively regard a 0 r ( z ) as having been set equal to the right hand side of (3) , and hence it would yield the Hubble constant for a decelerating universe with reduced speed of light, which is the interpretation proposed here. To resolve the conflict in the two interpretations, it will be shown next that when the CDL expression for H_{0} is taken to be that for a decelerating universe, it can be lowered in value to equal the BAO determination.

In [_{0}

log H 0 = M x 0 + 5 a x + 25 5 (4)

where M x 0 is a distance modulus, and from Equation (5) in [

a x = log ( c z { 1 + 1 2 [ 1 − q 0 ] z − 1 6 [ 1 − q 0 − 3 q 0 2 + j 0 ] z 2 + O ( z 3 ) } ) − 0.2 m x 0 (5)

where j 0 = 1 from prior deceleration, and m x 0 is another distance modulus. It will be convenient to combine the two distance moduli and the additional constant term into the combined distance modulus M defined by

M ≡ M x 0 − m x 0 + 25 5 (6)

and upon introducing the kinematic term k ( z , q 0 , j 0 ) defined by

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 } (7)

in which the correction term O ( z 3 ) has been omitted, one can rewrite log H 0 as

log H 0 = M + log ( c z ) + log k ( z , q 0 , j 0 ) (8)

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 = c z 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 = c z k ( z , q 0 , j 0 ) , 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 = − log D . However, the following argument does not make use of the value of M, since one is going to subtract from log H 0 for an accelerating universe, 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

log ( H 0 H ′ 0 ) = M − M ′ + log ( k ( z , q 0 , j 0 ) k ( z , q ′ 0 , j 0 ) ) (9)

After setting q 0 = − 0.55 for the accelerating universe as given in [

log ( H 0 ( − 0.55 ) H 0 ( 0.5 ) ) = M − M ′ + log ( 1 + 0.775 z − 0.274 z 1 + 0.25 z − 0.125 z 2 2 ) (10)

One can determine M from (8) as a function of redshift since the value of H 0 ( − 0.55 ) is known, or alternatively from its definition in (6). However, in order to obtain M ′ 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 ( 0.5 ) 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 ′ , one can proceed as follows: Set the ratio of the two Hubble constants, which will be denoted by η , equal to H 0 ( C D L ) / H 0 ( B A O ) ; in addition, for simplicity, take their mean values as fiducial, so that one has η = 73.24 / 67.6 = 1.083 . Next, find the behavior of M − M ′ as function of redshift by using (10) with η = 1.083 . Since it is mentioned in [

( z * ) 2 − 0.775 − η 0.250 0.274 − η 0.125 z * + η − 1 0.274 − η 0.125 = 0 (11)

For η = 1.083 , one has the two roots z * = 0.17 , and z * = 3.47 . Because terms of O ( z 3 ) 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 [

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 [

A prediction that fully supports the model, although it does not confirm it, was first discussed in [

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 [

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 Ω d e = Ω Λ . 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 [

The authors declare no conflicts of interest regarding the publication of this paper.

Tangherlini, F.R. (2018) A Possible Solution to the Disagreement about the Hubble Constant. Journal of Modern Physics, 9, 1827-1837. https://doi.org/10.4236/jmp.2018.99116