Evolutionary Physical Constants in Einstein's Field Equations and the Redshift

Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 10 April 2019 doi:10.20944/preprints201904.0119.v1

Evolutionary physical constants in Einstein’s field equations and the redshift

Rajendra P. Gupta

Macronix Research Corporation, 9 Veery Lane, Ottawa, ON Canada K1J 8X4; [email protected]

Abstract We have shown that the use of evolutionary gravitational constants G, the speed of light c, and cosmological constant Λ in the Einstein’s field equations leads to a very simple model that fits the supernovae Ia data with a single parameter as well as the standard ΛCDM model with two parameters, and has the predictive capability superior to the latter. We have shown that at the current time G and c both increase as dG/dt = 5.4GH0 and dc/dt = 1.8cH0 with H0 as the Hubble constant, and Λ decreases as dΛ/dt = -1.2ΛH0. Negative finding on the variation of G with time derived from the analysis of lunar laser ranging data in several studies has been attributed to the fact that these studies did not consider the variation of c simultaneously with G. Keywords: cosmology: theory, cosmological parameters – galaxies: distances and redshifts – variable physical constants

1 INTRODUCTION final section, Sec. 6, we will discuss our findings and present conclusions. Varying physical constant theories have been in existence The essence of this work is that the new model implicitly since time immemorial, especially since Dirac (1937, 1938) includes the cosmological constant (that is varying) and it can suggested such variation based on his large number fit the supernovae Ia data with a single parameter almost as hypothesis. The most comprehensive review of the subject well as the ΛCDM model with two parameter, and it has the was done by Uzan (2003) followed by his more recent review predictive capability superior to the latter. In addition, we not (Uzan 2011). Chiba (2011) has provided an update of the only establish that the physical constants vary, we provide observational and experimental status of the constancy of exactly how much they vary. physical constants. We therefore will not attempt to cover the subject’s current status except to mention a few of direct 2 EVOLUTIONARY CONSTANTS MODEL relevance to this work. Magueijo (2000, 2003) is a strong proponent of the variable speed of light theories. We will develop our model in the general relativistic domain Accordingly, the model developed here will be considered starting from the Robertson-Walker metric with the usual generally Lorentz invariant which is true for any theory based coordinates 푥휇 (푐푡, 푟, 휃, 휙): on varying constants. We will be using the null results of Hofmann and Muller’s (2018) attempts to determine the 푑푟2 푑푠2 = 푐2푑푡2 − 푎(푡)2[ + 푟2(푑휃2 + sin2 휃푑휙2)].(1) variation of gravitational constant 퐺 by analysing several 1−푘푟2 decades of LLR data to establish that when the speed of light variation is included, one finds that 퐺 and 푐 vary exactly in Here 푎(푡) is the scale factor and 푘 determines the spatial the proportion of 3 to 1. geometry of the universe: 푘 = −1 (closed), 0 (flat), +1 (open). We will follow Maharaj and Naidoo’s (1993) approach to The Einstein’s field equations may be written in terms of the 휇휈 휇휈 include variable gravitational constant 퐺 and cosmological Einstein tensor 퐺 , metric tensor 푔 , and energy-momentum 휇휈 constatnt Λ in the Einstein’s field equations, and generalize it tensor 푇 , as: to include the variable speed of light 푐, to obtain their 8휋퐺 solution for the Robert-Walker metric in Sec. 2. We will show 퐺휇휈 + Λ푔휇휈 = − 푇휇휈. (2) 푐4 that Λ̇ /Λ = −1.2퐻 and 퐻̇ /퐻 = −0.6퐻 where 퐻 is the Hubble parameter. In Sec. 3 we will establish that 퐺̇/퐺 = 5.4퐻 and When solved for the Robertson-Walker metric, we get the 푐̇/푐 = 1.8퐻. We will derive the expression for distance following non-trivial equations for the flat universe (푘 = 0) of modulus 휇 vs. redshift 푧 in Sec. 4. Sec. 5 is devoted to state interest to us here, with 푝 as the pressure and 휀 as the energy the methodology for fitting the supernovae Ia data (Scolnic et density: al. 2018) and apply the same to the new model and compare it with the fits obtained with the standard ΛCDM model. In the

푎̈ 1 푎̇ 2 4휋퐺 1 푎̈ 푎̇ 3+3푤−푘 푎̈ 푎 −1−3푤+푘 + ( ) = − 푝 + Λ (3) = ( ) (1 − ) ; and −푞 ≡ 2 = . (12) 푎 2 푎 푐2 2 푎̇ 푎 2 푎̇ 2 푎̇ 2 8휋퐺 1 = 휀 + Λ (4) 푎2 3푐2 3 Here 푞 is deceleration parameter that does not depend on time, i.e. 푞0 = 푞. As we know the radiation energy density is If we do not regard 퐺, 푐 and Λ to be constant and define negligible at present, so we need be concerned with the matter 퐾 ≡ 퐺/푐2, we may easily derive the continuity equation by only solutions, i.e. with 푤 = 0. taking time derivative of Eq. (4) and substituting in Eq. (3) The deceleration parameter 푞0 has been analytically determined on the premise that expansion of the universe and 3푎̇ 퐾̇ Λ̇ the tired light phenomena are jointly responsible for the 휀̇ + (휀 + 푝) + 휀 + = 0. (5) 푎 퐾 8휋퐾 observed redshift, especially in the limit of very low redshift (Gupta 2018). One could see it as if the tired light effect is It reduces to the standard continuity equation when 퐾 and Λ are superimposed on the Einstein de Sitter’s matter only universe held constant. And since the Einstein’s field equations require rather than the cosmological constant. By equating the that the covariant derivative of the energy-momentum tensor expressions for the proper distance of the source of the redshift 푇휇휈 be zero, we can interpret Equation (5) as comprising of two for the two, one gets 푞0 = −0.4. Then from Eq. (12) we get continuity equations (Maharaj & Naidoo 1993), viz 푘 = 1.8, and also 푙 = −1.2, and 푚 = −0.6. We thus have from Eq. (8) 퐾̇ /퐾 = 1.8퐻, Λ̇ /Λ = −1.2퐻 and 퐻̇ /퐻 = −0.6퐻. 3푎̇ 휀̇ + (휀 + 푝) = 0, and (6) 푎 3 VARYING G AND c FORMULATION 8휋휀퐾̇ + Λ̇ = 0. (7) Having determined the value of 푘 = 1.8, and since the Hubble parameter is defined as 퐻 = 푎̇/푎, we may write from Eqs. (8) This separation simplifies the solution of the field equations and (9) (Eqs. 3 and 4). Eq. (6) yields the standard solution for the −3(1+푤) energy density 휀 = 휀0푎 . Here 푤 is the equation of state 퐾̇ 퐾 = 퐾 푎1.8, and = 1.8퐻. (13) parameter defined as 푝 ≡ 푤휀 with 푤 = 0 for matter, 1/3 for 0 퐾 radiation and −1 for Λ. It has been explicitly discussed by Magueijo in several of his We may also write explicitly papers (e.g. Magueijo 2000) that this approach is not generally 퐾̇ 퐺̇ 2푐̇ Lorentz invariant albeit relativistic. Moreover, one should = − = 1.8퐻. (14) ideally use Einstein-Hilbert action to obtain correct Einstein’s 퐾 퐺 푐 equations with variable 퐺 and 푐. Our approach therefore may be Taking 퐻 at the current time as 퐻 ≃ 70 km s−1 Mpc-1 (2.27 considered quasi-phenomenological. 0 퐾̇ × 10−18 s−1) we get = 4.09 × 10−18 s−1 = 1.29 × Since the universe expansion is determined by 퐻(푡) ≡ 푎̇/푎, it 퐾 is natural to assume the time dependence of any time dependent 10−10 yr−1. parameter to be proportional to 푎̇/푎 – the so called Machian The findings from the Lunar Laser Ranging (LLR) data scenario (Magueijo 2003). Let us therefore write analysis provides the limits on the variation of 퐺̇/퐺 (7.1 ± 7.6 × 10−14) (Hofmann & Müller 2018), which is considered to be 퐾̇ 푎̇ Λ̇ 푎̇ 퐻̇ 푎̇ = 푘 ( ), = 푙 ( ) and = 푚 ( ), i.e. (8) about three orders of magnitude lower than expected. However, 퐾 푎 Λ 푎 퐻 푎 the LLR data analysis is based on the assumption that the speed of light is constant and non-evolutionary. If this constraint is 퐾 = 퐾 푎푘, Λ = Λ 푎푙 and 퐻 = 퐻 푎푚. (9) 0 0 0 dropped then the finding would be very different.

As is well known (Merkowitz 2010), a time variation of 퐺 Here 푘 (not the same as in Eq. (1)) 푙 and 푚 are the should show up as an anomalous evolution of the orbital period proportionality constants, and subscript zero indicates the 푃 of astronomical bodies expressed by Kepler’s 3rd law: parameter value at current epoch (푡 = 푡0). With this substitution in Eq. (4) we may write 4휋2푟3 푃2 = , (15) 퐺푀 푎̇ 2 8휋 1 = 퐻2푎2푚 = (퐾 푎푘)휀 푎−3(1+푤) + Λ 푎푙. (10) 푎2 0 3 0 0 3 0 where 푟 is semi-major axis of the orbit, 퐺 is the gravitational Comparing the exponents of a of all the terms, we may write constant and 푀 is the mass of the bodies involved in the orbital motion considered. If we take time derivative of Eq. (15), 2푚 = 푘 − 3 − 3푤 = 푙, and with 푤 = 0 for matter, we have 2 2푚 = 푘 − 3 = 푙. Thus, if we know 푘, we know 푙 and 푚. divide by 푃 and rearrange, we get We can now have a closed analytical solution of Eq. (10) as 퐺̇ 3푟̇ 2푃̇ 푀̇ follows (since 푎(푡 ) ≡ 1): = − − . (16) 0 퐺 푟 푃 푀

2 푟̇ 1 푐̇ 푎(푡) 푡 3+3푤−푘 푎̇ 2 −1 If we write 푟 = 푐푡 then = + . We may now rewrite Eq. 푎(푡) = = ( ) ; = 푡 ; (11) 푟 푡 푐 푎(푡0) 푡0 푎 3+3푤−푘 (16) as 2

Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 10 April 2019 doi:10.20944/preprints201904.0119.v1

퐺̇ 3푐̇ 3 2푃̇ 푀̇ − = − − . (17) The ratio of the two distances may be considered the 퐺 푐 푡 푃 푀 normalization factor 퐹 when using the variable 푐 in calculating Since LLR measures the time of flight of the laser photons, it is the proper distance of a source. Since 푎 ≡ 1/(1 + 푧), we may the right hand side of Eq. (17) that is determined from LLR data write for the source of redshift 푧 with emission time 푡푒 −14 analysis (Hofmann & Müller 2018) to be 7.1 ± 7.6 × 10 and 푡 푒 = 푎(푧)0.6 = (1 + 푧)−0.6 , and (23) not the right hand side of Eq. (16). 푡0 Then, taking the right hand side of Eq. (17) as 0 and ( ) [ ( )−0.6] ( )−2.4 combining it with Eq. (14), one can solve the two equations and 퐹 푧 = 4 1 − 1 + 푧 /[1 − 1 + 푧 ] (24) −10 −1 get 퐺̇/퐺 = 5.4퐻0 = 3.9 × 10 yr and 푐̇/푐 = 1.8퐻0 = Now the proper distance of the source with variable 푐 may be 1.3 × 10−10 yr−1 . It should be emphasized that both 퐺̇/퐺 and defined as (Ryden 2017) 푐̇/푐 are positive and thus both of them are increasing with time 1.8 푡0 푐 푡0 푐0푎 푡0 0.8 푑푃 = ∫ ( ) 푑푡 = ∫ ( ) 푑푡 = 푐0 ∫ 푎 푑푡 rather than decreasing as is generally believed (Barnes & Dicke 푐 푡푒 푎 푡푒 푎 푡푒 1961, van Flandern 1975). This may be considered the most 4 7 significant observational finding of cosmological consequences 푡0 푡 3 3 푡푒 3 = 푐0 ∫ ( ) 푑푡 = 푐0푡0[1 − ( ) ]. (25) just by studying the Earth-Moon system. 푡푒 푡0 7 푡0 −1 From Eq. (11) 퐻0 ≡ 푎̇/푎 = (2/1.2)푡0 . Therefore, 4 REDSHIFT VS. DISTANCE MODULUS 1 푑 = (푐 /퐻 )[1 − (1 + 푧)−1.4] . (26) 푃푐 1.4 0 0 The distance 푑 of a light emitting source in a distant galaxy is determined from the measurement of its bolometric flux 푓 and Thus the expression for 푑 to be substituted in Eq. (20) to comparing it with a known luminosity 퐿. The luminosity determine the luminosity distance of the source is 푑 = 푑푃푐퐹. distance 푑퐿 is defined as Since the observed quantity is distance modulus 휇 rather than the luminosity distance 푑퐿, we will use the relation 퐿 푑 = √ . (18) 휇 = 5 log(푑퐿) + 25, (27) 퐿 4휋푓 1 = 5log ( 푅 (1 − (1 + 푧)−1.4) + 5 log(퐹(푧)) In a flat universe the measured flux could be related to the 1.4 0 luminosity 퐿 with an inverse square relation 푓 = 퐿/(4휋푑2). However, this relation needs to be modified to take into account +5 log(1 + 푧) + 25. (28) the flux losses due to the expansion of the universe through the scale factor 푎, the redshift 푧, and all other phenomena which can Here 푅0 ≡ 푐/퐻0 and all the distances are in Mpc. result in the loss of flux. Generally accepted flux loss phenomena We will compare the new model, hereafter referred as VcGΛ are as follows (Ryden 2017): (variable 푐, 퐺 and Λ) model, with the standard ΛCDM model a. Increase in the wavelength causes the flux loss proportional to which is the most accepted model for explaining cosmological 1/(1 + 푧), and phenomena, and thus may be considered the reference models for b. In an expanding universe, an increase in detection time all the other models. Ignoring the contribution of radiation between two consecutive photons emitted from a source leads to a density at the current epoch, we may write the distance modulus reduction of flux proportional to 푎, i.e. proportional to 1/(1 + 푧). 휇 for redshift 푧 in a flat universe for the ΛCDM model as follows Therefore, in an expanding universe the necessary flux (Peebles 1993): correction required is proportional to 1/(1 + 푧)2. The measured 푧 bolometric flux 푓 and the luminosity distance 푑 may thus be 퐵 퐿 휇 = 5log [푅 ∫ 푑푢/ Ω (1 + 푢)3 + 1 − Ω ] written as: 0 √ 푚,0 푚,0 0 2 2 +5 log(1 + 푧) + 25. (29) 푓퐵 = 퐿/[4휋푑 (1 + 푧) ], and (19) 푑 = 푑(1 + 푧). (20) 퐿 Here Ω0,푚 is the current matter density relative to critical density How does 푑 compare with and without varying 푐? Let us first and 1 − Ω푚,0 ≡ ΩΛ,0 is the current dark energy density relative to consider the case of non-expanding universe. The distance from critical density.

the point of emission at time 푡푒 to the point of observation at 푡0 5 SUPERNOVAE Ia z-µ DATA FIT time 푡0 may be written as 푑푐 = ∫ 푐 푑푡. Therefore for constant 푡푒 We tried the VcGΛ model developed here to see how well it fits 푐 = 푐0 the best supernovae Ia data as compared to the standard ΛCDM 푡푒 푑푐0 = 푐0푡0(1 − ) (21) model. The data fit is shown in Fig. 1. The VcGΛ model 푡0 requires only one parameter to fit all the data (퐻0 = 68.90 ± 2 −1 −1 1.8 푡 1.2 0.26 km 푠 Mpc ) whereas the ΛCDM model requires two When 푐 = 푐0푎 , and since 푎 = ( ) from Eq. (11), we may −1 −1 푡0 parameters (퐻0 = 70.16 ± 0.42 km 푠 Mpc and Ω푚,0 = write 0.2854 ± 0.0245). 3 4 The data used in this work is the so called Pantheon Sample of 푡0 푡 푐0 푡0 3 1 푡푒 푑푐 = 푐0 ∫ ( ) 푑푡 = 3 ∫ 푡 푑푡 = 푐0푡0 (1 − 4). (22) 1048 supernovae Ia (Scolnic et al. 2018) in the range of 푡푒 푡0 푡0 푡푒 4 푡0 0.01 < 푧 < 2.3. To test the fitting and predictive capability of 3

Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 10 April 2019 doi:10.20944/preprints201904.0119.v1

the two models, we divided the data in 6 subsets: a) 푧 < 0.5; b) corresponding 휒2 probability 푃 (Press et al. 1992). Here 휒2 is 푧 < 1.0; c) 푧 < 1.5; d) 푧 > 0.5; and e) 푧 > 1.0; and f) 푧 > 1.5. the weighted summed square of residual of 휇: In addition, we considered the fits for the whole data. The models were parameterized with subsets a), b) and c). The 2 푁 2 휒 = ∑푖=1 푤푖[휇(푧푖; 푅0, 푝1, 푝2 … ) − 휇표푏푠,푖] , (30) parameterized models were then tried to fit the data in the subsets that contained data with z values higher than in the where 푁 is the number of data points, 푤푖 is the weight of the 푖th parameterized subset. For example if the models were data point 휇 determined from the measurement error 휎 in parameterized with data subset a) 푧 < 0.5, then the models were 표푏푠,푖 휇푂푏푠,푖 fitted with the data subsets d) 푧 > 0.5, e) 푧 > 1.0 and f) 푧 > 1.5. the observed distance modulus 휇표푏푠,푖 using the relation 푤푖 = 2 ( ) 1/휎휇푂푏푠,푖, and 휇 푧푖; 푅0, 푝1, 푝2 … is the model calculated distance modulus dependent on parameters 푅0 and all other model dependent parameter 푝1, 푝2, etc. As an example, for the ΛCDM models considered here 푝1 ≡ Ω푚,0 and there is no other unknown parameter. We then quantified the goodness-of-fit of a model by calculating the 휒2 probability for a model whose 휒2 has been determined by fitting the observed data with known measurement error as above. This probability 푃 for a 휒2 distribution with 푛 degrees of freedom (DOF), the latter being the number of data points less the number of fitted parameters, is given by:

∞ 푛 2 1 −푢 −1 푃(휒 , 푛) = ( 푛 ) ∫ 푒 푢2 푑푢 , (31) Γ( ) 휒2 2 2

FIG. 1. Supernovae Ia redshift 푧 vs distance modulus µ where Γ is the well know gamma function that is generalization of data fit using the VcGΛ model as compared to the fit the factorial function to complex and non-integer numbers. using the ΛCDM model. Lower the value of 휒2 better is the fit, but the real test of the 2 The Matlab curve fitting tool was used to fit the data by goodness-of-fit is the 휒 probability 푃; higher the value of 푃 for a minimizing 휒2 and the latter was used for determining the model, better is the model’s fit to the data. We used an online

TABLE I. Parameterizing and prediction table for the two models. This table shows how well a model is able to fit the -1 -1 data that is not used to determine the model parameters. The unit of R0 is Mpc and of H0 is km s Mpc . P% is the χ2 probability in percent that is used to assess the best model for each category; higher the χ2 probability 푃 better is the model fit to the data. R2 is the square of the correlation between the response values and the predicted response values. RMSE is the root mean square error. Highest P% value in each category is shown in bold and the cell highlighted.

Action / Item ΛCDM VcGΛ ΛCDM VcGΛ ΛCDM VcGΛ ΛCDM VcGΛ Parameterized Model dataset z<0.5; 832 points Model dataset z<1.0; 1025 points Model dataset z<1.5; 1042 points Model dataset all; 1048 points

R0 4259±34 4337±18 4269±27 4351±17 4271±26 4352±17 4273±26 4351±16

Ωm,0 0.2601±0.0457 NA 0.2793±0.0261 NA 0.2818±0.0249 NA 0.2845±0.0245 NA

H0 70.39±0.56 69.13±0.29 70.23±0.44 68.90±0.27 70.19±0.42 68.89±0.27 70.16±0.42 68.90±0.25 χ2 863.5 889.4 1018 1060 1033 1074 1036 1076 DOF 830 831 1023 1024 1040 1041 1046 1047 P% 20.39 7.83 53.82 21.15 55.53 23.26 58.11 26.02 R2 0.9961 0.9960 0.9969 0.9968 0.9970 0.9969 0.9970 0.9969 RMSE 1.020 1.035 0.9977 1.017 0.9965 1.016 0.9951 1.014 Model Fit Dataset z>0.5; 216 points Dataset z>0.5; 216 points Dataset z>0.5; 216 points Dataset z>0.5; 216 points χ2 176.9 190 DOF 216 NOT APPLICABLE SINCE NOT APPLICABLE SINCE NOT APPLICABLE SINCE THIS DATASET INCLUDES THIS DATASET INCLUDES THIS DATASET INCLUDES P% 97.59 89.84 2 THE DATASET USED TO THE DATASET USED TO THE DATASET USED TO R 0.9605 0.9575 PARAMETERIZE THE MODEL PARAMETERIZE THE MODEL PARAMETERIZE THE MODEL RMSE 0.905 0.938 Model Fit Dataset z>1.0; 23 points χ2 19.54 17.01 17.59 16.75 DOF 23 NOT APPLICABLE SINCE NOT APPLICABLE SINCE THIS DATASET INCLUDES THIS DATASET INCLUDES P% 66.94 80.43 77.93 82.13 2 THE DATASET USED TO THE DATASET USED TO R 0.8741 0.8904 0.8867 0.8921 PARAMETERIZE THE MODEL PARAMETERIZE THE MODEL RMSE 0.9216 0.86 0.8746 0.8533 Model Fit Dataset z>1.5; 6 points χ2 4.090 1.946 3.167 1.983 3.076 1.986 DOF 6 NOT APPLICABLE SINCE THIS DATASET INCLUDES P% 66.44 92.45 78.76 92.12 79.92 92.09 2 THE DATASET USED TO R 0.5993 0.8093 0.6897 0.8057 0.6986 0.8054 PARAMETERIZE THE MODEL RMSE 0.8256 0.5696 0.7265 0.5749 0.716 0.5754

Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 10 April 2019 doi:10.20944/preprints201904.0119.v1

calculator to determine 푃 from the input of 휒2 and DOF holistically for all the constants involved. We have (Walker 2018). Our primary findings are presented in Table established that the physical constants not only vary but I. The unit of the Hubble distance 푅0 is Mpc and of the also how much they vary: 푐̇/푐 = 1.8퐻, 퐺̇/퐺 = 5.4퐻, -1 -1 Hubble constant 퐻0 is km s Mpc . The table is divided in Λ̇ /Λ = −1.2퐻 and 퐻̇ /퐻 = −0.6퐻. Also, from the null four categories vertically and four categories horizontally. results on the variation of the fine structure constant Vertical division is based on the parameterizing data subset (Gohar 2017, Rosenband et al. 2008) we can show that indicated in the second row and discussed above. The ħ̇ /ħ = 1.8퐻. We urge that they be used in union rather parameters determined for each model are in the first than in isolation. horizontal category. The remaining horizontal categories show the goodness-of-fit parameters for higher redshift Acknowledgements Author is thankful to Franz subsets than used for parameterizing the models. Thus this Hofmann for providing his latest research paper and table shows the relative predictive capability of the two references on Lunar Laser Ranging. He is grateful to models. The model cells with the highest probability in each Dan Scolnic for providing the SNe Ia Pantheon data used category are shown in bold and highlighted. in this work.

6 DISCUSSION AND CONCLUSION References As should be expected, the two parameter ΛCDM model is able to fit any data set better than the one Barnes, C., Dicke, R. H. 1961 Phys. Rev. 124, 925. parameter VcGΛ model. What is unexpected is that when Chiba, T. 2011, Prog. Theor. Phys. 126, #6. parameterized with a relatively low redshift data the Dirac, P. A. M. 1937, Nature 139, 323. VcGΛ model is able to fit the higher redshift data better Dirac, P. A. M. 1938, Proc. Roy. Soc. A 165, 199–208. than the ΛCDM model. This shows that the second Gohar, H. 2017, Universe 3, 26. parameter in the latter, while trying to fit a limited Gupta, R. P. 2018, Int’l J. Astron. Astrophys. 8, 219; dataset as best as possible, compromises the model fit for Universe 4, 104. data not used for parameterizing. This means that the Hofmann, F, Müller, J. 2018, Class. Quant. Grav. 35, ΛCDM model does not have as good a predictive 035015. capability (i.e. the capability to fit the data that is not Magueijo, J. 2000, Phys. Rev. D 62, 103521. included for determining the model parameters) as the Magueijo, J. 2003, Rept. Prog. Phys. 66, 2025. Maharaj, S. D., Naidoo, R. 1993, Astrophys. Sp. Sci. 208, VcGΛ model despite having twice as many parameters as 261. the VcGΛ model. In addition the VcGΛ model has the Merkowitz, S. M. 2010, Liv. Rev. Rel. 13, 7. analytical expression for the distance 휇 unlike the ΛCDM Mould, J., Uddin, S. A. 2014, PASA 31, e015. model which must be evaluated numerically. Peebles, P. J. E. 1993, Principles of Physical Cosmology One would notice that while 푅 (and hence 퐻 ) values 0 0 (Princeton University Press, N.J.). are relatively stable with the parameterizing dataset Press, W. H., Teukolsky, S. A., Vetterling, W. T., Flannery, containing higher and higher redshift values, varying no B. P. 1992, Numerical Recipes in C – The Art of more than 0.35%, while the variation in the Ω is up to 푚,0 Scientific Computing, 2nd Ed. (Cambridge University 9.4%, i.e. 27 time higher. This confirms that Ω푚,0 Press, U. K.). parameter, and hence Λ through ΩΛ,0, is an artificially Rosenband, T., Hume, D. B., Schmidt, P. O., Chou, C. W., introduced parameter to fit the data rather than being Brusch, A., Lorini, et al. 2008, Science 319, 1808. fundamental to the ΛCDM model. In contrast Λ is an Ryden, B. 2017 Introduction to Cosmology (Cambridge 2 integral part of the VcGΛ model. Since 퐾 (≡ 퐺/푐 ) and University Press, U.K.). Λ are related through Eq. (7), one could easily derive that Scolnic, D. M., Jones, D. O., Rest, A., et al. 2018, ApJ. the Λ term contributes 60% for the VcGΛ model against 859, 101. 70% for the ΛCDM model. Uzan, J-P. 2003, Rev. Mod. Phys.75, 403. We have established that the supernovae 1a data is Uzan, J-P. 2011, Liv. Rev. Rel., 14, 2. compatible with the variable constants proposition. This van Flandern, T. C. 1975, Mon. Not. Roy. Astron. Soc. is contrary to the findings of Mould and Uddin (2014) 170, 333. who considered only the variation of 퐺 in their work. We Walker, J. 2018 Chi-square calculator - https://www.- believe most of the negative findings on the variation of fourmilab.ch/rpkp/experiments/analysis/chiCalc.html. physical constants are possibly due to one considering the variation of a constant in isolation rather than