Cosmology with relativistically varying physical constants

Rajendra P. Gupta* Department of Physics, University of Ottawa, Ottawa, Canada K1N 6N5

September 15, 2020

ABSTRACT We have shown that the varying physical constant model is consistent with the recently published variational approach wherein Einstein equations are modified to include the variation of the of , and cosmological constant using the Einstein-Hilbert action. The general constraint resulting from satisfying the local conservation laws and contracted Bianchi identitiesΛ provides the freedom to choose the form of the variation of the constants as well as how their variations are related. When we choose , / = 3/ , and , where is the scale factor and , we are able to show that the = resulting exp[ model:− 1 (a)] fits = the exp[3supernovae − 1a 1 ]observationalΛ = Λ exp[ data marginally − 1] better than the CDM model; (b) =determines 1.8 the first peak in the power spectrum of the cosmic microwave background temperature anisotropies at multipoleΛ value of ; (c) calculates the age of the as 14.1 Gyr; and (d) finds the BAO acoustic scale to be 145.2 Mpc. These numbers are within = 217.3less than 3% of the values derived using the CDM model. Surprisingly we find that the dark-energy density is negative in a universe that has significant negative curvature and whose expansionΛ is accelerating at a faster rate than predicted by the CDM model. Λ Key words: cosmology: theory, cosmic background radiation, cosmological parameters, dark energy, dark matter, distance scale.

Variable physical constants are introduced in most of the 1. INTRODUCTION proposed theories at the cost of either not conserving energy- momentum or violating Bianchi identity, which then leads to Despite the stunning success of the in explaining the breaking the covariance of the theory. Such theories may be cosmological observations, there remain many gaps that need to be considered inconsistent or ad hoc (Ellis & Uzan 2005) or quasi- bridged. We believe these gaps warrant looking at alternative phenomenological (Gupta 2019). An action principle to take into theories based on examining the foundation of physics that has been account the variation of the fundamental constants that are being developed over centuries from local observations extrapolated to the considered in a theory is required for generalization of Einstein universe at large. One of the foundations is that constants relating equations. Costa et al. (Costa et al. 2019; Franzmann 2017) have various observables are indeed constants everywhere and at every attempted this approach by considering the , in the universe. This is especially so because attempts to gravitational constant , and cosmological constant as scalar measure their variation has put very tight limit on their potential fields, and introducing an Einstein-Hilbert action that is considered variation. consistent with the Einstein equations and with the general Varying physical constant theories have been in existence constraint that is compliant with contracted Bianchi identities and since time immemorial (Thomson & Tate 1883; Weyl 1919; standard local conservation laws. Their approach is general Eddington 1934) especially since Dirac (1937; 1938) suggested covariant as it preserves the invariance of the . In variation of the constant based on his large number hypothesis. the work of Costa et al. (2019) the focus was to explain early Brans and Dicke (1961) developed the variation theory universe problems - the flatness problem and the horizon problem - compliant with general relativity in which constant was raised without invoking inflation, which they have successfully shown by to the status of scalar field potential. While Einstein developed simply promoting the constants and in Einstein’s equation to his ground breaking theory of based on the scalar fields. Here we will follow the same prescription. However, constancy of the speed of light, he did consider its possible we will limit ourselves to consider important direct cosmological variation (Einstein 1907). This was followed by the varying observations from now up to the time of cosmic microwave speed of light theories by Dicke (1957), Petit (1988) Moffatt background emission. (1993a; 1993b). More recently, Albrecht and Magueijo (1999) We will begin with establishing the theoretical background for and Barrow (1999) developed such a theory in which Lorentz our work in this paper in Section 2 starting from the equations invariance is broken as there is a preferred frame in which scalar presented in the paper of Costa et al. (2019). We will confine field is minimally coupled to gravity. Other proposals include ourselves to two considerations of the model: one with varying locally theories (Avelino & Martins 1999; Avelino, cosmological constant, and another with fixed cosmological Martins & Rocha 2000) and vector field theories that cause constant. The standard formulae used in cosmology are based on spontaneous violation of Lorentz invariance (Moffat 2016). fixed physical constants. We will go through carefully with the The most comprehensive review of the varying fundamental derivations of those we will use and determine how they are physical constants was done by Uzan (2003) followed by his modified when the constants are allowed to evolve. These involve more recent review (Uzan 2011). Chiba (2011) has provided an generalization of the luminosity distance, and therefore also of the update of the observational and experimental status of the distance modulus, as well as the scale factor for the surface of last constancy of physical constants. We therefore will not attempt to scattering and the deceleration parameter. cover the subject’s current status except to mention a few of Section 3 delineates the results. Firstly, we present the direct relevance to this work. parameters and curves obtained by fitting the redshift versus distance modulus Pantheon Sample data (Scolnic et al. 2018) for

*Email: [email protected] 1048 supernovae 1a to the CDM model and the two variable Here a dot on top of a variable denotes the time derivative of that constants models. The secondΛ test is to see how well a model variable, e.g. . Taking time derivative of Eq. (6), dividing computes the multipole moment of the first peak in the power by and equating ≡ / it with Eq. (7), yields the general continuity spectrum due to the temperature anisotropies in the cosmic equation:2 microwave background. The third test may be considered how well the models match the value of the BAO acoustic scale obtained (8) from the peak in the two point separation correlation function + 3 + = − − 4 + Λ determined from over a million galaxies. Another important test comprises determining the . Eq. (3) for the FLRW metric and perfect fluid stress-energy Section 4 discusses the findings of this paper and Section 5 tensor reduces to: presents our conclusions. , (9) − 4 + Λ = 0 2. THEORETICAL BACKGROUND therefore,

Following Costa et al. (2019) we may write the Einstein equations . (10) with varying physical constants (VPC) with respect to time - + 3 + = 0 speed of light , gravitational constant and cosmological constant = - applicable to the homogeneous = Using the equation of state relation with for matter = = 0 and isotropic universe, asΛ =follows: Λ and for relativistic particles, the solution for this equation = 1/3 is , where is the current energy density of all the = components of the universe when . . (1) = = 1 = − Λ Next we need to consider the continuity equation Eq. (9). The simplest solution is by assuming constant. Then i.e. Here is the Einstein tensor with the Ricci Λ = = 4 , = − . This is what Costa et al. (2019) used in their paper. We tensor and the Ricci scalar, and is the stress energy tensor. = Applying the contracted Bianchi identities, torsion free continuity will label this model as VcG model (varying and model). In fact and local conservation laws they dropped the cosmological constant altogether for their c-flation solution. However, one could choose any relationship between and , (2) ∇ = 0 ∇ = 0 and , say . Then from Eq. (9), by defining we = = , have one gets a general constraint equation for the variation of the physical constants , , 4 − = Λ ⇒ = ≡ . (3) (11) − − Λ = 0 ⇒ = 4 − Now the FLRW (Friedmann–Lemaître–Robertson–Walker) The parameter may be determined based on the physics or by metric for the geometry of the universe is written as: fitting the observations. We have determined in the past (Gupta

2019) that , i.e. and confirmed it by fitting the SNe 1a , = 3 = 3 = − + + + sin (4) data. Thus, we must have . We will label this model as = VcGΛ model (varying and model). , Λ with depending on the spatial geometry of the The most common way of defining the variation of the constant universe. = −1, 0, 1 is by using the scale factor powerlaw (Barrow & Magueijo 1999; The stress-energy tensor, assuming that the universe contents Salzano & Dabrowski 2017) such as which results in = can be treated as perfect fluid, is written as: . The advantage is that it results in very simple = = Freedmann equations. However, as the variable constant . (5) tends to zero or infinity depending on the → 0sign of . So, it yields = + + reasonable results when corresponds to relatively small = Here is the energy density, is the pressure, is the 4-velocity redshift , but not for large . What we have used in the past (Gupta vector with the constraint . (Unless necessary = − 2019) is the relations like i.e. , to avoid confusion, we will drop showing variation, e.g. is = , = exp[ − ] which leads to a limiting value of at . This approach was written as .) = 0 Solving the Einstein equation (such as by using Maplesoft very useful when solving problems with very small, such as − 2019) then yields VPC compliant Friedmann equations: for explaining astrometric anomalies. However, due to the involvement of time coordinate, it is difficult to use it in very simple Friedmann equations for their general solution. We therefore , , (6) tried here another relation that results in: ≡ = + − ⇒ = + − . = − + 3 + + = − + 3 + + ; ; and (7) = exp[ − 1] = exp[3 − 1] . (12) Λ = Λ exp[ − 1] 2

Their limitation is that can decrease in the past at most by a factor We could determine following a null geodesic from the time of and can decrease by a factor of (for positive a photon is emitted by the source to the time it is detected by the within = the 2.7183 region of their applicability). In the limit of , they observer with in Eq. (19) at constant and : reduce to the earlier exponential forms. For exampl → 1e = 0 = , or , which in the limit of exp − 1 × = = ⇒ =⇒= = is , or . It should be → 1 = = exp[ − ] . (21) mentioned that we found in an earlier work (Gupta 2018) that ⇒ = analytically . Using relations = 1.8 of Eq. (12), we can now write Eq. (11) for We may write , and = . / = / = // = / , , . Therefore = 1/1 + = −/1 + = − = 3 , and (22) . (13) = − = − = − ≡ = = The first Friedmann equation, Eq. (6), becomes [ ] . (23) = = 8 Λ 8 Here is obtained from Eq. (18) by substituting for = + − = + − 3 8 3 . 1/1 + . (14) Constant Model: Let us also consider the proposition of Costa = 1 + − Λ et al. (2019) that be taken as constant (albeit for the study of very Λ Dividing by , and since where early universe) - the VcG model. While they did not propose how subscript is for matter and = is for+ radiation = , (relativistic +, particles, and should vary in general, we will use the same form as for the VcG model so that the results of the two can be comparable. In e. g. photons and ), we get Λ that case, as mentioned above, we have . (15) = , +, 1 + − , , and . (24) = exp[ − 1] = exp[4 − 1] Λ = Λ At (current time), and . Therefore, Then, the first Friedmann equation becomes: = = 1 = 8 = exp [2 − 1] , + , + − 1 = , + , 1 + − 3 . (25) . (16) Then, as before, we get = Ω, + Ω, 1 + − = exp[2 − 1] Ω, +Ω, + Ω, + Ω, ≡ Here we have defined the current critical density as , = . (26) , and . Thus, by 3 /8 Ω, = ,/, Ω, = ,/, defining , we may write Eq. (16) Substituting we get . We can use Eq. (23) with Ω = Ω, + Ω, 1 + = 1/1 + this E(z) to obtain proper distance corresponding to the VcG model. ΛCDM Model: If we substitute in Eq. (26), we get back , (17) the case of physical constants not varying. = 0 We can then solve for Ω, ≡ − = 1 − Ω the standard CDM model parameters by fitting the observational Λ data, or put the parameter of our choice to see how well they fit the , , data. = exp[ − 1 ] Ω +Ω 1 + + . (18) , [ ] Ω exp 2 − 1 ≡ 2.1 Redshift vs. Distance Modulus Now, the FLRW metric, Eq. (4), can be written in its alternative form (Ryden 2017) as We will now consider the specifics of applying the theory to develop relationship between the redshift and distance modulus . First of all we will check if the relation holds when . (19) = 1/1 + = − + [ + + sin ] physical constants are varying. Here for (closed universe); Since light travels along null geodesics, we have as per Eq. (21) = sin/ = +1 = , or . A wave crest of the light for (flat universe), for (open = = / = 0 = sinh/ = −1 emitted from a galaxy at time and observed at time travels a universe), where is the parameter related to the curvature. The proper distance between an observer and a source is determined distance at fixed time by following a spatial geodesic at constant and . Then . (27) = = . (20) Next wave crest will be emitted at time and observed at = ⇒ = = + / time and will cover the same distance + /

/ . (28) = = / 3

(35) Therefore, = 5 log + 5 log 1 + + 25 + 2.5log

The last term is the extra correction term that must be included in / . (29) calculating the distance modulus when one is considering the = / varying speed of light irrespective of the model applied and whether

Subtracting from both sides the integral we get or not the space is flat. / 2.2 CMB Power Spectrum / / . (30) = Here we will be focussing on the first peak in the power spectrum of the cosmic microwave background (CMB) temperature Assuming and remain unchanged over the extremely short anisotropies and the related parameters such as its multipole time between two consecutive wave crests compared to the age of moment, sound horizon distance at the time universe became the universe, we have transparent, acoustic scale derived from baryonic acoustic oscillation, and angular diameter distance of the surface of last / / . scattering. = ⇒ = ⇒ = The sound horizon distance is the distance sound travels (31) at speed in photon-baryon fluid from the big-bang until the Since , we get from above time such plasma disappeared due to the formation of the atoms, i.e. ≡ − λ/ 1 + = / = 1/ the time of last scattering . Following Eq. (21) we may write with This result is the same as for the case when is not varying. ≡ 1. (Durrer 2008) Now the distance of a source is determined by measuring flux of radiation arriving from the source with known luminosity. We . (36) = therefore need to relate proper distance that relates to the redshift as per Eq. (23) to the luminosity distance which relates The speed of sound in terms of the speed of light in the to the flux. We will then be able to establish what we mean by photon-baryon fluid with baryon density and radiation density luminosity distance in the variable physical constant approach and is given by (Durrer 2008) Ω how it differs from the case when constants are indeed constants. Ω The photons emitted by a source at time are spread over the / sphere of proper radius and, referring to Eq. (19), proper . (37) ≈ 1 + surface area is given by (Ryden 2017) √ Substituting it in Eq. (36) and making use of Eq. (23), we get . (32) = 4 [ ] . (38) The flux is defined as luminosity divided by the area in a / = √ stationary universe. When the universe is expanding then the flux is reduced by a factor due to energy reduction of the photons. 1 + This distance represents the maximum distance over which the We need to also determine how the increase in distance between the baryon oscillations imprint maximum fluctuations in thermal emitted photons affects the flux. radiation that is observed as anisotropies in CMB. This represents The proper distance between two emitted photons separated by an angular observed at an angular diameter distance a time interval is whereas the proper distance between the given simply by same two photons when observed would become . Thus the proper distance would increase by a factor 1 + 1 + , and (39) . When is constant, i.e. , this ≡ /1 + ≡ / effect/ reduces = 1 the + flux /by a factor of . But when it is= not, there 1 + And the corresponding multipole moment is given by . is an additional factor that we have to consider to correct the = / / Next thing is to determine the correct value of corresponding flux. We thus have to account for an extra factor in the / to the last scattering surface when the physical constants are luminosity distance increase when calculating proper distance of a varying. The blackbody radiation energy density is given by source from its flux data. (Ryden 2017) ( being the Boltzmann The flux and the luminosity distance relation may = 8 /15ℎ therefore be written as constant), and the radiation energy density as per Eq. (10) is given by . Therefore, when the constants and are not = , ℎ varying then with . Using our earlier finding , (33) ∝ = 1 + = that the varies as (Gupta 2019), and that varies as . as ℎ discussed in Section 4 of this paper, we see that . Additionally, we have to see how does / / . (34) the ∝ ionization exp[0.25 energy − of 1 an] atom evolve since it is proportional to ≡ = 1 + , where with the mass of , ℎ ≡ /8 ℎ The distance modulus by definition is related to the its charge and the permittivity of space. Thus does luminosity distance : not vary and therefore ∝ 1/ . And then how does the thermal photon energy evolve, ∝since ℎ the ratio of the ionization energy and thermal energy determines the temperature of the last scattering ≡ 5log + 25 surface? This ratio can be seen to evolves as ℎ/ ∝ 1 . Cumulatively, the two effect lead to exp[0.75 − 1] ∝ 4

. Now for . (45) exp [ − 1] ≡ exp[1 + − 1] = 1/ . Therefore Thus ∴ = = / ≫ 1 . We = can now= see that when. = 1 + =, BAO Acoustic Scale: It relates to baryon acoustic oscillations i.e.1 + the /2.7183 last scattering surface is at the redshift = 400 1089,’ in our = VPC400 (BAO) that are linked directly to the CMB anisotropies. BAO is models. considered observable today through the correlation function of One may question that the blackbody spectrum will be affected galaxies’ distribution in space (Anderson et al. 1914). It is given by: due to the varying physical constants since the energy density for the photons in the frequency range and is given by + . (46) , which would evolve due to = 1 + = [/exp − 1] the variation of and . However, we do not know what exactly Here is given by Eq. (35) and for the VPC models. = 400 the spectrum of ,the ℎ distant cosmological objects is. We only know This parameter could be considered another test for the VPC what it is when it is observed. If we consider the peak photon models. energy of the spectrum, it is given by (Ryden 2017) ℎ = , or . ., . . 2.8 = = exp [1.25 − 1] / 3 RESULTS Then . . . . = exp1.25 − 1 2.25 − 1 We will now test the proposed model against observations. The Therefore . , which, near , i.e. most used test is to see how a model fits the redshift - distance = 2.25 − 1/ = = 1 modulus ( ) data from the observation on supernovae 1a is . Since − (standard candle). The data used in this work is the so-called = = 1.25 ≈ 71 km s Mpc = 2.3 × , we get . This is Pantheon Sample of 1048 supernovae Ia in the range of 0.01 < < 10 ≈ 3 × 10 ≈ 9 × 10 yr (Scolnic et al.). The data is in terms of the apparent magnitude very small shift to observe in the blackbody spectrum. Nevertheless, and2.3 we added 19.35 to it to obtain normal distance modulus it would be nice if one could come up with some observation or numbers as suggested by Scolnic (private communication). experiment which could detect the thermal spectrum variation with The Matlab curve fitting tool was used to fit the data by time. minimizing and the latter was used for determining the 2.3 Other Cosmological Parameters corresponding probability (Press et al. 1992). Here is the weighted summed square of residual of Let us now consider some other cosmological parameters. Deceleration Parameter: The second Friedmann equation, Eq. , (47) (7) is used to determine the deceleration parameter , which by = ∑ ; , , … − , definition (Narlikar 2002) where is the number of data points, is the weight of the th

data point determined from the error in , , . (40) the observed distance modulus using the relation = − = − , = , and is the model calculated distance 1/, ; , , .. Recalling that and in the limit of , we may modulus dependent on parameters and all other model write Eq. (7) at = 1 as / = = dependent parameter etc. As an example, for the CDM = models considered here, , , and there is no other unknownΛ ≡ Ω, , or parameter. = − 1 + 3 + + We then quantified the goodness-of-fit of a model by calculating the probability for a model whose has been . (41) determined by fitting the observed data with known measurement = − = + − − error as above. This probability for a distribution with degrees of freedom (DOF), the latter being the number of data Since , we get points less the number of fitted parameters, is given by: , = . (42) , (48) = Ω, + Ω, − Ω, − , = / The last term results from the varying physical constants approach. Age of the Universe: We need to calculate the parameter that where is the well know gamma function that is generalization of Γ is the current cosmic time relative to the big-bang time. From Eq. the factorial function to complex and non-integer numbers. The (18), for VcGΛ model lower the value of the better the fit, but the real test of the , goodness-of-fit is the probability ; the higher the value of for a model, the better the model’s fit to the data. We used an online = ⇒ = ⇒ = calculator to determine from the input of and DOF (Walker 2020). , (43) ⇒ = ≡ Our primary findings are presented in Figure 1 and Table 1. One additional thing we wished to explore was if the prior of

is the right choice for the VcG model. This can be = exp [ − 1] Ω, +Ω, 1 + confirmed = 3 by fitting the SNe Ia data. Thus,Λ with all the parameters , (44) ( free, i.e. no priors, we obtained results closed to +Ω, exp [2 − 1] the ,Ωvalues,, ,in ,Table 1 but with large 95% confidence bounds. When 5 we progressively constrained the parameters based on our prior fails this test then it may not be worthwhile to consider it for other knowledge, the 95% confidence bounds shrank and we obtained tests. those parameters still unconstrained as expected. The results are presented in Table 2. Table 2. This table shows the results of fitting the Pantheon Sample SNe Ia data in order to determine the value of the parameter that is used in the − VcGΛ model. The description of other parameters is the same as in Table −1 −1 1. The unit of H 0 is km s Mpc . 46 Paramete All free 4 free 3 free 2 free r parameters parameters parameters parameters 44 H0 Fixed Fixed 70 .47 70 .87 70 .87 70Fixed.87 42 Ωm,0 . . . . 0.5273 . 0.5211 . 0.5298 . 0.528 .

40 α . . Fixed Fixed 1.85 . 1.854 . 1.8 1.8 β . . . 38 −2.29 . −2.204 . −1.756 . Fixed−1.8

σ . . . . SNe Ia data 3.027 . 3.001 . 3.001 . 3. 36 VcG model χ2 VcG model 1032 1032 1032 1032 34 CDM model DOF 1043 1044 1045 1046 0.5 1 1.5 2 P% z 58 .98 59 .83 60 .67 61 .51 R-sq 0.997 0.997 0.997 0.997 Figure 1. Supernovae Ia redshift vs. distance modulus data fits RMSE using the VcG model and VcG model as compared to the fitµ using the 0.9948 0.9943 0.9938 0.9933 ΛCDM model.Λ All the curves are so close to each other that they appear to be superimposed. The second test is to see how well the model computes the multipole moment of the first peak in the power spectrum due to the Table 1 . Parameters for the three models determined by fitting the Pantheon temperature anisotropies in the cosmic microwave background. Sample SNe Ia data. This table shows all the models are able to fit the WMAP (Wilkinson Microwave Anisotropy Probe) and Plank data very well with − the VcGΛ model showing slight edge on others. The unit −1 −1 (Planck Collaboration 2019) have determined the multipole moment of H0 is km s Mpc . P% is the χ2 probability in percent; the higher the χ2 2 for the first peek. probability the better the model fits to the data. R is the square of the ≈ 220 correlation between, the response values and the predicted response values. A third test may be considered how well the models match the RMSE is the root mean square error and DOF is the degrees of freedom. value of the BAO acoustic scale and compare it with the measured (Anderson et al. 2014) and Planck estimate of this parameter. To be Parameter ΛCDM VcG VcGΛ consistent with the measured value using Hubble constant −1 −1 H0 70.18±0.42 70.79±0.70 70.87±0.50 (Anderson et al. 2014) of km s Mpc , we have normalized the two other models = to 70 km s −1 Mpc −1 for = 70 Ωm,0 0.2845±0.0245 1.053±0.1363 0.528±0.0.037 computing BAO acoustic scale only. Another important test comprises determining the age of the Ω 1-Ω 0.06441±0.22419 -Ω Λ,0 m,0 m,0 universe. We have determined it for the models and compared it

Ω0 1 1.117 0 with the well accepted value of 13.7 Gyr. Table 3 presents the above tests. It also includes deceleration α NA 1.25 1.8 parameter , as well as some other cosmological parameters for β NA NA -1.8 ready comparison. Both the VPC models estimate higher

2 acceleration than the ΛCDM model. χ 1036 1032 1032 DOF 1046 1045 1046 4. DISCUSSION P% 58.1 60.7 61.5 An objective of this paper is to see if the theory of Costa et al. R-sq 0.997 0.997 0.997 (2019) complements our quasi-phenomenological model and RMSE 0.9951 0.9938 0.9933 provides improvement on it by being covariantly relativistic. The resulting very simple Friedmann equations and continuity equation eliminates the arbitrariness of our earlier models (Gupta 2018; We notice that as the parameters are progressively fixed, the Gupta 2019). The continuity equation, Eq. (8) essentially breaks value does not change. This means that the constraint put on the down into three separate equations: the first for energy density , parameter is reasonable, otherwise value would have increased. the second relating speed of light to the gravitational constant , In fact, the probability slightly improves and so does the and the third relating the energy density to the dark-energy RMSE. SNe Ia data fit therefore confirms our choice of and density through the cosmological constant . As a result the for the VcG model based on our previous work. = 3 Λ = 1.8 Λ solution of the Freedmann equations is greatly simplified even The SNe Ia test is to see how good a model is for the directly though the physical constants and are allowed to vary. observable universe for which the redshift and light fluxes of Despite its simplicity, the VcGΛ, model Λ is highly satisfactory in galaxies are measurable. This test is the primary test. If the model

6 explaining the observations from to . It is therefore its value determined from SNe Ia data and CMB anisotropy worth examining it for other cosmological = 0 attribute = 1100s. spectrum. Their model comparison analysis determined that the CDM model is favoured over their negative cosmological constant Table 3. This table presents the results of other tests for the two models and model.Λ However, they did not consider the variation of speed of compare them to the ΛCDM fit derived from the CMB temperature light and gravitational constant in their model. anisotropy observations by Planck (Planck Collaboration 2019). The Hubble While on the surface it appears that fitting observation with a constant used for the other two models is the same as obtained for these minimum number of parameters is more satisfactory than with a models from SNe Ia data fit. VSL stands for varying speed of light. The unit −1 −1 larger number of parameters, the models that can determine larger of H 0 is km s Mpc . number of parameters from the same observation is more desirable Planck+BAO in order go deeper in understanding the universe. A significant Parameter ΛCDM VcG VcGΛ amount of work is required to understand how the new approach of Matter energy this paper could determine other parameters of the universe and if it density Ωm,0 0.3106 1.053 0.528 can resolve Hubble constant tension. One may notice that the Hubble constant values in Table 2, Dark-energy density Ω 0.6894 0.06441 -0.528 Λ,0 based on SNe 1a data fit, are 1% higher for the VcG and VcG Λ Curvature parameter Ωk,0 -0.0096 -0.1175 1.0000 models than its value for the CDM model. We have put the same values in Table 3 for the VcGΛ and VcG models as we have not yet Hubble constant H0 67.70 70.79 70.87 Λ Redshift at last determined the same using the CMB temperature anisotropy scattering surface z 1090 1090 1090 spectrum. Thus numbers in Table 3 are not comparable and VSL corrected should therefore not be seen as if they resolve the tension. redshift (see text) z’ 1090 400 400 It is even more staggering that the spatial curv ature of the 1st peak’s universe is strongly negative rather than flat and thus contradicts all multipole moment l 220.6 130 217.3 the theories and observations derived assuming physical constants Deceleration not varying. When analysing any observational data we need to be parameter q0 -0.5341 -0.788 -1.01 careful that it is not biased in favour of the physical constants that Age of the are constrained to their currently observed values, especially in universe t0 (Gyr) 13.787 6.068 14.108 view of the new definition of dependent on the constant BAO acoustic value of the speed of light as 299,792,458 meters per second (NIST scale (measured r 2020). As discussed below, physical constants other than and s , Λ value 149.28) (Mpc) 147.57 180.6 145.2 that are not directly involved in the models here, such as and also vary. This makes it even ℎ The first test both the VcG and VcG models passed is the SNe more difficult to reach conclusions based on observations that 1a test (which involves relatively low redshiftΛ data) as is clear from directly or indirectly involve many physical constants. studying Table 1 and Figure 1. However, as is obvious from Table It should also be pointed out that the matter density Ω, = 3, the VcG model did not pass other tests including the age of the , obtained by VcG model is significantly higher than that 0.528 Λ universe test and those which involve high redshift observations estimated by the Planck mission assuming CDM Ω, = 0.3153 Λ whereas VcG model passed them all with flying colors. Few cosmology (Planck Collaboration 2019). The Planck mission also noticeable thingsΛ about the VcG model in the table are: (a) the estimated the baryon density and dark-matter density sign of the dark-energy densityΛ is negative, (b) the curvature (Planck papersΩ, label= 0.049 it as . Since the baryon parameter meaning that the space is curved negatively Ωdensity, = has 0.2607 also been estimated directly by Ωaccount ing the visible unlike in theΩ ,CDM= 1 model that yields the space as flat, and (c) the and non-visible baryonic matter, we can assume it to be valid for deceleration parameterΛ is almost twice as much as for the CDM any cosmological model. Then the dark-matter density for the model meaning that the universe’s expansion is acceleratingΛ at VcG model is 84% higher than the CDM model. almost twice the rate predicted by the CDM model. These CosmologistΛ nowΩ, will= 0.479 have more freedom to play withΛ the dark- discrepancies are a result of the varying physicalΛ constants. matter. The staggering feature of the varying physical constant model The results presented here may be considered a continuation of VcGΛ is that it yields negative dark-energy density while still our previous work (Gupta 2019) wherein we explained three showing accelerated expansion of the universe. Possibility of such a astrometric anomalies and the null results on the variation of and scenarios has been explored by many researchers (e.g. Visinelli, the fine structure constant using the quasi-phenomenological Vagnozzi & Danielsson 2019; Dutta et al. 2020; Hartle, Hawking & version of the current VcGΛ model. Since our current model Hertog 2012; Wang et al. 2018) who considered the varying reduces to the quasi-phenomenological model in the limit of the cosmological constant to primarily resolve Hubble constant tension, scale factor close to its current value 1, and since all these which is due to the difference in its values obtained from CMB data problems relate to the space where scale factor is close to 1, we and SNe Ia data. Such a scenario is extremely interesting from the conclude that the current model is also able to resolve these perspective as obtaining a vacuum solution with a problems. Thus the findings of our previous work that the Planck positive value of within moduli-fixed consistent and stable string constant varies just like the speed of light (Gupta 2019), theory compactificationsΛ has been a formidable task (Maldacena & inferred fromℎ the null results on the variation of the fine-structure Nunez 2001; Kachru et al. 2003; Conlon & Quevedo 2007; constant , can also be considered valid under the VcG model. Λ Danielson & van Riet 2018). We have assumed in our work here that varies as . . This It is important to point out that the negative cosmological assumption leads to the redshift for the surface of last constant does not yield satisfactory outcome in most models. For scattering to be . This is confirmed = /from our results for the first example, Visinelli et al (2019) have considered one such model to peak of the CMB400 power spectrum and BAO acoustic scale which see if their negative cosmological constant model, that is consistent are within 3% of the standard CDM model and observations. One with string theory, could resolve Hubble constant tension between Λ 7 could be sure that the results would only improve with the very few parameters. Most difficult to accept findings of the current refinement of the model. work are (a) the evolutionary dark-energy density being negative, The variation of the Planck constant has been the subject of and (b) the universe having negative spatial curvature. Nonetheless, several studies (e.g. Davies et al. 2002; Kentoshℏ & Mohageg 2012; the VcG model (i) fits the supernovae Ia observational data Mangano et al. 2015; de Gosson 2017; Massood-ul-Alam 2018). marginallyΛ better than the CDM model; (ii) determines the first The interest in the variation of has been mainly due to its peak in the power spectrumΛ of the cosmic microwave background ℏ temperature anisotropies at the multipole value of against occurrence in the fine structure constant . If varies, = 217.3 = ℏ the observed value of 220 derived from CDM model; (iii) then one could ask which constant occurring in its expression does calculates the age of the universe as 14.1 Gyr Λagainst the accepted vary. The variation of the Boltzmann constant has not enjoyed such value of 13.7 Gyr; and (iv) finds the BAO acoustic scale to be 145.2 attention. However, the variability of any constant can be associated Mpc against the observed value of 149.3. Also, the analysis with a dynamical field that evolves due to the action of a presented here predicts the energy density peak in the blackbody background potential which drives the field towards its minimum. frequency spectrum at to shift with time as Thus it is implicitly assumed that the field reached a stable ≈ 9 × minimum very early in the Big-Bang era resulting in the constancy . We therefore conclude that the VcG model deserves to of the physical constant associated with the field. An evolutionary be10 consideredyr seriously for further work. It will rΛequire collaboration background potential could therefore cause the field minimum to to study more difficult cosmological problems such as fitting the also evolve. This may also explain why all the physical constants power spectrum of CMB temperature anisotropies. Current codes might evolve and why studying any constant in isolation might not such as CAMB (Lewis, Challinor & Lasenby 2000), CLASS be prudent. (Lesgourgues 2011) and CMBAns (Das & Phan 20190 are unable to It would be interesting to see how some of the well-known handle background and other changes required to properly handle constants vary when the fundamental constants are considered VPC models. It is challenging undertaking to modify them evolutionary. Consider for example the asexpressed by the lead authors of these codes in private = communications. where is the rest mass of the electron, is the elementary charge, and is the permittivity of space. If we consider and , which belong to baryonic matter, to not vary (at least in comparison ACKNOWLEDGEMENTS to the variation of the fundamental constants considered here), and This work has been supported by a generous grant from Macronix since and varies as , we infer that does not vary. Research Corporation. Thanks are due to Madhav Singhal at the ∝ 1/ ℎ Another example can be taken as the Stefan-Boltzmann constant University of Western Ontario who used his python code to verify . Again, since varies as and varies as the SNe 1a data fit by Matlab curve fitting app. The author is = 2 /15ℎ ℎ . , does not vary. Similarly, the fine-structure constant, given grateful to Krishnakanta Bhattacharya at the Indian Institute of by will be immune to the variation of and . As Technology, Guwahati, for his critical comments and helpful = ℏ discussions. He remains thankful to Dan Scolnic at Duke University already mentioned, ℏ it is the constancy of the fine-structure constant for providing the SNe 1a Pantheon Sample data used in this work, that led us to conclude in our previous work (Gupta 2019) that the and to Santanu Das at the University of Wisconsin, Madison, for his Planck constant varies as the speed of light. Any variations in these comments and discussion on the use and modification of CMB constant would mean either our assumptions are wrong or and codes. The author is grateful to Antony Lewis at the University of do vary to the extent of the measured variations in the Rydberg Sussex and Julien Lesgougues at Aachen University for advising constant and the fine structure constant. how it might be possible to modify their CMB codes to include We could think of physical constants belonging to two different variable physical constants and non-flat . He wishes to categories: constants that are independent of the Hubble expansion express his gratitude to the reviewer of the paper whose very of the universe - e.g. the fine structure constant α and proton to constructive critical comments helped improve the paper electron mass ratio µ; and the constants that may be tied to the significantly. Hubble expansion - e.g. the Newton's gravitational constant G, and possibly also the speed of light c, the Planck's constant h and the Boltzmann constant kB. The constants in the first category might be varying much more slowly, if varying at all, than those in the DATA AVAILABILITY second category. The problem we see is that, in the absence of any Data used is from references discussed in the manuscript (Scolnic, knowledge about how different constants vary, one tends to ignore et al., 2018). the possible variation of other constants in studying the variation of one constant. We have tried to study some constants in the second category and found that in many expressions and formulae they REFERENCES vary in such a way that they negate the variations of others and thus make their variations unobservable. Albrecht A., Magueijo J., 1999, Phys. Rev. D, 59, 043516. It should be explicitly stated that both the models VcG and Anderson L. et al., 2014, MNRAS, 441, 24. 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