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 speed of light , gravitational constant and cosmological constant using the Einstein-Hilbert action. The general constraint resulting from satisfying the local conservation laws and contracted Bianchi identities 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 ]observational = 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 multipole value of ; (c) calculates the age of the universe 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 expansion is accelerating at a faster rate than predicted by the CDM model. 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 standard model 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 speed of light , time 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 general relativity. 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 special relativity 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 invariant 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 second 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 G 5/5 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 & H # % % (8) from the peak in the two point separation correlation function C 7 3 H C 7 D = − / # − N %' C 7 !"# 2 determined from measurements over a million galaxies. Another important test comprises determining the age of the universe. 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) / − N ' & C 7 2 = . # % % 2. THEORETICAL BACKGROUND therefore, Following Costa et al. (2019) we may write the Einstein equations H . (10) with varying physical constants (VPC) with respect to time - C 7 3 C 7 D = . speed of light , gravitational constant and H cosmological constant = - applicable to the homogeneous = Using the equation of state relation with for matter D = OC O = . and isotropic universe, as =follows: and for relativistic particles, the solution for this equation O = 1/3 is JJP, where is the current energy density of all the C = C C components of the universe when . !"# $ . (1) = = 1 = %$& '( − ) Next we need to consider the continuity equation Eq. (9). The simplest solution is by assuming constant. Then # % i.e. + Here is the Einstein tensor with the Ricci = # = N % , = * − , ) * * # %&. This is what Costa et al. (2019) used in their paper. We tensor and the Ricci scalar, and is the stress energy tensor. & * ( #Q = %Q Applying the contracted Bianchi identities, torsion free continuity will label this model as VcG model (varying and model). In fact and local conservation laws they dropped the cosmological constant altogether for their c-flation solution. However, one could choose any relationship between and , (2) & - = . - ( = . and , say # % . Then from Eq. (9), by defining % K we = R CK = , have # % !"# one gets a general constraint equation for the variation of the physical constants & !"# % , % % K K + IT K , %& % N − RC = L % = !"# K' 1SI'G 1SI K + 1 !"# . (3) % K (11) & /# 0 − % 0 2 % '( − 30 4) = . L CK = N − RC % K Now the FLRW (Friedmann–Lemaître–Robertson–Walker) The parameter may be determined based on the physics or by metric for the geometry of the universe is written as: fitting the observations.R We have determined in the past (Gupta 2019) that , i.e. # % and confirmed it by fitting the SNe 1a + , , , , , , , , , , R = 3 # = 3 % 56 = − 5 7 /+89: 5; 7 ; 5< 7 sin <5@ 2 data. Thus, we must have % K . We will label this model as (4) CK = C VcGΛ model (varying and %model).K , with depending on the spatial geometry of the The most common way of defining the variation of the constant universe.A = −1, ., 1 is by using the scale factor powerlaw (Barrow & Magueijo 1999; The stress-energy tensor, assuming that the universe contents Salzano & Dabrowski 2017) such as which results in = can be treated as perfect fluid, is written as: % H . The advantage is that it results in very simple = = F Freedmann% H equations. However, as the variable constant + . (5) tends to zero or infinity depending on the U .sign of . So, it yields ( = %:$ C 7 DE E 7 D) reasonable results when + corresponds to relatively small = Here is the energy density, is the pressure, is the 4-velocity +VW C D E redshift , but not for large . What we have used in the past (Gupta vector with the constraint , . (Unless necessary X X ) E E = − 2019) is the relations like % i.e.