March 2020 Dark Energy and the Time Dependence of Fundamental Particle Constants Bodo Lampe II. Institut für theoretische Physik der Universität Hamburg Luruper Chaussee 149, 22761 Hamburg, Germany Abstract The cosmic time dependencies of G, α, h and of Standard Model parameters like the Higgs vev and elementary particle masses are studied in the framework of a new dark energy interpretation. Due to the associated time variation of rulers, many effects turn out to be invisible. However, a rather large time dependence is claimed to arise in association with dark energy measurements, and smaller ones in connection with the Standard Model. 1 I. Introduction Dirac was one of the first to suggest that fundamental physical constants may vary in time due to the expansion of the universe[1]. Dirac concentrated on Newton's constant G, but since then a time dependence of c, α, ~ and so on has been considered possible as well ([2]-[17]). From the 21st century perspective it is clear that if fundamental constants are time dependent in this way, the observed dark energy effect must have to do with it, because dark energy dominates the present expansion of the universe. In the course of the present work time dependencies will therefore be partly reduced to a ('Planck' or 'dark' or cosmological) energy dependence of physical quantities and constants. This energy dependence is completely separate from and not to be confused with the usual energy dependence from the renormalization group. The framework of the article will be the ordinary FLRW cosmology with a scale factor a(t) and a spatial curvature k, the latter assumed to be tiny (in accordance with observations). Furthermore, the so-called 'cosmic coordinate system' will be used, i.e. cosmic time t and proper distances r as parameters. This will prove to be optimal for the presentation. It is well known that the fundamental spacetime constants c, h and G can be used to define the Planck length, time and mass L, T and M which describe the basic properties of space[m], time[s] and matter[kg] L(t) = √ ~(t)G(t) c3 T (t) = √ ~(t)G(t) c5 M(t) = √ ~(t)c G(t) (1) One may invert these relations to obtain c = L(t) T (t) ~(t) = E(t)T (t) κ(t) = L(t) E(t) (2) where E = Mc2 is the Planck energy and κ = G/c4 the Einstein constant. A time dependence of these quantities has been anticipated here. t = 0 is taken to be the present, so we have the present day values L0 = L(0) = Planck length, T0 = T (0) = Planck time and E0 = E(0) = Planck energy. Numerical values are L0 = 1.6× 10−35m M0 = 2.2× 10−8kg T0 = 5.4× 10−44s (3) 2 No time dependence of c is indicated, because in the present model there is none at least if one uses the above mentioned cosmic coordinates t and r, in which case the FLRW solution of the Einstein equations has the line element ds2 = −c2dt2 + dr2/(1 + ...) (4) with a constant i.e. time-independent speed of light. c being constant, one only needs to consider time dependencies of G and h. Equivalently, since one has T(t)=L(t)/c one only needs to consider time dependencies of the Planck length L(t) and Planck energy E(t). Rewriting eq. (2) as ~(t)c = E(t)L(t) (5) G(t) = c4L(t)/E(t) (6) one sees that there are 2 really fundamental time dependencies to be considered: -L(t)=the time dependence of the fundamental measure of space -E(t)=the time dependence of the 'physically active' quantities the 'quantities of motion', as Isaac Newton called them. Remark: The time dependence of elementary particle couplings α, GF and so on is a different story. It will be treated in section V and will boil down to determine the time dependence of one other quantity: -J(t)=the time dependence of the 'internal exchange energy' to be defined in section V. II. Measure-of-Space Equation To determine L(t) and E(t) I introduce 2 equations: L = −4π 3 GρL− ω2(L− Ls) + Λ 3 c2L (7) The idea behind this is that the universe is an elastic medium which consists of elementary constituents called tetrons[18, 19], and the bond length of these constituents is given by the Planck length L(t), while the Planck energy E(t) measures the binding energy of every 2 bound constituents. It is to be noted that the 3 Figure 1: Schematic depiction of the universe as an elastic medium made of tiny constituents. Shown are the binding lines between nearest neighbour constituents. The 2 lattices represent the expanding universe at 2 times t1 < t2. tetrons are invisible to us. All (ordinary and dark) matter particles and radiation we know are quasi-particles/wave-excitations of them and can propagate on the elastic medium.1 We ourselves are wave-excitations, too, and because of this, the world appears Poincaré invariant to us, without a preferred rest system. Within such a picture, in an expanding universe, L and E will vary with time (and so will h and G as well as all particle physics constants), and the next step is to make the most straightforward ansätze for these variations. First of all, when the universe (=the elastic medium) expands, the variation of the Planck or bond length L(t) must reflect the general expansion as described by the FLRW expansion parameter a(t). Eq. (7) relies on the simple assumption that on the average the bond length between 2 tetrons is always proportional to the scale parameter, i.e. a ∼ L or equivalently a(t) a0 = L(t) L0 (8) At first sight this equation may not seem reasonable to all readers. After all, the FLRW scale a(t) describes changes at cosmological distances, while the bond length 1In the tetron-model[18] our universe is embedded in a higher-dimensional space, and as an elastic medium it can thus acquire the full 3+1 GR curvature within this space, including the timely curvature related to expansion. 4 L(t) between tetrons is microscopic in origin. The idea behind (8) is depicted schematically in Figure 1. The 2 lattices represent the expanding universe at 2 times t2 > t1. a(t) corresponds to the full extension of the lattice, while L(t) is the lattice spacing, i.e. the distance between 2 nearest neighbors. In Figure 1 the ratio a(t2) : a(t1) is given by 2. The same is true for the ratio L(t2) : L(t1), because the average bond length between the lattice points grows in the same way as the lattice as a whole. This simple consideration is at the heart of (8) and also (through the FLRW equation) of (7). Thus, the first term in (7) arises from the general relativistic deceleration of the universe through its matter content ρ, while the second term accounts for the dark energy phenomenon, however, not quite in the usual form of a cosmological constant (indicated in green, not utilized in the present work), but of a harmonic force −ω2(L − Ls), that expands the elastic medium towards an equilibrium value Ls of the bond length L. Eq. (7) tells us that linear forces are acting, one induced by (ordinary and dark) matter and driving the system towards L = 0, the other induced by the ('dark energy') tetron binding and driving it towards the equilibrium binding distance Ls. Presently we are in the region L0 < Ls, so that −ω2(L − Ls) really is an expanding force. The value of ω can and will be determined from a fit to dark energy measurements. In the course of time, i.e. with increasing L, the matter force becomes smaller because the matter density dilutes according to ρ = ρ0L 3 0/L 3. This is a well known effect and makes the first term on the RHS of (7) behave like ∼ 1/L instead of ∼ L. The differential equation (7) can be solved using initial values L(0) = L0 L(0) = H0L0 (9) where L0 is the (present day) value of the Planck length and H0 the Hubble constant (=present day value of the Hubble parameter H(t) = ȧ/a = L/L). The solution will be given later in (20). From the initial conditions it is immediately clear that ω is naturally of the order of H0, In section IV this will be confirmed by fitting with observations. ω and H(t) are extremely small frequencies corresponding to an approximately harmonic movement 5 of the universe as a whole and a priori have little to do with the Planck frequency 1/TP which is the local response frequency of a single tetron in the elastic medium. H0 TP ≈ 1.18× 10−61 (10) So seemingly, there are 2 very different fundamental scales in the universe: one is the single tetron binding energy/Planck energy E and the other is the collective dark energy of the universe as a whole, which drives it to its equilibrium value. However, this collective drive is just a reflection of the microscopic tetron binding energy having a minimum at bond length Ls. Therefore, although the values of E and H are vastly different, the time dependencies E(t) and H(t) are connected. See eq. (18) later. In other words, due to the homogeneity of the elastic medium, the time behaviour of the microscopic tetron energy E(t) and that of the cosmological frequency H(t) can be related, as will be seen in eq. (18). III. Quantities-of-Motion Equation If one thinks it over, a time dependent L(t) has long been observed, namely in the form of the cosmolgical redshift. Usually this time dependence is not put into L, G or h, as in eqs. (5) and (6), but into the redshifted photon frequency f and the expansion paramter a. This is possible, because these quantities always appear in products h*f and G*a, respectively. So one can choose whether to absorb the time dependence of L in h and G or in f and a. The conventional choice is to keep G and h constant. We shall follow this choice as far as the variation of the Measure-of-Space equation is concerned. From this point of view, the ansatz of a time dependent L(t) is not so much new [apart from the modified cosmological constant approach to dark energy with −ω2(L− Ls) instead of a Λ-term]. As for the time dependence of the Planck energy E, the situation is different, i.e. there will be something new: E can be interpreted as the binding energy among the constituents of the elastic medium which is our universe. Not too far away from the equilibrium L = Ls it has 6 Figure 2: The binding energy E of 2 constituents as a function of their bond length L. At present t=0 one has the Planck energy E0 and the Planck length L0. The expansion of the universe through dark energy corresponds to the elastic bonds expanding towards equilibrium values Es and Ls. In the neighbourhood of Ls the quadratic dependence E(L) of eq. (12) is a good approximation. a quadratic dependence on L E(L) = C +D(L− Ls)2 +O(L− Ls)4 (11) The constants C and D can be determined from the conditions that E(L0) = E0 and E(Ls) = Es. One obtains E(L) = Es − (Es − E0)( L− Ls L0 − Ls )2 = Es[1− (1− E0 Es )( 1− L/Ls 1− L0/Ls )2] (12) As will turn out, the energy difference E0 − Es triggers the harmonic dark energy term ∼ ω2 in eq.(7), i.e. the accelerated expansion of the universe. The approximate behavior of E(L) is that of a parabola and together with the solution L(t) to (7) one deduces the time dependence E(t) as needed in eqs. (5) and (6). Since we have absorbed the factors L(t) in eqs. (5) and (6) into the redshift description, we only have to consider time dependencies according to h(t) ∼ E(t) G(t) ∼ 1/E(t) (13) or equivalently h(t) = h0 E(L) E(L0) G(t) = G0 E(L0) E(L) (14) 7 with E(L) to be taken from (12). Considered as a binding energy, E(t) is negative, so one should better write h(t) ∼ |E(t)| and G(t) ∼ 1/|E(t)|. Since E(t) is negative and presently becomes more negative as it approaches its minimum value Es, one concludes that Plancks constant presently goes up with time, whereas the gravitational coupling is decreasing. At this point one may worry, whether a varying E has a problem with energy conservation. Actually, this question also arises in connection with the redshift, and is usually answered by saying that energy 'goes into the metric'. Interpreting the universe as an elastic medium one can reformulate this by stating that energy goes into the total binding energy of the universe. Setting aside the problem of overall cosmological energy conservation[20], one can at least attribute the ω-term in (7) to an 'energy' W (L) = ω2s 2 − ω 2 2 ( L− Ls L0 )2 = ω2s 2 [1− (1− ω 2 0 ω2s )( 1− L/Ls 1− L0/Ls )2] (15) with ω2 = ω2s − ω20 (1− Ls/L0)2 (16) Note the similarity between (12) and (15). Since the dark energy phenomenon is a smooth collective effect of all tetron binding energies E having a minimum at bond length Ls, i.e. the behaviour of W is a reflection of the tetron bond length driving towards its equilibrium value Ls (=the point where the tetron binding energy E is having a minimum value Es), the time evolution of W and E is absolutely parallel, in an analogous way as the time evolution of a(t) is parallel to that of L(t). In other words, E(L) ∼ W (L) holds similarly as L(t) ∼ a(t) for the cosmic scale factor a and the bond/Planck length L, cf. (8) and Figure 1, and one comes up with W (L) W (L0) = E(L) E(L0) (17) The physical difference between W and E is that -E is the microscopic tetron binding energy and is roughly of the order of the present day Planck energy to be measured in Joule. 8 -the ω's are frequencies of the universe as a whole and measured in Hertz, and they are of the order of the Hubble parameter. A direct consequence of (17) is ω20 ω2s = E0 Es (18) IV. Comparison with Astrophysical Data In the laboratory it is more or less impossible to observe time variations of G and h, because via (1) these quantities define our rulers for mass and energy. While the universe expands, the rulers will expand, too. In case of the redshift, astronomers were able to obtain relevant information on L(t) from observations of distant galaxies. In contrast, it seems difficult to measure the time variation (12) of energy from such observations, because any process, which took place in the past in some distant galaxy, will do so with the energy/rulers relations valid at that time, and when the produced particles arrive on earth they will interact with the detectors with the energy/rulers relations valid now; so that the observer will see no difference between processes now and then. As a consequence, time variations of h and G will generally not be visible. Not testable in particle processes, it turns out, however, that E(t) from eq. (12) can be directly observed in dark energy measurements. Dark energy observations do not usually concern the very early universe, so that the parabolic approximation (12) should be good enough2. They are in effect testing eq. (7), and E(t) in (12) not only governs the ω-term but according to (14) and (13) also enters the G-term on the RHS of (7). In order to check this idea with astrophysical data, we go over from L(t) to the redshift z defined by z(t) = a a0 − 1 = L L0 − 1 (19) 2For considering time variations of h and G in the very early universe, an approximation of the form (12) is not sufficient, because at small bond length L a typical binding energy is expected to be governed by a power behaviour of the form E(L) ∼ L−n. 9 The most precise measurement of the dark energy effect comes from the study of type-Ia supernovae in distant galaxies. I shall compare my redshift prediction to those data in a small-t approximation. This is justified because on cosmic scales the times involved are not too large. Under this condition, up to O(t4), the solution to (7) can be written as z = tH0 + t2H20 2 [−Ω 0 M 2 + ω20 H20 Es EP − 1 Ls L0 − 1 + Λc2 3H20 ] + t3H30 6 [Ω0M(1 + Es EP − 1 Ls L0 − 1 )− ω 2 0 H20 Es EP − 1 (Ls L0 − 1)2 + Λc2 3H20 ] (20) The term indicated in red is the contribution from the time dependent Newton constant, the terms in blue come from the harmonic dark energy ω contribution, and the terms in green from a cosmological constant (the latter to be ignored in the present model). Ω0M = 4π 3 G0ρ0 H20 (21) is the present day density parameter of matter in the universe, frequently used in this type of analysis. In the dark energy interpretation with a cosmological constant it comes out as roughly 0.3, which is usually considered a reasonable value. As for any parabola, hidden in the parabolic dark energy (12) and (15) are 3 parameters, which need to be determined from observations. They may be chosen as (i) Es E0 = ω 2 s ω20 > 1 = the ratio of the Planck energies resp dark energies at cosmic equilibrium and at present (ii) Ls L0 > 1 = the ratio of the tetron binding lengths at cosmic equilibrium and at present (iii) ω20 H20 = the ratio of the present dark energy over the present value of the Hubble constant. Since there are more parameters than in the ansatz of a cosmological constant, the observations will only give relations between i, ii and iii. Furthermore, an estimate for Ω0M has to be taken from other sources. Nevertheless, our next aim is to see what the observations allow to say. A fit to the redshifts of supernovae yields[21] z = tH0 + t2H20 2 (1.00± 0.05) + t 3H30 6 (0.54± 0.05) (22) 10 Comparing with (21) one finds that it is easy to accommodate the data with the help of the quantities i, ii and iii. For example, choosing Ω0M = 0.3 and -Ls = 10L0 one obtains Es = 1.34E0 and ω 2 0 = 3.4H 2 0 -Ls = 2L0 one obtains Es = 5.6E0 and ω 2 0 = 0.25H 2 0 At first sight, the fact that data can be fitted this way so easily, seems to be a big surprise. After all, we are fitting numbers which usually are explained with an expontential increase due to a cosmological constant. The essential feature here is the contribution from the time variation of Newton's constant (red) which in combination with the harmonic dark energy contribution (blue) leads to an agreement with observations. The point is that since G(t) is going down with time, the retarding effect of (ordinary and dark) matter becomes smaller, and no exponential increase of the dark energy term as in the cosmological constant approach is needed. In other words, although the harmonic force ansatz corresponds to a more moderate re-acceleration of the universe than the cosmological constant term, this is compensated by the time variation of energy as a whole which affects Newton's constant. V. Cosmic Time Dependence of Particle Physics Parameters The analysis will now be extended to the 'constants', which describe the particle physics interactions. All parameters of the Standard Model (SM) of particle physics will be considered, i.e. -the 3 dimensionless gauge couplings: the weak and electromagnetic fine structure constants αweak and α together with the QCD scale parameter ΛQCD. -the 2 parameters of the Higgs potential: the Higgs mass mH and the vacuum expectation value v of the Higgs field. Note that using v is equivalent to using the Fermi coupling GF = 1/[ √ 2v2], and the quartic Higgs coupling is given by λ = m2H/v 2. -the Yukawa couplings, which are all proportional to v. Except for αweak and α, all these parameters have dimension of energy. If one looks at the definition of the fine structure constant α = e2 4πε0~c (23) 11 it is the only dimensionless combination which can be built from the quantities e2/ε0, h and c. As dimensionless, it is independent of the choice of rulers for time, length and energy3. This is good news, because in looking for a cosmic time dependence of α one circumvents all the problems usually encountered in determining the time dependence of dimensionful quantities like E(t). The bad news in considering ratios like α is that most effects tend to drop out between numerator and denominator (see later). An interesting point is that although α itself is not an energy, it can be written as a ratio of forces or energies. Namely one can rewrite (23) as α = e2 4πε0r(2) / G0M 2 0 r(2) (24) i.e. as the ratio of the electrostatic Coulomb (force) energy and the gravitational (force) energy of 2 point particles with elementary charge e and Planck mass M0 at an arbitrary distance r. From this point of view the gravitational force is by no means small as compared to the electric force, but for such tetron-like test particles is 137 times stronger! The key relation here is ~c = G0M20 = L0E0 (25) which follows from (2). Defining Q2 = e2/[4πε0] and introducing time dependencies, one has α(t) = Q2(t)/L(t) E(t) (26) whereQ2 comprises the electromagnetic effect in a measurement-system independent way. Obviously, Q2 has the dimension of length×energy. Since measurements and astrophysical observations show almost no time variation of α, the time dependence of Q2/L must be the same as that of E(t) to a very good approximation. Referring once again to the tetron model, this has to do with the fact, that the time dependence of Q2 is determined by that of the binding energy E(t)[18], so that any time dependence of α drops out between numerator and denominator in (26). 3The dependence on the Planck/tetron binding energy E is not to be confused with the Wilsonian running of coupling constants, i.e. the dependence of α on the energies of particles in a scattering process. 12 To understand this point in detail, one should note that the tetron model is more than a microscopic theory for the cosmic elastic medium. The tetrons actually appear in the form of tetrahedrons which extend into a 3-dimensional internal space and whose excitations can be shown to represent the complete 3-family quark and lepton spectrum[18, 19]. The internal interactions among tetrons are typical quantum interactions in the sense that one always has 'exchange' energies in addition to 'direct' energies, simply because for 2 (or more) identical particles tetrons in this case with single wave functions f1 and f2 their total wave functions are either symmetric or antisymmetric of the form f1(x1)f2(x2)± f1(x2)f2(x1). Correspondingly, the relevant 2-point function of the tetron Hamiltonian can be described as the sum of the Planck(=binding) energy E(t) and a function J(t) usually called the exchange energy. In the present case it may be called 'internal exchange energy' because it arises as an integral including the internal space, in which the tetrahedrons are living. E = ∫ d6x1 ∫ d6x2f1(x1)f2(x2)V (1− 2)f1(x1)f2(x2) (27) J = ∫ d6x1 ∫ d6x2f1(x1)f2(x2)V (1− 2)f1(x2)f2(x1) (28) where the integrals are actually 6-dimensional, because they extend over both internal and physical space. V(1-2) is the potential between 2 tetrons with wave functions f1 and f2. 4 In a 6-dimensional environment the Green's function of the Laplace operator is r−4, instead of r−1 in the 3-dimensional case. Therefore, the most promising choice seems to be V (1− 2) = N |x1 − x2|4 (29) 4If one looks into the details of the tetron model[18], the situation is a bit more complicated than described here. First of all, f1 and f2 are the wave functions of tetron-antitetron pairs, and V(1-2) is the potential between these 2 pairs. Secondly, to really calculate E and J from the 6-dimensional integrals one has to take the configuration of 2 adjacent tetrahedrons with at least 8 tetrons into account. Furthermore, there are actually 2 types of exchange integrals, one corresponding to the inter-tetrahedral interactions, which gives rise to the Fermi scale and is responsible for the large masses mt, mW and mH of order 100 GeV, and another one corresponding to the inner-tetrahedral interactions, which gives rise to the lighter fermion masses and the QCD scale. 13 with some coupling constant N. A rough estimate of N can be obtained by equating V(1-2) at the Planck length to the Planck energy. This gives a value for the fundamental tetron coupling N: N L40 ≈ E0 =⇒ N ≈ 10−130 m6kg s2 (30) When trying to calculate E and J according to (27) and (28), one naturally runs into the so-called hierarchy problem of physics. Namely the question, why the relevant energy scales of gravity (E0 ≈ 1019GeV) and of particle physics (J0 = 1− 100GeV) are so much different. In the framework of the tetron model, the question can be reformulated: why is the exchange energy J so much smaller than the direct energy E? Looking at (27) and (28), one sees that J ≪ E can happen, if the tetron wave functions are strongly localized. In the extreme case of delta functions one even finds, that the exchange integral vanishes, while the direct integral attains the value (30). Such an extreme localization is of course unnatural. In order to get J ≈ 10−17E, it is enough to demand that f(x) drops from its maximum value at x = 0 by about a factor of 10 at x = L0. This is because J is a multidimensional integral and to integrate the product dx2f2(x2)V (1 − 2)f1(x2) will give a suppression factor of roughly ∼ 0.1 for each of the 6 dimensions. Similarly for the x1-integration. Except for α, which is constant, I will argue that J(t) gives a universal time dependence for all internal/particle interactions in a similar way as does the Planck energy E(t) for the spacetime quantities of motion. In other words, while the time-dependence of all dimensionful spacetime quantities is dictated by E(t), the time dependencies of dimensionful SM particle properties like v, mH , mW and all quark and lepton masses can be described in terms J(t). To see how this works in detail, one should relate J to the electroweak symmetry breaking scale. This was already done in [18], where it was shown that the critical energy of the electroweak phase transition is given by an exchange integral J of the form (28). This is because in the tetron model the electroweak phase transition corresponds to an alignment of the tetrahedrons in the internal spaces, and the Curie energy of this phase transition is given by J. Since the critical energy of the 14 electroweak phase transition is approximately given by the Higgs vev v, one has v = J or, equivalently GF (t) = 1√ 2J2(t) (31) It is well known that all particle masses in the SM are proportional to v. Therefore, J enters all dimensionful parameters of the electroweak SM the fermion masses, the Higgs vev and the masses of the weak gauge bosons in a linear way. All these quantities are ∼ J(t). Just as for E, the time-dependence of J arises through the time variation of the bond length L(t), i.e. through the expansion of the universe. If one would calculate the integrals J and E (27) and (28) as a function of L, then knowing L(t) according to (7), one could deduce from that the time-dependence of v, of the Fermi constant (31) and of all other parameters, and compare it to present upper limits[22, 2]. Unfortunately, the situation is not that simple. First of all, as mentioned in footnote 4, the integrals are difficult to calculate. Secondly, in everything we do, in every experiment we undertake, we encounter the Planck energy E(t) as a ruler, whose time dependence influences our perception of dimensionful quantities like v, GF , mW and so on. To say it plainly, the time dependence we can perceive is not that of J(t) but that of the ratio J(t)/E(t). This means: if we consider, for example, a matter particle with mass m0 in the present epoch, our perception of the time development m(t) of m0 does not follow 5 m(t) = m0 J(t) J0 (32) but m(t) = m0 J(t)/J0 E(t)/E0 (33) In the ideal case, that J and E would have an identical time dependence, the time dependence of m or of a dimensionful SM parameter like (31) could never be measured. 5Note there is no problem with the principle of equivalence because the heavy mass and the inert mass are both developing with J(t). 15 By analyzing the structure of the direct and the exchange integrals E and J in some detail, one can indeed show, that their dependence on the bond length L is quite similar, both with an extremum at nearly the same value Ls. Making an ansatz for J(L) analogous to that for E(L) in (12) J(L) = Js − (Js − J0)( L− Ls L0 − Ls )2 = Js[1− (1− J0 Js )( 1− L/Ls 1− L0/Ls )2] (34) one sees that the crucial part is the ratio J0/Js. To the extent that the equality Js J0 = Es E0 (35) holds, a time dependence of SM parameters cannot be measured. Conversely, any observed time dependence in a SM parameter can be traced back to a deviation from (35). The integrals J and E can be analysed on a qualitative level, and according to this analysis the relation (35) is approximately true. On the other hand there is no a priori reason, why it should be exactly true. First of all, the integrals (27) and (28) are definitely distinct. Secondly, particle physics interactions have to do with inner symmetries not contained in the energetic analysis of the elastic universe [governed by E(t)]. Therefore, although present observations only give upper limits on time dependencies of SM parameters, their cosmic time dependence at least in principle follows its own rule, given by J(t). VI. Discussion In this study a theory concerning the time dependence of all known fundamental physical parameters has been developed. It rests on the idea that dark energy is a harmonic rather than an exponential effect, which is furthermore related to the binding energy of the underlying constituents of the universe. As has been shown, one is led to a time-dependence of Newton's and Planck's constant. These effects, however, are usually impossible to measure except in the dark energy itself and in certain paricle physics properties. Furthermore -microsopic (L) und cosmic (a) length scales are connected in a simple linear kind 16 of way ('the universe expands in the same manner as the tetron bonds expand'), cf. eq. (8) and Figure 1. -In an analogous fashion, Planck energies E(t) and dark energies ω are linearly related ('the total dark energy of the universe increases proportional to the single tetron binding energy') via (17). Since the universe is rather cool by now and apparently expands in a rather homogeneous way, these assumptions are expected to be very good approximations. This expectation was substantiated in section IV by proving that it leads to agreement with present day dark energy observations. Thereby it has turned out that there is a significant contribution to the observed dark energy effect from the time variation of Newton's constant. Since G(t) is going down with time, the retarding effect of ordinary matter becomes smaller, and no exponential increase of the dark energy effect as in the cosmological constant approach is needed. The Planck energy E0 and its time-dependent generalization E(t) play a central role in the considerations presented here, see (5), (6) and (12). Actually, E0 has been used in this paper with 2 meanings: -it represents the gravitational energy of the interaction of 2 matter particles with Planck mass M0 at Planck distance L0, i.e. E0 = G0M 2 0/L0. -it describes the binding energy of 2 tetrons bound at distance L0. Concerning the fundamental parameters of particle physics, it was shown that they depend on cosmic time via the internal exchange function J, whose dependence on L is similar but not exactly the same as that of E. With the advent of higher precision observations, this effect may become observable. A remaining problem is the calculation of E (Planck energy) and J (internal exchange energy) from first principles, i.e. from fundamental tetron interactions. Another problem is the question of energy conservation in a theory with a varying G(t). Energy conservation is an uneasy business in GR anyhow[20], but assuming a time varying G makes the set of the ordinary FLRW equations 1 2 ȧ2 − 4π 3 Gρa2 = 0 (36) aä+ 1 2 ȧ2 = 0 (37) which comprises a 'force' equation for ä and an 'energy' equation for ȧ2/2, incon17 sistent. [The FLRW equations have been written down here in a simplified form taking Λ = 0, p = 0 and k = 0]. (36) and (37) are 2 differential equations for one function a(t) and are only consistent, as long as the product Gρ behaves like ∼ a−3 corresponding to a uniformly diluting mass density and no variation of the Newton constant at all. The underlying reason is that Einstein's theory itself relies on a constant, timeindependent G. This has to do with the fact that it is a theory for a medium with curvature whose basic properties and couplings do not change when the medium expands. For the large expansion factors, however, which we encounter on the cosmic scale, such an assumption seems unrealistic. In order to solve the conflict between (36) and (37) in case of a varying G, I am therefore retreating to the point of view that a Hooke-type 'force' ä = −4π 3 G(a)ρ(a)a (38) is induced on the elastic medium by a matter density ρ, with non-constant coefficients ρ(a) ∼ a−3 and G(a) ∼ 1/E(a), and use this as the basic starting point for (7). In a similar way as it does not allow a time-dependent G, general relativity does not include an ω-term like in (7). In other words, the harmonic expansion describing the behavior of the elastic medium for L → Ls is not part of Einstein's theory. This is not a big surprise, because GR is a theory of local curvature induced by energy-momentum and does not know about the equilibrium of the unterlying elastic medium. References [1] P. A. M. Dirac, Nature 139, 323 (1937); Proc. Roy. Soc. A165, 199 (1938). [2] see the review by J.P. Uzan, Living Reviews in Relativity 14 (2011) 2. [3] M. J. Duff, L. B. Okun and G. Veneziano, JHEP 0203, 023 (2002). [4] J. W. Moffat, Int. J. Mod, Phys. D2, 351 (1993). 18 [5] J. D. Barrow, Phys. Rev. Lett. 82 (1999) 884. [6] J. D. Bekenstein, J. D. Phys.Rev. D25 (1982) 1527. [7] J. Magueijo, Rept. Prog. Phys. 66 (2003) 2025. [8] J. K. Webb et al. Phys.Rev.Lett. 87 (2001) 091301. [9] J. A. Belinchon et al., Int. J. Mod. Phys. D12 (2003) 1113. [10] J. C. 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