43 2025.06.16 2025.11.25 2025.11.26 article M. Ljubičić (Amenoum)108. brigade ZNG 43, 35252 Sibinj, Croatia (completerelativity.org)mljubicic99{EAT}gmail.com Solving cosmological anomalies though massive graviton-photon coupling. physics gravity, graviton, photon, anomaly, dark energy, Hubble tension /authors/Amenoum.html#credits 1 Abstract All deep-space probes seem to experience small anomalous acceleration. Although original thermal modelling of Pioneer spacecraft effectively ruled it out, newer modelling suggests that the anomalies do arise from the anisotropic thermal radiation, mainly associated with radioisotope thermoelectric generators. However, although the results are convincing, alternative solutions are not ruled out. Here, it is proposed that the thermal anisotropy is generally only a part of the solution and may not be a dominating part in all cases. The other contribution is hypothesized to come from a non-zero rest mass associated with gravitational or gravitational-like coupling. The exact solution for the spacecraft anomalies is presented, in remarkable agreement with measurements, and with the original thermal modelling. In addition, the solution provides an alternative interpretation of the Schwarzschild radius of the observable universe, explains Hubble tension and could potentially explain other cosmological anomalies. Introduction The analysis of motion of deep-space probes shows anomalous constant weak long-range acceleration on the order of -10^-10 m/s² (deceleration relative to the Sun). A solution has been proposed in the form of anisotropic thermal radiation and it has even been claimed eventually that that this could completely resolve the Pioneer anomaly, by different authors. However, although convincing, the result is not absolutely definitive and it is questionable to what degree the new models match physical reality. A large uncertainty exists in RTG surface degradation, for example. Although apparently explained within uncertainties, thermal estimate of about 80% of the Doppler-inferred acceleration is not a perfect match. The apparent onset of the anomaly only once the giant planets have been reached (unexplainable by thermal anisotropy) is assumed to be a result of mismodelling of the solar thermal contribution, but this is not certain. The observed diurnal/annual oscillation in the anomalous acceleration (unexplainable by thermal anisotropy) is also assumed to be the artefact of mismodelling (including that of the spacecraft orientation). The best fit to the original data is a temporally constant anomaly, while the thermal anisotropy implies temporal decay of the anomaly. Although the decay cannot be ruled out - especially with the extended Pioneer data sets, there's a good probability that the obtained marginal signs of decay are a result of mismodelled solar radiation pressure. The newer thermal modelling is significantly different from the original modelling (which effectively ruled out thermal anisotropy), and others have criticized it (e.g., for the extensive parametrisation, potential bias, reliance upon an inferior quality extension of the highest quality data set to overturn the consensus view of a [mean] constant anomaly, the fact that only the "acceleration" value is considered, etc.). Considering the involved assumptions, uncertainties, criticism, and the potential scientific value of a different explanation, alternative solutions should not be easily dismissed. Here, an exact solution in the form of a graviton-photon coupling is proposed, a solution that could, in addition, explain some other cosmological anomalies. The hypothesis Photons in motion across the universe experience a frequency redshift, which is assumed to be a consequence of negative pressure of dark energy. The universe is also filled with positive pressure energy, which, like dark energy, is homogeneous and isotropic on large scales. This pressure is, however, assumed to significantly affect photons only as they pass through strong gravitational wells. I propose that the positive pressure has the equivalent effect on photons as dark energy on large scales (effective constant acceleration, but differently signed), producing a frequency blueshift in photons. Total coupling is then a superposition of positive and negative pressure coupling so the observed effect on frequency is a superposition of blueshift and redshift. If the positive coupling is larger the coupling has a range - equal to reduced Compton wavelength, per Yukawa formalism: $\displaystyle r = {\hbar \over {m c}} \tag{1}$ where ℏ is the reduced Planck constant, c is the standard vacuum speed of light, and m is the coupling carrier rest mass. Here, the mass m should be a sum of two parts, one associated with positive pressure coupling, and the other associated with negative pressure coupling (which should be negatively signed in this context). I propose that the effects of redshift and blueshift are a result of constant effective radial acceleration equivalent to positive and negative Newtonian free fall over the range, respectively: $\displaystyle a = - {1 \over 2} {c^2 \over r} = - {1 \over 2} c^2 {c \over \hbar} m \tag{2}$ where r is the coupling range and rest mass m is a superposition of positive and negative coupling rest mass. This can then be expressed in terms of clock acceleration: $\displaystyle a_t = {a \over c} \tag{3}$ and from this the frequency drift is obtained as: $\displaystyle \dot f = f_0\, a_t \tag{4}$ with f₀ representing the original/expected communication frequency (e.g., in case of Pioneer probes, the S-band is used, ~2.1/~2.3 Ghz for uplink/downlink, respectively). Now, different interpretations are possible of the acceleration. Here, I do not propose that the photon is changing velocity during motion, but one plausible explanation is trajectory curvature - the blueshift being a consequence of increasing curvature, redshift a consequence of decreasing curvature or flattening (conventionally interpreted as expansion). Different interpretations of the curvature are possible as well. One can interpret it as virtual, however, as it will be shown later, it appears that for the hypothesized mass eigenstates the positive rest mass solution yields a range equal to the Schwarzschild radius of the mass-energy enclosed by the wavefront. This strongly suggests that the energy distributed across the propagating spherical wavefront is sensitive to the enclosed mass-energy (with the assumption of homogeneous/isotropic distribution, per the shell theorem, the outer mass would have no net effect), but the strength of coupling to both, positive and negative pressure, is running (being inversely proportional to scale - wavefront radius, or the enclosed mass-energy). Conventionally, for a virtual (off-shell) particle, the range is interpreted as distance at which the associated field "dies out", when the particle has to annihilate or recombine (although the process here is assumed to be purely mathematical). The range is, thus, considered a consequence of the limited lifetime of the carrier particle (correlated with the uncertainty principle). For a real particle, the range is interpreted as a distance at which its observable probability magnitude (wavefunction) "dies out". I propose, however, that the coupling of wave-like force carrier particles (and field disturbances such as real gravitons/photons) is generally a superposition of coupling to negative and positive pressure, where imbalance can lead to confinement. Thus, in case of dominating positive pressure coupling even photons can be confined, and instead of "dying out", the non-absorbed photon should bounce upon reaching the range - equivalent to the bounce of a photon circle/sphere near an event horizon of a black hole. In any case, the negative radial acceleration of photons (or, more precisely, the associated blueshift due to curvature) will then be misinterpreted as a decrease in velocity (Doppler shift) of the moving source of emission if the rest mass of the coupling is assumed to be zero but is instead larger and positive.
With the proposed coupling, the onset of the anomaly at particular distance can be explained with settling of positive and negative parts of associated rest mass in different mass eigenstates, so the positive and negative pressure couplings are imbalanced. The transition to different eigenstates should be correlated with the transition in conditions - e.g., from the density/pressure of the Solar System to that of the galaxy, or the cosmological vacuum. Interestingly, the anomaly onset seems to occur as the spacecraft themselves decouple from the Solar System (pass from an elliptic orbit into an hyperbolic escape trajectory, following a gravity-assist Jupiter/Saturn flyby). In fact, since the hypothesis involves coupling of photons to gravity, the anomalies could occur with changes in gravitational coupling in general. Note that anomalous kinetic energy has been detected in several spacecraft even during gravity-assist Earth flyby. One can thus assume that emissions associated with stable gravitationally bound objects in the Solar System are coupled with zero total rest mass (balanced positive/negative pressure coupling), or, the rest mass in these cases is simply too small for the effect to be detectable. The detected annual periodicity could then be explained by the motion of Earth about the Sun, as this is the place of photon absorption. Since the distance to the spacecraft is oscillating annually due to Earth's motion, the maximal radius the photon can have (and associated curvature) before absorption is oscillating as well - hence the annual oscillation of the anomaly (this correlation, however, is not further analysed in this paper). Note that the favoured interpretation provided here (sensitivity to enclosed mass) also goes in favour of physical interpretation of the wavefunction associated with the photon - the collapse of the wavefunction is the collapse (localization) of the mass-energy forming the propagating spherical waveform to the point of absorption. Note also that limited ranges in this interpretation imply limits on the observability of the universe regardless of the dominating pressure - in case of dominating positive pressure the photon is confined, in case of dominating negative pressure the positive radial velocity at some point effectively exceeds the speed of light. Conventionally, the Schwarzschild radius of the observable universe is interpreted as a mathematical coincidence - it can be shown easily that, for a spatially flat expanding universe (where the observable radius is a Hubble radius), the observable radius is equal to the Schwarzschild radius of the total mass-energy inside the observable universe, which seems to be the case with our universe. The hypothesis presented here, however, doesn't require an expanding universe for the observable radius to be a Schwarzschild radius, although the two interpretations are not mutually exclusive. Interpretations of constant acceleration How to interpret the constant acceleration? As noted before, one possibility is a running gravitational coupling (note that the photon is changing scale during propagation). E.g., with the gravitational constant G proportional to 1/r and enclosed mass M proportional to r³, the acceleration becomes constant, effectively independent of the radius r. One explanation for this is the dependence of dimensionality of gravitational coupling on the dimensionality of interacting bodies. The photon is 2-dimensional (spherical surface), while the enclosed mass is 3-dimensional, resulting in the 1/r (r²/r³) ratio. For localized particles/bodies the ratio would be 1/1. Note, however, that this implies asymmetry in coupling - the force from the photon acting on the enclosed mass would have a G proportional to r, not 1/r. While the asymmetry cannot be ruled out (note that the asymmetry is obvious in more complex forces - one person, for example, can be attracted by another, but at the same time the latter may be repelled by the former), other explanations are possible. Note that the anomalous acceleration is on the order of 10^-10 m/s², the same order as the inferred values of the critical a₀ parameter in MOND - below a₀ the gravitational acceleration becomes proportional to 1/r. MOND however is a pure mathematical description of measurements, providing no explanation for the cause of transition. Another solution is the alternative distribution of photon's gravitationally interacting mass. If this mass would be concentrated in a ring on the spherical wavefront , the enclosed mass would grow with r² and the acceleration remains constant. This explanation doesn't require a running G for the constant acceleration (under the assumption of constant density): $\displaystyle a = {GM \over r^2} = G {{r^2 \pi \rho} \over r^2} = G \pi \rho \tag{5}$ However, this solution doesn't seem to be compatible with observations. In any case, assuming invariant G, obviously the range, and thus the coupling rest mass, depend on the enclosed mass-energy density. The question is, however, is the rest mass changing dynamically - as the density is constant only on larger scales? Here, under that assumption, angular momentum would be conserved (as increase in rest mass is equal to decrease in range), but energy conservation would imply exchange of energy with the traversing medium. Alternative - and more easily acceptable - solution is the dependence of rest mass on initial conditions (at the point of emission) only, with either a dynamic acceleration on smaller scales or a changing coupling strength (with changing density) to conserve the fixed acceleration across all scales. Instead of the ring-like form of the gravitationally interacting photon mass, however, the same effect (proportionality of interacting non-photon mass to r²) can be produced assuming the acceleration depends on the mass traversed by the wavefront, rather than the total mass enclosed by it. Note, however, that, even if G is invariant here, it is not equal to the Newton's gravitational constant, as the coupling strength at the start of propagation must be significantly higher to produce the observed acceleration. Limiting the G to the Newton's gravitational constant, the correct order of acceleration can only be produced from the gravitational acceleration of the total mass enclosed by the radius of the observable universe (or, more precisely, by the particle range). In that case, one would have to accept the notion that the photon is, upfront, aware of the total mass within its range and is reacting to that mass from the start. While the coupling rest mass/range is known upfront, it is not intuitively easy to correlate the total enclosed mass within the range with the conditions at the point of emission. Thus, the photon is probably reacting to the density of enclosed or traversed mass-energy at its current radius, with the strength of coupling changing in such way to ensure a radial acceleration that conserves the mass-range relation - or to ensure that the range is effectively a Schwarzschild radius. The solution and discussion In established quantum physics photon is assumed to have zero rest mass, just as the hypothetical graviton. However, this doesn't have to be the case in reality. Localized photon mass in local experiments is on the order of 10^-50 to 10^-54 kg. This is usually interpreted as the upper limit on its rest mass. However, it can also be interpreted as the effective photon mass from a specific reference frame (associated specific density/pressure). Similarly, the upper limit for the graviton rest mass is on the order of 10^-59 kg. In any case, if the carrier of gravity or electro-magnetic force has rest mass, the associated potential is a Yukawa potential and both, the carrier and field disturbances, have a range, per Eq. (1). Assuming the range should be at least equal to the observable universe for cosmological photons/gravitons, this translates to the upper limit on the order of ~10^-69 kg. Note, however, that this should be interpreted as an upper limit only under the assumption that the universe is expanding. If the universe would be static but with the observable radius equal to the range associated with this mass, then this mass becomes the actual rest mass, not the upper limit. Others have calculated the photon's rest mass, with the assumption of dS vacuum and a Ricci scalar of 4Λ (where Λ is a positive cosmological constant), to be ≈2 × 10^-69 kg. Using matter density and pressure of the Solar System (Sun magnetosphere) in the Ricci scalar instead, and a zero cosmological constant, one obtains photon mass of ≈2 × 10^-72 kg. The best estimates for the rest mass of cosmological gravitons are also on the order of ~10^-69 kg. The author has obtained similar values in a different approach (see chapter \chr_derivation_of_p_mass), where 3 mass eigenstates are hypothesized as well. In example, the mass of 6.335 × 10^-69 kg was obtained as the tau equivalent eigenstate, and 1.822 × 10^-72 kg as the lowest mass eigenstate (e). Assuming the positive pressure coupling pair is a tau/e combination, with negligible or zero negative pressure coupling, one obtains the hypothesized radial acceleration (negative Newtonian free fall) of: $\displaystyle a = - {1 \over 2} {c^2 \over r} = - {1 \over 2} c^2 {c \over \hbar} \left( M_{\gamma \tau} + M_{\gamma e} \right) \approx - {1 \over 2} c^2 {c \over \hbar} M_{\gamma \tau} = -8.093 \times 10^{-10} {m \over s^2} \tag{6}$ c = standard vacuum speed of light = 2.99792458 × 10⁸ m/s
ℏ = reduced Planck constant = 1.054573 × 10^-34 Js
M_γτ = tau coupling rest mass = 6.335 × 10^-69 kg which is in remarkable agreement with the observed Pioneer 10 anomaly of -8.09±0.20 × 10^-10 m/s². Similarly, if one assumes the coupling pair is a superposition (sum) of tau and muon eigenstates, one obtains: $\displaystyle a = - {1 \over 2} c^2 {c \over \hbar} \left( M_{\gamma \tau} + M_{\gamma \mu} \right) = -8.574 \times 10^{-10} {m \over s^2} \tag{7}$ M_γτ = tau coupling rest mass = 6.335 × 10^-69 kg
M_γμ = muon coupling rest mass = 3.767 × 10^-70 kg which is in remarkable agreement with the observed Pioneer 11 anomaly of -8.56±0.15 × 10^-10 m/s². The obtained results are also in agreement with the anomalous Galileo acceleration of -8±3 × 10^-10 m/s². Note that, for the lowest mass eigenstate, the contribution to acceleration is relatively negligible (on the order of 10^-13 m/s²). The onset of the anomaly at particular distance can then be explained as transition to higher mass eigenstates (which, however, may be more correlated with the decoupling from the Solar System, rather than with the distance itself). Alternatively, with coupling to negative pressure taken into account (explored in the following chapter) the acceleration prior to the onset may be effectively 0, and this probably is a proper interpretation, as the former may be in conflict with other measurements. However, even though the results above suggest the thermal anisotropy in Pioneer spacecraft is likely to be negligible (in agreement with the original modelling, where the value of 0.55±0.55 × 10^-10 m/s² has been obtained for the thermal anisotropy, with reasons given to consider this an upper bound), higher anisotropy of thermal dissipation in the probes generally cannot be ruled out. In fact, thermal anisotropy could, in some cases, dominate over the proposed effect. Coupling proper thermal modelling with the proposed solution (potentially taking into account the other mass eigenstates), one could possibly obtain results in agreement with anomalies of other probes - e.g., detected Ulysses anomaly of -12±3 × 10^-10 m/s², New Horizons anomaly of -13.2±0.6 × 10^-10 m/s². However, even in these cases, solutions can be obtained without requiring thermal contribution. In example, allowing different combinations of mass eigenstates in virtual particles (see \chr_derivation_of_p_mass for coupling rest mass calculation). In the calculations above only the pairs of equal particles as precursors were used (the precursor for M_γτ, for example, is a virtual tau/tau pair). Using a tau/muon precursor in combination with a tau/tau precursor, one obtains: $\displaystyle a = - {1 \over 2} c^2 {c \over \hbar} \left( M_{\gamma \tau} + M_{\gamma \tau\mu} \right) = -12.380 \times 10^{-10} {m \over s^2} \tag{8}$ M_γτμ = tau/muon coupling rest mass = 3.356 × 10^-69 kg a value in agreement with the Ulysses anomaly. So far, a single pair of particles has been associated with the coupling. However, depending on its nature, multiple pairs may be allowed. Adding a muon/muon pair to the pair above, one obtains -13.343 × 10^-10 m/s², a value in agreement with the New Horizons anomaly.

Note that in the obtained results positive pressure coupling dominates (negative pressure coupling is assumed to be negligible) but this is obviously not generally the case, as for the cosmological photons the redshift dominates. The proposed coupling can potentially explain some other anomalies as well. By the hypothesis, a photon emitted from an celestial object may be bounced back towards the original point of emission upon reaching the range. Since celestial objects are generally in motion, an observer receiving both direct light from the object and reflected light will observe two images of the same object at different points in time, which could then be interpreted as two different but highly correlated objects even if they appear far away from each other. This could explain, for example, the alignment of many quasar polarization vectors over extremely large regions of the sky - billions of light-years apart, even though the quasars are not gravitationally bound. Another potential example is the Huge Large Quasar Group. Here, at least a part of the group may be formed by reflections, so the actually physical group is smaller. However, detailed exploration of the plausibility of the proposed explanation for these particular cases is beyond the scope of this paper. Obviously, reflections cannot act as gravitational lenses and, assuming they are equivalent to reflected photon circles, should exhibit specific polarization. Also, reflections cannot happen in cases where the coupling to negative pressure is stronger than the coupling to positive pressure, as the range is indefinitely extended. \ch_added Evidence in dark energy If the range is correlated not only with the coupling's particle rest mass but also with the enclosed mass what are the implications in a dynamic universe? In an expanding universe, where the density of matter causing positive pressure of gravity is decreasing, the range should be increasing. Assuming that the universe expands at the speed of light at the distance equal to the range, the blueshift will be completely cancelled in regions dominated by dark energy. Thus, any signs of decreasing anomalous acceleration in space probes with distance could indicate increasing effect of dark energy (or the increasing coupling strength to negative pressure). Detected redshift of distant objects, however, suggests not only that the blueshift is cancelled but that the coupling to negative pressure is significantly larger than the coupling to positive pressure. Note, however, that the redshift in this model can also be produced differently. If the universe is contracting instead, the distance between the source of emission and the observer is decreasing with each photon emitted. This can produce a net redshift if the density of energy associated with positive pressure would not increase with contraction - implying loss of such energy. This, however, doesn't seem to be the case in reality. Thus, the universe probably is expanding, as conventionally assumed. The highest rest mass eigenstate, per the hypothesis, is equal to 6.335 × 10^-69 kg. Per Eq. (2), this produces the acceleration of -8.093 × 10^-10 m/s². Expressing this in a value analogous to the Hubble constant, one obtains: $\begin{aligned}\displaystyle A_0 = {a \over r} \sqrt{{2 r} \over a} 3.086 \times 10^{22} {m \over Mpc} &= - {c \over r} 3.086 \times 10^{22} {m \over Mpc} \\ &= - {m_p c^2 \over \hbar} 3.086 \times 10^{22} {m \over Mpc} = -166.61\, km\, s^{-1}\, Mpc^{-1}\end{aligned}\tag{9}$ This is the average radial velocity decrease per Mpc across the whole range. The absolute value here is suspiciously close to the inferred values of the Hubble constant (H₀). This is not surprising if the universe is at the coupling range expanding at the speed of light, but this is also suggesting that the same or similar rest mass may be associated with both, the blueshift and the redshift. Assume now that, as hypothesized, the rest mass is a sum of positive and negative mass eigenstates associated with coupling to positive and negative pressure, respectively. The Hubble constant can then be interpreted as a superposition of expansion and contraction (blueshift, as a product of the proposed model). To produce the superposed velocity at the range equal to the speed of light, the expansion of the universe would have to be equal to twice the calculated value (and thus can be associated with the rest mass that is a sum of two mass eigenstates with each having a mass equal to the already associated mass with positive pressure, albeit negatively signed). Over the same range, the expansion is then: $\displaystyle B_0 = - {{2 a} \over r} \sqrt{{2 r} \over {2 a}} 3.086 \times 10^{22} {m \over Mpc} = - {2 \over \sqrt{2}} A_0 = +235.63\, km\, s^{-1}\, Mpc^{-1} \tag{10}$ And the superposition gives: $\displaystyle H_0 = B_0 + A_0 = 69.02\, km\, s^{-1}\, Mpc^{-1} \tag{11}$ A value that is about the average of the measured values, and is even within uncertainty of some measurements. Note that, with this model, deviation from this value with distance is expectable as there are two competing forces (one producing blueshift and the other redshift) and 3 different mass eigenstates involved. And the measurements, interestingly, seem to clump about 3 values (~67, ~70 and ~73 km s^-1 Mpc^-1). Indeed, doing the same calculation for the muon eigenstate one obtains A₀ of 9.91 km s^-1 Mpc^-1, and H₀ of 4.10 km s^-1 Mpc^-1. Thus, a tau/muon superposition gives a H₀ value of 73.12 km s^-1 Mpc^-1. It should be noted, however, that recent age-bias corrections to supernova cosmology lower the inferred Hubble constant of ~73 km s^-1 Mpc^-1 to a value in agreement with the original calculation (~69 km s^-1 Mpc^-1). If correct, this would decrease the Hubble tension but not entirely - problems with ΛCDM remain. If one now inverts the muon coupling (associating 2 × muon mass with positive pressure, 1 with negative), one obtains a H₀ of -4.10 km s^-1 Mpc^-1, and a tau/muon superposition of 64.92 km s^-1 Mpc^-1. Thus, the changing coupling can be behind the Hubble tension.
However, one now must ask - if the blueshift is an illusion (in a sense that is an effect on the photon, not the change in velocity of the source of emission), is the conventional interpretation of redshift (expansion) an illusion as well? Indeed, the result above suggests a possibility that the amount of negative pressure is at least roughly equal to the amount of positive pressure, with the redshift (currently interpreted as ~2/3 negative pressure energy in the universe) being mainly a consequence of stronger coupling of the photon to negative pressure (2 tau mass eigenstates associated with negative pressure and 1 tau mass eigenstate associated with positive pressure). On the other hand, as the higher mass eigenstates do not only imply higher acceleration but also higher density of traversed mass-energy, the ratio between coupling strengths may actually mirror the ratio between amounts of positive and negative pressure. However, the only reason the photon is experiencing this kind of coupling is assumed to be its expanding spherical form - the coupling between localized bodies can be different. Note that the 2/3 ratio can be obtained from the calculation of the speed of light from the density of the observable universe (ρ) and its pressure on the spherical wavefront, at the radius of the observable universe: $\displaystyle c = {2 \over 3} \sqrt{{M a} \over {4 \pi R^2 \rho}} = {2 \over 3} \sqrt{{a R} \over 3} = {2 \over 3} \sqrt{{G M} \over {3 R}} = 2.99792458 \times 10^8\, m/s \tag{12}$ where M is the total mass of the observable universe (~4 × 10⁵⁴ kg), a is the gravitational acceleration obtained from that mass (~13.78 × 10^-10 m/s²), and R is the radius of the observable universe (4.4 × 10²⁶ m). The same result can be obtained using energy density of 9.9 × 10^-27 kg/m³ (mass of 3.53 × 10⁵⁴ kg) and a ratio of 0.71 instead of 2/3. Obviously, the relation satisfies the conventional estimates of energy density and the amount of dark energy. Thus, it can certainly be interpreted as evidence that the photon is indeed interacting with the mass-energy enclosed by the wavefront. Derivation of coupling rest mass The theory of Complete Relativity postulates self-similarity of universes. Thus, each universe is of a different scale - which can be interpreted as a different dimension. This obviously requires running coupling of the dominating force, where the coupling must get stronger with decreasing scale. And this is evident in reality. Cosmological scales are dominated by the weakest force (gravity), atomic scales are dominated by a stronger force (electro-magnetism) and nuclear scales are dominated by an yet stronger force. We've been unable to break particles like electron, suggesting even stronger forces on smaller scales - in agreement with the postulate. Each universe is then associated with a discrete vertical energy level (scale) where energies are most stable. This is also evident in reality, e.g., atoms are more stable than molecules, these are more stable than larger composites, etc. To preserve this stability, transition between the vertical energy levels must require sufficiently high energies - lower energies lead to less stable excitations. The energy levels must also not be symmetric (from a reference frame with fixed metric), i.e., their progression must grow exponentially. Thus, these vertical energy levels are somewhat analogous to the energy levels of particles bound to atomic nuclei. One major difference is that the progression is much stronger (logarithmic) so the difference between energies is in the order of magnitude (vertical), rather than dominantly in the value (horizontal). Another salient difference is that the equivalence between a particle on one energy level and the other is very relative. And this is due to the associated running coupling and energy transformation between scales (vertical energy levels). On one vertical energy level, for example, the particle may be electrically charged with its gravitational mass being negligible, on the other energy level gravity may dominate (one type of energy is always exchanged for the other with the transition between levels). Thus, transition between vertical energy levels does not occur with simple removal or addition of energy, transformation (e.g., through annihilation) is required. The vertical energy levels can explain many unexplained features of the observable universe, and more evidence for their existence can be found in other works of the author. Given the postulates, one can formulate the discrete vertical energy levels through conservation of momentum of wave-like energy (allowing only sufficiently distant harmonics): $\displaystyle m_{(n-1)}\, c\, r_{(n-1)} = 10^{(n-1)n}\, \hbar \tag{13}$ where n is an integer. For n = 1 one now obtains the reduced Compton wavelength for r_n-1 = r₀, associated with mass m_n-1 = m₀. Conservation of the nature of r (reduced Compton wavelength) through energy levels then implies: $\displaystyle 10^{(n-1)n}\, m_{(n-1)}\, c\, {r_{(n-1)} \over 10^{(n-1)n}} = \hbar \tag{14}$ from this follows: $\displaystyle m_n = 10^{(n-1)n}\, m_{(n-1)} \tag{15}$ The allowed masses/wavelengths are thus solutions of the quadratic equation in the integer logarithmic metric, which, for n > 0 can be expressed as: $\displaystyle (n-1)n = n^2 - n = \sum_{k=1}^{n} 2 \times (k-1) \tag{16}$ and can be interpreted as a generalization of the Fibonacci sequence - something expectable for the postulated self-similarity. Assuming now that standard particles occupy level n = 7, electron's [equivalent] mass on level n = 8 is: $\displaystyle {M_e}_8 = 10^{\log(M_e) + 7 \times 8} = 0.910938356 \times 10^{26}\, kg \tag{17}$ M_e = standard electron mass = 9.10938356 × 10^-31 kg This is on the order of Neptunian planets, and very close to the mass of Neptune (which, the author argues is not a coincidence). Similarly, using masses of atoms and other standard particles, masses on the order of stars and other stable celestial objects can be obtained. Calculating now the electron mass on level n = 6, one obtains: $\displaystyle {M_e}_6 = 10^{\log(M_e) - 6 \times 7} = 9.10938356 \times 10^{-73}\, kg \tag{18}$ One evidence for this progression of vertical energy levels comes from the obvious preservation of their asymmetry. Consider the following: $\displaystyle {M_e}_6 = {{M_e}_7 \over {M_e}_8} K_A = {M_e \over {M_e}_8}\, 1.02413 \times 10^{-16}\, kg \tag{19}$ The quantum of mass K_A (1.02413 × 10^-16 kg = 5.7 × 10¹⁹ eV = 57 EeV) here is the mass (energy) of asymmetry. If the standard electron mass would be equal to K_A and M_e6 would have a mass equal to M_e6/K_A, the adjacent energy levels to n = 7 would be symmetric relative to that level. Symmetry could also be achieved by decreasing M_e6 mass to 8.1 × 10^-87 kg (setting K_A = M_e), which requires addition (in logarithmic metric) of 8.9 × 10^-15 kg (4.99 × 10²¹ eV = 4988 EeV). One can now interpret the first limit as the 2nd order limit, and the second limit as the 1st order limit. Particles exceeding these energies should not be impossible, but should be very rare. And studies indeed confirm these limits. The 2nd order limit has been found to be the cut-off energy for intra-galactic sources. And the limit is also in agreement with measurements of the GZK (Greisen-Zatsepin-Kuzmin) energy limit (cut-off) for protons - 5.6±0.5±0.9 × 10¹⁹ eV (uncertainties are statistical and systematic, respectively). Interestingly, particles with higher energy than the 2nd order limit have been detected, on the order of 10²⁰ eV, but they are rare and no particle has been produced or observed exceeding the 1st order limit. How to interpret the M_e6 particle? Well, there are not many known candidates for such mass, but it is within the orders of mass expected for photons/gravitons, and while it is not the photon/graviton itself it shouldn't be surprising that the electron equivalent on level n = 6 is involved in the formation of a photon, and - even if indirectly, gravitons. Note that the difference between the electron mass and the obtained M_e6 mass is 42 orders of magnitude, exactly the difference in strength between the electro-magnetic force and gravitational force between two electrons/positrons. This is probably not a coincidence - if the M_e6 particle is a fermion, this suggests the exchange between the Yukawa coupling to Higgs field and some other coupling (as noted before, such exchanges are expectable with changes in vertical energy levels). Different interpretations of the exchange are possible. One possibility is that on the level (scale) n = 6, the strength of gravitational and electro-magnetic coupling is equal, but with the inflation to n = 7, the strength of gravitational coupling is exchanged for the coupling to the Higgs field (effectively implying the conservation of the GM product between the scales). In any case, assuming the particles are fermions, pairs of these particles should be involved in photon or graviton formation. Assuming the photon is a composite particle, the simplest composite structure of a photon involves 2 fermions. Assuming now that one component is electron on n = 6, and the other is a positron on the same level, and assuming negligible binding energy compared to component masses, one obtains a photon mass that is 2 × M_e6. However, it is unlikely that photon is a standard composite particle (as experiments suggest). Most likely, photon's rest mass is a result of annihilation of an even number of n₆ fermions (fermions occupying the vertical energy level 6). Again, the simplest case involves a single electron/positron pair, which, with energy conserved and single photon production (assuming recoil momentum is absorbed by a 3rd body - such as the atomic nucleus, or, its equivalent on n₆) gives the same outcome: $\displaystyle M_{\gamma} = {M_{\gamma}}_0 = 2 \times {M_e}_6 = 1.821876712 \times 10^{-72}\, kg \tag{20}$ Now, since the equation for vertical energy levels is the same for particles of different mass, oscillation between vertical energy levels of muon and tau electrons should, obviously, result in muon and tau photons, respectively - with the same ratio between the 3 mass eigenstates. Therefore, assuming the obtained photon mass is the mass of the e eigenstate, masses of other eigenstates can be easily obtained: $\displaystyle M_{\gamma \mu} = {M_{\gamma}}_1 = {M_{\mu} \over M_e} M_{\gamma} = 3.767 \times 10^{-70}\, kg \tag{21}$ $\displaystyle M_{\gamma \tau} = {M_{\gamma}}_2 = {M_{\tau} \over M_e} M_{\gamma} = 6.335 \times 10^{-69}\, kg \tag{22}$ M_μ = standard muon mass = 105.6584 MeV/c²
M_τ = standard tau mass = 1776.86±0.12 MeV/c²
M_e = standard electron mass = 0.510999 MeV/c² If photon is a product of annihilation, however, the obtained masses for single-photon production should be understood as minimal rest masses of these photon eigenstates. Generally, the rest mass of a photon should be: $\displaystyle M_{\gamma} = \sum_{0}^{m-1} {1 \over 2} \left( {M_{\gamma}}_i + {M_{\gamma}}_j \right) \tag{23}$ i, j ∈ {0, 1, 2} where m is the number of precursor (virtual) pairs contributing to photon mass, and M_γi, M_γj correspond to mass eigenstates, as calculated above. The m higher than 1 or 2, however, may be unlikely. Note that the equation allows for combinations where i ≠ j, however, whether this is allowed or not will depend on formation mechanism. While single-photon production is rare in annihilation of standard particles (n₇ particles), any kind of photon production probably involves annihilation of n₆ pairs (whether it involves n₇ annihilation or not). In any case, since particles forming these pairs are unobservable, they may be interpreted as virtual. Note also that this calculation involves the part of rest mass associated with positive pressure, however, the rest mass associated with negative pressure should have mass eigenstates with the same absolute values. Total rest mass is then the sum of the two, and will be zero in some cases. The obtained particle may be interpreted as a photon but also as a half-graviton (virtual graviton), as the annihilation of pairs of these can create spin-0 and spin-2 bosons (standard gravitons). Now it becomes clear why in the solutions above the two photon mass eigenstates are involved in the gravitational coupling. Not only should the photons couple to gravitons in pairs, the mass eigenstates of half-gravitons are equal to the mass eigenstates of photons.
Note that, since the coupling to negative pressure here is equivalent to positive pressure coupling (the only difference between produced acceleration is in the sign), one could interpret this as dipolar gravitational coupling. However, although gravitational-like, the simplest solution in the context of established theories would require a spin-0 carrier for the negative pressure coupling. Assuming the positive carrier is also a spin-0 carrier, this cannot be a solution in conventional GR framework, rather an extended one (e.g., generalized Brans–Dicke theory). Obviously, the massive carrier implies a Yukawa-type contribution to the acceleration, however, considering the masses involved, this is negligible. The strength of the spin-0 coupling may be scale dependent and, obviously, it won't be detectable in cases where the positive coupling is cancelled with a negative one (even though some asymmetry between the two probably exists, it is also probably negligible). This is one potential explanation for it being unobservable between the bound large bodies in the Solar System, for example. Inflation of spherical waveforms, such as the photon, can be interpreted as the inflation of the dipolar coupling (or the inflation of the coupling dimension) to large scales. The existence of the spin-1 gravitational-like coupling cannot be ruled out either, but it is more complex. In any case, non-zero rest masses and similar or equal masses between photons and gravitons are not expectable in standard 4D physics, as well as the running coupling, however, this symmetry breaking (including the spin-1 gravitons) is predicted by the Kaluza-Klein (KK) theory. KK solutions to the Pioneer anomalies have been proposed before and a detailed analysis has been published, although the approach is significantly different (the anomalous acceleration is assumed to be a result of the effect on spacecraft, not the effect on the photon itself). However, while the solution proposed here could be compatible with the KK framework, it doesn't stem from the KK theory and the analysis in that framework is beyond the scope of this paper. Conclusion It has been shown that anomalous acceleration of Pioneer probes can be fully explained with the proposed gravitational or gravitational-like coupling, going in favour of the original thermal modelling of Pioneer spacecraft. Although the results of newer thermal modelling seem convincing, it remains questionable whether it is the proper solution of the anomalies. The solution provided here can explain multiple aspects of the anomaly, not limited to the value of "acceleration". It can also explain anomalies in other deep-space probes, although in some cases it may represent only a partial solution (the other being thermal recoil). The solution provides an alternative interpretation for the Schwarzschild radius of the observable universe - without excluding the established one. Additionally, it explains the Hubble tension and could potentially explain some other anomalies. In any case, experimental verification of the proposed solution is desirable, as any deviation from conventional physics could have large implications for cosmology, astrophysics and particle physics. A deep-space probe on a hyperbolic trajectory, for example, designed in such way to minimize or rule out anisotropic thermal dissipation could provide an unambiguous confirmation or refutation of alternative solutions, so it would be very useful even in case the solution proposed here does not hold up to scrutiny. Paper updated/revised.