The all-inclusive axiom allows us to describe any system with a general energy-level diagram, where all entities comprise quanta (Figure 1), and to derive the thermodynamics of dissipative systems constituting the evolving universe [31,33,45].

The equation of state can be derived by considering what it takes for an entity, indexed with j, for example, a grain of dust, to exist. Its probability, ₁ P _j = ϕ ₁ ϕ ₂ ϕ ₃⋯ = ∏ _k ϕ _k , is a product, ∏ _k , of the substrates, say atoms, indexed with k. If any k-substrate is missing, the product form ensures that ₁ P _j = 0. The factor, density in energy [38], ϕ _k = N _k exp[( − ΔG _jk + iΔQ _jk )/k _B T], denotes the k-substrate in numbers, N _k , having an energy difference, ΔG _jk = G _j − G _k , with respect to the j-product. The energy, iΔQ _jk , in the form of the vector potential, absorbed or emitted in the jk-transformation, is distinguished by the prefactor, i, from the scalar potential bound in matter, for example, to deal with decay that violates the presumed, stationary-state conservation of probability [46]. The self-similar exponential form, de ^x /dt = e ^x , applies for a statistical system, where a transformation perturbs the system’s average energy, k _B T, insignificantly [38].

The probability of a population, as a product, P j = ( P j 1 ) ( P j 1 ) ( P j 1 ) ⋯ / N j ! = ( P j 1 ) N j / N j ! , enumerating N _j interchangeable entities, ensures that P _j = 0 if one entity is missing entirely. Since the indistinguishable configurations, i.e., microstates, are not proper states differing by energy, P _j is divided by permutations, N _j !

The system’s probability, P, is the product, ∏ _j , of all P _j , P = ∏ j P j = ∏ j ∏ k ϕ k N j / N j ! . (1)

The logarithm of P multiplied by k _B is entropy: S = k B ⁡ ln ⁡ P = k B ∑ j ln P j = 1 T ∑ , k N j − Δ μ j k + i Δ Q j k + k B T , (2) where ln N _j ! ≈ N _j ln N _j − N _j approximates the factorial and Δμ _jk = μ _j − μ _k denotes the difference between the substrate potential, μ _k = k _B T ln[N _k exp (G _k /k _B T)], and the product potential, μ _j . When multiplied with T, S equals the system’s total energy, TS. It comprises the bound, ∑N _j k _B T, and free, ∑N _j ( − Δμ _jk + iΔQ _jk ), forms of energy.

As the system evolves with time, t, toward balance with its surroundings, free energy, −Δμ _jk + iΔQ _jk , is consumed in dissipative transformations from N _k to N _j . For example, in stars, hydrogen is transformed into helium.

For a statistical system, the population changes can be approximated by continuous differentials, dN _j , to give the following: T d S d t = T ∑ j d S d N j d N j d t = ∑ j , k d N j d t − Δ μ j k + i Δ Q j k , (3) where the variation in average energy, k _B T, is contained in dS. Since entropy is a function of energy (Eq. 2), S attains at dS/dt = 0 the steady-state maximum, S = ∑k _B N _j , when free energy attains its minimum, −Δμ _jk + iΔQ _jk = 0.

Free energy drives the population changes d N j d t = 1 k B T ∑ k σ j k − Δ μ j k + i Δ Q j k , (4) through mechanistic factors, σ _jk > 0, facilitating jk-transformations [47]. For example, a high-density stellar core can be regarded as a mechanism enabling nuclear reactions.

Substituting dN _j /dt in Eq. 3 with Eq. 4 and squaring −Δμ _jk + iΔQ _jk in the orthogonal jk-basis show dS ≥ 0. The fact that entropy cannot decrease follows from the conservation of quanta. If it were to be violated, the quanta emitted from matter would be lost into nowhere instead of being absorbed into the surrounding vacuum.

A continuous form of the equation of motion (Eq. 3) is useful for understanding the scale-free astronomical data. It is obtained by approximating the scalar, μ _j = (∂U/∂N _j ), and vector, Q _j = (∂Q/∂N _j ), potentials by continuous scalar, U, and vector, Q, potentials: T d N j d t = ∑ j k d N j d t − ∂ U ∂ N j + i ∂ Q ∂ N j = − ∂ U ∂ t + i ∂ Q ∂ t = d d t 2 K , (5) where TdS equals the change in kinetic energy d2K. As mentioned previously, the prefactor, i, serves to distinguish dissipated energy from the concomitant change in scalar potential [46].

The integral form of Eq. 5 with vanishing variation of the integrand p·dx = 2Kdt leads to Maupertuis’ principle of least action where quanta with momenta, p, propagate on geodesics, i.e., optimal paths, x, [48]. Unlike the Lagrangian, based on the conservation of energy, Maupertuis’ form is open for evolution. Mathematically speaking, the integration limit moves while integrating due to dissipation, ∂Q/∂t ≠ 0. Thus, Eq. 5 is non-integrable, yet not arbitrary, but bound by free energy. According to Eqs 3–5, entropy does not only increase but also increase in the least time. Thus, bodies move along optimal, i.e., geodesics, rather than arbitrary paths. For example, a stone falls straight down.

Also multiplying Newton’s second law, i.e., the change, d _t p, in momentum, p = m v, with velocity, v, gives the equation of motion: F = d d t p = m a + v d m d t | ⋅ v v ⋅ F = v ⋅ d d t p = d x d t ⋅ m a + v ⋅ v d m d t = − d U d t + v 2 c 2 d E d t = − d U d t + i d Q d t , (6) since kinetic energy, 2K = v ⋅ p = ∑v _j mv _k , vanishes for j ≠ k and d _t v ⋅p = 0 for acceleration a = d _t v ⊥ v. The relativistic mass–energy equivalence, E = mc ², derives from the total action, nh = Et = mc ² t = px, of a system with n quanta. Defined in dissipative terms, the change in mass, d _t m = d _t E/c ² = id _t Q/v ², denotes changes in geodesics, the quantized trajectories opening up for absorption of quanta from the surroundings or emission into the surroundings [49]. For example, in annihilation, quanta bound in matter transform into quanta of the vacuum.

When the system attains thermodynamic balance with its surroundings, Eqs. 5 and 6 reduce to the integrable virial theorem, 2K + U = 0, relating the kinetic energy of a self-gravitating system, 2K, to the gravitational potential energy, U, of the system. The time invariance, corresponding to constant energy by Noether’s theorem, 2Kt = nh, means that the quanta orbit geodesics with characteristic periods, t, of the motional modes [42,50].

The ubiquitous, nearly lognormal, skewed distributions summing up in a power-law manner can be deduced from Eq. 4 [32]. Initially, when there is a lot of free energy, e.g., for star formation, mechanisms, ∑ _k σ _jk , constrain the free energy consumption, d _t ( − Δμ _jk + iΔQ _jk ). Then, the change d d t 1 k B T ∑ k − Δ μ j k + i Δ Q j k = d N j d t d d N j 1 k B T ∑ k − Δ μ j k + i Δ Q j k ≈ ∑ k σ j k ⇒ d N j d t = ∑ k σ j k N j , (7) is approximately exponential. For example, the weak gravity of dilute gas curtails the initial star formation; however, soon, the increasing gravity of aggregating gas accelerates further aggregation. It should be noted that, in Eq. 7, the change reduces to dμ _j /dN _j = d(G _j + k _B T ln N _j )/dN _j = k _B T/N _j because μ _k , Q _j , and Q _k do not depend explicitly but stoichiometrically on N _j . Conversely, the final phase dies out almost exponentially when the free energy has been nearly exhausted. For example, the star formation decreases as free gas is exhausted.

The bulk distribution between a process’s initial and final phases follows a power law [32]. This becomes apparent using the atomistic axiom, N j = ∏ k ϕ k = α j N 1 j . It should be noted that the j-entities, in numbers N _j , assemble from the elemental constituents, N ₁, through various mn-transformations, amassed in α _j = ∏ _mn exp[∑ _j N _j (−Δμ _mn + iΔQ _mn )/k _B T]. Thus, the change d N j d t = j α j N 1 j − 1 d N 1 d t = j N j N 1 d N 1 d t ⇒ d N j N j = j d N 1 N 1 , (8) when integrated, follows a power law, ln N _j = j ln N ₁ + constant. The form is familiar from various self-organizing processes [51,52]. Also, Newton’s second law (Eq. 6), when divided with momentum, p, and integrated, delivers the power law, d ln p = d ln v + d ln m. Thus, it is not surprising that power laws are ubiquitous, including gravitational phenomena.

When the system evolves gradually, the change in energy is small compared with the average energy, i.e., |−Δμ _jk + iΔQ _jk |/k _B T ≪ 1. As the variation, n, is small, i.e., n ≪ j, around the average factor, ϕ _j , the distribution of factors, ln ϕ j − n … j + n = ln ϕ j + ∑ n n ⁡ ln ϕ 1 , (9) can be given in terms, lnϕ _j = j ln ϕ ₁, of the elemental factor, ϕ ₁. By the central limit theorem, data distribute around the typical form, j, in a skewed, approximately lognormal manner [23,53,54]. For example, all red giants look like red giants and not white dwarfs—the red giant distribution peaks on the typical giant and the white dwarf distribution on the typical dwarf. Logarithmic spirals are also approximately lognormal distributions in polar coordinates, i.e., energetically optimal forms [32].

Equations 7–9 show that the skewed distributions sum up along sigmoid curves [55]. Thus, data follow closely, but not precisely, the power law, deviating at low and high ends [56]. Moreover, many data display a series of power laws whose slope changes when a mechanism of transformations changes to another. For example, in the main sequence, the most massive stars are the most effective at releasing energy through nuclear reactions, whereas the stars with the lowest mass are the least effective. Likewise, the cosmic ray flux turns from one slope to another when one mechanism of dissipation changes to another [57,58].

The background spectrum of free space can be derived from the atomistic axiom by considering how photons of energy, ɛ _i , relative to the average, k _B T, distribute in numbers, n _i , among rays of paired photons (Figure 2). The number of ways, in-phase and antiphase, the photons populate the numerous rays intersecting in degenerate directions, g _i , is the product of n _i combinations of the sets with n _i + 2g _i − 1 elements. W = ∏ i n i + 2 g i − 1 ! n i ! 2 g i − 1 ! exp − n i ε i / k B T ≈ ∏ i n i + 2 g i ! n i ! 2 g i ! exp − n i ε i / k B T . (10)

Taking logarithm and using Stirling’s approximation for the factorials yields, we obtain the following: ln ⁡ W = ∑ i n i + 2 g i ln n i + 2 g i − n i ⁡ ln n i − 2 g i ⁡ ln ⁡ 2 g i − n i ε i / k B T . (11)

The most probable state is the stationary state: d ⁡ ln ⁡ W d t = ∑ i d ⁡ ln ⁡ W d n i d n i d t = ∑ i d n i d t ln n i + 2 g i n i − ε i k B T = 0 , (12) where quanta distribute according to the Bose–Einstein distribution, n i = 2 g i e ε i / k B T − 1 , (13) having two states for each photon, in-phase and antiphase, per locus in the paired ray (Figure 2) [59]. Since the spectral density of the paired-photon vacuum has precisely the form of the black body spectrum, we consider that the void is basically as physical as matter comprising quanta.

In agreement with Planck’s law of radiation, the photon pairs open up with increasing temperature and pair up with decreasing temperature. Moreover, the paired-photon vacuum responds to an accelerating charge by unpairing and radiating photons. Thus, the photons do not appear out of vacuum nor disappear into it but comprise the vacuum.

The proposed substance of space also complies with electromagnetism. Explicitly, the paired-photon vacuum, embodying the four-potential of the electric scalar potential, φ, and magnetic vector potential, A, satisfies the Lorenz gauge[64], ∇ φ + ∂ A ∂ t = 1 c ∂ φ ∂ t + c ∇ ⋅ A = 0 , (14) as a seemingly continuous and indestructible substance whose wave nature gives rise to interference phenomena. For example, in the Aharonov–Bohm experiment [65], a phase difference Δ ϕ = ϕ 1 − ϕ 2 = e ℏ ∫ A 1 ⋅ d x 1 − e ℏ ∫ A 2 ⋅ d x 2 , (15) develops between waves when charges, e, traverse A ₁ and A ₂ along paths x ₁ and x ₂, respectively

Gradients in the vacuum are fields [66–68]. The electric field, E = − ∇ φ − ∂ A ∂ t , (16) for example, around an electron is a gradient in the paired-photon rays, whose winding tallies the electron winding number, a topological quantum number [69]. Therefore, it takes the full 4π, rather than only 2π, rotation of the electron, e⁻, to return the vacuum to its original state [70,71]. This SU(2) _L symmetry is disclosed by monodromy of the vacuum around the electron, e ⁻ = ∫ρdV = −ɛ _o ∫∇⋅∇ϕdV = −ɛ _o ∫∇ϕdS, enclosed in a volume, V, or, as by Gauss’ law, bordered by an area, S.

The vacuum polarization is similar to the polarization of a dielectric material; the paired-photon rays wind up, i.e., twist, to counteract introduced charges. Conversely, when charges neutralize, the unwinding of the paired-photon rays gives rise to a time-varying electric field, i.e., displacement current, J _D = ɛ _o ∂ _t E.

The magnetic field B = ∇    × A , (17) is a vortex of the twists in the paired-photon rays, for example, encircling a line of moving charges (Figure 3).

The electromagnetic fields (Eqs. 16 and 17) give rise to the Lorentz force, F = q E + v × B , (18) experienced by a charge, q, moving with velocity, v. Thus, from the proposed perspective, it is not that charges themselves would attract or repel each other; instead, the charges move as they couple with the physical vacuum that is leveling off its gradients.

The proposed substance of space also complies with gravitation. The gradient in the gravitational four-potential of the scalar, ϕ _g , and vector, A _g , components is the gravitational field [72–76], i.e., acceleration. E g = − ∇ ϕ g − ∂ A g ∂ t . (19)

For example, at a distance, r, from a body of mass, M _o , the density gradient of the paired-photon potential, ϕ = GM _o /r, tallies the geodesic curvature, the Euler characteristic, χ, proportional to M _o (Figure 4) [77]. The density gradient is observed as the bending of light rays, a gravitational time delay, and as a gravitational frequency shift, f e / f o = 1 + z = ( 1 − G M o / c 2 r e ) − 1 / 2 , between emission, f _e , at the radius, r _e , and absorption, f _o , at the detection [73,78]. In turn, the leveling of a density gradient is detected as the Doppler shift of light emitted by the moving body coupled to the moving vacuum.

Similar to B = ∇ × A (Eq. 17), the rotational field, B g = ∇    × A g . (20) for example, at a distance, r, from a body with inertia, I, rotating with the angular velocity, ω, revolves at the rate, Ω = GI/c ² r ³∇ × ( ω ×r) [79]. This vortex in the paired-photon density displays itself as frame-dragging precession [73] (Figure 4) that was examined, for example, with Gravity Probe B [80,81].

Moreover, similar to Gauss’ law, the divergence of the density gradient in the vacuum potential relates to the mass density, ρ _M = −(1/4πG)∇² ϕ _g . Explicitly, ∇ ⋅ ∇ϕ _g = ∇ ⋅ a _R = ∂ _t GM/R ² c = −1/t ², of the all-embracing gravitational potential, ϕ _g , due to all mass, M, of the expanding universe of radius, R = ct, relates inversely to the age, t, squared using the mass–energy equivalence, Mc ² = GM ²/R. In other words, the average mass density of the universe, ρ _M = 1/4πGt ², decreasing with time, t, is the source of space embodied by the paired photons.

While all photons are bound to the universe they constitute, a paired photon is not a bound state because the vacuum’s energy spectrum is continuous. Also, the difference in the photon phases may change continuously, for example, in response to a moving charge. Despite its dynamics, the paired-photon ground-state vacuum does not break apart easily because the electromagnetic force is huge, e.g., compared with gravity.

The vacuum moves toward thermodynamic balance so that gradients in the vacuum energy, e.g., electromagnetic and gravitational fields, decrease, which is seen as charges attracting or repelling, magnets reorienting, and bodies moving.

Since light cannot but propagate at the speed of light, c, in the relativistic medium comprising photons, the proposed physical vacuum makes sense of the classical experiments carried out by Arago [82], Fizeau [83], Michelson and Morley [84], Trouton and Noble [85], and Sagnac [86], and modern measurements named after Casimir [87] and Aharonov and Bohm [65,88]. Moreover, the dynamic Casimir effect yields photons out of the vacuum, showing two by two at a time [89].

In agreement with measurements [61] and theory [39], the vacuum and matter comprising quanta are in thermodynamic balance Mc ² = GM ²/R, where the energy of all mass, M, in the universe of radius, R = ct, and age, t, tallies with all gravitational energy. This is in line with general relativity. The vacuum quanta embodying the Ricci curvature tensor and metric tensor are in balance with the quanta bound in the sources of gravitation denoted by the stress–energy tensor.

On the largest scale, gravitation is geometrized by the cosmological principle so that the metric, ds ² = a(t)² dx ² − c ² dt ², sums up by Pythagoras’ theorem the infinitesimal distance, ds, from the spatial, dx, and temporal, dt, coordinates, which correspond to the photon wavelength and period. While the scale factor, a(t), parameterizes the expansion, the thermodynamic balance between all matter and space implies that the universe expands physically rather than [para]metrically. As the quanta of matter transform into the paired photons of the vacuum, the energy density decreases. Redshifts from distant sources document this density change from the past to the present [7,10]. In geometric terms, the curvature of spacetime not only corresponds to but also spacetime itself emerges from the stress–energy tensor [90].

Also, consistent with quantum field theory, the physical vacuum comprising quanta of light, both paired and single, in balance with matter can be modeled with the mean-energy density photon wave function, ψ r , t = ∫ E d p h 3 / 2 γ p , t e i p ⋅ r / h , (21) as an integral over the photon probability amplitudes, γ, of momentum, p, and energy, E, in the unit volume, h ³. The dependence on the position, r, is formal because the massless photon does not localize [91,92].

Accordingly, a particle perturbing the vacuum can be presented with a wave function. This wave packet of the vacuum wave functions follows the Schrödinger equation and disperses as the particle with mass couples to the vacuum. In other words, mass is the measure of the coupling, i.e., inertia. In contrast, a photon as part of the vacuum (Fig. 2) abides by the dispersionless wave equation. Without the physical vacuum, particles would not have mass just as without the Higgs field. While, in this sense, the proposed paired-photon boson resembles the Higgs boson, the massless photon does not decay, whereas the massive Higgs boson is short-lived.

Assuming that the paired-photon strings or filaments form the fabric of space, their density gradients are gravitational fields [93–97]. Since the substance of space seeks thermodynamic balance in the least time, the bodies move coupled to the moving vacuum along the least-time paths, i.e., geodesics. Conversely, orbits are closed at a stationary state of the vacuum, where the influxes and effluxes of quanta tally. Then, the kinetic energy, 2K = mv ², of an orbiter of mass, m, and velocity, v, balances gravitational potential energy, U = GmM/r, at a distance, r, due to a central mass, M, as according to Kepler’s third law.

The photon qualifies as the ground-state substance because it does not decay. In our judgment, the paired photon, a massless spin-2 particle, is indistinguishable from the theorized graviton, the carrier of gravitation [73]. Correspondingly, the photon, a massless spin-1 particle, is the carrier of electromagnetism. Moreover, gravitation and electromagnetism cannot but share the same 1/r-form when the photon-embodied vacuum mediates both forces [63], as was thought already early on [66].

To an excellent approximation, a spiral galaxy is a stationary system. At the dynamic steady state, integrated over its characteristic orbital period, t, ∫ v ⋅ d t p + v ⋅ ∇ U d t = 0 , (23) momentum, p, and acceleration, a, are orthogonal. The orbital velocity, v, at a distance, r, balances by v ²/r the galactic acceleration, a _o = GM _o /r ², due to the central mass, M _o , within the circumference, 2πr, and the universal acceleration, a = GM/2πR ², due to all mass, M, within the radius, R, of the expanding universe [112], v 2 r = a + a o = a o 1 + a a o = G M o r 2 1 + 1 2 π M r 2 M o R 2 , (24)

near the galactic center, a ≪ a _o . Therefore, the feeble universal acceleration is hardly detectable in motions of strongly bound bodies, such as stars in clusters [113].

Conversely, far away from the luminous edge, a ≫ a _o , where v ² a _o /r ≈ aGM _o /r ², Eq. 24 limits to the Tully–Fisher relation, v ⁴ = aGM _o , for example, orbital velocities of dwarf galaxy profile Eq. 24 due to their low amounts of baryonic rather than high amounts of dark matter [114,115]. The flat tail of the orbital velocity curve signifies that distantly, the universal curvature, 1/R, dominates the curvature, 1/r = a/v ², of a bound geodesic. A closer match with a given observed rotation curve beyond the point-mass approximation (Eq. 24) would employ a detailed mass distribution, M _o (r), instead of an interpolation [116,117], from the dense galactic surroundings to the sparse universal voids.

Moreover, satellites of the Milky Way, Andromeda Galaxy, and other spirals can be understood to home in on the galactic plane [118] under the central force of the expansion because space, the substance of gravitation, zeros in on the free energy minimum of the least curvature. So, a galaxy punched by another realigns for the same reason as a poked top reorients. The substance of space also shapes bodies into spheroids.

Just as stars in galaxies orbit around, galaxies also move about in clusters to balance the local gravity and the universal gravitation [16,119]. The Faber–Jackson relation, ranging from tiny dwarf satellite galaxies through giant spiral galaxies to rich clusters of galaxies [120], even encompassing the whole expanding universe, displays the universal acceleration, a _R = c ²/R = u ²/r = 2πa = 2πv ²/r, in terms of speed of light, c, radial, u, and orbital, v, velocity. Since u ²/r = 2πv ²/r, the mass vs. radial velocity line on a log–log plot, relevant to galaxy velocity distribution, is offset by the factor 2 π ≈ 2.5 from the mass vs. orbital velocity line, relevant to galaxy rotational curve [8]. So, there is no need to extend MOND [121] or invoke the dark matter hypothesis to account for the velocity dispersion of galaxies in clusters.

While the universal acceleration due to the expansion shares the mathematical form (Eq. 24) of MOND [116], the thermodynamic tenet is intrinsically a relativistic formulation. The paired-photon void is a relativistic medium complying with the Bose–Einstein statistics (Eq. 13).

According to concordance cosmology, gravitational lensing provides yet another evidence of dark matter. This belief is based on images of background galaxies lensed through foreground galaxy gravitation because the images are more distorted than expected from the lensing masses of visible, ordinary matter [122].

We follow Einstein, who stated early on that light going through a gravitational potential, such as through an optical medium, gets deflected and delayed [93]. The stronger the gravity, the more curved the geodesic, i.e., the least-time path. A local gravitational potential, GM _o /r, at a distance, r, from a mass, M _o , is part of the vacuum’s total potential due to M within the universe of radius, R. Thus, near a celestial body of mass, M _o , the refractive index, n 2 = c 2 / v 2 = ( 1 − G M o / c 2 r ) − 1 ≈ 1 + G M o / c 2 r [81,94,95].

According to Eq. 22, the space curving around a body, for example, the Sun, causes an angular acceleration, α. It integrates over time, t, to an angle of precession, φ, d L d t = I α = I d d t d φ d t = m r 2 ω 2 = 2 π 2 m v 2 = 2 π 2 G m M o r ⇒ φ = ∬ 2 π 2 G M o r 3 d t d t = 2 π 2 G M o c 2 r , (25) where L is the angular momentum, G gravitational constant, r = ct the distance to the center of the Sun of mass, M _⊙ = 1.99 ⋅ 10³⁰ kg, and the mass, m, corresponds to light’s energy, hf, per the speed of light squared, c ². Eq. 25 gives φ = 8.65” for the ray that tangents the Sun’s surface at r _⊙ ≈ 695,700 km. Accordingly, it takes an excess time, Δt = 2r _⊙ φ/c = 196 μs, for the light to make a round trip along the geodetic line than along the night-sky straight line, in agreement with measurements [78].

The least-time principle reproduces bending and delay consistently with the same equation (Eq. 25), as the two phenomena are identical. In contrast, derivations from the Einstein field equations deliver the delay and bending in terms of different equations, which leads to a discrepancy. By Eq. 25, light passing a galaxy deflects 2π ²/4 ≈ 5 times more than by general relativity. So, there is neither a need nor room for dark matter to account for the galaxy image distortions due to gravitational lensing [78].

The amount of bending due to the Sun is determined by the difference between the line of sight to a star during an eclipse and the night-sky line 6 months later. Although obvious, it is essential to note that the ray ends up at a different point on the ground when it has passed through the gravitational lens than when coming directly from the night sky. So, to trace the same ray, the telescope should be displaced by the distance corresponding to the change in direction.

It is easy to grasp the relation between the change in direction and the telescope’s displacement. For example, looking at your finger with one eye and then switching to the other seems to shift it relative to the background. If the parallax is to be averted, the position of the eye, like the location of the telescope, has to be displaced to match the angle from which the object is seen. Unfortunately, as Eddington did not consider this, the bending appeared smaller than it was [78]. Thus, based on gravitational lensing, there are no compelling arguments to purport dark matter.

Since the gravitational time delay, Δt, relates to the local mass, M _o , as the age of the universe, t, relates to the mass of the universe, M, i.e., Δt/t = (2π)² M _o /M [123], Δt between two images of a variable signal [124] can be used to determine the Hubble parameter, H = 1/t = (Δθ)2d _l d _s /d _ls RΔt, from the image separation, Δθ, the observer–source, d _s , observer–lens, d _l , and lens–source, d _ls distances [125]. However, as contemporary inconsistencies in H imply, interpretations of galaxy lensing, time delay data [126], cosmic background radiation, and Type 1a supernovae data are predicated on the amount of postulated dark matter [127]. Thus, the bending of light passing by the Sun, seemingly free of dark matter, provides the crucial experimental reference to judge the hypothesis.

Considering a photon emitted with the standard-candle energy, hf _e , the photon shifts red on its least-time path from the dense past to the sparse present to balance the decreasing universal gravitation. Otherwise, the photon would gain energy out of nothing [128] relative to the surrounding photon distribution, diluting, and cooling due to the expansion (Eq. 13).

The optical distance, D _L , to the supernova is deduced from the flux, F = L / 4 π D L 2 , i.e., the observed luminosity, L = hf _e f _o , that scales down as light spreads across an ever-larger area, 4 π D L 2 . For calibration, an explosion at a known distance, e.g., a nearby one without a marked shift in the observed frequency, i.e., f _e ≅f _o , serves as a reference flux, F _r , [78]. Thus, F = L 4 π D L 2 = F r L / 4 π D L 2 L r / 4 π D r 2 ≅ F r h f e f o r r 2 h f e f e D L 2 = F r 1 1 + z r r 2 R 2 1 + z 2 z 2 = F r r r 2 R 2 1 + z z 2 , (26) where frequencies are in terms of the redshift, z = (f _e − f _o )/f _o , and the optical distance, D _L = Rz/(1 + z), relative to the radius of the universe, R.

On its way, light shifts to red due to the supernova’s recession velocity and the expanding universe’s decreasing gravitational energy density. More specifically, the gravitational frequency shift [73,129] is not the hypothetical tired light [130]. However, if the physical process is parametrized with the scale factor, as it is in standard cosmology, instead of using Eq. 26, the distance seems longer than it is. Then, the Type Ia supernova data give a false impression of the universe expanding at an accelerating rate [78].

Due to the same ΛCDM parametrization, the angular size redshift relation is non-monotonous. When z > 1.5, objects ever further away appear peculiarly, i.e., larger, contrary to empirical evidence [131]. In contrast, as quanta bound in matter transform into the vacuum quanta, the total energy density, ρ = c ²/4πGt ², decreases with time, t, and the angular size, θ ∝ 1/t [123].

On the logarithm scale, the brightness of a star (Eq. 26), the distance modulus, μ = − 2.5 ⁡ log L / 4 π D L 2 L r / 4 π D r 2 + K = − 2.5 ⁡ log R r r 2 + 2.5 ⁡ log z 2 1 + z + K , (27) is a function of two terms, 5log(z) and − 2.5log(1 + z). So, μ(z) does not follow a straight line but curves at z ≈ 1. The instrument factor, K ≈ 5log(1 + z), corrects for sensitivity, i.e., the more light shifts to red, the less is detected [131,132].

The calculated moduli (Eq. 27) follow the measured ones closely (Figure 6) [78], the difference mostly being within the scatter (Figure 7). Thus, the least-time propagation of light explains the supernovae data. There is no room for purporting dark energy. In other words, the expansion rate does not increase. On the contrary, it decreases at the rate, −d _t H = 1/t ² = 4πGρ _M , with decreasing density, ρ _M , as quanta in matter transform into quanta of space.

The uniformity of the cosmic microwave background temperature and the evenness of the distant galaxy distribution are customarily ascribed to a hypothetical inflationary scenario, a nascent exponential expansion [133].

However, by the thermodynamic theory, the correlation beyond causal connection follows naturally and necessarily from the least-time quest for balance (Eq. 5), namely, Newton’s second law of motion states that the bigger a force, the faster the change in motion (Eq. 6), which means, for instance, that the higher the temperature difference, the faster the rate of cooling. Thus, there will be minute temperature differences in due course, regardless of how massive the differences were initially.

Although the early differences in temperature and density have by now nearly evened out, the process is still ongoing. A large difference in energy decreases rapidly and a small one evens out slowly. For example, the most massive stars shine the brightest and for the shortest time, well below 100 million years; however, small stars glow over 100 billion years. Thus, there will be only small differences over time, as is observed. The derived distributions of quanta in matter (Eq. 2) and space (Eq. 13) imply that eventually, as T → 0, everything will become even, regardless of how uneven the earliest universe was.