While the H₀ tension lower values are strongly constrained around 67.4 ± 0.5 Km/s/Mpc [2] , its higher values are rather scattered along the scale, grossly ranging between 73 and 76 km/s/Mpc [3] . Also, S₈ tension showed a constrained value at one end, while scattered at the other, though in reverse order with respect to H₀. The measurements by PLANCK, TT, TE, EE have reported a higher tip of 0.831 ± 0.013, while the lower end ranges from about 0.721 ± 0.043 [4] to about ${0.800}_{-0.027}^{+0.029}$ [5] . Both tensions grossly show a discrepancy of 10% [6] .

The classification of the measurements in function of the universe age, as early and late sources, showed no apparent proportion with time.

A further classification has been based on the methods applied, as indirect observations, inferred from the Microwave Cosmic Background (MCB), and direct observations, from EM signals, which on the contrary exhibit the tensions as a common thread. To explore a possible reason, we have focused on the nature of the measurements, which characterize the lower, the middle and the higher values:

1) The measurements extrapolated from the micro-differences in temperature of the CMB, and from the Baryonic Acoustic Oscillation (BAO) were derived from an estimation of baryon density and of power spectral density parameters. It is important to keep in mind that both expansion parameter and the fluctuation of matter density parameter might have emerged much earlier than the formation of the CMB, which dates 300,000 years from the Big Bang (BB). They might have arisen already in the first second of universe life, at least at the time of the formation of the Cosmic Neutrinos Background (CνB) [7] . Those concern respectively the H₀ tension lower values, and the S₈ tension higher values (BOSS, Planck).

2) The values in the midway have been gained by taking an average value through a joint method from multiple observations [8] , as also from standard sirens [9] , aka multi-messenger, based on the signals received by gravitational waves.

3) The values at the opposite end of the scale were obtained by the so-called cosmic ladders, mainly based on electromagnetic (EM) signals, like luminosity, brightness, cosmological redshift, and lensing. They correspond respectively to the higher values of the H₀ tension (MIRAS, HOLiCOW, SH0ES, etc.), and to the lower values of the S₈ tension (KiDs, DES, VIKING, etc.).

4) There is a unique case, in which a single method has provided with measurements from the bottom to the end of the scale, as in the tip of the red giant branch (TRGB). Indeed, under the same umbrella of the TRGB, different measurements apply distinct approaches, choosing different types of sources with the relevant calibrations, with the result of a higher uncertainty in the measurements [10] [11] . If the different calibrations do not directly lead to a bias, they are not certainly free from it, due to the differently parametrized matter density and luminosity of the sources.

This analysis enhances that density parameters dominate at one end of the scale, while EM signals at the other, leaving the middle values to a mixing of the two. An aspect, which apparently had not been raised in recent proceedings [12] .

Concerning the H₀ tension, we have averaged between the two extremes of the higher tip of the scale 73.2 ± 1.3 km/s/Mpc [13] and 76 ± 2.3 [14] , obtaining 74.6 km/s/Mpc (fine-tuned to 0.0359, profitably aligned with the S8 tension down below). Then we have applied the formula of the fractional change:

$\frac{\Delta x}{x}$ (1)

and assigned to x the higher value for each tension. Thus:

$\frac{74.6359-67.4}{74.6359}=0.096949$ (2)

Concerning S₈, we have fine-tuned the higher value to 0.836(963); and chose for the lower a fair representative ${0.755}_{-0.021}^{+0.019}$ from KV450+DES-Y1 [15] , which also works well as an average between the two tips of S₈ tension lower values. Again, fine-tuning to 0.00082, always within the margins of error, and applying the fractional change formula in the same way, we have obtained:

$\frac{0.836963-0.75582}{0.836963}=0.096949$ (3)

Even applying further slight adjustments, the result would have kept on the same magnitude. Two identical results from distinct phenomena appeared to us as the two facets of a single medal, supporting the hypothesis that the origin of the two tensions must be common.

We have decided then to focus on the relation running between the different nature of data, i.e. mass density and EM signals, though more radically than in the stellar mass-to-light ratio. Since G, and c, are respectively directly related to mass and EM signals, we have put the two fundamental constants in relation. There already exists a precedent in Planck’s natural units [16] . Planck’s equations showed intriguing relations between G and c, i.e. the difference of c⁻² between length l_P and time t_P, or the interchange of c⁵ and G between time t_P and energy E_P, denoting an inverse proportionality, a central aspect in the concepts of evolution and entropy [17] . Since cosmological tensions obviously lie on a cosmological scale, we decided to let aside ħ, typical of the quantum mechanical scale, and to focus only on the correlation of G with c. We noted that both constants stand on the opposite sides of a fraction bar. Despite their distinct nature and the huge scale difference, it would not come naturally to relate them. Instead, we have followed the examples of Planck’s length, and Planck’s time (mimicking together the essentials of spacetime), though without exponentiation:

$\frac{G}{c}=?$ (4)

which through dimensional analysis becomes:

$\frac{{\text{m}}^3\cdot {\text{kg}}^{-1}\cdot {\text{s}}^{-2}}{\text{m}\cdot {\text{s}}^{-1}}={\text{m}}^2\cdot {\text{kg}}^{-1}\cdot {\text{s}}^{-1}$ (5)

We have obtained an unknown quantity, unless it is decomposed into the ratio of specific surface area (A) over time (t):

${\text{m}}^2\cdot {\text{kg}}^{-1}\cdot {\text{s}}^{-1}=\frac{{\text{m}}^2\cdot {\text{kg}}^{-1}}{\text{s}}$ (6)

which leads us to conclude:

$\frac{G}{c}=\frac{A}{t}$ (7)

We have obtained that the ratio between G and c equals, at least dimensionally, the ratio between A (surface over its volume or mass, or also the reciprocal of surface density ρ_a) and t. Assumed A corresponded to the amplitude of the tensions, then we could infer t, and discover what it could be:

$\frac{cA}{G}=t$ (8a)

$\frac{299792458\times 0.096949}{6.67430\times {10}^{-11}}=4.3547\times {10}^{+17}$ (8b)

in light-years the factor time ends up being 13,799,213,880, within the margins of error of the age of universe currently adopted in 13,787,000,000 ± 0.020.

Extrapolating the fractional change from the tensions, and identifying it with an A, a massive quantity, we have ended up to the age of universe confirmed by distinct sources (WMAP, Planck). A good result from just the empirical measures of G, c, H₀, and S₈, and independent from any theoretical model.

Since uncertainties on the measurements of the tensions have been ruled out with several sigma, we argued that the discrepancy is physically real. Given the common factor, our hypothesis of a common origin has grown stronger. The two fundamental constants G and c might have diverted from a common stem, shaping also the discrepancy of the tensions.

In the search for a contextualization of the factor A, we have looked at Newton’s universal law of gravitation. Its mass over the distance squared reproduces fairly well the dimensional units of a surface density (ρ_a), which is the reciprocal of A. Therefore, instead of deriving G from Newton’s formula, we made it to derive from an acceleration (a) over ρ_a:

$G=\frac{a}{{\rho }_a}$ (9)

or also, better fitting our case, from the product between a and A:

$G=aA$ (10)

Assuming A as the factor inferred from the tensions, then a should amount to:

$\frac{G}{A}=a$ (11a)

$\frac{6.67430\times {10}^{-11}}{0.096949}=6.88434\times {10}^{-10}$(11b)

We will see further on, whether this value may be of any cosmological relevance.

Speaking of gravitation, anomalies have been observed even at galactic scale, particularly in the spiral galaxies’ rotational velocity. Vera Rubin’s discovery [18] of the galaxies flat rotation curves signed the discrepancy between the expected radial acceleration, according to the Newtonian gravity, and the actual observations. The anomalous radial velocity in spiral galaxies led to the hypothesis of a mass discrepancy, today ascribed to the dark matter (DM) invisible mass. While DM is at the base of the current ΛCDM model, the alternative theories of Modified Newtonian Dynamics (MOND) modify G in various ways. It is important to note that the radial acceleration scale value g_† 1.20 × 10⁻¹⁰ ± 0.026 m·s⁻² [19] - [21] of MOND theories equals that of the radial acceleration relation (RAR) [22] . The RAR concerns another important tension between baryon density and EM luminosity. We have decided to explore whether the relation between G and g_† could hopefully provide with a hint to the eventual cosmological PT at the origin of the tensions. With g_† fine-tuned to +0.040 (well inside its margins of error) we have found:

$\frac{G}{g_{†}}=\tan \left({\phi }_W\right)$ (12a)

$\frac{6.67430\times {10}^{-11}}{1.20\left(40\right)\times {10}^{-10}}=0.5543439$ (12b)

a value, which coincides with the tangent of an angle of 29.00153˚. Speaking of PTs, the weak mixing angle, or the Weinberg angle (φ_W) of the electroweak theory, ranges between 28.7˚ and 29.3˚. We could therefore obtain the scale acceleration value simply from:

$g_{†}=\frac{G}{\tan \left({\phi }_W\right)}$ (13)

Instead of a mere empirical value, for the first time, the scale acceleration value has found a physical reason to be, originating directly from the ratio between G and tan(φ_W). Both gravity and the essential element φ_W of the EW symmetry break theory, seem to share an intimate relation. After all, mass represents a common actor for both the EW theory and the gravitational interaction. To be noted that a comparable value, that we would identify with the tangent of φ_W, has also been observed in the constant stellar surface density [23] and, at planetary level, in the low-mass-to-radius relation [24] .

Notwithstanding the character of acceleration of G, we have demonstrated that A can be intimately linked to the gravitational constant. However, whether a might concern dark energy (DE) remains an open question. In such a case, DE dimensional units, alias energy density over a given volume, must be equaled with a over A:

$\frac{E}{V}=U=\frac{a}{A}$ (14a)

$\frac{{\text{m}}^2\cdot \text{kg}\cdot {\text{s}}^{-2}}{{\text{m}}^3}={\text{m}}^{-1}\cdot \text{kg}\cdot {\text{s}}^{-2}=\frac{\text{m}\cdot {\text{s}}^{-2}}{{\text{m}}^2\cdot {\text{kg}}^{-1}}$ (14b)

$\frac{6.88434\times {10}^{-10}}{0.096949}=7.10099\times {10}^{-9}$ (14c)

Nevertheless, the DE density estimations from PLANCK and WMAP range within the same magnitude of 6 × 10⁻¹⁰ as in a.

Finally, we could even notice that sin φ_W over A:

$\frac{\sin \left({\phi }_W\right)}{A}$ (15a)

$\frac{0.4848}{0.096949}\cong 5$ (15b)

returns the number of times, by which DM exceeds baryonic matter. As if A becomes the mass, hidden or refracted or multiplied by the electroweak PT (in the time domain).